Advances in Applied Mechanics. v. 39

Preface We are pleased to present another volume, the thirty-ninth, in the Advances in Applied Mechanics series, the f...

Author: Erik van der Giessen | Hassan Aref

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Preface

We are pleased to present another volume, the thirty-ninth, in the Advances in Applied Mechanics series, the first by the present editorial team. Our labors are naturally divided such that Erik van der Giessen looks after the papers in solid mechanics and Hassan Aref looks after fluid mechanics. We take over – with some trepidation – from the team of John Hutchinson and Ted Wu, and fully cognizant of the giants in whose footsteps we are treading: von Mises, von Ka´rmań, Dryden, Howarth, Prager, and Yih, to mention just some of those who have made this publication what it is today. Advances in Applied Mechanics (AAM) has always been a review series, but the articles published in it have seemed to retain a distinct flavor over the years. While in many quarters the art of writing a review has degenerated into producing what amounts to an annotated bibliography, most articles in AAM have retained the flavor of the genuine item. That is, they have surveyed a field with authority, providing enough detail and exposition that a bright graduate student, or a researcher looking to get into the field, could use it to gain a firm foundation. At the same time an AAM article was not bashful about articulating a clear point of view, telling the reader where the author saw the real advances in the subject, what developments could be expected in the future, where further refinements were most needed, and so on. The reader came away with a feeling of having spent time with an articulate expert who had an opinion about the field in question and was willing to state it and present the case for it. This is the kind of spirit that we would like to keep alive. We hope the reader will agree that our field of mechanics will always be enriched by such articles, and that they may be needed at the present time as much as ever. Articles in AAM are, in general, by invitation only but we do invite readers who have a particularly good idea for a review to contact us and either volunteer their services or point us to an appropriate author. Someone who has recently given a short course for graduate students, or a workshop of some kind, may well have material that would be ideal as the basis for an article. There are few restrictions on length – the ability to capture an audience is much more important ix

x

Preface

than the page count. Our ultimate goal is to provide a collection of review articles of lasting value on advances in our field for use by the research community and, maybe most particularly, future generations who wish to enter our field. The present volume contains four articles that we hope meet this exacting standard. In the first Aref and co-authors have attempted to pull together aspects of the literature on vortex patterns that move without change of shape or size. This subject has been of paramount concern for different reasons since Kelvin’s theory of “vortex atoms” in the 19th century. It continues to bring forth intriguing problems, the solution of which would enrich both mechanics and mathematics. One of the grand challenges for scientific computation is the accurate simulation of flows with material of more than one phase, in the sense of what the computational fluid dynamicist calls a “direct numerical simulation”. That is, what is required is a numerical solution of the full dynamical equations, where the interfaces between the various phases are tracked accurately in time and space. In the article by Tryggvason and co-workers we find a state-of-the-art review of what is possible in this subject today. As the Vol. 38 article by Zhang et al. on piezoelectric materials has already shown, there is a much interest in the fracture mechanics of these kind of ‘smart’ materials. The present article by Chen and Lu gives a supplementary view on this subject, emphasizing the coupling between the mechanics and the electric effects in the nonlinear regime. Finally, Ted Wu, who for many years has so ably edited this serial, has arranged a unique article on research on environmental mechanics in China written by Li Jiachun and co-workers. While this paper may not be as “basic” as some of the others, it provides very valuable insights into a subject which by its nature varies from region to region throughout the World. It is particularly valuable to have the perspective on this subject from the People’s Republic of China. We hope this volume will as well be received as its predecessors. The editors are always eager and anxious to hear comments from readers, and we invite you to contact us. HASSAN AREF

AND

ERIK VAN DER GIESSEN

List of Contributors

Numbers in parentheses indicate the pages on which the authors’ contributions begin.

NABEEL AL -RAWAHI (81), Department of MIE, Sultan Qaboos University, Al-Khode, Oman HASSAN AREF (1), Department of Theoretical and Applied Mechanics, University of Illinois at Urbana-Champaign, USA BERNARD BUNNER (81), Coventor Inc., 625 Mount Auburn Street, Cambridge, MA 02138, USA YI -HENG CHEN (121), School of Civil Engineering and Mechanics, Xi’an Jiao-Tong University, Xi’an 710049, China ASGHAR ESMAEELI (81), Mechanical Engineering Department, WPI, Worcester, MA 01609, USA LI JIACHUN (217), Department of Engineering Sciences, Institute of Mechanics, Chinese Academy of Sciences, Beijing 100080, China ZHOU JIFU (217), Department of Engineering Sciences, Institute of Mechanics, Chinese Academy of Sciences, Beijing 100080, China TIAN JIAN LU (121), Department of Engineering, University of Cambridge, Trumpington Street, Cambridge CB2 1PZ, UK PAUL K. NEWTON (1), Department of Aerospace Engineering and Department of Mathematics, University of Southern California, USA LIU QINGQUAN (217), Department of Engineering Sciences, Institute of Mechanics, Chinese Academy of Sciences, Beijing 100080, China MARK A. STREMLER (1), Department of Mechanical Engineering, Vanderbilt University, USA vii

viii

List of Contributors

TADASHI TOKIEDA (1), De´partement de Mathe´matiques, Universite´ de Montreál, USA GRETAR TRYGGVASON (81), Mechanical Engineering Department, WPI, Worcester, MA 01609, USA DMITRI L. VAINCHTEIN (1), Department of Mechanical and Environmental Engineering, University of California at Santa Barbara, USA

ADVANCES IN APPLIED MECHANICS, VOLUME 39

Vortex Crystals HASSAN AREF,a PAUL K. NEWTON,b MARK A. STREMLER,c TADASHI TOKIEDAd, and DMITRI L. VAINCHTEINe a

Department of Theoretical and Applied Mechanics, University of Illinois at Urbana-Champaign, USA

b

Department of Aerospace Engineering and Department of Mathematics, University of Southern California, USA c

Department of Mechanical Engineering, Vanderbilt University, USA d

De´partement de Mathe´matiques, Universite´ de Montreál, Canada e

Department of Mechanical and Environmental Engineering, University of California at Santa Barbara, USA

I. Vortex Statics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

2

II. The Classification Problem of Vortex Statics . . . . . . . . . . . . . . . .

5

III. Collinear Equilibria of Three Vortices . . . . . . . . . . . . . . . . . . . . .

13

IV. Identical Vortices on a Line. . . . . . . . . . . . . . . . . . . . . . . . . . . . .

17

V. Vortex Polygons . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

20

VI. Beyond Vortex Polygons . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

22

VII. Morton’s Equation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

33

VIII. Stationary Vortex Patterns . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

39

IX. Translating Vortex Patterns . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

47

X. Vortex Crystals on Manifolds . . . . . . . . . . . . . . . . . A. Vortices on a Sphere . . . . . . . . . . . . . . . . . . . . . B. Two- and Three-Vortex Equilibria on the Sphere. C. Multi-Vortex Equilibria on a Sphere . . . . . . . . . . D. Vortices in a Periodic Strip. . . . . . . . . . . . . . . . . E. Vortices in a Periodic Parallelogram . . . . . . . . . . F. Vortices on the Hyperbolic Plane . . . . . . . . . . . .

. . . . . . .

52 52 56 59 63 71 73

XI. Concluding Remarks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

74

Acknowledgments. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

75

References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

75

ADVANCES IN APPLIED MECHANICS, VOL. 39 ISSN 0065-2156

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Copyright q 2003 by Elsevier USA. All rights reserved.

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H. Aref et al. Abstract

Vortex crystals is one name in use for the subject of vortex patterns that move without change of shape or size. Most of what is known pertains to the case of arrays of parallel line vortices moving so as to produce an essentially two-dimensional (2D) flow. The possible patterns of points indicating the intersections of these vortices with a plane perpendicular to them have been studied for almost 150 years. Analog experiments have been devised, and experiments with vortices in a variety of fluids have been performed. Some of the states observed are understood analytically. Others have been found computationally to high precision. Our degree of understanding of these patterns varies considerably. Surprising connections to the zeros of ‘special functions’ arising in classical mathematical physics have been revealed. Vortex motion on 2D manifolds such as the sphere, the cylinder (periodic strip) and the torus (periodic parallelogram) has also been studied because of the potential applications, and some results are available regarding the problem of vortex crystals in such geometries. Although a large amount of material is available for review, some results are reported here for the first time. The subject seems pregnant with possibilities for further development.

I. Vortex Statics In the years 1878 – 1879, the American physicist Alfred M. Mayer published accounts of experiments with needle magnets placed on floating pieces of cork in an applied magnetic field, intended as a didactic illustration of atomic interactions and forms. The different steady states displayed by such floating magnets and described in Mayer’s (1878a) paper were immediately seized upon by Thomson (1878), the later Lord Kelvin, as an illustration of his theory of vortex atoms, where each atom was assumed to be made up of vortices in an ideal fluid ‘ether’ of some kind (Thomson, 1867). Thomson’s interest spurred on a series of investigations, some of which refined or further systematized the study of Mayer’s magnets-on-corks, such as Warder and Shipley (1888) and Derr (1909), while others explored related equilibria in similar systems, e.g., the study of ‘electrified cylinders attracted by an electrified sphere’ (Monkman, 1889), of iron spheres floating in mercury (Wood, 1898), or of floating particles interacting through capillary forces (Porter, 1906). For a historical review see Snelders (1976). Mayer (1878b,c) himself wrote further papers on the transformations and relationships between the various stationary patterns that he observed. The vortex atom theory was taken seriously well into the 20th century when it was replaced by the achievements of J.J. Thomson (who was also an early contributor to the vortex atom theory), E. Rutherford and N. Bohr. Even in Thomson’s (1897) seminal paper, entitled ‘Cathode rays’, in which the discovery of the electron is announced there are allusions to vortex atom theory: “If we regard the system of

Vortex Crystals

3

magnets as a model of an atom, the number of magnets being proportional to the atomic weight,…we should have something quite analogous to the periodic law…” [i.e., the periodic table of the elements]. The interactions between the various pattern-forming particles are, of course, different in detail from the interactions between parallel, columnar vortices that interested Kelvin. However, as Kelvin showed, and as we shall see subsequently, steadily rotating patterns of identical vortices arise as solutions to a variational problem in which the interaction energy (vortex Hamiltonian) is minimized subject to the constraint that the angular impulse be maintained. Many of the other systems mentioned are, presumably, governed by analogous variational principles, although the detailed mathematical expression for the ‘free energy’ to be extremized will, undoubtedly, be different. Nevertheless, the various systems should, and in fact do, share many equilibrium patterns. This observation was the basis for Kelvin’s initial enthusiastic and sweeping claims regarding the analogy. Only much more recently anything approaching a steadily rotating configuration of vortices has been realized: Experiments by Yarmchuk et al. (1979) on vortices in superfluid 4He showed stable configurations (see Section VI and Fig. 5) of much the same kind that Mayer had observed with his magnets. In this case quantum mechanics assures us that the vortices are, indeed, point-like from a macroscopic perspective, and that they all have exactly the same circulation, h=m; where h is Planck’s constant and m is the mass of a He atom. Interest in such pattern-forming systems continues: Experiments using mm-sized rotating disks by Grzybowski et al. (2000) once again found patterns very similar to those formed by the vortices (see Fig. 1), and suggested that the exhibited spontaneous organization might be useful for ‘self-assembly’ of novel materials. We owe the term vortex statics, used as the heading for this section, to Kelvin, who introduced it in the title of a paper published in 1875 to designate the study of vortex configurations that move without change of shape or form. Kelvin’s agenda included both two-dimensional (2D) and three-dimensional (3D) configurations. As one interesting outgrowth of this work, P.G. Tait produced an early topological classification of knots while thinking about the various forms that a stationary vortex filament might assume. Of course, since the objects in question are vortices, vortex statics is really a topic in dynamics—the vortices just happen to be so configured that they do not change their relative positions or configuration. Clearly, in such cases the equations of vortex dynamics are much simplified. We shall refer to configurations that move without change of shape or size as vortex equilibria. Since the patterns in question are usually quite regular and contain a relatively small number of vortices, the term vortex crystals has also been used, and we have adopted that term as the title of the paper. These configurations are typically in a state of uniform rotation or translation and so are

4

H. Aref et al.

Fig. 1. Equilibrium patterns of 2 mm diameter, 400 mm thick magnetized disks floating at a liquid–air interface in the field of a rotating bar magnet. After Grzybowski et al. (2000) with permission.

what in celestial mechanics would be called relative equilibria. Sometimes the vortices all remain in place, and we then say that the configuration is stationary. In this article, we shall confine ourselves to the problem of 2D vortex motion where the richest set of results appears to be available. We shall, almost exclusively, consider point singularities, known as point vortices. Given an equilibrium of point vortices it is often possible to ‘soften’ the core of each vortex into a small region of constant vorticity and then find corresponding equilibria of finite-area vortices (although usually only numerically). In some cases, smooth vorticity distributions can also be found which in one limit converge on a point vortex configuration. Much further work seems to be possible on the general theme of the correspondence between point vortex equilibria and smoother solutions to the 2D Euler equation or even the Navier –Stokes equation. For a recent, very promising approach see Crowdy (1999) and subsequent papers by this author. There are important connections to related problems in other fields of mechanics, where one is again concerned with some kind of equilibrium of a set of interacting point or line singularities. Thus, in the subject of plastic behavior of solids interpreted in terms of dislocations Eshelby et al. (1951) considered identical dislocations situated in the same slip-plane, and the problem of what positions they ‘will take up under the combined action of their mutual repulsions and the force exerted on them by a given applied shear stress, in general a function of position along the plane.’ Better known, and also of considerable importance, are the related investigations of central configurations in the N-body problem of celestial

Vortex Crystals

5

mechanics (cf. Wintner, 1941). This topic goes back at least to Lagrange, who proved that for any three finite masses, attracting one another according to Newton’s law of universal gravitation, four distinct configurations exist such that, under proper initial conditions, the ratios of the mutual distances remain constant. This condition includes equilibria as a special case, but more generally allows for self-similar collapse and expansion of the configuration as well. In three of the four solutions the masses are on a straight line; in the fourth they are at the vertices of an equilateral triangle. The case of collinear masses was generalized by Moulton (1910) in a well-known paper. Moulton formulated the two essentially different problems that one can consider for any of the aforementioned systems, which we now state in the context of vortex statics: (1) Given a system of point vortices with prescribed circulations, find the configurations that will move without change of shape or size; and (2) given a geometrical pattern or configuration of points in the plane, find the set of circulations that will turn this configuration into a point vortex system that moves without change of shape or size. It is the first of these problems that tends to be of greater physical interest. For two vortices all configurations will lead to motions of the type sought. For three vortices we shall see that the equilateral triangle configuration will be an equilibrium regardless of the circulations assigned to the vortices. Variants of these statements continue to hold as we place the vortices on other 2D manifolds. For larger numbers of vortices such simplicity, of course, no longer arises.

II. The Classification Problem of Vortex Statics The point vortex equations for N interacting vortices a ¼ 1; 2; …; N with circulations Ga and positions in a complex flow plane, za ; are (Aref, 1983; Newton, 2001) N Gb dza 1 X 0 ¼ : 2pi b ¼ 1 za 2 zb dt

ð2:1Þ

The overbar denotes complex conjugation. The prime on the summation sign reminds us to omit the singular term b ¼ a: We assume that the configuration of vortices is instantaneously moving as a rigid body, i.e., that the velocity of every vortex is made up of a translational part and a rotation: dza ¼ V þ ivza ; dt where V is complex and v is real, and both are the same for all vortices.

ð2:2Þ

H. Aref et al.

6

Substituting the Ansatz (2.2) into (2.1), we obtain in place of the ODEs a set of algebraic equations N Gb 1 X 0 : 2pi b ¼ 1 za 2 zb

V 2 ivza ¼

ð2:3Þ

Turning first to the issue of existence of solutions to Eq. (2.3), Kelvin noticed that vortex equilibria are subject to a variational principle: If we seek extrema of H¼2

N 1 X 0 G G loglza 2 zb l 4p a;b ¼ 1 a b

ð2:4Þ

under the subsidiary conditions N X

N X

Ga xa ¼ const:;

a¼1

N X

Ga ya ¼ const:;

a¼1

Ga lza l2 ¼ const:; ð2:5Þ

a¼1

we obtain Eq. (2.3). (The physical significance of the three quantities introduced in Eq. (2.5) will become clear in what follows.) To see this we introduce the subsidiary conditions through three, real Lagrange multipliers, u, v and v, and are then concerned with the extrema of Hþv

N X

N X

Ga x a 2 u

a¼1

Ga ya þ 12 v

a¼1

N X

Ga lza l2 :

a¼1

Differentiating with respect to xa ; ya in turn, we obtain:

›H þ vGa þ vGa xa ¼ 0; ›x a

›H 2 uGa þ vGa ya ¼ 0: ›y a

But H, Eq. (2.4), is the Hamiltonian for point vortex motion on the infinite plane (cf. Aref, 1983; Newton, 2001). Thus,

Ga

dxa ›H ¼ ; dt ›y a

Ga

dya ›H ¼2 ; dt ›x a

and we obtain v þ v xa ¼

dya ; dt

2u þ vya ¼ 2

dxa ; dt

where the time derivatives of xa and ya are to be written out in terms of the coordinates and strengths of all the vortices in the system, i.e., as the ‘right hand sides’ of the equations of motion (2.1). Combining these relations yields u 2 iv 2 ivðxa 2 iya Þ ¼

dxa dy 2i a; dt dt

Vortex Crystals

7

which, when the time derivatives are written out, gives us Eq. (2.3) once again. With the wisdom of hindsight, the notation for the Lagrange multipliers has been chosen so that u and v are the components of the translational velocity, and v is the angular velocity. This result is Kelvin’s variational principle for vortex statics. For some sets of vortex strengths, e.g., if all the Gs are of the same sign, one can show explicitly that at least one solution must exist. The argument can probably be extended. We are not aware of any set of vortex strengths for which it has been shown that no solutions to Eq. (2.3) may be found. Assuming we have a solution to the vortex statics problem, we multiply Eq. (2.3) by Ga and Ga za in turn and sum each time. Thus, multiplying by Ga and summing we obtain SV 2 ivðX 2 iYÞ ¼ 0;

ð2:6Þ

where S¼

N X

ð2:7Þ

Ga ;

a¼1

and X and Y are the components of linear impulse, X¼

N X

a¼1

Ga x a ;

Y¼

N X

Ga y a ;

X þ iY ¼

a¼1

N X

Ga z a :

ð2:8Þ

a¼1

In general, X and Y are integrals of Eq. (2.1), related to the components of linear momentum of the fluid motion. Their conservation implies that the center of vorticity, defined for S – 0 as the point with coordinates ðX; YÞ=S; remains invariant during the evolution of the vortices. Further, multiplying Eq. (2.3) by Ga za and summing we get VðX þ iYÞ 2 ivI ¼

K ; 4pi

ð2:9Þ

where I is the angular impulse given by I¼

N X

Ga lza l2 ;

ð2:10Þ

a¼1

and K¼

N X 0

a; b ¼ 1

G b Ga :

ð2:11Þ

H. Aref et al.

8

The angular impulse is also a general integral of Eq. (2.1) related to the angular momentum of the fluid motion. We note for future reference that the quantities S, Eq. (2.7), and K, Eq. (2.11), are related by S2 ¼

N X

G 2a þ K:

ð2:12Þ

a¼1

Equations (2.6) and (2.9) are key to classifying solutions of the problem of point vortex statics. The form of these equations is very simple—two linear equations in two unknowns, V and v. The condition for a unique solution to exist is that the determinant of the coefficient matrix on the left hand side be non-zero, i.e., that SI 2 ðX 2 þ Y 2 Þ – 0:

ð2:13Þ

This condition may be re-stated in terms of the important quantity L¼

1 2

N X 0

Gb Ga lza 2 zb l2 :

ð2:14Þ

a;b ¼ 1

The condition (2.13) is equivalent to L – 0, since in Eq. (2.14) we can extend the summation to be over all a and b, and then it is clear that L ¼ SI 2 X 2 2 Y 2 : In other words, L ¼ SIcv ; where Icv means the angular impulse, Eq. (2.10), calculated with the center of vorticity taken as the origin. For L – 0, we find unique solutions for V and v depending only on combinations of the vortex strengths, S and K, and on the integrals of motion X, Y and I. Thus, if the Ansatz (2.2) is valid at some instant, it will be valid for all time and V and v will be constants. The actual values of V and v found by solving Eqs. (2.6) and (2.9) are 0 2iðX 2 iYÞ 1 K X 2 iY ; ð2:15Þ V¼ ¼i K 2 iL 4p L 2iI 4pi S 0 1 SK : ð2:16Þ v¼ K ¼ 2 iL X þ iY 4pL 4pi We now reason as follows (maintaining the assumption L – 0): for S – 0, we may assume X ¼ Y ¼ 0; since an inconsequential shift of the origin of coordinates will otherwise assure this result. Then V ¼ 0 and L ¼ SI so that

Vortex Crystals

9

the vortices rotate as a rigid body about the center of vorticity with an angular velocity given by N K 1 X 0 ¼ G G : 4p 4p a;b ¼ 1 b a

Iv ¼

ð2:17Þ

Equation (2.17) includes the possibility, for I – 0 and K ¼ 0, that the vortex configuration is stationary. For S ¼ 0; the motion consists of pure translation (since v ¼ 0) with velocity N X

V¼

Ga2

a¼1

4pi

X þ iY X2 þ Y 2

ð2:18Þ

where we have used Eq. (2.12) for S ¼ 0. We may take further moments of Eq. (2.3). Thus, if we multiply Eq. (2.3) by Ga z2a and sum, we obtain V

N X

Ga za2 2 iv

a¼1

N X

Ga za lza l2 ¼

a¼1

2 N 1 X 0 Ga Gb z a : 2pi a;b ¼ 1 za 2 zb

The right hand side may be re-written by noting that 2 N N N N X X X X za2 2 zb2 0 G a Gb z a 0 ¼ 1 G G ¼ S G z 2 Ga2 za : a b a a 2 z 2 z z 2 z b a b a;b ¼ 1 a a;b ¼ 1 a¼1 a¼1

Now, the first term vanishes trivially if S ¼ 0, and if S – 0, we can shift the center of vorticity to the origin so that the sum vanishes. In other words, we may always assume that V

N X

a¼1

Ga za2 2 iv

N X

a¼1

Ga za lza l2 ¼ 2

N 1 X G 2z : 2pi a ¼ 1 a a

ð2:19Þ

We shall make use of this identity on occasion in what follows. The general case L ¼ 0 remains to be considered. Equations (2.6) and (2.9) are no longer independent. For S ¼ 0, L ¼ 0 implies X ¼ Y ¼ 0: Equation (2.6) is then satisfied identically and Eq. (2.9) becomes Eq. (2.17). Since for S ¼ 0, we have K – 0, we must also have I – 0. Thus, the configuration rotates with the constant angular frequency given by Eq. (2.17). However, the center of rotation now needs to be determined as part of the analysis. In order to determine it we must return to the point vortex equation (2.1). For S – 0 we may assume X ¼ Y ¼ 0 or arrange for this to be so by a shift of the origin of coordinates. When the origin is so chosen, L ¼ 0 implies I ¼ Icv ¼ 0: It

10

H. Aref et al.

then follows from Eq. (2.6) that V ¼ 0 and from Eq. (2.9) that we must have K ¼ 0. The angular velocity is indeterminate in this case. In particular, it appears to be unknown at present whether completely stationary configurations with L ¼ 0 exist. As an elementary illustration of the classification obtained let us consider the case of two point vortices. Since L ¼ G1 G2 s2 ; where s is the distance between the vortices, all two-vortex motions belong to the class of vortex statics and L – 0 (the vortices are not allowed to coincide). For S – 0 the vortices orbit the center of vorticity—the point with coordinates ðX=S; Y=SÞ—with angular velocity

v¼

G1 þ G 2 : 2ps2

ð2:20Þ

This follows from Eq. (2.17) since K ¼ 2G1 G2 and L ¼ Ks2 =2: For S ¼ 0; i.e., G1 ¼ 2G2 ¼ G; the vortices translate with the common velocity (2.18). The direction of translation is perpendicular to the impulse, X þ iY, in other words, perpendicular to the line connecting the vortices. The speed of propagation of the pair is G=2ps; which we write in the somewhat artificial way qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ffi 2 2 1 G þ G 1 2 2 lVl ¼ : ð2:21Þ 2ps We may illustrate further aspects of the classification with examples from threevortex motion. Denote the distances between the vortices s1 ¼ lz2 2 z3 l; s2 ¼ lz3 2 z1 l; s3 ¼ lz1 2 z2l. An elementary calculation (Gro¨bli, 1877; Aref, 1979) starting from Eq. (2.1) then shows that ds12 2 s2 2 s2 ¼ G1 D 3 2 2 2 ; p dt s2 s3 ds22 2 s2 2 s2 ¼ G2 D 1 2 2 3 ; p dt s3 s1

ð2:22Þ

ds32 2 s2 2 s2 ¼ G3 D 2 2 2 1 ; p dt s1 s2 where D is the area of the vortex triangle 123 with the þ sign if 123 appear in counter-clockwise order in the plane, and with the 2 sign if 123 appear in clockwise order. The magnitude of D is related to the length of the sides in the vortex triangle by 16D2 ¼ 2s22 s23 þ 2s23 s21 þ 2s21 s22 2 s41 2 s42 2 s43 ; which is just Heron’s formula for the area of a triangle (cf. Coxeter and Greitzer, 1967). It follows from Eq. (2.22), by setting the left hand sides to zero, that in a threevortex equilibrium the vortices are either collinear or are at the vertices of an

Vortex Crystals

11

equilateral triangle. The case of three collinear vortices will be treated in detail in Section III, so here we concentrate on the case where the vortices form an equilateral triangle. For that particular geometry L ¼ Ks2 =2; where s is the side of the triangle. For L – 0 and S – 0, our analysis tells us that the vortices must rotate about the center of vorticity with angular frequency given by Eq. (2.17), or in this particular case by

v¼

G1 þ G2 þ G 3 : 2ps2

ð2:23Þ

For L – 0 and S ¼ 0, the vortex triangle translates without rotation. The velocity of translation is given by Eq. (2.18). For this particular case, a simple calculation gives qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 2 2 1 2 G1 þ G2 þ G3 lVl ¼ : ð2:24Þ 2ps The analogies between Eqs. (2.20), (2.21) and (2.23) and (2.24) will not have escaped the attentive reader. For L ¼ 0, the equilateral triangle configuration implies K ¼ 0 and thus S – 0. The ambiguity of Eq. (2.9) is resolved by returning to the point vortex equation (2.1). These equations show that the equilateral triangle with K ¼ 0 rotates about the center of vorticity with the angular velocity (2.23). If a configuration of three vortices is a stationary equilibrium, it must satisfy K ¼ 0 from Eq. (2.9). We can then arrange that X ¼ Y ¼ 0; i.e., modulo a shift of coordinates

G1 z1 þ G2 z2 þ G3 z3 ¼ 0: The relation (2.19) with V and v both equal to zero gives

G12 z1 þ G22 z2 þ G32 z3 ¼ 0: These two linear relations between z1, z2 and z3 have the solutions z1 ¼ j

G2 2 G3 ; G1

z2 ¼ j

G3 2 G1 ; G2

z3 ¼ j

G1 2 G2 ; G3

ð2:25Þ

where j is a complex parameter. A stationary equilibrium of three vortices, then, must have the three vortices on a line. From Eq. (2.25), one finds I ¼ 3ðG1 þ G2 þ G3 Þljl2 (recalling that K ¼ 0). In particular, for all stationary configurations of three vortices I ¼ Icv – 0; and hence L – 0. To resolve the open problem mentioned above, i.e., whether stationary equilibria with K ¼ 0 and L ¼ 0 are possible, we must look to configurations with four or more vortices.

12

H. Aref et al.

In summary, point vortex patterns that move without change of shape or size will rotate uniformly if the sum of the circulations is non-zero. This includes the possibility of completely stationary patterns for K ¼ 0. When the pattern rotates, the center of rotation is, in general, the center of vorticity and a general formula, Eq. (2.17), is available for the angular frequency of rotation. The case L ¼ 0 requires special consideration by returning to the equations of motion. When the sum of the circulations vanishes, an invariant vortex pattern will translate uniformly. A general formula, Eq. (2.18), involving the linear impulse is available for the velocity of translation. For vanishing linear impulse a neutral vortex system will rotate. The formula for the angular frequency of rotation is again Eq. (2.17), but determining the center of rotation requires special considerations, since the center of vorticity is indeterminate. Figure 2 illustrates some of the two- and three-vortex equilibria we have been discussing.

Fig. 2. Relative equilibria for two and three vortices. (a) Two identical vortices orbit their midpoint. (b) Two opposite vortices translate along parallel lines perpendicular to the line segment connecting them. (c) Three vortices on a line, of strengths 2G, 2 G and 2G, respectively, form a stationary equilibrium (K ¼ 0). (d) The same three vortices as in (c) placed at the vertices of an equilateral triangle; this configuration has L ¼ 0 but rotates with the angular frequency (2.23) about the center of vorticity (shown as þ ). (e) Three vortices on a line of strengths G, 22G and G, respectively, i.e., S ¼ 0, rotate about the center vortex. This is a simple model of the tripole in Fig. 3.

Vortex Crystals

13

III. Collinear Equilibria of Three Vortices Given three vortices on a line, we may assume the line to be the x-axis of coordinates and the instantaneous positions of the vortices to be x1, x2 and x3. The right hand side of any of Eq. (2.3) is then purely imaginary. Comparing this to the left hand side, we see that we are seeking solutions of the following system of equations: G2 G3 a þ bx1 ¼ þ x1 2 x2 x1 2 x3 G1 G3 a þ bx2 ¼ þ ð3:1Þ x2 2 x1 x2 2 x3 G1 G2 a þ bx3 ¼ þ x3 2 x1 x3 2 x2 where a and b are certain real constants: a is proportional to the translational velocity, b to the angular velocity. We have already dealt with the special case a ¼ b ¼ 0: Solutions for this case exists only if G1 G2 þ G2 G3 þ G3 G1 ¼ 0 and are given by Eq. (2.25). If b ¼ 0 but a – 0, solutions will only exist for G1 þ G2 þ G3 ¼ 0: This follows from the general theory, but is not hard to verify independently by multiplying the first of Eq. (3.1) by G1, the second by G2, the third by G3, and adding. If we subtract the second of Eq. (3.1) from the first, we find still assuming b¼0

G2 G3 G1 G3 þ 2 2 ¼ 0; x1 2 x2 x1 2 x3 x2 2 x1 x2 2 x3 or, using the result that the three strengths sum to zero 1 1 1 þ þ ¼ 0: x1 2 x2 x2 2 x3 x3 2 x1

ð3:2Þ

This equation, however, has no solutions. For assume that x1 . x2 . x3 : Set u ¼ x1 2 x2 . 0; and v ¼ x2 2 x3 . 0: Then Eq. (3.2) becomes 1 1 1 þ ¼ u v uþv or ðu þ vÞ2 ¼ uv or u2 þ v2 ¼ 2uv;

H. Aref et al.

14

which is, of course, impossible. Similar contradictions are reached regardless of how we assume the vortices to be arranged along the line. We conclude that the case b ¼ 0, a – 0 has no solutions at all. For a neutral vortex triple there are no translating, collinear equilibria. If b – 0, we may assume a ¼ 0, since we can always shift the origin of coordinates by a=b and thus eliminate any non-zero a from Eq. (3.1). We now have the equation

G1 x1 þ G2 x2 þ G3 x3 ¼ 0:

ð3:3Þ

Multiplying the first of Eq. (3.1) by ðx2 2 x3 Þ21 ; the second by ðx3 2 x1 Þ21 ; the third by ðx1 2 x2 Þ21 ; and adding—a trick due to Gro¨bli (1877)—we obtain an equation that does not contain the vortex strengths at all: x1 x2 x3 þ þ ¼ 0: x2 2 x3 x3 2 x1 x1 2 x2

ð3:4Þ

We have the solutions x3 ¼ 0, x1 ¼ 2x2 for vortex triples with G1 ¼ G2 ¼ G and any value of G3. They include the stationary configuration for G3 ¼ 2G=2; a special case of Eq. (2.25). Although, from the point of view of the theory, these may not be the most ‘exciting’ solutions, they are among the most important physically and they certainly are among the most celebrated. Three vortices on a line became a very fashionable topic in geophysical fluid dynamics with the discovery of the ‘tripole’ by van Heijst and Kloosterziel (1989) (see also Kloosterziel and van Heijst, 1991; van Heijst et al., 1991). The configuration, which evolves spontaneously from an unstable, axisymmetric ‘vortex’—we put this in quotes, since the state used, in fact, has zero net circulation—is shown in Fig. 3. It complements the ‘monopole’ and ‘dipole’ (vortex pair) that had been extensively studied in the geophysical context. In the third paper mentioned above the authors discuss the possibility of modeling the tripole, which in reality has distributed vorticity, by a set of three collinear point vortices in the type of configuration we have just considered. Since the initial state from which the tripole emerges has no net circulation, it is natural to assume that the circulations of the central and two satellite vortices also sum to zero, i.e., that G1 ¼ G2 ¼ 2G3 =2 ¼ G: If R denotes the distance between the central vortex, 3, and either satellite, we have I ¼ 2GR2 in Eq. (2.17). Since S ¼ 0, K ¼ 2 6G 2 . Thus, v ¼ 23G=4pR2 : The configuration rotates in the direction of the circulation of vortex 3 (clockwise in our case, since G3 , 0; in the experiments G3 was actually positive, G1 and G2 negative). Consider the variables

j1 ¼ x2 2 x3 ;

j2 ¼ x3 2 x1 ;

j3 ¼ x1 2 x2 ;

ð3:5Þ

Vortex Crystals

15

Fig. 3. The vortex ‘tripole’ found experimentally by van Heijst et al. (1991). Courtesy of G.J. van Heijst; reproduced with permission.

which are not independent since

j1 þ j2 þ j3 ¼ 0: Equation (3.4) may be written x1 x x þ 2 þ 3 ¼ 0; j1 j2 j3

ð3:6Þ

and from Eq. (3.5) we have

G2 j3 2 G3 j2 ; G1 þ G2 þ G3 G j 2 G1 j3 x2 ¼ 3 1 ; G1 þ G2 þ G3 G j 2 G2 j1 x3 ¼ 1 2 : G1 þ G2 þ G3 x1 ¼

Equation (3.3) is now identically satisfied, and Eq. (3.6) becomes j j j j j j G1 2 2 3 þ G2 3 2 1 þ G3 1 2 2 ¼ 0: j3 j2 j1 j3 j2 j1

ð3:7Þ

ð3:8Þ

H. Aref et al.

16

Choose one of the ratios between the js as a new independent variable, e.g., j x 2 x3 ; ð3:9Þ z¼ 1 ¼ 2 j3 x1 2 x2 and note that we then have j2 j þ j3 ¼2 1 ¼ 21 2 z; j3 j3

j1 j j z : ¼ 1 3 ¼2 1þz j2 j3 j2 Thus, Eq. (3.8) becomes 1 1 z 1 2 z þ G3 2 þ1þ G1 21 2 z þ þ G2 ¼ 0; 1þz z 1þz z which is a cubic equation for determining z: ðG1 þ G2 Þz3 þ ð2G1 þ G2 Þz2 2 ðG2 þ 2G3 Þz 2 ðG2 þ G3 Þ ¼ 0:

ð3:10Þ

A related equation appears in the paper by Borisov and Lebedev (1998). Equation (3.10) will always have at least one real solution. According to (3.9) this solution must be – 0 and – 2 1, i.e., the two cases G1 þ G3 ¼ 0 and G2 þ G3 ¼ 0 require special consideration. We discuss only the latter in detail. The former proceeds similarly. For G2 ¼ 2G3 we discard the solution z ¼ 0 and Eq. (3.10) reduces to the quadratic equation ðG1 þ G2 Þz2 þ ð2G1 þ G2 Þz þ G2 ¼ 0:

ð3:11Þ

Without loss of generality we may assume G1 $ G2 ; so this equation always has two real solutions. By way of example, for G1 ¼ G2 ¼ 22G3 ¼ G; Eq. (3.10) takes the form: 4z3 þ 6z2 2 1 ¼ 0: In this case,we have in addition to the configuration (i)

x1 : x2 : x3 ¼ 1 : ð21Þ : 0;

found previously, the two collinear equilibria: pffiffi pffiffi (ii) x1 : x2 : x3 ¼ ð1 þ p3ffiffiÞ : ð1 2 pffiffi3Þ : 4; (iii) x1 : x2 : x3 ¼ ð1 2 3Þ : ð1 þ 3Þ : 4: The equilibrium (i) has I – 0 and so, by Eq. (2.17), must be stationary, since we have K ¼ 0. The equilibria (ii) and (iii) have I ¼ 0 and so the angular velocity of rotation must be obtained directly from the equations of motion.

Vortex Crystals

17

In the general case the number of real solutions (one or three) is determined by the discriminant of the cubic equation (3.10). No simple criterion, e.g., in terms of symmetric functions of the vortex strengths, seems available to determine when we have just one and when we have three such equilibria.

IV. Identical Vortices on a Line There is a surprising connection between the problem of how to place N identical vortices on a line such that the configuration rotates like a rigid body and the zeros of a well-known family of orthogonal polynomials. This connection was first found by Stieltjes (1885) using the analogy with interacting line charges (see Szego¨, 1959; Marden, 1949, for further details and subsequent developments). It has later been re-discovered and utilized many times, e.g., by Eshelby et al. (1951) for a model of dislocation pile-up. Here we show how this result enters the N-vortex problem. Assume N identical vortices are given, and that they are placed on a line, for convenience taken as the x-axis of coordinates, such that the configuration rotates rigidly. At issue is to determine x1 ; …; xN given that they obey equations of the form

lx1 ¼

1 1 1 þ þ ··· þ ; x1 2 x2 x1 2 x3 x1 2 xN

lx2 ¼

1 1 1 þ þ ··· þ ; x2 2 x1 x2 2 x3 x2 2 xN

ð4:1Þ

.. .

lxN ¼

1 1 1 þ þ ··· þ ; xN 2 x1 xN 2 x2 xN 2 xN21

where, in physical units, l ¼ 2pv=G; with v the angular frequency of rotation and G the common circulation of the vortices. Since l will be positive according to the general formula (2.17), we can scale all the xs by l1/2 and it suffices to consider Eq. (4.1) with l ¼ 1. To solve this problem we embed x1 ; …; xN as the roots of a polynomial of degree N: PðxÞ ¼ ðx 2 x1 Þðx 2 x2 Þ· · ·ðx 2 xN Þ:

ð4:2Þ

This polynomial satisfies an ODE of second order which is obtained as follows.

H. Aref et al.

18

The first derivative of P is P0 ¼ P

N X

a¼1

1 : x 2 xa

ð4:3Þ

A second differentiation gives N X

P 00 ¼ P 0

a¼1

2

N X 1 1 2P 2 x 2 xa a ¼ 1 ðx 2 xa Þ

N X

¼ P4

a;b ¼ 1

3 N N X X 1 1 1 1 1 5¼P 2 : 2 x 2 xa x 2 xb x 2 xa x 2 xb a ¼ 1 ðx 2 xa Þ a;b ¼ 1 a–b

The summand can be re-written: 1 1 ¼ x 2 xa x 2 xb

"

1 1 2 x 2 xa x 2 xb

#

1 : xa 2 xb

ð4:4Þ

In the double sum we then get, according to Eq. (4.1) with l ¼ 1, N X

a;b ¼ 1 a–b

N N X X 1 1 1 1 xa 0 ¼2 ¼2 : x 2 xa x 2 xb x 2 xa xa 2 xb x 2 xa a;b ¼ 1 a¼1

Thus, P 00 ¼ 2P

N X

a¼1

xa ¼ 22NP þ 2xP 0 : x 2 xa

ð4:5Þ

We recognize this equation as the differential equation satisfied by the Nth Hermite polynomial HN ðxÞ: Since the Hermite polynomial is the unique polynomial solution to this second order ODE, we have established that the solutions to Eq. (4.1) with l ¼ 1 are the roots of the Nth Hermite polynomial. The result is both intriguing and disappointing. It is intriguing because it suggests a link between point vortex dynamics and other areas of applied mathematics with which the subject a priori would seem to have no connection whatsoever. Further links between vortex statics and families of polynomials that solve apparently unrelated equations will emerge later, so the ‘intrigue’ will deepen! It is disappointing because, of course, we have accomplished little in terms of finding solutions to our problem—we have simply related one set of unknown mathematical objects, viz. the vortex positions along a line, to another, viz. the roots of the Nth Hermite polynomial. It is a matter of taste whether one feels more information is conveyed by saying that the vortex positions satisfy

Vortex Crystals

19

Eq. (4.1), or that they are roots of HN : The larger question, however, is whether the idea of a generating function for the vortex positions, such as the polynomial PðxÞ introduced in Eq. (4.2), that will satisfy a relatively simple differential equation, carries further. Such generating functions have proven extremely powerful in other areas of mathematics, for example in combinatorics, and it would be very interesting if PðxÞ, or some generalization thereof, satisfied an ODE or PDE that allowed non-trivial results to be obtained concerning vortex motion. For the present we leave these thoughts as speculations. In Sections VIII and IX we review some results that are available in this direction. To help a bit with the disappointment, let us note that Eq. (2.17) with 2pv=G ¼ 1 tells us that the sum of the squares of the roots of the Nth Hermite polynomial must satisfy the ‘sum rule’ N X

x2a ¼

1 2

NðN 2 1Þ:

ð4:6Þ

a¼1

This result is known independently. Many readers are likely to know about Hermite polynomials but not know that this simple identity holds for the sum of the squares of their roots. It is, indeed, pleasing to have a proof of Eq. (4.6) ‘by vortex dynamics’, i.e., as a corollary of the correspondence with vortex statics. It is possible to generalize the results just obtained somewhat. For odd N ¼ 2n þ 1 there will always be a vortex at the origin, and one can consider that this vortex might have a different circulation from the other N 2 1. If the central vortex has circulation pG; where G is the common value of the circulations of the remaining vortices, then the positions of the vortices are given, up to a scaling ðp21=2Þ 2 factor, by the roots of the Laguerre polynomial Ln ðx Þ: For p ¼ 1, we return to the case of N identical vortices and, indeed, H2nþ1 ðxÞ is proportional to ð1=2Þ Ln ðx2 Þ: For p ¼ 0, we return to the case of 2n identical vortices and H2n ðxÞ is ð21=2Þ 2 known to be proportional to Ln ðx Þ: There is a restriction (cf. Aref, 1995) to p . 2 1/2 for N . 3. For the threevortex problem, p can have any value and the state is always an equilibrium. For p ¼ 2 1/2, we have the stationary ‘tripole’ already mentioned following Eq. (3.4) and in the discussion of Eq. (3.10). As N increases it is known that the roots of the Nth Hermite polynomial become more and more uniformly spaced. One would expect this from the connection with vortex statics, since in the limit N ! 1 the collinear equilibria should converge to the infinite line of equally spaced vortices, a time-honored model of a vortex sheet. In the limit, one can again consider one vortex to have a different circulation, pG; from the rest. The vortices are then, of course, not equally spaced and one can view this as the problem of finding the equilibrium

H. Aref et al.

20

spacing of a row of vortices with an ‘inhomogeneity’. The pleasing solution is that the vortex positions are given as the zeros of the Bessel function Jp21=2 ðxÞ: It is quite remarkable that the linearized stability analysis for these configurations can be carried out analytically to a large extent. We shall not elaborate further on this here, since a rather accessible account exists elsewhere (Aref, 1995). Many of the underlying mathematical results were obtained by Calogero and co-workers in the 1970s (cf. Calogero, 2001, and references therein).

V. Vortex Polygons Perhaps the best known equilibria for identical vortices are the vortex polygons, first studied by Kelvin stimulated by Mayer’s experiments on the floating magnets mentioned earlier, and later by many others, most notably perhaps by Thomson (1883) in his Adams Prize essay. With such strong proponents as Kelvin and Thomson the analogy of vortex equilibria to atoms was a powerful, motivating force for fundamental physics. Much time and energy was spent analyzing such states. Since the polygons are stable for small numbers of vortices, they have been sought in experimental systems that approximately realize the point vortex equations, such as superfluids and electron plasmas. The basic configuration has identical vortices at the corners of a regular N-gon. This state rotates uniformly as we may see by the following considerations: We use the Ansatz za ¼ R expði2pa=NÞ; a ¼ 0; …; N 2 1: Equations (2.3) (with V ¼ 0) then yield 2pR2 v ¼

N21 X

a¼1

G : 1 2 exp½2pia=N

ð5:1Þ

To evaluate the sum we return to the ideas in Eqs. (4.2) and (4.3). Let PðzÞ be a polynomial of degree N in the complex variable z with distinct roots z1 ; …; zN : Then PðzÞ ¼ ðz 2 z1 Þ · · · ðz 2 zN Þ; P 0 ðzÞ ¼ PðzÞ

N X

a¼1

1 : z 2 za

ð5:2Þ ð5:3Þ

In particular, for PðzÞ ¼ z N 2 g N ; with g a complex number that is not an Nth root of unity, the roots are z1 ¼ g; z2 ¼ g1; …; zN ¼ g1N21 ; where 1 ¼ expð2p i=NÞ; and so Nz N21 ¼ ðz N 2 g N Þ

N X

a¼1

1 : z 2 g1a

Vortex Crystals

21

For z ¼ 1, this tells us that N X

a¼1

1 N ¼ ; a 1 2 g1 1 2 gN

ðg N – 1Þ:

ð5:4Þ

For g ¼ 1, we proceed as above but with P1 ðzÞ ¼ ðz 2 1Þ· · ·ðz 2 1N21 Þ ¼

zN 2 1 ¼ 1 þ z þ · · · þ z N21 : z21

We now have P 01 ðzÞ ¼ 1 þ 2z þ · · · þ ðN 2 1Þz N22 : Hence, P1 ð1Þ ¼ N; P 01 ð1Þ ¼

1 2

NðN 2 1Þ; and Eq. (5.3) becomes

N21 X

a¼1

1 ¼ 1 2 ðN 2 1Þ; 1 2 1a

ð5:5Þ

which also arises by taking the limit g ! 1 of Eq. (5.4). Combining Eq. (5.5) with Eq. (5.1) produces

v¼

G ðN 2 1Þ: 4pR2

ð5:6Þ

If we place a vortex at the center of a regular N-gon, we obtain an equilibrium of N þ 1 identical vortices. The vortices at the corners of the N-gon rotate as before, each with a velocity augmented by the presence of the central vortex. The central vortex is stationary by symmetry. The angular velocity of rotation is from Eq. (5.6)

v¼

G G G ðN 2 1Þ þ ¼ ðN þ 1Þ: 2 2 4pR 2pR 4pR2

ð5:7Þ

Note that this is a configuration of N þ 1 vortices. The vortices at the corners of a centered, regular N-gon rotate with the same angular frequency as the vortices at the corners of an open, regular (N þ 2)-gon. Actually, to have an equilibrium the central vortex need not have the same circulation as the corner vortices. Thus, consider N identical vortices of circulation G at the corners of a regular N-gon with a vortex of circulation pG at the center, where p is any real number. This configuration rotates steadily with angular velocity

v¼

G ðN 2 1 þ 2pÞ: 4pR2

ð5:8Þ

In particular, the configuration is stationary for p ¼ 2ðN 2 1Þ=2: For N ¼ 3, the central vortex is then simply opposite to the three vortices at the corners of the equilateral triangle. For general N, p ¼ 2ðN 2 1Þ=2 is equivalent to K ¼ 0.

H. Aref et al.

22

It is a classical result of Thomson (1883), corrected by Havelock (1931), and later given in modified forms by Dritschel (1985) and Aref (1995), that the regular N-gon with six or fewer identical vortices is linearly stable, the heptagon is neutrally stable in linear theory, while the open N-gon with eight or more vortices is linearly unstable. Khazin (1976) considers the non-linear stability of regular vortex polygons. The linear stability of centered, regular polygons, with a central vortex of a different strength than the ones in the polygon itself, has been studied by Morikawa and Swenson (1971) and Mertz (1978). Cabral and Schmidt (1999) prove the following result for centered regular N-gons: If the central vortex has circulation pG; where G is the common circulation of the N vortices making up a regular N-gon, then the configuration is Liapunov stable for ðN 2 2 8N þ 8Þ=16 , p , ðN 2 1Þ2 =4 when N is even; ðN 2 2 8N þ 7Þ=16 , p , ðN 2 1Þ2 =4 when N is odd: For N , 6, these ranges include the possibility that the circulation of the central vortex is of the opposite sign to the vortices making up the polygon, i.e., p may be negative. VI. Beyond Vortex Polygons For N identical vortices we now have, at least, three equilibria by direct construction: the N-vortices-on-a-line (Section IV), the regular N-gon, and the centered, regular (N 2 1)-gon (Section V). For this set of vortex strengths Kelvin’s variational principle guarantees existence of a stable equilibrium for each N. Since the aforementioned three families include stable equilibria only for small N, we need to explore further possibilities. If we place N1 vortices of circulation G1 on one regular polygon of radius R1, and N2 vortices of circulation G2 on a second, concentric, regular polygon of radius R2, i.e., set za ¼ R expði2pa=N1 Þ; if

zb ¼ R e expði2pb=N2 Þ;

a ¼ 0; …; N1 2 1; b ¼ 0; …; N2 2 1;

then Eq. (2.3) (with V ¼ 0) require 2p R21 v ¼

NX 1 21

G1 1 2 exp½2pia=N1

a¼1 NX 2 21

þ

b¼0

G2 ; 1 2 j exp½2piðb=N2 2 a=N1 Þ

ð6:1Þ

Vortex Crystals 2p R22 v ¼

NX 1 21

a¼0

þ

12j

NX 2 21

b¼1

21

23

G1 exp½2piða=N1 2 b=N2 Þ

ð6:2Þ

G2 : 1 2 exp½2pib=N2

The phase f describes how much one vortex polygon is turned relative to the other and j ¼ ðR2 =R1 Þeif : The sums are done using Eqs. (5.4) and (5.5) to yield 2p R21 v ¼

2p R22 v ¼

1 2

ðN1 2 1ÞG1 þ

N2 G; 1 2 j N2 exp½22piaN2 =N1 2

N1 G þ 1 2 j 2N1 exp½22pibN1 =N2 1

1 2

ðN2 2 1ÞG2 :

ð6:3Þ

ð6:4Þ

These relations must hold for all a and b, i.e., exp½iðf 2 2pa=N1 ÞN2 must be real for a ¼ 0; …; N1 2 1; and exp½iðf þ 2pb=N2 ÞN1 must be real for b ¼ 0; …; N2 2 1: Thus, f must be a multiple of p/N1 (set b ¼ 0) and of p/N2 (set a ¼ 0). Without loss of generality, we see from the definition of f that we may choose it to be either 0 or p=N1 : With either choice we see that both 2N1 =N2 and 2N2/N1 must be integers. Since the product of these two quantities is 4, we have just three possibilities: (a) N1 ¼ N2; (b) 2N1 ¼ N2; (c) N1 ¼ 2N2. However, the right hand sides in Eqs. (6.3) and (6.4) must be independent of the index, a or b. This rules out (b) and (c), and we conclude that for two nested, regular polygons to form an equilibrium, they must have the same number of vertices. This is not true if we allow polygons that are not regular, e.g., there are known equilibria consisting of nested polygons of identical vortices with different numbers of vertices, but the polygons are no longer regular. For example, there is an equilibrium of nine identical vortices consisting of an equilateral triangle nested within a hexagon, but the hexagon is not regular. For N1 ¼ N2 ¼ n; where the total number of vortices is even, N ¼ 2n; we refer to the case f ¼ 0 as the symmetric configuration (cf. Fig. 4(a) where n ¼ 3) and f ¼ p=n as the staggered configuration (cf. Fig. 4(c) where n ¼ 4). Equations (6.3) and (6.4) now simplify even further: 2p R21 v ¼

1 2

2p R22 v ¼

nrn G þ n r 21 1

ðn 2 1ÞG1 þ

n G; 1 2 rn 2 ð6:5Þ

1 2

ðn 2 1ÞG2 ;

H. Aref et al.

24

Fig. 4. Nested polygons: (a) symmetric 3 –3; (b) symmetric 4 –4; (c) staggered 4– 4; (d) symmetric 3–3 –3; (e) staggered 3–3–3; and (f) symmetric 5–5. Vortices are shown as solid dots; þ indicates the center of vorticity.

for the symmetric configuration, and 2p R21 v ¼ 2p R22 v

1 2

ðn 2 1ÞG1 þ

nrn ¼ n G þ r þ1 1

n G; 1 þ rn 2 ð6:6Þ

1 2

ðn 2 1ÞG2 ;

for the staggered configuration. In these equations r is the real parameter R2/R1. The ratio of the second equation in Eq. (6.5) to the first gives 2n 2n þ g rn 2 1 þ rnþ2 2 g r2 þ g ¼ 0; ð6:7Þ n21 n21 where g ¼ G2 =G1 : Similarly, the ratio of the second equation in Eq. (6.6) to the first gives 2n 2n nþ2 n þg r þ 1þ r 2 g r2 2 g ¼ 0: ð6:8Þ n21 n21

Vortex Crystals

25

For g ¼ 1, Eq. (6.8) has the solution r ¼ 1 corresponding to the regular N-gon. Analogously, for odd n, Eq. (6.7) has the solution r ¼ 2 1, since the regular N-gon in this case may be thought of as a ‘symmetric’ configuration with a negative ratio of radii. Substituting r ¼ e2r ; we may rewrite Eq. (6.7) as ðn 2 1Þcosh½ðn þ 2Þr ¼ ð3n 2 1Þcosh½ðn 2 2Þr;

ð6:9Þ

which shows that there is a unique, positive solution for r when n $ 2. Similarly, we may rewrite Eq. (6.8) as ðn 2 1Þsinh½ðn þ 2Þr ¼ ð3n 2 1Þsinh½ðn 2 2Þr;

ð6:10Þ

which gives a unique, positive solution for r when n $ 4. For n ¼ 3 there is no nested equilateral triangle solution beyond the regular hexagon. Itpisffiffiinteresting to note that both Eqs. (6.9) and (6.10) lead to the same limit, r ! 3 as n ! 1. The existence and stability of these ‘double-rings’ was discussed by Havelock (1931) in an important paper, where particular attention was paid to the staggered case for g ¼ 2 1, a circular counterpart of the Ka´rmań vortex street. With the notation introduced above, the equation determining the ratio of ring radii in this case is ðn 2 1Þcosh½ðn þ 2Þr ¼ ðn þ 1Þcosh½ðn 2 2Þr:

ð6:11Þ

For large n, the ratio of radii, r, will tend to 1 as 1 þ n 21. The width-to-spacing ratio of the resulting vortex street would asymptotically be ðr 2 1Þ : p=n or 1=p or 0.32, somewhat larger than von Ka´rmań’s ratio for the least unstable point vortex street. If we add to a double-ring of identical vortices a vortex at the center, we obtain yet another family of equilibria with a total of N ¼ 2n þ 1 vortices. With the notation above, the ratio of radii must satisfy

rnþ2 2

3n þ 1 n 3n þ 1 2 r 2 r þ1¼0 nþ1 nþ1

ð6:12Þ

for the symmetric arrangement, or

rnþ2 2

3n þ 1 n 3n þ 1 2 r þ r 21¼0 nþ1 nþ1

ð6:13Þ

for the staggered arrangement. Equation (6.13) has the solution r ¼ 1 and, for odd n, Eq. (6.12) has the solution r ¼ 2 1, corresponding to the centered, regular 2n-gon. For each n there is a unique solution to Eqs. (6.12) and (6.13) corresponding to nested, centered, double-polygon configurations of identical

26

H. Aref et al.

vortices, both symmetric and staggered, with an odd number of vortices in total. By way of example, for n ¼ 2, Eq. (6.12) gives the solution for five vortices on a line in the form 3r4 2 14r2 þ 3 ¼ 0:

ð6:14Þ

On the other hand, from Section IV we know independently that the positions of the vortices are given by the zeros of the Hermite polynomial of degree 5. This polynomial is H5 ðuÞ ¼ 32u5 2 160u3 þ 120u ¼ 4uð8u4 2 40u2 þ 30Þ: pffiffiffi The squares of the roots are 0 and ð5 ^ 10Þ=2: Thesepffiffiffi satisfy Eq.p(4.6), ffiffi ffiffiffi aspthey 2 must. The ratio of radii is given by r ¼ ð5 þ 10 Þ=ð5 2 10Þ ¼ ð 5 þ pffiffi pffiffi pffiffi pffiffiffi 2Þ=ð 5 2 2Þ ¼ ð7 þ 2 10Þ=3; which is, in fact, a solution of Eq. (6.14). It is possible to continue ‘nesting’ polygons in this way, although a systematic investigation does not appear to have been done. A selection of possibilities is shown in Fig. 4. The only comprehensive study of such states of which we are aware is the paper by Lewis and Ratiu (1996). So long as the numbers of vertices in the polygons being nested are commensurate, the algebra appears to work out, but a precise statement is lacking in the literature. We shall return to this notion of ‘commensurability’, and its apparent role in defining allowable patterns, below. Campbell and Ziff (1978, 1979) made Kelvin’s variational principle (Section II) the basis of a numerical algorithm to determine equilibria of N identical vortices. Their 1978 report is often referred to as the Los Alamos Catalog. This study, in turn, was stimulated by the experimental investigation of Yarmchuk et al. (1979) of vortex patterns in superfluid Helium. We are concerned here with the ‘classical’ superfluid 4He, not the more exotic superfluids, 3He and the Bose– Einstein condensates (BEC) that have since also been discovered to display vortex patterns. For vortices in 3He see Lounasmaa and Thuneberg (1999). For vortices in BECs (see Butts and Rokhar, 1999, Abo-Shaer et al., 2001 and Anglin and Ketterle, 2002).1 It would take us too far afield to describe the details of the Yarmchuk et al. (1979) experiment and all the physics behind it. Suffice it to say that in a sample of superfluid Helium, held in a cylindrical container and rotated about its axis at a sufficiently high angular velocity, vortices will nucleate. These vortices will, 1

See http://cua.mit.edu/ketterle_group/Projects_2001/Vortex_lattice/GrayLattice.jpg for pictures of vortex lattices in BECs.

Vortex Crystals

27

predominantly, be aligned with the axis of rotation. Quantum theory restricts vortices in superfluid Helium to being lines with cores of atomic dimensions, and restricts the circulation of the vortices to be a multiple of the quantum of circulation, h=m; where h is Planck’s constant and m the mass of a Helium atom. The value of h=m is approximately 0.001 cm2/s. Usually each vortex carries just a single quantum of circulation; in other words, the vortices are identical. To excellent accuracy the dynamics of the interacting vortices is given by the ideal fluid theory that we have been pursuing and the various states we have been discussing should arise. See the monograph by Donnelly (1991) for a discussion of vortex dynamics in a superfluid. Figure 5 reproduces experimental pictures from the paper by Yarmchuk et al. (1979). The white spots are ‘flow visualizations’ of the vortices, a rather complex affair in a superfluid deep within a cryostat: Ions are directed towards and trapped on the vortices. They are then pulled by an electric field along the vortex, ultimately hitting a phosphorescent screen. The pictures are of the dots on this screen, and thus are much more diffuse than the actual location of the vortices. The pattern is of macroscopic dimensions, as is the frequency of rotation of the sample. Equations such as Eq. (5.6) or (5.7), or generally Eq. (2.17), now take on added significance. Consider Eq. (5.6) for a moment. On the left hand side, we have an angular frequency set by the frequency of rotation of the sample. On the right, we have a radius, R, and a number of vortices, N, both measurable at the macroscopic level. The only unknown is G ¼ h=m; the quantum of circulation, which by its nature we may think of as a microscopic quantity and certainly one that belongs to the realm of quantum physics. The experiment just described, then, becomes one of an elite class of ‘macroscopic quantum measurements’, which have yielded many of the constants of nature with unprecedented accuracy. Unfortunately, the accuracy in determining the geometry of the patterns has not so far been high enough to compete with other techniques for arriving at a value of h=m: However, this connection does provide further motivation for understanding the detailed geometry of rotating vortex patterns. Campbell and Ziff (1978) investigated the range 2 # N # 30 in considerable detail and also did some exploratory calculations for larger N. They claim to have found all linearly stable equilibria for N # 30, a claim that has so far stood the test of time. Stable vortex equilibria were of interest from the start, since if the vortex patterns were to be interpreted as ‘atoms’, as Kelvin and Thomson argued, it was presumably essential that they orbit stably. Similar arguments are applied when predicting the vortex patterns likely to be seen in experiments: The inevitable small amount of dissipation will lead the vortices to form an equilibrated pattern. On the other hand, ‘noise’ from sources not included in Eq. (2.1) will act to perturb

28

H. Aref et al.

Fig. 5. Vortex patterns in a rotated sample of superfluid Helium with 1,…,11 vortices. Note the two more complex equilibria ‘2– 8’ in panel (k) and ‘3–8’ in panel (l). After Yarmchuk et al. (1979) with permission.

any such state, so that only vortex equilibria that are stable should be expected as the persistent patterns. The patterns in Fig. 5, for example, should correspond to stable N-point-vortex equilibria (as indeed they do). However, in a Hamiltonian system stable equilibria are by their nature isolated in the sense that other states with the same values of the integrals of motion cannot evolve to them. Thus, while a stable equilibrium may be the natural end result of an evolutionary process, unstable equilibria are often more important to the dynamical evolution of an N-vortex system. Indeed, unstable equilibria can

Vortex Crystals

29

appear spontaneously during the evolution of the system, and since they are equilibria, these particular configurations, or configurations close to them, will maintain themselves for a relatively long time. In the overall appearance of the system, and in any time averaging, unstable equilibria may therefore play a dominant role. One can often view the evolution of an N-vortex system as a succession of ‘visits’ to the vicinity of the unstable equilibria, where a degree of regularity both in the spatial pattern and the temporal evolution prevails. Between these ‘visits’ the motion evolves more rapidly and characteristic configurations are not in evidence. The exclusive focus on stable equilibria that has historically dominated the subject of vortex statics may, thus, be overly restrictive. As an intriguing case in point, Fig. 6 shows the results of an experiment on the evolution of a magnetized electron plasma in a so-called Malmberg – Penning trap (cf. Durkin and Fajans, 2000a,b). The plasma displays point-vortex-like structures. Once again, we have to omit the details of the experiment. O’Neil (1999) provides an account for a general audience. We must also omit an assessment of how good the analogy between plasma excitations and vortices in an ideal fluid is. Under the right experimental conditions the plasma column behaves two-dimensionally in any plane perpendicular to the applied, magnetic field, which is along the column axis. The plasma evolves through the interaction of its self-electric field with the applied magnetic field. It may be shown that this evolution is governed by equations identical to the 2D Euler equation. The electron density is the counterpart of the fluid vorticity, providing enviable experimental access to this important and usually somewhat inaccessible field. A strongly magnetized electron column is the counterpart of a 2D vortex. These plasma experiments may currently be the best physical laboratory realization in existence of the point vortex equations. In the experiments shown in Fig. 6 the strong vortices (the dots in the figure) are immersed in a low-level ‘background’ vortical fluid. They equilibrate by exchanging energy with this ‘background’. During the transient, which ultimately leads to the seven-vortex pattern settling into the centered, regular hexagon seen in Fig. 6(c), a certain pattern, Fig. 6(b), was observed to ‘linger’ for a considerable time. This pattern is very similar to the one found numerically by Aref and Vainchtein (1998), Fig. 10(c), by a method to be discussed below. The state in Fig. 10(c) is unstable, yet it clearly plays an important role in the dynamical evolution of the system, and it is essential to understand the experiment. The state in Fig. 10(c) does not appear in the Los Alamos Catalog. Incidentally, it was found already by Glass (1997)—see Fig. 1(b) of her paper— who explored equilibrium equations for various many-body systems from a different vantage point.

30

H. Aref et al.

Fig. 6. Stages in the evolution of a seven-vortex system in an electron plasma during ‘cooling’. (a) Initial state; (b) state after 2.5 ms; (c) state at 100 ms. The ‘slow transient’ (b) corresponds to an equilibrium found below (Fig. 10c). The asymptotic state is the centered hexagon. Courtesy of D. Durkin.

In Fig. 7 we have reproduced some of the stable states identified and catalogued by Campbell and Ziff (1978). As in the earlier studies of floating magnets and other systems, and since most of their patterns look as though the vortices are arranged on concentric rings, they chose a convenient and visually suggestive labeling scheme, assigning to each vortex pattern a set of ‘ring numbers’. The resemblance to electron ‘shells’ in pictures of atoms found in elementary chemistry and physics texts would surely have pleased Kelvin! This notion of rings has led to various conjectures about the geometry of equilibria for identical vortices. First, one might conjecture that the symmetry group of an equilibrium is always some finite set of rotations with a ‘generator’ given by the smallest angular deviation between two vortices. This is not true. Closer inspection of the computed results shows that in many cases the vortices are not arranged on exact circles at all. Only when the ring numbers are ‘commensurate’ does one find exact rings. Thus, in Fig. 7(a) neither the two central vortices nor the seven outer vortices form exact rings. Rather, the symmetry of this state is that the two inner and one of the outer vortices are on a line (which is horizontal in the figure), and the remaining six vortices are pairwise symmetrically placed with respect to this

Vortex Crystals

31

Fig. 7. Linearly stable equilibria appearing in the Los Alamos Catalog (Campbell and Ziff, 1978) and there labeled (a) 92; (b) 101 (cf. Fig. 5(k)); (c) 111 (cf. Fig. 5(l)); (d) 121; (e) 131; (f) 141. Vortices are shown as solid dots; þ indicates the center of vorticity.

line. In Fig. 7(b), the two inner vortices are at equal distances from the center of vorticity. The remaining eight are organized into two rectangles not inscribed in a single circle. This state has two perpendicular reflection axes. Figure 7(c), the counterpart of Fig. 5(l), again has just an axis of symmetry— it is vertical in this illustration with one of the inner and two of the outer vortices situated on it. Similar remarks apply to the other configurations shown in Fig. 7. When an axis of symmetry exists, we can rotate the entire configuration such that the complex coordinates of the vortices can be listed as n real values followed by ðN 2 nÞ=2 pairs of complex conjugate positions. Of course, n must be odd for odd N, as in Fig. 7(c) where N ¼ 11 and n ¼ 3, and even for even N, as in Fig. 7(f ) where N ¼ 14 and n can be taken to be 2 (vertical axis) or 4 (horizontal axis) since this configuration has two perpendicular axes of symmetry. These empirical observations suggest that the vortex positions can

32

H. Aref et al.

arise as the roots of a ‘generating polynomial’, in the sense of Eq. (4.2), with real coefficients. Ideally, the equation determining such a polynomial would have arisen in another branch of mathematical physics, or the polynomials for various N would obey a recursion formula, or might otherwise be ‘known’. This idea fits neatly with the appearance of the Hermite polynomials for vortices on a line (Section IV), and with the appearance of the Adler –Moser polynomials for stationary configurations, to be discussed in Section VIII. The vortex polygons, of course, may be associated with the simple polynomials z N 2 1 and zðz N 2 1Þ for the open and centered cases, respectively. Unfortunately, this appealing approach has not thus far led to the desired breakthrough in our analytical understanding of equilibria of identical vortices. Recently, the conjecture that all equilibria have an axis of symmetry was shown to be incorrect by the discovery through numerical computation of completely asymmetric equilibria (Aref and Vainchtein, 1998). Figure 8 shows some of these states, which have been found for 8 # N # 14, but probably exist for all N $ 8. The balance between induced velocities at work in the states in Fig. 8 is quite remarkable and not at all understood analytically. The numerical algorithm used to find the states in Fig. 8 is quite different from algorithms based on Kelvin’s variational principle. We discuss it in the next section since it requires some preparatory considerations that are of independent interest. Figure 8 reproduces two equilibria from Aref and Vainchtein (1998) but also includes other states that were found as part of that study but due to space limitations imposed by the journal, not published. To establish the correspondence, let AVðxÞ denote panel (x) of Fig. 1 in Aref and Vainchtein (1998). Then Fig. 8(a) is a 9-vortex counterpart of AV(d) which, in fact, appears already in Fig. 1(c) of Glass (1997); Fig. 8(b) is a 10-vortex counterpart of AV(e); Fig. 8(c) is AV(f ) (rotated 908); Fig. 8(d) is AV(g) which has a clear ‘family relationship’ to Fig. 8(c); Fig. 8(e) is a representative of yet another ‘family’ of asymmetric equilibria not previously published; and Fig. 8(f ) is an 11-vortex counterpart of AV(h). All these equilibria are linearly unstable. In summary, as we move beyond the regular polygon equilibria for N identical vortices, both open and centered, we encounter a large number of states. Some of these, such as the nested polygons, are sufficiently regular that an analytical understanding may be established. Others, including many of the more complex stable equilibria found by Campbell and Ziff (1978), are at best understood as local minima of the ‘free energy’ in the sense of Kelvin’s variational principle. They give the general impression of having the vortices arranged on concentric circles, but unless the ‘ring numbers’ are commensurate, this is not an accurate description of these states. Their symmetry is lower than one would expect from such a characterization. The general case, and currently

Vortex Crystals

33

Fig. 8. Asymmetric equilibria of identical vortices as found by Aref and Vainchtein (1998). See the text for additional details on how these figures relate to those published in the paper cited. Vortices are shown as solid dots; þ indicates the center of vorticity.

the only viable conjecture, appears to be that linearly stable equilibria have an axis of symmetry. Including unstable equilibria opens up a Pandora’s box of possibilities, including many states that are not ‘round’ at all. Many of these states have the axis of symmetry (Section VII and Fig. 10) of the stable equilibria, but completely asymmetric equilibria have also been found for eight or more vortices (Fig. 8). There is at present no analytical understanding of such states.

VII. Morton’s Equation Consider a uniformly rotating configuration of N vortices. We focus attention on particles in the flow that move as if rigidly attached to the vortex configuration, i.e., particles for which the orbit, zðtÞ; has the form zð0ÞexpðivtÞ;

34

H. Aref et al.

where v is the same angular frequency that enters the equation for the vortex pattern. The vortex pattern satisfies Eq. (2.3) with V ¼ 0: 2ivza ¼

N Gb 1 X 0 ; 2pi b ¼ 1 za 2 zb

ð7:1Þ

and we seek the points z that satisfy the equation 2ivz ¼

N 1 X Ga : 2pi a ¼ 1 z 2 za

ð7:2Þ

There is no prime on this last sum, since the particle feels the velocities induced by all the vortices. Given an equilibrium, Eq. (7.1), we shall refer to points that satisfy Eq. (7.2) as co-rotating points relative to that configuration. We call Eq. (7.2) Morton’s equation, since problems of this type seem to have been first studied systematically in the paper by Morton (1933). For identical vortices of strength G, arranged in a regular N-gon of radius R, Morton’s equation takes the form N X 2pv 1 ; z ¼ G z 2 R1a a¼1

ð7:3Þ

where 1 ¼ expð2p i=NÞ: Using Eq. (5.4) this becomes 2pv N 1 ; z ¼ G z 1 2 ðR=zÞN or, introducing z ¼ z=R and using Eq. (5.6), ðN 2 1Þlzl2 ¼

2N : 1 2 z2N

ð7:4Þ

We see from this equation that z2N must be real. If we write z ¼ r eiw ; we must then either have w ¼ 2pn=N or w ¼ ð2n þ 1Þp=N; n ¼ 0; 1; …; N 2 1: In the former case z2N ¼ r2N : In the latter z2N ¼ 2r2N : Now Eq. (7.4) produces two equations ðN 2 1Þr2 ¼

2N ; 1 2 r2N

ð7:5Þ

ðN 2 1Þr2 ¼

2N : 1 þ r2N

ð7:6Þ

and

Vortex Crystals

35

For N ¼ 3, for example, we get r ¼ 0 as a solution, and two cubics

r3 2 3r 2 1 ¼ 0;

and

r3 2 3r þ 1 ¼ 0;

ð7:7Þ

with real solutions

r ¼ 2 cos

p ; 9

pffiffi p p 2 3 sin ; 9 9

2cos

pffiffi 2p 2p 2 3 sin ; 9 9

2cos

2cos

p pffiffi p þ 3 sin ; 9 9

and

r ¼ 2 cos

2p ; 9

2cos

2p pffiffi 2p þ 3 sin ; 9 9

respectively.2 Only the positive solutions are physically meaningful, so we arrive at the following nine co-rotating points for the equilateral triangle with three identical vortices: pffiffi 2p 2p ip=3 2p ip=3 p 2 cos e 3 sin 2 cos 22 cos eip=3 e 9 9 9 9 pffiffi 2p 2p 2p p þ cos 22 cos 2 cos 2 3 sin 9 9 9 9 pffiffi 2p 2p i2p=3 2p i2p=3 p 2 cos e 2 3 sin 22 cos 2 cos ei2p=3 e 9 9 9 9 Including the origin, we have 10 solution points in all for Eqs. (7.5) and (7.6) with N ¼ 3. Figure 9 depicts the rotating equilateral triangle and its co-rotating points. Equations (7.5) and (7.6) for general N give r ¼ 0 or r N 2 ð2N=ðN 2 1ÞÞrN22 ¼ ^1: The polynomial on the left hand side vanishes at r ¼ 0, is negative for smallffi r, and positive for large r. It has a minimum at r ¼ pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2ðN 2 2Þ=ðN 2 1Þ: The minimum value is less than 2 2 for N $ 3. There is a pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi unique, positive zero at r ¼ ð2N=ðN 2 1Þ: For r ¼ 2, the polynomial is larger than 2 when N $ 3. For w ¼ 2pn=N; n ¼ 0; 1; …; N 2 1; that is, the case described by Eq. (7.5), there will always be a unique solution, r3, for r, with p ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ð2N=ðN 2 1Þ , r3 , 2; and N co-rotating points all with r ¼ r3 : For w ¼ ð2n þ 1Þp=N; n ¼ 0; 1; …; N 2 1; wepare considering Eq. are now ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ffi (7.6). There pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi two solutions, r1 and r2, 0 , r1 , 2ðN 2 2Þ=ðN 2 1Þ , r2 , 2N=ðN 2 1Þ for r. (Actually, one can easily verify that r1 , 1.) We thus obtain 2N co-rotating points for a total of 3N þ 1, including the origin. 2

Since ðr3 2 3r 2 1Þðr3 2 3r þ 1Þ ¼ r6 2 6r4 þ 9r2 2 1 contains only even powers, the roots of the two polynomials arise as pairs of opposites. This may not be immediately clear from the formulae. p It is, nevertheless, true that 2 cosðp=9Þ ¼ cosð2p=9Þ þ 3 sinð2p=9Þ; 2 cosð2p=9Þ ¼ cosðp=9Þ þ p p p 3 sinðp=9Þ; and 2cosðp=9Þ þ 3 sinðp=9Þ ¼ cosð2p=9Þ 2 3 sinð2p=9Þ:

H. Aref et al.

36

Fig. 9. Equilateral triangle of three identical vortices (solid dots) and its co-rotating points (open circles).

Fig. 10. Symmetric vortex equilibria for 6, 7 and 8 identical vortices found via Eqs. (6.5), (6.7) and (6.9) as part of the study reported in Aref and Vainchtein (1998). The state in (c) is the equilibrium counterpart of the transient observed in the experiment of Fig. 6(b). Vortices are shown as solid dots; þ indicates the center of vorticity.

Vortex Crystals

37

One might have thought by counting powers in Eq. (7.2) that it should yield at most N þ 1 solutions, but the complex conjugation on the left hand side throws off this count. Thus, for the N-gon of identical vortices we found 3N þ 1 solutions. A count of the number of co-rotating points as a function of the vortex strengths in the ‘general case’ has apparently not been performed but should be possible using Be´zout’s theorem. Calculations show that there are usually more co-rotating points than one might immediately have surmised, and certainly more than the number of vortices in the configuration. Consider now the system comprised of N vortices in an equilibrium configuration and one of the co-rotating points. For simplicity let the vortices be identical. These equations are precisely what we would write down if we were to find an equilibrium configuration of N þ 1 ‘vortices’, where the first N are identical and the last is a ‘vortex’ of circulation zero! Now, let this last ‘ghost’ vortex acquire a tiny circulation dG; where d ,, 1. Then the equilibrium equations for the system of N þ 1 vortices are N X 2pv 1 d 0 za ¼ þ ; a ¼ 1; …; N: G z 2 z z 2 zNþ1 b a b¼1 a

ð7:8Þ

N X 2pv 1 zNþ1 ¼ : G z 2 za a ¼ 1 Nþ1

ð7:9Þ

In general, the positions za ; a ¼ 1; …; N þ 1; that solve Eqs. (7.8) and (7.9) will be slightly different from the position za ; a ¼ 1; …; N þ 1; and z that solve Eqs (7.1) and (7.2). Equations (7.8) and (7.9) thus describe an algorithm in which d is incremented step-by-step and the system is solved at each step. In this way the (N þ 1)st vortex, which started ‘life’ as a co-rotating point of the N-vortex equilibrium, may gradually be ‘grown’ to the same strength, G, as the remaining N vortices, i.e., d is gradually incremented from 0 to 1. There is, of course, no guarantee that a smooth, parametric evolution will result. Indeed, numerical experiments with Eqs. (7.8) and (7.9) suggest that bifurcations sometimes occur and that various procedures need to be used to negotiate bifurcation points. These may be as crude as relaxing the convergence criterion on the solution and augmenting d further until the bifurcation value has been over-stepped, and then tightening the convergence criterion once again so that a specific solution branch is selected. Computational experience suggests that the net displacement of any of the original vortices in Eqs. (7.8) and (7.9), or even the co-rotating point that is being ‘grown’ into a vortex, is quite small as d is increased from 0 to 1. Thus, given equilibrium states for N þ 1 vortices, one can often guess quite reliably

H. Aref et al.

38

which of these states will be produced from a given N-vortex state and its co-rotating points. Many variations on the method just outlined are possible. More than one co-rotating point can be ‘grown’ simultaneously and, if desired, different points can be ‘grown’ at different rates. Conversely, a vortex in an (N þ 1)-vortex equilibrium can be decreased in strength until it becomes a zero-circulation ‘ghost’, and a co-rotating point, in an N-vortex equilibrium. The end result is that using Eqs. (7.8) and (7.9) a large number of new vortex equilibria are found, many of them quite different in nature and appearance from the concentric ring solutions in the Los Alamos Catalog. These new solutions have so far all been unstable, as one might expect from their construction. In Fig. 10 we provide a sampling of symmetric equilibria that have been obtained by this method. The state in Fig. 10(c) will be recognized as the ‘slow transient’ seen in the experiment of Fig. 6. All of these equilibria have an axis of symmetry. The solution method based on Eqs. (7.8) and (7.9) was also the one that led to the asymmetric equilibria mentioned previously (Fig. 8). We should remark that although there are many co-rotating points to use in this algorithm, there are also usually symmetries that reduce the number of these points that are intrinsically different. Thus, in Fig. 9, we see that by symmetry there are only four intrinsically different co-rotating points that can be used in Eqs. (7.8) and (7.9). There are yet other ways of solving Eq. (7.1) for identical vortices. Consider, for example, the following recursion in which N new vortex positions {^za } are obtained from the current positions {za }: N X 0

1 z 2 zb b¼1 a zâ ¼ vffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2ffi : u uX N X N 1 u 0 t z 2 zm l ¼ 1 m ¼ 1 l

ð7:10Þ

The construct in Eq. (7.10) assures that

N X

a¼1

zâ ¼ 0;

2 N X N X 1 0 N N z 2 zb X X a ¼ 1 b ¼ 1 a 2 l^za l ¼ zâ z^ a ¼ 2 ¼ 1: N X N X a¼1 a¼1 1 0 z 2 zm l ¼ 1 m ¼ 1 l

ð7:11Þ

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39

Numerical experiments show that the iteration (7.10) converges to a fixed point for many different initial conditions. A fixed point of the iteration will satisfy

V za ¼

N X 0

b¼1

1 ; za 2 zb

vffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2ffi u uX N X N u 1 0 : V¼t z 2 zm l ¼ 1 m ¼ 1 l

ð7:12Þ

It is not difficult to show that in this case, because of Eq. (7.11), we must have V ¼ NðN 2 1Þ=2: The final state may be renormalized to produce the desired solution to Eq. (7.1) for N identical vortices. Starting with randomly chosen initial positions, this algorithm yields many equilibria not easily found by methods based on Kelvin’s variational principle. The states shown in Fig. 7 were produced using this method. We comment in closing that the dynamical stability of a vortex pattern and the ability of a numerical algorithm to produce it as a solution, e.g., as a fixed point of an iterative scheme, have no simple relationship, in general. Thus, the method of Campbell and Ziff (1978) did capture some dynamically unstable equilibria even though it was designed to seek out minima of the ‘free energy’ in the sense of Kelvin’s variational principle. The other methods we have highlighted capture further dynamically unstable equilibria. Thus far no method has claimed to capture all equilibria for N less than some limit.

VIII. Stationary Vortex Patterns It may sound surprising that configurations of vortices exist where the total circulation is non-zero, yet every vortex in the pattern remains at rest. A single vortex, of course, has precisely this property. Furthermore, when K ¼ 0, Eq. (2.17) tells us that the only possible equilibria are stationary patterns, and K ¼ 0 assures us that the sum of the circulations, S, is non-zero by Eq. (2.12). Simple examples, such as an equilateral triangle of identical vortices with an opposite vortex at the center, and the collinear state mentioned in Eq. (2.25), tell us that ‘non-trivial’ stationary equilibria exist. The question, then, is how to find solutions to Eq. (2.3) of this particular type. As a point of reference we may recall that an equilateral triangle configuration of three vortices with K ¼ 0 is not stationary but rotates about its center of vorticity, so K ¼ 0 is a necessary but not a sufficient condition for a relative equilibrium to be stationary. For general N, the most elegant and far-reaching developments have been obtained for the case when all the vortices have the same absolute value of the circulation, i.e., we have Nþ vortices of circulation þ G and N2 of circulation 2 G.

H. Aref et al.

40

With this ‘quantization’ of the strengths, Eq. (2.12) at once stipulates that if K ¼ 0, then ðNþ 2 N2 Þ2 ¼ Nþ þ N2 : Denote Nþ 2 N2 by n, so that N ¼ Nþ þ N2 ¼ n2 ; and solve the two resulting, linear equations for Nþ and N2 together to obtain N2 ¼ 12 nðn 2 1Þ;

Nþ ¼ 12 nðn þ 1Þ; n ¼ 1; 2; …

ð8:1Þ

Thus, the number of vortices of the two circulations must be successive triangular numbers (and we have, arbitrarily, chosen the majority population to be the vortices with positive circulation). The total number of vortices is a square. The counting in Eq. (8.1) captures the single vortex for n ¼ 1, which formally constitutes the smallest vortex ‘system’ with K ¼ 0! The equilateral triangle of identical vortices with an opposite vortex at the center will turn out to be the unique solution for n ¼ 2. In the general case, we return to the ideas of Section IV and set PðzÞ ¼ ðz 2 z1 Þ· · ·ðz 2 zNþ Þ;

QðzÞ ¼ ðz 2 z1 Þ· · ·ðz 2 zN2 Þ:

ð8:2Þ

Here z1 ; …; zNþ are the complex positions of the positive vortices, and z1 ; …; zN2 the positions of the negative vortices, where N2 and Nþ are as in Eq. (8.1). The equations determining these positions in this case are Eq. (2.3) with zeros on the left hand sides. Thus, Nþ X 0

b¼1

N2 X 1 1 ¼ ; za 2 zb z 2 zl l¼1 a

Nþ X

a¼1

N2 X 1 1 0 ¼ : zl 2 za z 2 zm m¼1 l

ð8:3Þ

Now calculate as in Section IV Nþ X

Q 0 ðzÞ ¼ QðzÞ

1 1 ; z 2 za z 2 zb

Q 00 ðzÞ ¼ QðzÞ

a¼1

P 00 ðzÞ ¼ PðzÞ

Nþ X 0

a; b ¼ 1

P 00 ðzÞ ¼ 2PðzÞ

Nþ X 0

a;b ¼ 1

N2 X

1 ; z 2 za

P 0 ðzÞ ¼ PðzÞ

1 1 ; z 2 za za 2 zb

l¼1

1 ; z 2 zl

N2 X 0

l;m ¼ 1

Q 00 ðzÞ ¼ 2QðzÞ

N2 X 0

l;m ¼ 1

At this point we use Eq. (8.3) to re-write P 00 ðzÞ and Q 00 ðzÞ as P 00 ðzÞ ¼ 2PðzÞ

Nþ X

a¼1

1 1 ; z 2 zl z 2 zm

N2 X 1 1 ; z 2 za l ¼ 1 za 2 z l

1 1 : z 2 zl zl 2 zm

Vortex Crystals

41

and N2 X


Nþ X 1 1 : z 2 zl a ¼ 1 zl 2 za

l¼1

From these relations 00

00

QP þ PQ ¼ 2PQ

Nþ X N2 X

a¼1 l¼1

¼ 2PQ

Nþ X

a¼1

1 za 2 z l

1 1 2 z 2 za z 2 zl

N2 X 1 1 ; z 2 za l ¼ 1 z 2 zl

i.e., we have QP 00 þ PQ 00 ¼ 2P 0 Q 0 :

ð8:4Þ

We shall call this result Tkachenko’s equation, since it was first derived by Tkachenko (1964) in his (unpublished) thesis. Assume the center of vorticity of the entire configuration is chosen as the origin of coordinates such that Nþ X

za 2

a¼1

N2 X

zl ¼ 0:

l¼1

Then, from Eq. (2.19) we have the result Nþ X

za þ

a¼1

N2 X

zl ¼ 0;

l¼1

since all the vortices have the same absolute strength, and the entire configuration is stationary. Thus, Nþ X

za ¼

a¼1

N2 X

zl ¼ 0:

ð8:5Þ

l¼1

For example, for n ¼ 2 we now have z1 þ z2 þ z3 ¼ 0;

z1 ¼ 0;

hence PðzÞ ¼ z3 þ bz þ a;

QðzÞ ¼ z;

where a, b are coefficients to be determined. Substitution in Eq. (8.4) gives 6z2 ¼ 2ð3z2 þ bÞ;

H. Aref et al.

42

i.e., b ¼ 0, and we have PðzÞ ¼ z3 þ a: This implies that the three positive vortices form an equilateral triangle with the negative vortex at the center as the unique solution for n ¼ 2. To go further we note the following simple algebraic result: Let P, Q be polynomials (not identically zero) that satisfy Eq. (8.4), and let R be a polynomial that satisfies R0 Q 2 RQ 0 ¼ P2 :

ð8:6Þ

Then P and R satisfy Eq. (8.4). The advantage of this is, of course, that Eq. (8.6) is a first order differential equation for R (given P and Q), whereas Eq. (8.4) is a second order differential equation for P given Q. This result permits the polynomials for successive n to be found recursively, as we shall see below. The result just stated is not difficult to prove: From the relation Eq. (8.6) it follows that R 00 Q 2 RQ 00 ¼ 2PP 0 : Thus, QðR 00 P þ RP 00 Þ ¼ Pð2PP 0 þ RQ 00 Þ þ QRP 00 ¼ 2P2 P 0 þ RðP 00 Q þ PQ 00 Þ: Since P, Q satisfy Eq. (8.4), the last term can be replaced by 2RP 0 Q 0 ; and we have QðR 00 P þ RP 00 Þ ¼ 2P 0 ðP2 þ RQ 0 Þ ¼ 2QR 0 P 0 ; where Eq. (8.6) has been used in the last step. Dividing by Q gives the desired result. As just mentioned, this result allows Tkachenko’s equation (8.4) to be solved recursively. Consider a sequence of polynomials, Pn ; generated as follows: P 0nþ1 Pn21 2 P 0n21 Pnþ1 ¼ ð2n þ 1ÞP 2n ; n ¼ 1; 2; …

ð8:7Þ

starting from P0 ¼ 1, P1 ¼ z: For n ¼ 1 the recursion formula (8.7) with these starting polynomials tells us that P 02 ¼ 3z 2 ; i.e., P2 ¼ z3 þ const:; so the start of the recursive construction is in order. Assume we have constructed polynomials through Pn recursively from Eq. (8.7). We wish to verify that Pnþ1 ; constructed from Eq. (8.7), and Pn satisfy Eq. (8.4). We shall also verify that the normalization factor, 2n þ 1; on the right hand side of Eq. (8.7) leads to the coefficient of the highest power in Pn being 1.

Vortex Crystals

43

pffiffiffiffiffiffiffiffiffi We first use the result Eq. (8.6) with Q ¼ Pn21 ; P ¼ 2n þ 1 Pn ; and R ¼ Pnþ1 : By assumption Pn21 and Pn ; or equivalently Q and P, satisfy Tkachenko’s equation. Since Q, P and R satisfy Eq. (8.6), we conclude that P and R, or equivalently Pn and Pnþ1 ; satisfy Eq. (8.4). Next, we assume that Pn21 and Pn both have 1 as the coefficient of their highest order terms. Denote the coefficient of the highest degree term in Pnþ1 by Anþ1 : If the degree of Pn is denoted by dn ; equating terms of highest degree in the recursion formula (8.7) (both matching the degree itself and considering the coefficient) tells us that dnþ1 2 1 þ dn21 ¼ 2dn ; Anþ1 ðdnþ1 2 dn21 Þ ¼ 2n þ 1: The first of these relations tells us that dnþ1 2 dn ¼ dn 2 dn21 þ 1, i.e., since d0 ¼ 0, d1 ¼ 1 that dnþ1 2 dn ¼ n þ 1; or dn ¼ nðn þ 1Þ=2; as we would have expected from Eq. (8.1). The second relation then shows that Anþ1 ¼ 1: All this is quite straightforward. What is considerably less clear is that Eq. (8.7), viewed as a first order ODE for Pnþ1 ; with Pn21 and Pn already determined polynomials of the appropriate degrees, will again yield a polynomial! A detailed proof may be found in Adler and Moser (1978). Elements of the proof are indicated below. Let us first show the next step of the recursion. We set P1 ðzÞ ¼ z; P2 ðzÞ ¼ z3 þ a and seek P3 ðzÞ: From Eq. (8.5) and our other considerations we know it has the form P3 ðzÞ ¼ z6 þ Ez4 þ Dz3 þ Cz2 þ bz þ A; where A, b, C, D and E are coefficients to be determined. From Eq. (8.7) with n¼2 ð6z5 þ 4Ez3 þ 3Dz2 þ 2Cz þ bÞz 2 ðz6 þ Ez4 þ Dz3 þ Cz2 þ bz þ AÞ ¼ 5ðz3 þ aÞ2 ; or 5z6 þ 3Ez4 þ 2Dz3 þ Cz2 2 A ¼ 5z6 þ 10az3 þ 5a2 ; i.e., P3 ðzÞ ¼ z6 þ 5az3 þ bz 2 5a2 ;

ð8:8Þ

44

H. Aref et al.

where b can have any value and thus is a second, free parameter. Similarly, one finds P4 ðzÞ ¼ z10 þ 15az7 þ 7bz5 þ cz3 2 35abz2 þ 175a3 z 2

7 2 b þ ac; 3

ð8:9Þ

where c is a third, free parameter. In each step of the recursion we introduce a new parameter, so that given a configuration of the minority species via the polynomial, Pn ; there is a oneparameter family of polynomials, Pnþ1 ; whose roots give the positions of the majority species. Note that the parameter b enters Eq. (8.8) through a term bP1 ðzÞ and the parameter c enters Eq. (8.9) through cP2 ðzÞ: It is clear from Eq. (8.7) that, in general, Pnþ1 is only determined up to the addition of a multiple of Pn21 : The parameter a has dimensions of ‘length’ cubed, the parameter b dimensions of length to the fifth power, the parameter c ‘length’ to the seventh power, and so on. Stationary patterns may now be displayed by simply specifying values for a; b; c; … and solving the resulting polynomials. Some particularly symmetrical examples are shown in Fig. 11. Thus, Fig. 11(a) shows the solution, unique up to scale, for n ¼ 2. In Fig. 11(b) we show the 3 –6 configuration for a ¼ 2 1, b ¼ 0,

Fig. 11. Examples of stationary equilibria generated from pairs of Adler– Moser polynomials.

Vortex Crystals

45

i.e., P2 ðzÞ ¼ z3 2 1; P3 ðzÞ ¼ z6 2 5z3 2 5: Figure 11(c) shows the 6 – 10 configuration for a ¼ 0, b ¼ 2 1, c ¼ 0, i.e., P3 ðzÞ ¼ zðz5 2 1Þ; P4 ðzÞ ¼ z10 2 7z5 2 7=3: The negative vortices form a centered, regular pentagon. The positive vortices form two nested, staggered pentagons. For 21 negative and 28 positive vortices Kadtke and Campbell (1987) find the generating polynomials: z21 þ 15z14 2 66z7 =5 2 11=25; z28 þ 55z21 2 2211z14 =5 2 9438z7 =25 þ 1573=125: (We have rescaled the variable z in order to simplify the coefficients.) These correspond to three nested, regular heptagons of negative vortices, staggered with respect to one another, and four, staggered, regular heptagons of positive vortices, all nested as shown in Fig. 11(d). From our present vantage point such diagrams are, of course, trivial to generate. The reader may compare this situation to our discussion of relative equilibria of identical vortices, where the calculation and classification of patterns was of necessity left in a rather incomplete state. The polynomials generated by Eq. (8.7) are known as the Adler –Moser polynomials. They arose in studies of rational solutions of the Korteweg – deVries equation and their related ‘pole decomposition’ equations (Airault et al., 1976). Bartman (1983) was the first to recognize the connection between that work and Tkachenko’s equation. See also Kadtke and Campbell (1987). The Adler – Moser polynomials have a remarkably simple construction: Consider the recursion w1 ¼ z; w00n ¼ wn21 for n $ 2. This clearly leads to a sequence of polynomials, w2 ¼ w3 ¼

1 5 5! z

1 3 3! z

þ az þ b;

þ 3!1 az3 þ 2!1 bz2 þ cz þ d;

and so on, where a; b; c; d; … are arbitrary constants (but not the same as the arbitrary constants designated by the same symbols in Eqs. (8.8) and (8.9) above). Now consider the Wronskians W1 ðw1 Þ ¼ w1 ; w1 W2 ðw1 ; w2 Þ ¼ 0 w 1 w1 W3 ðw1 ; w2 ; w3 Þ ¼ w01 w00 1

w2 ¼ w1 w02 2 w01 w2 ; 0 w 2

w2 w02 w002

w3 w03 ; w00 3

46

H. Aref et al.

and so on. (Of course, the third row of W3 can be written in terms of the entries in the first row because of the construction of the wn :) The key insight is that the nth Adler –Moser polynomial is proportional to the nth Wronskian in this sequence: Pn ¼ 1n 3n21 5n22 · · ·ð2n 2 1Þ1 £ Wn ðw1 ; w2 ; …; wn Þ:

ð8:10Þ

Adler and Moser (1978) acknowledge developments by Crum (1955) a dozen years before in establishing this elegant formula. Many interesting features of these solutions undoubtedly remain to be discovered. Let us just note one that is apparently new (although the idea behind it was suggested by Khanin, 1982). Consider replacing every positive vortex in one of these stationary equilibria by an infinitesimal version of Fig. 11(a). Similarly replace every negative vortex by an infinitesimal version of that same configuration with the circulations reversed (so that it has a net negative circulation). This should, once again, be an equilibrium, since at short range we have the dominant balance that prevails within Fig. 11(a), and at long range we have the balance that prevailed in the original, stationary equilibrium. The counting of vortices also works out: If we had Tn21 ¼ nðn 2 1Þ=2 negative and Tn ¼ nðn þ 1Þ=2 positive vortices before the replacements, we will have a total of 3Tn þ Tn21 ¼ 2n2 þ n ¼ 12 2nð2n þ 1Þ ¼ T2n positive and Tn þ 3Tn21 ¼ 2n2 2 n ¼ 12 2nð2n 2 1Þ ¼ T2n21 negative vortices after the replacements. (Elegant geometrical proofs of these identities for the triangular numbers are given by Nelsen, 1993.) In terms of the generating polynomials the replacement implies that if a pair of polynomials P, Q satisfy Eq. (8.4), then the pair p ¼ P3 Q; q ¼ PQ3 should also satisfy this relation. It is straightforward, albeit slightly tedious, to check this: p0 ¼ 3QP2 P 0 þ P3 Q 0 ;

q 0 ¼ 3PQ2 Q 0 þ Q3 P 0 ;

p00 ¼ 6P2 P 0 Q 0 þ 6QPðP 0 Þ2 þ 3QP2 P00 þ P3 Q00 ; q00 ¼ 6Q2 Q 0 P 0 þ 6PQðQ 0 Þ2 þ 3PQ2 Q00 þ Q3 P00 : Thus, qp00 þ pq00 ¼ 12P3 Q3 P 0 Q 0 þ 6P2 Q2 ½ðQP 0 Þ2 þ ðPQ 0 Þ2 þ 4P3 Q3 ðQP00 þ PQ00 Þ:

Vortex Crystals

47

In the last term we now use Eq. (8.4) for P and Q to obtain qp00 þ pq00 ¼ 6P2 Q2 ½ðQP 0 Þ2 þ ðPQ 0 Þ2 þ 20P3 Q3 P 0 Q 0 : This is easily seen to equal 2p 0 q 0 : IX. Translating Vortex Patterns The vortex pair—two opposite vortices translating side by side, a 2D counterpart of the circular vortex ring—has inspired attempts to construct configurations of several point vortices, with the sum of all circulations equal to zero, that translate like a rigid body. Three vortices with sum of circulations equal to zero, placed at the vertices of an equilateral triangle, produce such a translating state according to our classification from Section II. Let us again consider the case of vortices of the same absolute circulation. Since the total circulation must vanish, the total number of vortices, N, is even, N ¼ 2n; and there are n vortices of either circulation. We again let z1,…,zn designate the complex positions of the n positive vortices, and z1,…,zn the positions of the n negative vortices. The vortex pair, with one vortex of either sign, corresponds to n ¼ 1. If each vortex is translating with velocity V (a complex number giving both speed and direction of propagation), Eqs. (2.3) determining the vortex positions become: n n X X 2pi 1 1 0 2 ; V¼ z 2 zb z 2 zl G b¼1 a l¼1 a

ð9:1Þ

n n X X 2pi 1 1 2 0 : V¼ G z 2 za z 2 zm a¼1 l m¼1 l

ð9:2Þ

The quantity on the left hand side of Eqs. (9.1) and (9.2) will appear frequently, and so we introduce the abbreviation

x¼

2pi V: G

ð9:3Þ

For this set of circulations K ¼ 22nG 2 ; [see Eq. (2.12)], and Eqs. (2.15), (2.16) or (2.18) gives ! n n X X x za 2 zl ¼ 2n: a¼1

l¼1

ð9:4Þ

H. Aref et al.

48

Next, consider this state from the point of view of Eq. (2.19), which takes the form ! n n n n X X X X 2 2 x za 2 zl ¼ 2 za 2 zl : ð9:5Þ a¼1

l¼1

a¼1

l¼1

Eliminating x between Eqs. (9.4) and (9.5) now gives ! ! n ! n n n n n X X X X X X 2 2 za 2 zl ¼ za 2 zl za þ zl ; n a¼1

l¼1

a¼1

l¼1

a¼1

l¼1

or n 1 X z2 2 n a¼1 a

n 1 X z n a¼1 a

!2

n 1 X ¼ z2 2 n l¼1 l

n 1 X z n l¼1 l

!2 :

ð9:6Þ

Formally, the quantities on both sides of Eq. (9.6) are ‘variances’ of the complex coordinates of the vortices in the two populations. The relation (9.6) suffices to show that Eqs. (9.1) and (9.2) have no solution for n ¼ 2. This is somewhat surprising, since one might have thought that two pairs, placed at a great distance from one another, would translate independently with minimal mutual influence and so would approximate a translating state for n ¼ 2. One might further have assumed that slight adjustments to this state would make it exactly satisfy Eqs. (9.1) and (9.2), and that it would then be possible to move the two pairs closer and in that way generate a one-parameter family of translating states. As we shall now show, none of this ‘physical intuition’ is correct. For n ¼ 2, Eq. (9.6) becomes 1 2

ðz21 þ z22 Þ 2 14 ðz1 þ z2 Þ2 ¼ 12 ðz21 þ z22 Þ 2 14 ðz1 þ z2 Þ2 ;

or ðz1 2 z2 Þ2 ¼ ðz1 2 z2 Þ2 i.e., z1 2 z2 ¼ ^ðz1 2 z2 Þ:

ð9:7Þ

xðz1 þ z2 2 z1 2 z2 Þ ¼ 22:

ð9:8Þ

From Eq. (9.4) we also have

Equation (9.7) with the þ sign, taken together with Eq. (9.8), will then give

xðz1 2 z1 Þ ¼ xðz2 2 z2 Þ ¼ 21:

Vortex Crystals

49

Now, Eq. (9.1) for z1 becomes

x¼

1 1 1 1 1 2 2 ¼ þx2 ; z1 2 z2 z1 2 z 1 z1 2 z2 z1 2 z2 z1 2 z2

implying z2 ¼ z2 ; which is unacceptable. Equation (9.7) with the 2 sign runs into a similar contradiction. Thus, for n ¼ 2 there are no solutions to Eqs. (9.1) –(9.3). For a general discussion we again introduce ‘generating polynomials’ PðzÞ ¼ ðz 2 z1 Þ· · ·ðz 2 zn Þ;

QðzÞ ¼ ðz 2 z1 Þ· · ·ðz 2 zn Þ;

ð9:9Þ

and calculate as before, following Eq. (8.3): P 0 ðzÞ ¼ PðzÞ

n X

a¼1

P 00 ðzÞ ¼ 2PðzÞ

n X 0

a; b ¼ 1

1 ; z 2 za

1 1 ; z 2 za za 2 zb

n X

Q 0 ðzÞ ¼ QðzÞ

l¼1


1 ; z 2 zl

n X 0

l;m ¼ 1

1 1 : z 2 zl zl 2 zm

Next, use Eqs. (9.1) – (9.3) to re-write P 00 ðzÞ and Q 00 ðzÞ as 00

P ðzÞ ¼ 2PðzÞ

n X

a¼1

and 00

Q ðzÞ ¼ 2QðzÞ

n X

l¼1

! n X 1 1 ; xþ z 2 za z 2 zl l¼1 a

! n X 1 1 : 2x þ z 2 zl z 2 za a¼1 l

From these relations we get [cf. the derivation of Eq. (8.4)]: QP 00 þ PQ 00 ¼ 2xðP 0 Q 2 PQ 0 Þ þ 2P 0 Q 0 :

ð9:10Þ

Equating coefficients of the highest order terms gives us back Eq. (9.4). The vortex pair corresponds to n ¼ 1, i.e., PðzÞ ¼ z 2 z1 ; QðzÞ ¼ z 2 z1 ; which solves Eq. (9.10) if 0 ¼ 2xðz 2 z1 2 z þ z1 Þ þ 2; i.e., if

xðz1 2 z1 Þ ¼ 21: This is the result for the translation velocity of the vortex pair in another guise. For n ¼ 2 we have QP 00 þ PQ 00 ¼ 2ðz 2 z1 Þðz 2 z2 Þ þ 2ðz 2 z1 Þðz 2 z2 Þ ¼ 4z2 2 2ðz1 þ z2 þ z1 þ z2 Þz þ 2ðz1 z2 þ z1 z2 Þ;

H. Aref et al.

50

2xðP 0 Q 2 PQ 0 Þ ¼ 2x½ð2z 2 ðz1 þ z2 ÞÞðz 2 z1 Þðz 2 z2 Þ 2 ðz 2 z1 Þðz 2 z2 Þð2z 2 ðz1 þ z2 ÞÞ ¼ 2x½ðz1 þ z2 2 z1 2 z2 Þz2 2 2ðz1 z2 2 z1 z2 Þz þ z1 z2 ðz1 þ z2 Þ 2 z1 z2 ðz1 þ z2 Þ; 2P 0 Q 0 ¼ 2½2z 2 ðz1 þ z2 Þ½2z 2 ðz1 þ z2 Þ ¼ 8z2 2 4ðz1 þ z2 þ z1 þ z2 Þz þ 2ðz1 þ z2 Þðz1 þ z2 Þ: Balancing coefficients of z 2 returns (9.8). The coefficients of z give 2xðz1 z2 2 z1 z2 Þ ¼ 2ðz1 þ z2 þ z1 þ z2 Þ:

ð9:11Þ

Finally, balancing the constant terms requires 2ðz1 z2 þ z1 z2 Þ ¼ 2x½z1 z2 ðz1 þ z2 Þ 2 z1 z2 ðz1 þ z2 Þ þ 2ðz1 þ z2 Þðz1 þ z2 Þ:

ð9:12Þ

We write the square bracket on the right hand side of Eq. (9.12) as 1 2

ðz1 þ z2 þ z1 þ z2 Þðz1 z2 2 z1 z2 Þ 2 12 ðz1 þ z2 2 z1 2 z2 Þðz1 z2 þ z1 z2 Þ;

and then use Eqs. (9.8) and Eq. (9.11) to eliminate x in Eq. (9.12). This gives 2ðz1 z2 þ z1 z2 Þ ¼ 2 12 ðz1 þ z2 þ z1 þ z2 Þ2 þ 2ðz1 z2 þ z1 z2 Þ þ 2ðz1 þ z2 Þðz1 þ z2 Þ; i.e., z1 þ z2 ¼ z1 þ z2 ; which contradicts (9.8). Again, we conclude that no solution exists for n ¼ 2. For n ¼ 3, on the other hand, we can easily give explicit solutions. For example, set PðzÞ ¼ ðz þ cÞ3 2 4c3 ;

QðzÞ ¼ ðz 2 cÞ3 þ 4c3 :

ð9:13Þ

Then P 0 ðzÞ ¼ 3ðz þ cÞ2 ; Q 0 ðzÞ ¼ 3ðz 2 cÞ2 ; P 00 ðzÞ ¼ 6ðz þ cÞ; and Q 00 ðzÞ ¼ 6ðz 2 cÞ: Thus, QP 00 þ Q 00 P ¼ 6ðz2 2 c2 Þ½ðz 2 cÞ2 þ ðz þ cÞ2 þ 24c3 ½ðz þ cÞ 2 ðz 2 cÞ ¼ 12ðz4 þ 3c4 Þ; QP 0 2 Q 0 P ¼ 3ðz2 2 c2 Þ2 ½z 2 c 2 ðz þ cÞ þ 12c3 ½ðz þ cÞ2 þ ðz 2 cÞ2 ¼ 6cð2z4 þ 6c2 z2 þ 3c4 Þ; 2P 0 Q 0 ¼ 18ðz2 2 c2 Þ2 ¼ 18ðz4 2 2c2 z2 þ c4 Þ;

Vortex Crystals

51

i.e., P, Q satisfy Eq. (9.10) if 2cx ¼ 1: These configurations consist of two equilateral triangles each with vortices of one sign that are mirror images of one another. Dividing by PQ Eq. (9.10) may be written 2 3 00 0 2 0 P P P 4 5 log þ log 22x log z ¼ 22ðlog QÞ00 : Q Q Q Thus, if we set

cðzÞ ¼ expð2xzÞ

P ; Q

ð9:14Þ

from which follow 2 3 0 P0 Q0 P 5 c; 2 c ¼ 2x þ c ¼ 42x þ log Q P Q 0

2

P c 00 ¼ 42x þ log Q we obtain

0

32

00 5 c þ log P c; Q

c 00 ¼ 22ðlog QÞ00 c þ x 2 c:

ð9:15Þ

If we rotate our axes so that V is real, then x 2 is a negative real number, 2 E, and the equation may be written 2c 00 þ U c ¼ Ec;

ð9:16Þ

where U ¼ 22 ðlog QÞ00 : In other words, c describes an eigenstate of Schro¨dinger’s equation for the potential U! This potential is derived from the polynomial Q, and the wavefunction c is, except for the exponential prefactor, a rational function with Q in the denominator. According to Bartman (1983), the analysis of Adler and Moser (1978) shows that solutions of the desired form can only arise when Q is one of the Adler – Moser polynomials. The corresponding polynomial in the numerator is then proportional to PðzÞ ¼ expðxzÞWnþ1 ðw1 ; w2 ; …; wn ; expð2xzÞÞ:

ð9:17Þ

We thus arrive at the intuitively surprising conclusion that only when the number of vortices of either species is a triangular number can one find configurations that translate uniformly without change in the relative positions of the vortices. It would be nice to understand this restriction independently of the full solution to the problem, as we did in Eq. (8.1) for the case of stationary configurations. At present such an understanding, unfortunately, appears to be lacking. Some

52

H. Aref et al.

examples of uniformly translating patterns are shown in the paper by Kadtke and Campbell (1987). X. Vortex Crystals on Manifolds Our knowledge of equilibrium configurations of point-vortices on general 2D surfaces is less complete than for the unbounded plane. There are compelling reasons for studying both vortex statics and dynamics in such domains. For example, vortices in a periodic strip, which may be thought of as ‘vortices on a cylinder’ in the topological sense (sometimes called a ‘flat cylinder’), is a problem that has entered the theory of shear layers and wakes for decades. Vortices in a doubly periodic parallelogram, topologically equivalent to a torus (i.e., a ‘flat torus’), is the canonical domain for both theory and simulation of homogeneous 2D turbulence, a topic of considerable importance to geophysical fluid dynamics. Vortex dynamics on a sphere, both rotating and non-rotating, has direct applications to the flow in a planetary atmosphere on such large scales that the curvature of the planet plays a role. This problem, then, is potentially of importance to processes in the atmosphere and oceans of Earth and to the large vortices, including the Red Spot, observed in the atmosphere of Jupiter. Even though the actual vortex structures themselves are not spread over distances comparable to the planetary radius, the velocity fields they generate, and the associated global streamline patterns, are influenced by the compact nature of the sphere, as shown in Kidambi and Newton (2000). When analyzing equilibrium configurations, it is not only the long-range interactions between the vortices that determine the structure of the equilibria, but also the detailed shape of the surface. For this reason, a complete classification of equilibria with arbitrary vortex strengths on a general 2D surface seems, at the moment, a monumental task. Nonetheless, it offers a classical setting in which techniques from both geometry and dynamics play a central role. The monograph of Aranson et al. (1996) provides a general mathematical introduction to some techniques that can be used for these problems.

A. Vortices on a Sphere Vortices on a spherical shell seem first to have been studied by Gromeka (1885). Special cases were taken up by Bogomolov (1977, 1979). The general solution for three vortices on a sphere was first given by Kidambi and Newton (1998, 1999) a full 120 years after Gro¨bli’s treatment of the planar problem! Just

Vortex Crystals

53

as the planar problem is related to other pattern-forming systems, as we have seen, there are direct connections between vortices on a sphere and the large and growing literature on the ‘charge-on-a-sphere’ problem, where one considers the equilibrium configurations of N equal point charges confined to the surface of a sphere, repelled by their mutual Coulomb interactions. This is also called the ‘dual problem for stable molecules’. Interaction laws ranging from ‘soft’ logarithmic potentials (Bergersen et al., 1994) to ‘hard’ contact forces have been studied, and lead to the so-called Tammes problem of how to optimally pack disks on the surface of a sphere. The discovery of stable carbon-60 molecules in 1985 by Curl, Kroto, and Smalley, later recognized by the Nobel Prize in Chemistry, with atoms arranged in a soccer-ball pattern on a sphere, has stimulated more abstract mathematical studies of these buckminsterfullerenes via group-theoretic methods (cf. Chung et al., 1994). For general treatments of n-body problems on a sphere see Lim (1998) and Montaldi (2000). The question of how to uniformly distribute points on the surface of a sphere, generalizing the obvious solution of having them equally spaced around the periphery of a circle, has relevance to Monte-Carlo computational algorithms that rely on randomly sampled data points used to approximate area integrals by taking averages over these points. An introduction can be found in Saff and Kuijlaars (1997). All of these problems have their own separate literatures, yet it seems likely that many of the techniques and ideas developed in the different contexts will be more generally applicable. The equations of motion for N point vortices on a sphere of radius R are N nb £ ðxa 2 xb Þ dxa 1 X 0 ¼ G ; 2p b ¼ 1 b lxa 2 xb l2 dt

ð10:1Þ

where xa and xb are vectors originating at the center of the sphere, pointing to vortices with strengths Ga and Gb ; respectively, on the surface. The unit normal vector on the spherical surface at the position of vortex g is ng ¼ xg =R; g ¼ 1; …; N: Equation (10.1) is the natural generalization of Eq. (2.1), written in Cartesian rather than complex coordinates. They reduce to Eq. (2.1) if we assume the unit normal to be independent of position. Since the sphere is a closed surface, the total vorticity of the flow on a sphere, that is, the integral over the surface of the sphere of the normal component of vorticity, must vanish. Equations (10.1) embody this constraint in the sense that the sum of the circulations explicitly shown in Eq. (10.1) and a uniform background vorticity, which is not directly discernible in Eq. (10.1) but enters through the nature of the interaction term, satisfy the constraint of vanishing total vorticity (cf. Kimura and Okamoto, 1987). Thus, if the sum of the strengths, Ga ; is zero, the

54

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point vortices carry all the vorticity in the flow. If not, one has to remember the existence of a largely passive, uniform vorticity in the background. (This is, unfortunately, not the same as the background vorticity associated with uniform rotation about an axis!) Using nb ¼ xb =R in the numerator, and lxa 2 xb l2 ¼ 2ðR2 2 xa ·xb Þ in the denominator, Eq. (10.1) may be recast as N xa £ xb dxa 21 X 0 ¼ G : 4pR b ¼ 1 b R2 2 xa ·xb dt

ð10:2Þ

Another interesting reformulation arises by projecting the surface of the sphere stereographically onto the plane tangent to the sphere at its north pole. This is pursued by Newton (2001; Section 4.5) and the result is 2 3 N Gb dza ð1 þ lza l2 Þ2 4 X S za 5 0 ¼ 2 : ð10:3Þ dt z 2 zb 8piR2 1 þ lza l2 b¼1 a Here za is the complex position in the tangent plane of the projection of vortex a from the sphere, and S is again the sum of the vortex strengths [Eq. (2.7)]. The first term in the square brackets is, of course, the planar interaction term from Eq. (2.1), so Eq. (10.3) may also be viewed as a planar system with a modified interaction law between the vortices. This is intriguing in view of the several related pattern-forming systems that we have mentioned previously, and the question of how ‘universal’ the vortex crystal patterns are when the law of interaction is modified. Hally (1980) gave a discussion of the modification of the equations of motion when vortices are placed on more general surfaces of revolution. See also Kimura (1999). As R ! 1 the problem of point vortices on a sphere reduces to point vortices in a plane, as considered in the previous sections. Thus, apart from the direct physical motivations for understanding this problem, such as the dynamics of planetary atmospheres, point vortices on a sphere may also shed light on exact solutions in the plane, ‘lifting’ certain degeneracies of the planar problem for finite values of R. Such connections are only beginning to be understood at the present time. The generalization of the linear impulse integral, Eq. (2.8), is still an integral of Eq. (10.2), i.e., these equations conserve M¼

N X

Ga x a :

ð10:4Þ

a¼1

The generalization of the angular impulse in Eq. (2.10) is also conserved, but in a trivial way, since the length of each position vector is R, so that the sum becomes

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55

SR 2. The algebra that led to the conservation of L, as given by Eq. (2.14), however, holds virtually word for word in three dimensions. Thus, L¼

N X

1 2

Gb Ga lxa 2 xb l2 ¼ S2 R2 2 M 2

ð10:5Þ

a; b ¼ 1

is obviously a constant of the motion of Eqs. (10.1) –(10.3). More significant is the analog of the Hamiltonian [Eq. (2.4)], which for vortices on a sphere takes the form H¼2

N X 1 0 Ga Gb log lxa 2 xb l2 : 4pR2 a; b ¼ 1

ð10:6Þ

The equations determining equilibria, the counterparts of Eq. (2.3), are v £ xa ¼

N xa £ xb 21 X 0 Gb 2 4pR b ¼ 1 R 2 xa ·xb

ð10:7Þ

for each a, where v, the angular velocity vector common to all the vortices, is to be determined. Similarly to what we did in deriving Eq. (2.6), we multiply Eq. (10.7) by Ga and sum on a ¼ 1; …; N: This gives v £ M ¼ 0:

ð10:8Þ

Next, as in the derivation of Eq. (2.9), we take the cross product of Eq. (10.7) with Ga xa and sum on a. This gives N X

Ga xa £ ðv £ xa Þ ¼ 2

a¼1

N X xa £ ðxa £ xb Þ 1 0 G a Gb 2 : 4pR a; b ¼ 1 R 2 xa ·xb

ð10:9Þ

The numerator in the sum on the right is xa ðxa ·xb Þ 2 R2 xb : Using the symmetry of the other factors in the summand in a and b this may just as well be written xa ðxa ·xb Þ 2 R2 xa ¼ xa ðxa ·xb 2 R2 Þ: The right hand side of Eq. (10.9) now simplifies dramatically: 1 4pR

N X

Ga G b x a ¼

a; b ¼ 1 a–b

1 8p

N X

Ga Gb ðna þ nb Þ:

a; b ¼ 1 a–b

For stationary equilibria, we then immediately have the necessary condition N X

a; b ¼ 1 a–b

Ga Gb ðna þ nb Þ ¼ 0:

ð10:10Þ

H. Aref et al.

56

This is a generalization of the result K ¼ 0 in the planar case to which it reduces when all the normal vectors na can be assumed equal. For equilibria with a finite angular frequency of rotation we can take the scalar product of Eq. (10.9) with v to obtain N X

Ga ðv £ xa Þ2 ¼

a¼1

1 8p

N X

Ga Gb ðna þ nb Þ·v

a; b ¼ 1 a–b

If M is non-zero, Eq. (10.8) tells us that v is along the fixed direction of M. It is convenient to take this fixed direction as the z-axis of coordinates. Then N X

Ga ðv £ xa Þ2 ¼ v2

a¼1

N X

Ga ðx2a þ y2a Þ ¼ I v2

a¼1

where the symbol I is used, literally, as in Eq. (2.10) since the sum is only over the coordinates in the xy-plane. On the right hand side, we substitute na ·v ¼ v cos ua ; where ua is the polar angle of vortex a. Finally, then, Iv ¼

1 8p

N X

Ga Gb ðcos ua þ cos ub Þ:

ð10:11Þ

a;b ¼ 1 a–b

This is the analog of Eq. (2.17) for vortices on the surface of a sphere. We note that both I on the left hand side and the double sum on the right hand side of Eq. (10.11) are constants for an equilibrium state. The configuration moves without change of shape so the distances between vortices dab ¼ lxa 2 xb l are all constants. Expanding the square of this length, we see that all scalar products xa ·xb are constants, since each position vector has length R. Thus, M·xa ¼ Mza ¼ MR cos ua is a constant for each a. Each vortex orbits the z-axis at constant latitude.

B. Two- and Three-Vortex Equilibria on the Sphere For two vortices we obtain from Eq. (10.7) v £ x1 ¼

G2 x2 £ x1 ; 4pR R2 2 x1 ·x2

ð10:12Þ

v £ x2 ¼

G1 x1 £ x2 : 4pR R2 2 x1 ·x2

ð10:13Þ

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57

A necessary and sufficient condition for a stationary equilibrium, then, is that x1 ¼ 2x2 : The vortices are antipodal. They may have any strengths. In the general case v £ x1 and v £ x2 ; and thus x1 and x2 ; point in the same or in opposite directions according as G1 and G2 have opposite signs or have the same sign. Geometrically this means that vortices 1 and 2 orbit on latitude circles with the vortices on the same longitude when the strengths are of opposite sign, and with a difference of 1808 in longitude when the vortex strengths are of the same sign. Figure 12(a) illustrates the general case of two vortices of opposite sign. If the sign of G2 was changed, that vortex would appear on the same latitude circle but at the diametrically opposite point. Figure 12(b) shows the special case of two identical vortices, and Fig. 12(c) shows the special case of opposite strengths. The angular frequency of rotation is given by Eq. (10.11), or in this case directly from Eqs. (10.12) and (10.13): Use Eq. (10.8) to write v ¼ vM=M; then Eq. (10.12) gives

v G x £x G x £ x1 ¼ 2 2 2 1 ; M 2 2 2pR d12 or

v¼

M ¼ 2pRd 212

qffiffiffiffiffiffiffiffiffiffiffiffiffi S2 2 L=R2 2pd 212

:

ð10:14Þ

The last form follows from Eq. (10.5) and makes it obvious that in the limit R ! 1 this formula reduces to that of an orbiting pair on the plane.

Fig. 12. Two-vortex equilibria on the sphere: (a) two vortices of the same sign but different strengths; (b) identical vortices; (c) opposite vortices. After Newton (2001) with permission.

58

H. Aref et al. For three vortices on a sphere Eqs. (10.7) become " # 1 G 2 x2 £ x1 G3 x3 £ x1 v £ x1 ¼ ; þ 2pR d 212 d 231 " # 1 G 1 x1 £ x2 G3 x3 £ x2 v £ x2 ¼ ; þ 2pR d 212 d 223 " # 1 G 1 x1 £ x3 G2 x2 £ x3 v £ x3 ¼ : þ 2pR d 231 d 223

ð10:15Þ

ð10:16Þ

ð10:17Þ

Taking the scalar product of Eq. (10.16) with x3, we see that a necessary condition for a stationary equilibrium is that D ¼ x1 £ x2 ·x3 vanishes. The three vortices are co-planar, i.e., in terms of the sphere they are on the same great circle. Sufficient conditions for stationary equilibria, and for equilibria in the case of vanishing M, are more complicated (cf. the discussion of collinear equilibria of three vortices in the plane in Section III). The known results may be found in Section 4.2 of Newton (2001). In the general case we write v ¼ vM=M and Eqs. (10.15) –(10.17) become " # " # v 1 v 1 2 2 x 2 £ x1 þ G3 x3 £ x1 ¼ 0; ð10:18Þ G2 M M 2p Rd 212 2pRd 231 " # " # v 1 v 1 2 2 x 1 £ x2 þ G3 x3 £ x2 ¼ 0; ð10:19Þ G1 M M 2p Rd 212 2pRd 223 " # " # v 1 v 1 2 2 x 1 £ x3 þ G2 x2 £ x3 ¼ 0; G1 ð10:20Þ M M 2pRd 231 2pRd 223 from which we conclude (by taking scalar products with x1, x2, and x3) that equilibria with non-vanishing D occur only if d12 ¼ d23 ¼ d31 ; i.e., if the vortices are at the vertices of an equilateral triangle (in space). In this case, regardless of the vortex strengths, the configuration is a relative equilibrium. The angular frequency of rotation, as we see immediately from Eqs. (10.18) –(10.20), is

v¼

M ; 2pRs2

ð10:21Þ

where s is the side of the equilateral triangle. For this case L ¼ Ks2 =2, so using Eq. (10.5) we may write qffiffiffiffiffiffiffiffiffiffiffiffiffi K S2 2 L=R2 : ð10:22Þ v¼ 4pL

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59

In the planar limit R ! 1 the numerator becomes KS, the denominator 4pSI; and we return to Eq. (2.17). The ‘degenerate’ case, M ¼ 0, requires special considerations and we again refer the reader to Newton (2001; Section 4.2) for what is known. For vortices on a sphere one can derive formulae similar to Eq. (2.22), viz. ds21 1 s2 2 s2 ¼2 G1 D 2 2 2 3 ; pR dt s2 s3 ds22 1 s2 2 s2 ¼2 G2 D 3 2 2 1 pR dt s3 s1

ð10:23Þ

ds23 1 s2 2 s2 ¼2 G3 D 1 2 2 2 ; pR dt s1 s2 in essence as these are derived in the planar case. In Eq. (10.23) s1 ¼ lx2 2 x3 l; s2 ¼ lx3 2 x1 l; s3 ¼ lx1 2 x2 l and, as before, D ¼ x1 £ x2 ·x3 : We see immediately from Eq. (10.23) that in order for three vortices to form a relative equilibrium we must either have D ¼ 0, or have the vortices form an equilateral triangle. The analogy to the planar case is obvious.

C. Multi-Vortex Equilibria on a Sphere Equilibria with more than three vortices on the sphere are currently an active area of research. Thus, we will simply highlight what is known to date and point the motivated reader to the relevant literature. One tried-and-true method is to take known ‘classical’ configurations in the plane and look for their analogues on the sphere. The vortex polygons, simple, centered, and nested, both in the symmetrical and the staggered configurations, that we considered in Sections V and VI have been studied on the sphere, as have vortex street configurations (Hally, 1980). Polvani and Dritschel (1993) revisit these configurations of N equal strength vortices, placing them on a sphere at a fixed latitude and using the longitude angle 0 # u # 908 as a parameter. Their main finding is that the regular polygons are ‘more unstable’ on the sphere than in the plane, with the following linear stability ranges: For N ¼ 3: 0 # u # 908; for N ¼ 4: 0 # u # 558; for N ¼ 5: 0 # u # 458; for N ¼ 6: 0 # u # 278; for N ¼ 7 there is only linear (neutral) stability at u ¼ 08. Thus, at the pole, as on the plane, the heptagon gives the dividing point between stable, regular N-gons and unstable ones. As the latitude increases, the critical number required to produce a neutrally stable, regular N-gon decreases. The result of Pekarsky and Marsden (1998) extends the linear stability to non-linear stability on the whole sphere for N ¼ 3.

60

H. Aref et al.

Boatto and Cabral (2003) have recently extended the linearized stability results to the non-linear regime for general N. An approach currently being exploited is to use group-theoretic methods to generate equilibria on the sphere, restricting the vortex strengths in such a way that discrete symmetries of the equations of motion are respected. Lim et al. (2001) consider relative equilibria made up of equal strength vortices, while Laurent-Polz (2002) considers systems comprised of n vortices of strength þ G and n of strength 2 G. In both studies, polar vortices of equal and opposite strength can (sometimes) be added (since they lie on the axis of rotation), and these, in general, do not need to be of the same strength as the other vortices in the configuration. These papers also consider the interesting problem of bifurcation of the relative equilibria as a function of H and M. Lim et al. (2001) are able to find relative equilibria made up of rings where all vortices have the same latitude, and configurations made up of two such rings at different latitudes with the vortices on one ring staggered in longitude with respect to those on the other. For stability analyses regarding these states see Laurent-Polz et al. (2003). Laurent-Polz (2002) considers both identical and opposite strength vortices, and is able to find a separate class of equilibria, including staggered equal and opposite rings symmetrically placed across the equator (‘vortex streets’), as shown in Fig. 13. In Fig. 13(a) the vortices all lie on the equator, while in Fig. 13(b) they are symmetrically placed on either side of the equator. Further generalizations of the polygonal and staggered planar ring configurations and their spherical analogs arise by placing identical vortices at the vertices of each of the five Platonic solids inscribed in the sphere (Tokieda, 2001; Khushalani and Newton, 2003a). Thus, the tetrahedron, octahedron, cube, icosahedron and dodecahedron yield configurations with 4, 6, 8, 12 and 20 vortices, respectively. On each of these one can, furthermore, place vortices also at the vertices of the ‘dual’ polyhedron, i.e., at the points on the sphere corresponding to the midpoints of the faces of the original polyhedron. Or one can place vortices at the midpoints of the edges of the original polyhedron. All these configurations are stationary equilibria for vortices on a sphere. Perhaps more interesting is the possibility of choosing vortices with both positive and negative circulations of the same absolute magnitude to arrive at relative equilibria that rotate about M. Relative equilibria for each of the Platonic solids are shown in Fig. 14. The tetrahedron shown in Fig. 14(a) is made up of a (non-equatorial) ring of three equally spaced identical vortices, with a single vortex of opposite strength placed at the north pole. It rotates about the axis shown with angular frequency 3G=8pR2 : The cube (hexahedron) in Fig. 14(b) is formed from two rings at fixed latitude symmetrically placed across the equator, each with four equally spaced vortices placed around the ring. The signs of the vortices

Vortex Crystals

61

Fig. 13. Single and double, multi-vortex ring equilibria on the sphere.

in the northern hemisphere are opposite to those in the southern hemisphere. It p rotates about the axis shown with angular frequency 3 3G=4pR2 : The octahedron, Fig. 14(c), is made up of an equatorial ring of four identical, equally spaced vortices, together with a vortex at the north pole and one at the south pole. In general, these two polar vortices could have any strength. We show the special case of opposite polar vortices whose strengths match those on the equator. The angular frequency of rotation is G=2pR2 (which follows readily from Eq. (10.11) because the latitude angles are 0 and p/2). The icosahedron, Fig. 14(d), is formed from two fixed-latitude, staggered rings of five vortices each, with opposite signs on the two rings, symmetrically placed on either side of the equator, and a vortex at each of the poles. As for the octahedron, the polar vortices can have any strength. We show the special case in which the six vortices in either hemisphere have equal strengths, and the six vortices in the northern hemisphere are of opposite strength to the six in the southern hemisphere. The angular

H. Aref et al.

62

Fig. 14. Equilibria with vortices at the vertices of the Platonic solids. See the text for further details.

p frequency of rotation then is 5ð1 þ 5ÞG=8pR2 : Finally, the dodecahedron, Fig. 14(e), is formed by placing two sets of staggered rings symmetrically about the equator. This case is somewhat more complicated in that the vortex strengths on the outer two rings and those on the inner two rings must be related in such a way that each of the ring pairs rotates about M with the same angular frequency. This is achieved by letting G1 ¼ G sin u1 ; G2 ¼ G sin u2 ; where u1 and u2 are the latitudes of the two rings. The angular frequency of rotation is 9G=4pR2 : Undoubtedly, much more complex patterns can be formed by collections of vortices on the sphere, presumably more exotic classes such as the spiral

Vortex Crystals

63

configuration shown in figure 4 of Saff and Kuijlaars (1997) and, possibly, some with no symmetries at all as generated in the plane by Aref and Vainchtein (1998). The Archimedean solids have also been used recently by Khushalani and Newton, (2003b) to generate interesting structures, such as ‘vortex buckyball’ states, and it seems as if this approach is only the tip-of-the-iceberg. A promising numerical technique based on Monte-Carlo algorithms that locates states of lowest energy has been developed by Assad et al. (2003). D. Vortices in a Periodic Strip This time-honored problem includes well-known models for the shear layer and the vortex street which may be thought of, respectively, as a single vortex and a vortex pair in a periodic strip. The quickest way to derive the equations of motion for vortices in a periodic strip, which was apparently first done by Friedmann and Poloubarinova (1928),3 is to ‘periodize’ the planar theory. Thus, consider N vortices placed at z1 ; …; zN in a strip of width L and with circulations G1 ; …; GN : The term periodic strip means that we consider at the same time all nL-translates of these vortices for all integers n. Formally, the Hamiltonian (2.4) becomes H¼2

1 N X X 1 þ 0 G G loglza 2 zb 2 nLl; 4p n¼21 a; b ¼ 1 a b

which diverges as it stands. However, since additive constants in the Hamiltonian may be omitted, we subtract off a constant, divergent series to produce a finite H with the same dynamics. Thus, we consider H¼2

1 1 N N X X X X 1 þ 1 þ 0 0 Ga Gb loglza 2 zb 2 nLl þ G G loglnLl; 4p n¼21 a; b ¼ 1 4p n¼21 a; b ¼ 1 a b

and rearrange the sum by pairing terms with ^ n, to produce " # 1 N Y za 2 zb 2 1 X 0 H¼2 G G logðza 2 zb Þ 12 4p a;b ¼ 1 a b nL n¼1 ¼2

3

N p 1 X 0 Ga Gb logsin ðza 2 zb Þ: 4p a;b ¼ 1 L

ð10:24Þ

A. Friedmann is probably best known for his cosmological solutions to Einstein’s field equations of general relativity published in two papers from 1922 to 1924.

64

H. Aref et al.

The equations of motion corresponding to Eq. (2.1), and arising from those equations by inclusion of all the periodic images, are therefore N dza 1 X 0 ¼ G F ðz 2 zb Þ; 2pi b ¼ 1 b cyl a dt

where

Fcyl ðzÞ ¼

p p cot z : L L

ð10:25Þ

ð10:26Þ

One interesting feature of these equations is that as za 2 zb ! i1 (corresponding to infinite vertical separation) the velocity induced by vortex b on vortex a does not decay to zero as in the case of the infinite plane theory, but converges to Gb =2L: This is obvious upon calculating the circulation around a tall box of width L enclosing the point zb : Another way to interpret this feature is to note that the limit of infinite vertical separation is, up to rescaling, equivalent to the narrowing of the period L to 0; this limit produces a continuous vortex sheet, which induces a velocity field constant everywhere above the sheet (with the opposite constant below the sheet) independently of the distance to the sheet. This is a general result of 2D potential theory, possibly more familiar in the case of electrostatics. The quantities X and Y in Eq. (2.8) are still integrals (Birkhoff and Fisher, 1959), but rigorously speaking there is a subtlety involved. Indeed, each xa in X being defined only modulo L (i.e., on the manifold a vortex that ‘leaves’ the strip at x ¼ L; ‘reappears’ at x ¼ 0), X is not well-defined as a function on the whole periodic strip (physically, one would have to keep track of how many times each vortex went through the strip—the instantaneous position of the vortices, of course, gives no hint of this). Nor is it much use treating X as a multi-valued function, for in general the strengths will not be rationally dependent and so the set of values LSa Ga na giving the ‘ambiguity’ in X will be dense in the real numbers. Nevertheless, X is well defined locally, i.e., so long as the vortices remain within the strip, and this suffices for our purposes. First integrals that are only locally defined are apt to arise when the surface on which we develop the vortex theory has non-zero first homology (or, in mathematical jargon, when the symplectic action of the symmetry group is not Hamiltonian). Such is the case with a periodic strip (topologically a cylinder) and a periodic parallelogram (topologically a torus), or any surface of genus . 0. The sphere, being simply connected, involves no such subtlety. This issue is similar to the familiar one in ideal hydrodynamics of having to introduce ‘barriers’ to define the potential for multiply connected domains.

Vortex Crystals

65

There is a partial classification result for equilibria in a periodic strip and a periodic parallelogram (Montaldi et al., 2003). We shall describe the results for a periodic strip first. Unlike the plane, a periodic strip (cylinder) does not have rotational symmetry. It is not difficult to see that rotating equilibria are impossible. Thus, the only possibility for equilibria is that the entire configuration translates without change of relative positions of the vortices, i.e., in Eq. (10.25) all the left hand sides equal the same complex number. Multiplying Eq. (10.25) by Ga and summing, and using the antisymmetry of the summand, produces the result SV ¼ 0; where S is the sum of circulations, and V is the common velocity of translation. Equilibria of vortices in a periodic strip, then, require either that S ¼ 0 or, if S – 0, that the equilibrium is stationary. For two vortices in a periodic strip with S ¼ 0 the equations of motion and Fcyl being an odd function guarantees that the vortices will translate uniformly regardless of their initial separation. These configurations are the vortex streets studied by von Ka´rmań (1911, 1912) in the cases where the direction of propagation is along the horizontal axis, and by Dolaptschiew and Maue (cf. Maue, 1940) in the general case. According to Eq. (10.25) the complex velocity is given by 1 p V ¼ u 2 iv ¼ cot ðz 2 z2 Þ ; 2Li L 1

ð10:27Þ

i.e., the direction of propagation is given by v : u ¼ 2sin½2p=Lðx1 2 x2 Þ : sinh½2p=Lðy1 2 y2 Þ: This direction is not, in general, perpendicular to the line segment connecting the vortices, although in the ‘deperiodizing limit’, L ! 1, it converges to that direction. A stationary pair with S ¼ 0 requires y1 ¼ y2 and x1 2 x2 ¼ ^L=2; i.e., the vortices are uniformly spaced along the x-axis, the cylinder counterpart of the state illustrated in Fig. 13(a) for vortices on a sphere. Stationary pairs with S – 0 are also covered by this analysis. The uniformly spaced vortices need not have the same absolute magnitude. For three vortices with S – 0 the conditions for a stationary equilibrium require

c1 c c ¼ 2 ¼ 3 G1 G2 G3

ð10:28Þ

H. Aref et al.

66

(Stremler, 2003). Here we have used the abbreviations

p ðz 2 z3 Þ ; c1 ¼ cot L 2 p ðz 2 z2 Þ : c3 ¼ cot L 1

p ðz 2 z1 Þ ; c2 ¼ cot L 3

ð10:29Þ

Cases of two vortices having opposite circulations, such as G1 ¼ 2G3 ; need to be considered separately. If G1 ¼ 2G3 ; we must have c1 ¼ 2c3 or cot

p p ðz2 2 z3 Þ ¼ 2cot ðz1 2 z2 Þ ; L L

p p p p ðz 2 z2 Þ cos ðz 2 z2 Þ sin ðz 2 z3 Þ þ cos ðz 2 z3 Þ ¼ 0; sin L 1 L 1 L 2 L 2 p ðz1 2 z3 Þ ¼ 0 sin or z1 2 z3 ¼ nL; n an integer: L But this means that vortex 1 coincides with vortex 3, or with one of its periodic images, which is unacceptable. The other cases of opposite vortices lead to similar contradictions. Hence, we conclude that when two vortices are opposite, there are no three-vortex equilibria. Except for these special cases we may use that c1, c2 and c3 are related by the addition formula for the cotangent, viz. c1 ¼ cot

p 1 2 c2 c3 ðz2 2 z1 2 ðz3 2 z1 ÞÞ ¼ ; L c2 þ c3

i.e., by c1 c2 þ c2 c3 þ c3 c1 ¼ 1:

ð10:30Þ

If the common value of c1 =G1 ; c2 =G2 and c3 =G3 ; cf. Eq. (10.28), is denoted by j, Eq. (10.30) states that K j2 ¼ 2; where K was given by Eq. (2.11). To have a solution we must require K – 0 and then qffiffiffiffiffiffi qffiffiffiffiffiffi qffiffiffiffiffiffi c 1 ¼ ^ G1 K=2; c 2 ¼ ^ G2 K=2; c 3 ¼ ^ G3 K=2; ð10:31Þ where the same sign must be used for each of c1, c2 and c3.

Vortex Crystals

67

The nature of the resulting equilibria depends on the sign of K. For K . 0, the quantities c1, c2 and c3 are all real and the vortex separations z2 2 z3 ; z3 2 z1 and z1 2 z2 must also all be real. In other words we must have y1 ¼ y2 ¼ y3 : The vortices are on a line parallel to the x-axis. Solutions are given by 1 p 1 p 1 p ðx2 2 x3 Þ ¼ ðx3 2 x1 Þ ¼ ðx1 2 x2 Þ cot cot cot L L L G1 G2 G3 qffiffiffiffiffiffi ¼ ^1 K=2: For example, let vortices 1 and 2 both have strength G and place them at a distance x1 2 x2 ¼ a apart on a horizontal line. They are immobilized if we add half-way between them a vortex of strength

G3 ¼ 2G cot

pa L

pa 1 pa ¼G sec2 21 : cot 2L 2 2L

In the deperiodizing limit, L ! 1, this converges to 2 G/2 as it should from the unbounded plane results (cf. Section III). On the other hand, G3 vanishes when a ¼ L=2 : the two vortices are antipodal on the periodic strip and are already stationary by themselves. As a ! L; the two vortices nearly meet ‘in the back’ and it takes a stronger and stronger vortex at their midpoint to prevent them from moving. For K , 0 the quantities c1, c2 and c3 in Eq. (10.31) are all pure imaginary. Thus, we must have 2p 2p 2p sin ðx1 2 x2 Þ ¼ sin ðx2 2 x3 Þ ¼ sin ðx3 2 x1 Þ ¼ 0; L L L which implies that the vortices must be on lines parallel to the y-axis offset by a multiple of L/2. If the vortices are on a vertical line, x1 ¼ x2 ¼ x3 ; we have the solutions 1 p 1 p 1 p ðy2 2 y3 Þ ¼ ðy3 2 y1 Þ ¼ ðy1 2 y2 Þ coth coth coth L L L G1 G2 G3 qffiffiffiffiffiffi ¼ ^1 K=2: For further discussion see Stremler (2003) and Montaldi, Soulie`re and Tokieda (2003). By way of example, let vortices 1 and 2 both have strength G but place them at a distance y1 2 y2 ¼ b apart on the same vertical line. They are immobilized if we add at their midpoint a vortex of strength

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G3 ¼ 2G coth

pb L

coth

pb ¼G 2L

1 pb sech2 21 : 2 2L

In the ‘deperiodizing limit’, L ! 1, this again converges to 2 G/2. On the other hand, in the ‘vortex sheet limit’, b ! 1, this converges to 2 G, also as it should. Aref and Stremler (1996) have shown how to obtain the general solution of the integrable dynamical problem of three vortices with S ¼ 0 in a periodic strip. The approach consists of ‘mapping’ the original three-vortex problem, with vortices at locations z1, z2 and z3 of circulations G1, G2 and G3, respectively, in a strip of width L, onto a simpler problem, viz. the problem of advection of a fictitious, passive particle in a field of certain fixed vortices derived from the strengths and first integrals X and Y of the original problem. The details of this construction are as follows: The ‘mapped’ particle is situated at z1 2 z2 and moves in the field of three rows of vortices of strengths 1=G1 ; 1=G2 and 1=G3, respectively. The vortices of the first ‘family’, all of strength 1=G3 , are located at nL, where n runs through the integers, i.e., their spacing is that of the original strip width. The vortices of the second ‘family’, all of strength 1/G2, are located at 2ðX þ iYÞ=G2 þ nLG3 =G2 ; n ¼ 0; ^1; ^2; … The vortices of the third ‘family’, all of strength 1/G1, are located at ðX þ iYÞ=G1 þ nLG3 =G1 ; n ¼ 0; ^1; ^2; … If the ratio of any two of G1, G2 and G3 is rational, then all such ratios are rational since G1 þ G2 þ G3 ¼ 0; and the three rows repeat with a period that is a multiple of the original strip width L. Note that although the circulations of the three original vortices sum to zero, the net circulation of the advecting system is non-zero. The problem of advection of the fictitious particle is readily solved, by constructing the steady streamline pattern for the three-vortex rows. Vortex equilibria correspond to stagnation points in the derived, advecting flow. These must all be saddle points. The advecting vortices themselves are the only elliptic points and correspond to two of the original vortices 1, 2, 3 coinciding. That is, all the equilibria are unstable. Several are obtained. Remarkably, they can all be calculated explicitly, as we show below. The ‘generic’ number of equilibria can be counted by topological considerations. Figure 15 shows an example of the derived streamline pattern for the case of three vortices of relative strengths 2:1:(2 3). In this case the derived advection problem ‘lives’ in a periodic strip of width 3L. The solid dots are the advecting vortices. The equilibria correspond to the several saddle points seen in the pattern of separatrix streamlines. We may find these points explicitly by considerations similar to those for the stationary equilibria given previously. Thus, from the equations of motion (10.25) and (10.26) we find (Stremler, 2003):

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Fig. 15. Streamline pattern for the advection problem in a strip. For this particular choice of vortex strengths the strip for the advection problem is three times as wide as the original strip of the three-vortex problem. The various saddle points produce equilibria with three vortices per period of sum zero. The advecting vortices themselves (dots) produce an equilibrium with six vortices per period and non-zero sum. (After Aref and Stremler (1996)).

2LiV ¼ G2 c3 2 G3 c2 ; 2LiV ¼ G3 c1 2 G1 c3 ;

ð10:32Þ

2LiV ¼ G1 c2 2 G2 c1 : Taking the difference of any two of these, and using S ¼ 0, gives c1 þ c2 þ c3 ¼ 0;

ð10:33Þ

which is the analog of Eq. (3.2). Conversely, Eq. (10.33) and S ¼ 0 assure that the right hand sides of Eq. (10.32) are all equal. We can now solve Eqs. (10.30) and (10.33) for two of the cs given a value of the third. For example, choosing the position of vortex 2 relative to vortex 1 gives c3, and then c1 and c2 are the two roots of the polynomial ðc 2 c1 Þðc 2 c2 Þ ¼ c2 2 ðc1 þ c2 Þc þ c1 c2 ¼ c2 þ c3 c þ 1 þ c23 ; i.e., c1 ¼

1 2

qffiffiffiffiffiffiffiffiffiffi ð2c3 ^ i 4 þ 3c23 Þ;

c2 ¼

1 2

qffiffiffiffiffiffiffiffiffiffi ð2c3 7 i 4 þ 3c23 Þ;

where the signs are to be chosen such that c1 – c2 : Thus, for any allowable value of c3, that is for any value of c3 for which the vortices and their periodic images are all distinct, we can determine c1 and c2 such that we have a translating equilibrium. What is quite remarkable is that these configurations are independent of the values of the vortex strengths. We can populate the three points with vortices of any circulations, subject to the constraint that S ¼ 0, and we arrive at a relative equilibrium. We have seen something of this sort

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before: For the equilateral triangle on the infinite plane we could place any three vortices at the vertices and always have an equilibrium. If the sum of the strengths was zero, the triangle would translate uniformly. The solutions just mentioned give rise to a large family of translating ‘vortex streets’ with three vortices, whose strengths sum to zero, per period. But there is more. As observed by Aref and Stremler (2003), the three-vortex rows in the mapping construction that we have described are themselves multivortex equilibria with total non-zero circulation (and, thus, stationary patterns)! This follows from the construction: Any advecting vortex corresponds via the mapping to the coincidence of one of the original vortices with another or with any one of its periodic images (based on the original strip of width L). But if two of the original vortices coincide, the three-vortex problem reduces to a twovortex problem, the solution of which is simply that the two vortices translate in parallel. This implies that the difference z1 2z2 is constant. In other words, the rate of change of z1 2z2 vanishes at any advecting vortex. But this rate of change is just the velocity at the position of the advecting vortex (and by extension at all its periodic images). Hence, each advecting vortex finds itself at a point of vanishing advection velocity due to all the other vortices. When z1 2z2 coincides with an advecting vortex of one of the three aforementioned ‘families’, it follows that z2 2z3 and z3 2z1 coincide with advecting vortices of the other two ‘families’. The upshot is that all the advecting vortices form a stationary pattern, which then has the simple parametrization already given (restated more ‘generically’ below). This can, of course, also be verified by direct calculation using Eqs. (10.25) and (10.26). There is a three-parameter family of stationary equilibria consisting of three-vortex rows. The three parameters are X, Y, and a ratio of two of the original vortex strengths G1, G2 and G3. In other words, take three real numbers, G1, G2 and G3, that sum to zero. For general complex Z place vortices of strength 1/G3 at nL, n integer; vortices of strength 1/G2 at ð2Z þ nLÞG3 =G2 ; vortices of strength 1/G1 at ðZ þ nLÞG3 =G1 : (The only constraint on Z is that vortices of the three different families do not coincide.) The states just given are stationary equilibria. If the ratios of G1, G2 and G3 are rational, the pattern repeats after some multiple of L. If these ratios are irrational, we obtain the possibility of stationary vortex patterns without longrange periodicity. With a bit of poetic license we call such patterns vortex quasicrystals. It is presently unknown whether such states can be realized experimentally.

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For even N . 3, say N ¼ 2k; and S ¼ 0, the vortex street equilibria found for two vortices per strip can, of course, be recovered ‘in a wider strip’. The system with k ¼ 2, two positive and two negative vortices all of the same absolute circulation, was studied by Domm (1956) in a seminal paper on the stability of vortex streets. No other families of equilibria appear to be known for N . 3. In summary, we have obtained several families of vortex equilibria, both stationary and translating, for N ¼ 3 vortices in a periodic strip. For N . 3 we found an explicit construction of a large family of stationary equilibria with S – 0.

E. Vortices in a Periodic Parallelogram Proceeding to a periodic parallelogram we use the notation of the theory of elliptic functions for the sides, denoting them by 2v1 and 2v2, respectively, where t ¼ v2 =v1 has positive imaginary part. Periodic squares have been used extensively for numerical simulations of homogeneous 2D turbulence. Such simulations reveal the flow to be dominated by strong discrete vortices. Hence, it is of considerable interest to study the problem of interacting vortices in this geometry. In these applications the periodic nature of the flow requires the vanishing of the total circulation for all vortices in the basic periodic parallelogram. It is also of interest to consider the problem of infinite, regular lattices of vortices since such structures arise in superfluids and, approximately, in superconductors. For the simplest case of one vortex per cell (identical vortices) Tkachenko (1966a) showed that all lattices give configurations that rotate uniformly. The angular frequency of rotation, V, depends on the shape of the basic cell (parallelogram), i.e., on the parameter t. The simplest way to establish this dependence is to think of a large section of the lattice as a ‘discretized’ patch of fluid with uniform vorticity v. Thus, on the one hand, from the general relation between uniform vorticity and angular velocity we would have v ¼ 2V: On the other hand, the total circulation of the flow around the patch, which by Stokes’ theorem is vA (with A the area of the patch) also equals N G; where N is the number of vortices in the patch, all of which have the common circulation G. That is, V ¼ nG=2; where n is the area density of vortices. If the area of the basic cell of the lattice, i.e., the area of the periodic parallelogram, is denoted by D, we have V ¼ G=2D: Tkachenko (1966b) also examined the stability of the various simple lattices and concluded that for small perturbations (i.e., in a linearized stability analysis) the triangular lattice is the only stable one.

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The Hamiltonian of the vortex motion is found to be " # N ½Imðza 2 zb Þ2 1 X 0 H¼2 ; G G logq1 ðza 2 zbjtÞ 2 p 4p a;b ¼ 1 a b D

ð10:34Þ

where q1 is the first Jacobian theta function and D ¼ 2iðv1 v2 2 v2 v1 Þ ¼ 4lv1 l2 Im t is the area of the parallelogram (O’Neil, 1989; Stremler and Aref, 1999; Tokieda, 2001). The counterparts of Eqs. (2.1), (10.25) and (10.26) are N dza 1 X 0 ¼ G F ðz 2 zb Þ; 2pi b ¼ 1 b torus a dt

ð10:35Þ

where

Ftorus ðzÞ ¼ zðzÞ þ

pv1 h p 2 1 z 2 z ; D Dv1 v1

ð10:36Þ

where the Weierstrass z-function has half-periods v1 and v2, and h1 ¼ zðv1 Þ: The two-vortex problem with two vortices of opposite circulation again leads to uniform translation for all initial conditions. For three vortices (with the sum of the circulations equal to zero) a complete solution was provided by Stremler and Aref (1999). It follows the analysis for the periodic strip (cylinder) closely. One has to verify that the ‘mapping’ idea will work once again, which is more complicated since the induced velocities are given by the Weierstrass z-function rather than by cotangents, but the end results are very similar. One finds, once again, that the dynamics of the three interacting vortices can be ‘mapped’ onto an advection problem for a fictitious particle at z1 2 z2 which moves in the field of three lattices of vortices of strengths 1/G1, 1/G2 and 1/G3, respectively. The vortices of the first ‘family’, all of strength 1/G3, are located at Vmn ¼ 2mv1 þ 2nv2 ; where m and n are integers, i.e., on a lattice with the same periods as the original problem. The vortices of the second ‘family’, all of strength 1/G2, are located at 2ðX þ iYÞ=G2 þ Vmn G3 =G2 : The vortices of the third ‘family’, all of strength 1/G1, are located at ðX þ iYÞ=G1 þ Vmn G3 =G1 : If the strength ratios are rational, the three lattices repeat with periods that are a multiple of the original 2v1 by 2v2 parallelogram. In this case the sum of the circulations of the advecting vortices situated within the extended parallelogram is, indeed, zero as it must be for a periodic flow. Figure 16 shows advecting vortices and dividing streamlines for the case G1 : G2 : G3 ¼ 2 : 1 : ð23Þ: Each saddle point in this figure gives a value of z1 2 z2 that corresponds to an equilibrium of the threevortex problem in the original periodic parallelogram (a square of side L in this case). If z1 2 z2 is known, then given the vortex strengths, and values of X and Y, the three-vortex positions are known up to translation.

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Fig. 16. Streamline pattern for the advection problem in a square. For this particular choice of vortex strengths the side of the square for the advection problem is three times the side of the original square of the three-vortex problem. The various saddle points produce equilibria with three vortices per square. The advecting vortices themselves (dots) produce an equilibrium with 14 vortices per square. The sum of the circulations in any of these equilibria must be zero by the periodicity of the flow. (After Stremler and Aref (1999)).

Furthermore, as in the case of the strip and by virtually the same argument, the three interwoven lattices of advecting vortices form a stationary equilibrium (Aref and Stremler, 2001, 2003). For rational vortex ratios these are again periodic patterns. For irrational ratios they form 2D quasi-crystals. These seem to be the only known exact results for 2D vortex patterns.

F. Vortices on the Hyperbolic Plane Apart from the study by Kimura (1999) this topic is still largely unexplored. After developing the theory of point of vortices on the plane, on a sphere, in a periodic strip, and in a periodic parallelogram, it is natural to try to do as much on other surfaces, in particular on Riemann surfaces of genus . 1. Chief interest in the hyperbolic plane stems from the fact that every orientable surface of genus . 1 is a discrete quotient of the hyperbolic plane. Here we confine ourselves to deriving the Hamiltonian for point vortices on the hyperbolic plane and mentioning a class of equilibria. Place a vortex of unit circulation at a point, and draw a circle of radius r (measured, of course, in the hyperbolic metric) around that point. This circle has length 2p sinh r (compare 2p sin r in the spherical case). The Hamiltonian H contributed by the vortex should be rotationally symmetric, so that H is a function

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of r alone. The circulation of the induced velocity field around the circle is 2

›H 2p sinh r ¼ 1; ›r

which upon integration yields H¼2

1 logltanhðr=2Þl: 2p

For N vortices, the Hamiltonian is a weighted sum over all pairs: H¼2

N 1 X 0 G G logltanhðrab =2Þl; 4p a;b ¼ 1 a b

ð10:37Þ

where rab is the hyperbolic distance between vortices a and b. As in the planar case, a regular N-gon of identical vortices is an equilibrium. The basic reason is that the rotation by 2p/N about the center of the N-gon is both a symmetry of H and an isometry of the hyperbolic plane.

XI. Concluding Remarks The topic of vortex crystals, and vortex statics in general, has been pursued for almost a century and a half. Even confining attention to the very simplest case of 2D motion, we were surprised to find how spotty our knowledge of this subject really is. Very basic questions about the existence of solutions to these problems remain open. This is all the more puzzling when one realizes that some of the states in question correspond to frequently studied flows such as vortex street wakes. For vortices on the sphere, the cylinder, or the torus—three often encountered manifolds in applications—the topic of vortex crystals brings us right up to the frontlines of current investigations. Problems of vortices in planar regions enclosed by solid boundaries, with or without symmetries, or on more ‘exotic’ manifolds, such as the hyperbolic plane, potentially of considerable mathematical interest, have hardly been touched. The links between point vortex positions in certain equilibria and the roots of families of polynomials, some of them ‘classical’, others arising in problems that appear totally unrelated to vortex dynamics, is very intriguing and suggests that a more encompassing theoretical understanding of vortex equilibria is possible. As the theory is extended to vortices on manifolds, we may hope that profound connections will arise between mathematical entities not commonly thought to be related.

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In recent years, experimental ingenuity has produced images of vortex equilibria of great beauty and appeal. The analytical problem of vortex crystals seems poised to produce results with similar qualities in the near future.

Acknowledgments We thank Russ Donnelly for comments on vortices in superfluids, and GertJan van Heijst, Richard Packard and Dan Durkin for sending us original graphics from their experiments (Figs. 3, 5 and 6, respectively) and allowing us to reproduce these interesting images in this article. We thank Kristjan Onu for, on very short notice, computing vortex coordinates of the states shown in Fig. 7. PKN acknowledges the support of National Science Foundation grant DMS9800797.

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Mayer, A. M. (1878a). A note on experiments with floating magnets; showing the motions and arrangements in a plane of freely moving bodies, acted on by forces of attraction and repulsion; and serving in the study of the directions and motions of lines of magnetic force. Am. J. Sci. Arts 15, 276 –277. [Also published as: Floating magnets. Nature 17, 487 –488.]. Mayer, A. M. (1878b). Experiments with floating and suspended magnets, illustrating the action of atomic forces, molecular structure of matter, allotropy, isomerism, and the kinetic energy of gases. Scientific Am. 2045–2047. Mayer, A. M. (1878c). On the morphological laws of the configuration formed by magnets floating vertically and subjected to the attraction of a superposed magnet; with notes on some of the phenomena in molecular structure which these experiments may serve to explain and illustrate. Am. J. Sci. 16, 247–256. [Also published in Philos. Mag. 7, 98 –108.]. Mertz, G. J. (1978). Stability of body-centered polygonal configurations of ideal vortices. Phys. Fluids 21, 1092–1095. Monkman, J. (1889). On the arrangement of electrified cylinders when attracted by an electrified sphere. Proc. Cambridge Philos. Soc. 6, 179–181. Montaldi, J. (2000). Relative equilibria and conserved quantities in symmetric Hamiltonian systems. In Peyresq Lectures on Nonlinear Phenomena (J., Montaldi and R., Kaiser, eds.), World Scientific, Singapore. Montaldi, J., Soulie`re, A., and Tokieda, T. (2003). Vortex dynamics on a cylinder. SIAM J. Applied Dyn. Syst. Morikawa, G. K., and Swenson, E. V. (1971). Interacting motion of rectilinear geostrophic vortices. Phys. Fluids 14, 1058– 1073. Morton, W. B. (1933). On some permanent arrangements of parallel vortices and their points of relative rest. Proc. R. Irish Acad. A 41, 94–101. Moulton, F. R. (1910). The straight line solutions of the problem of N bodies. Ann. Math. Ser. II 10, 1–17. Nelsen, R. B. (1993). Proofs without Words—Exercises in Visual Thinking. The Mathematical Association of America, 152 pp. Newton, P. K. (2001). The N-Vortex Problem—Analytical Techniques. Applied Mathematical Sciences, Vol. 145, Springer, Berlin. O’Neil, K. A. (1989). On the Hamiltonian dynamics of vortex lattices. J. Math. Phys. 30, 1373– 1379. O’Neil, T. (1999). Trapped plasmas with a single sign of charge. Phys. Today February, 24–30. Pekarsky, S., and Marsden, J. E. (1998). Point vortices on a sphere: Stability of relative equilibria. J. Math. Phys. 39, 5894–5907. Polvani, L., and Dritschel, D. G. (1993). Wave and vortex dynamics on the surface of a sphere. J. Fluid Mech. 255, 35– 64. Porter, A. W. (1906). Models of atoms. Nature 74, 563– 564. Saff, E. B., and Kuijlaars, A. B. J. (1997). Distributing many points on a sphere. Math. Intell. 19, 5–11. Snelders, H. A. M. (1976). A. M. Mayer’s experiments with floating magnets and their use in the atomic theories of matter. Ann. Sci. 33, 67–80. Stieltjes, T. J. (1885). Sur certains polynoˆmes qui verifient une e´quation diffe´rentielle. Acta Math. 6–7, 321–326. Stremler, M. A. (2003). Relative equilibria of vortex arrays. Preprint. Stremler, M. A., and Aref, H. (1999). Motion of three point vortices in a periodic parallelogram. J. Fluid Mech. 392, 101–128. Szego¨, G. (1959). Orthogonal Polynomials. American Mathematical Society Colloquium Publications, Vol. 23, revised ed., American Mathematical Society, Providence, RI. Thomson, J. J. (1883). On the Motion of Vortex Rings. Macmillan, New York (Adams Prize Essay). Thomson, J. J. (1897). Cathode rays. Philos. Mag. 44, 293–316. Thomson, W. (1867). On Vortex Atoms. Proc. R. Soc. Edinburgh 6, 94 –105. Thomson, W. (1878). Floating magnets [illustrating vortex systems]. Nature XVIII, 13–14. Tkachenko, V. K. (1964). Thesis, Institute of Physical Problems, Moscow.

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Computations of Multiphase Flows GRETAR TRYGGVASON,a BERNARD BUNNER,b ASGHAR ESMAEELIa, and NABEEL AL-RAWAHIc a

Mechanical Engineering Department, WPI, Worcester, MA 01609, USA

b

Coventor Inc., 625 Mount Auburn Street, Cambridge, MA 02138, USA c

Department of MIE, Sultan Qaboos University, Al-Khode, Oman

I. Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

82

II. Little Bit of History . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . A. Simple Flows (Re ¼ 0 and Re ¼ 1) . . . . . . . . . . . . . . . . . . . . B. Finite Reynolds Number Flows . . . . . . . . . . . . . . . . . . . . . . .

83 84 86

III. ‘One-field’ Methods . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . A. Mathematical Formulation . . . . . . . . . . . . . . . . . . . . . . . . . . . B. Numerical Implementation. . . . . . . . . . . . . . . . . . . . . . . . . . .

88 88 92

IV. Disperse Flows . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

96

V. Flows with Phase Change . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . A. Solidification. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . B. Boiling . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

104 106 108

VI. Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

112


114

References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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Abstract Computational studies of multiphase flows go back to the very beginning of Computational Fluid Dynamics. It is, however, only during the last decade that direct numerical simulations of multiphase flow have emerged as a major research tool. It is now possible, for example, to simulate the motion of several hundred bubbles and particles in simple flows and to obtain meaningful averaged-quantities that can be compared with experimental results. Much of this progress has been made possible by methods based on the ‘one-fluid’ formulation of the governing equations, in addition to rapidly increasing computational power. Here, we review computations of multiphase flows with particular emphasis on finite Reynolds number flows and methods using the ‘one-fluid’ approach. After an overview of the mathematical formulation and the various ‘one-fluid’ methods, ADVANCES IN APPLIED MECHANICS, VOL. 39 ISSN 0065-2156

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the state-of-the-art is reviewed for three problems: Dispersed bubbly flows, microstructure formation during solidification, and boiling. For the first example numerical methods have reached the maturity where they can be used in scientific studies. For the second and third examples, major numerical development is still taking place. However, progress is rapidly being made and it is realistic to expect large-scale simulations of these problems to become routine within a few years.

I. Introduction Multiphase flows play an essential part in the working of Nature and the enterprises of men. Our every day encounter with liquids is nearly always at a free surface, such as drinking, washing, rinsing and cooking. Similarly, multiphase flows are in abundance in industrial applications: Heat transfer by boiling is the preferred mode in both conventional and nuclear power plants and bubble driven circulation systems are used in metal processing operations such as steel making, ladle metallurgy and the secondary refining of aluminum and copper. The combustion of liquid fuels usually relies on the introduction of the fuel as a spray, and sprays are central to most painting and coating processes. Indeed, except for drag (including pressure drops in pipes) and mixing of gaseous fuels we would not be far off to assert that nearly all industrial applications of fluids involve a multiphase flow of one sort or another. Of natural multiphase flows, rain is perhaps the experience that first comes to mind, but bubbles play a major role in the exchange of heat and mass between oceans and the atmosphere and in volcanic explosions, for example. Living organisms are essentially large and complex multiphase system. Understanding the dynamics of multiphase flows is of critical engineering and scientific importance and the literature is extensive. Much of what we know has, however, been obtained by experimentation and scaling analysis. From a mathematical point of view, multiphase flow problems are notoriously difficult. In addition to the nonlinearity of the governing equations, the position of the phase boundary must generally be found as a part of the solution. Exact analytical solutions therefore exist only for the simplest of problems such as the steady state motion of bubbles and drops in Stokes flow, linear inviscid waves, and small oscillations of bubbles and drops. Experimental studies of multiphase flows are, however, not easy. For many flows of practical interests the length scales are small, the time scales are short, and optical access to much of the flow is limited. The need for numerical solutions of the governing equations has therefore been felt by the multiphase research community since the origin of computational fluid dynamics in the late fifties and early sixties. Although much has been accomplished, simulations of multiphase flows have remained far behind

Computations of Multiphase Flows

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homogeneous flows where direct simulations have become a standard tool in turbulence research. Considerable progress has, however, been made in the last few years and it is clear that direct numerical simulations will play a leading role in multiphase flow research in the next decade. The goal of the present review is to summarize the recent progress made in direct numerical simulations of multiphase flow, with a particular emphasis on the so-called ‘one-field’ formulation. While the one-field formulation has been used for a long time, the accuracy of early predictions has often been suspect. Recent progress, however, has yielded methods that produce solutions of accuracy comparable to what can be obtained by the best of other methods for simple problems. Since the one-field formulation is generally not limited to simple problems, it is now possible to solve fairly complex problems accurately. We will first give a description of the one-field formulation and a short review of the various numerical implementations. Then we will describe three specific applications in detail. We have studied the first one, homogeneous bubbly flows in several papers and the simulations have led to major new insight. The other problems, microstructure formation during solidification, and boiling are much more recent investigation and our understanding of those problems is still relatively primitive. Several other reviews can be found that cover various aspects of the topics discussed here. Early reviews include Hyman (1984), and Floryan and Rasmussen (1989), and more recent reviews are given by Scardovelli and Zaleski (1999) who discuss volume of fluid methods, and Anderson et al. (1998) who review phase field methods. Several up-to-date articles about various aspects of computations of multiphase systems and related problems can be found in a special issue of the Journal of Computational Physics (volume 169, 2001). The book by Shyy et al. (1996) also discusses several aspects of computations of multiphase flows. For discussions of the role of numerical predictions for industrial problems, see Crowe et al. (1998), for example.

II. Little Bit of History Computations of multifluid and multiphase flows are nearly as old as computations of constant density flows. As for such flows, a number of different approaches have been tried and a number of simplifications used. In this section, we will attempt to give a brief but comprehensive overview of the major attempts to simulate multifluid flows. We make no attempt to cite every paper, but have selected a few critical and/or representative papers.

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G. Tryggvason et al. A. Simple Flows (Re ¼ 0 and Re ¼ 1)

In the limit of high and low Reynolds numbers, it is sometimes possible to simplify the flow description considerably by either ignoring inertia completely (Stokes flow) or by ignoring viscous effects completely (inviscid, potential flow). Most success has been achieved when the particles are undeformable spheres where in both these limits, it is possible to reduce the governing equations to a system of coupled ordinary differential equations for the particle positions. For Stokes flow the main contributor is Brady and his collaborators (see Brady and Bossis, 1988, for a review of early work) who have investigated extensively the properties of suspensions of particles in shear flows, and other problems. For inviscid flows, see Sangani and Didwania (1993), and Smereka (1993) for simulations of the motion of many bubbles in periodic domains. For both Stokes flows as well as potential flows, problems with deformable boundaries can be simulated with boundary integral techniques. One of the earliest attempts was due to Birkhoff (1954) where the evolution of the interface between a heavy fluid initially on top of a lighter one (the Rayleigh – Taylor instability) was followed by a method where both fluids were assumed to be inviscid and irrotational, apart from baroclinic generation of vorticity at the interface. This allowed the evolution to be reformulated as an integral equation along the boundary between the fluids. Both the method and the problem later became a stable of multiphase flow simulations. A boundary integral method for water waves was presented by Longuet-Higgins and Cokelet (1976) and used to examine breaking waves. This paper was enormously influential and was followed by a large number of very successful extensions and application, particularly for water waves (Baker et al., 1982; Vinje and Brevig, 1981; Schultz et al., 1994 and others). Other applications include the evolution of the Reyleigh – Taylor instability (Baker et al., 1980), the growth and collapse of cavitation bubbles (Blake and Gibson, 1981; Robinson et al., 2001), the generation of bubbles and drops due to the coalescence of bubbles with a free surface (Oguz and Prosperetti, 1990; Boulton-Stone and Blake, 1993), the formation of bubbles and drops from an orifice (Oguz and Prosperetti, 1993), and the interactions of vortical flows with a free surface (Yu and Tryggvason, 1990), just to name a few. All boundary integral (or boundary element, when the integration is element based) methods for inviscid flows are based on following the evolution of the strength of surface singularities in time by integrating a Bernoulli type equation. The surface singularities give one velocity component and Green’s second theorem yields the other, thus allowing the position of the surface to be advanced in time. Different surface singularities allow for a large


85

number of different methods (some that can only deal with a free surface and others that are suited for two-fluid problems) and different implementations multiply the possibilities even further. For an extensive discussion and recent progress see Hou et al. (2001). Although continuous improvements are being made and new applications continue to appear, two-dimensional boundary integral techniques for inviscid flows are by now—a quarter century after the publication of the paper by Longuet-Higgins and Cocklet—a fairly mature technology. Fully three-dimensional computations are, however, still rare. Chahine and Duraiswami (1992) have computed the interactions of a few inviscid cavitation bubbles and Xue et al. (2001) have simulated a three-dimensional breaking wave. The key to the reformulation of inviscid interface problems in terms of a boundary integral is the linearity of the field equations. In the opposite limit, where inertia effects can be ignored and the flow is dominated by viscous dissipation, the Navier– Stokes equations become linear (Stokes flow) and it is also possible to recast the governing equations into an integral equation on a moving surface. Boundary integral simulations of unsteady two-fluid Stokes problems appear to have been originated by Youngren and Acrivos (1976) and Rallison et al. (1978) who simulated the deformation of a bubble and a drop, respectively, in an extensional flow. Subsequently, several authors have examined a number of problems. Pozrikidis and collaborators have examined several aspects of the suspension of drops, starting with a study by Zhou and Pozrikidis (1993) of the suspension of a few two-dimensional drops in a channel. Simulations of fully three-dimensional suspensions have been done by Loewenberg and Hinch (1996) and Zinchenko and Davis (2000). The method has been described in detail in the book by Pozrikidis (1992) and Pozrikidis (2001) gives a very complete summary of the various applications. In addition to inviscid flows and Stokes flows, boundary integral methods have been used by a number of authors to examine two-dimensional two-fluid flows in Hele-Shaw cells. Although the flow is completely viscous, the flow away from the interface is a potential flow. The interface can be interpreted in terms of standard singularities used for inviscid flows (de Josselin de Jong, 1960) but the evolution equation for the singularity strength is different. This was used by Tryggvason and Aref (1983, 1985) to examine the Taylor–Saffman instability between two fluid, where they used a fixed grid to solve for the normal velocity component, instead of Green’s theorem. Green’s theorem has, however, been used by several authors subsequently. See, for example, DeGregoria and Schwartz (1985) and Meiburg and Homsy (1988), and the review by Hou et al. (2001). Under the heading of simple flows we should also mention simulations of the motion of solid particles, in the limit where the fluid motion can be neglected

86


and the dynamics is governed only by the inertia of the particles. Several authors have followed the motion of a large number of particles that interact only when they collide with each other. Here, it is also sufficient to solve a system of ODEs for the particle motion. Simulations of this kind are usually called ‘granular dynamics’ and a discussion can be found in Louge (1994), for example. While these methods have been enormously successful in simulating certain types of dispersed multiphase flows, they are limited to a very small class of problems.

B. Finite Reynolds Number Flows For intermediate Reynolds numbers it is necessary to solve the full Navier– Stokes equations. Nearly ten years after Birkhoff’s effort to simulate the Rayleigh – Taylor problem by a boundary integral technique, the Marker-AndCell (MAC) method was developed at Los Alamos by Harlow and collaborators. In Harlow and Welch (1965), the method was introduced and two sample computations of the so-called dam breaking problem are shown. Several papers quickly followed: Harlow and Welch (1966) examined the Rayleigh – Taylor problem and Harlow and Shannon (1967) studied the splash when a drop hits a liquid surface. As originally implemented, the MAC method assumed a free surface so there was only one-fluid involved. This required boundary conditions to be applied at this surface and the fluid in the rest of the domain to be completely passive. The Los Alamos group realized, however, that the same methodology could be applied to two-fluid problems. Daly (1969a,b) computed the evolution of the Rayleigh – Taylor instability for finite density ratios and Daly and Pracht (1968) examined the initial motion of density currents. Surface tension was then added by Daly (1969a,b) and the method again used to examine the Rayleigh –Taylor instability. The MAC method quickly attracted a small group of followers that used it to study several problems: Chan and Street (1970) applied it to free surface waves, Foote (1973, 1975) simulated the oscillations of an axisymmetric drop and the collision of a drop with a rigid wall, and Chapman and Plesset (1972) and Mitchell and Hammitt (1973) simulated the collapse of a cavitation bubble. While the Los Alamos group did a number of computations of various problems in the sixties and early seventies, and Harlow described the basic idea in a Scientific American article (Harlow and Fromm, 1965), the enormous potential of this newly found tool did not, for the most part, capture the fancy of the fluid mechanics research community. Bradley and Stow (1978) who compared their experimental results for collision of drops with Foote’s results and found excellent agreement, commented that ‘this complicated treatment gives little insight into the physical processes involved.’ Although the MAC


87

method was designed specifically for multifluid problems (hence the M for Markers!) it was also the first method to successfully solve the Navier –Stokes equation using the primitive variables (velocity and pressure). The staggered grid used was also a novelty and today it is a common practice to refer to any method using a projection based time integration on a staggered grid as a MAC method. The next generation of methods for multifluid flow evolved gradually from the MAC method. It was already clear in the Harlow and Welch (1965) paper that the marker particles could cause inaccuracies and among the number of algorithmic ideas explored by the Los Alamos group, the replacement of the particles by a marker function soon became the most popular alternative. Thus the volume-offluid (VOF) method was born. VOF was first discussed in a refereed journal article by Hirt and Nichols (1981), but the method apparently originated a few years earlier (DeBar, 1974; Noh and Woodward, 1976). The VOF method has been extended in various ways by a number of authors. It has also been distributed widely as the NASA SOLA-VOF code and as FLOW3D from Fluid Sciences Inc. In addition, many commercial CFD codes now include the option of simulating free surface or multiphase flows using the VOF method. For a review of VOF methods, see Scardovelli and Zaleski (1999). While the MAC methodology and its successors were being developed, other techniques were also being explored. Hirt et al. (1970) described one of the earliest use of structured boundary fitted Lagrangian grids. This approach is particularly well-suited when the interface topology is relatively simple and no unexpected interface configurations develop. In a related approach, a grid line is aligned with the fluid interface, but the grid away from the interface is generated using standard grid generation techniques such as conformal mapping or other more advanced elliptic grid generation schemes. The method was used by Ryskin and Leal (1984), to compute the steady rise of buoyant, deformable, axisymmetric bubbles. Ryskin and Leal assumed that the fluid inside the bubble could be neglected, but Dandy and Leal (1989) and Kang and Leal (1987) extended the method to two-fluid problems and unsteady flows. Several authors have used this approach to examine relatively simple problems such as the steady state motion of single particles or moderate deformation of free surfaces. Fully three-dimensional simulations are relatively rare and it is probably fair to say that it is unlikely that this approach will be the method of choice for very complex problems such as the three-dimensional unsteady motion of several particles. A much more general approach to continuously represent a fluid interface by a grid line is to use unstructured grids to resolve the fluid motion. This allows grid points to be inserted and deleted as needed and distorted grid cells to be reshaped. While the grid was moved with the fluid velocity in some of the early applications of this method, the more modern approach is to either move only the interface


88

points or to move the interior nodes with a velocity different from the fluid velocity in such a way that the grid distortion is reduced but adequate resolution is still maintained. A large number of methods have been developed that fall into this general category, but we will refer to a few examples only. Oran and Boris (1987) simulated the breakup of a two-dimensional drop; Shopov et al. (1990) examined the initial deformation of a buoyant bubble; Feng et al. (1994, 1995) and Hu (1996) computed the unsteady two-dimensional motion of several particles, and Fukai et al. (1995) did axisymmetric computations of the collision of a single drop with a wall. Although this appears to be a fairly complex approach, Johnson and Tezduyar (1997) and Hu et al. (2001) have recently produced very impressive results for the three-dimensional unsteady motion of many spherical particles. Several hybrid methods combine the ideas discussed above in a variety of ways. Front tracking where the interface is marked by connected marker points but a fixed grid is used for the fluid within each phase have been particularly successful. In the method of Glimm and collaborators (for a recent review see Glimm et al., 2001) the fixed grid is modified near the front to make a grid line follow the interface, but in the method of Tryggvason and collaborators (Unverdi and Tryggvason, 1992a,b; Tryggvason et al., 2001) the interface is spread out slightly and represented on a fixed grid. The most recent addition to the collection of methods capable of simulating finite Reynolds number multiphase flows is the lattice boltzman method (LBM). Although there have been some doubts about the accuracy and correctness of the LBM, it seems now clear that they can be used to produce results of accuracy comparable to more conventional methods. It is still not clear whether the LBM is significantly faster or simpler than other methods (as sometimes claimed), but most likely these methods are here to stay. For a discussion see, for example, Shan and Chen (1993) and Sankaranarayanan and Shan (2002).

III. ‘One-Field’ Methods A. Mathematical Formulation The flow of multiphase flow is governed by the Navier – Stokes equations with the appropriate boundary conditions. The key to several methods used to simulate multiphase flow is the use of a single set of conservation equations for the whole flow field. In addition to accounting for differences in the material properties of the different phases, interfacial phenomena such as surface tension must be included by adding the appropriate interface terms to the governing equations.


89

Since these terms are concentrated at the boundary between the different fluids, they are represented by d-functions. When the equations are discretized, the d-functions must be approximated along with the rest of the equations. The material properties and the flow field are, in general, discontinuous across the interface and all variables must be interpreted in terms of generalized functions. Before we write down the equations governing multiphase flow it is useful to discuss a few elementary aspects of the representation of a discontinuous function by generalized functions. The various fluids can be identified by a step (Heaviside) function H which is 1 for one particular fluid and 0 elsewhere. Since H remains constant in each fluid, we can identify H with a marker function that is advected by the flow

›H þ u·7H ¼ 0: ›t

ð2:1Þ

The interface itself is marked by a nonzero value of the gradient of the step function. To find the gradient it is most convenient to express H in terms of an integral over the product of one-dimensional d-functions. For a two-dimensional field: Hðx; y; tÞ ¼

ð

dðx 2 x0 Þdðy 2 y0 Þda0 :

ð2:2Þ

A

In two-dimensions the integral is over an area A bounded by a contour S. H is obviously 1 if ðx; yÞ is enclosed by S and 0 otherwise. To find the gradient of H we note first that since the gradient is with respect to the unprimed variables, the gradient operator can be put under the integral sign. Since the d-functions are anti-symmetric with respect to the primed and unprimed variables, the gradient with respect to the unprimed variables can be replaced by the gradient with respect to the primed variables. The resulting area (or volume in threedimensions) integral can be transformed into a line (surface) integral by a variation of the divergence theorem for gradients. Symbolically: 7H ¼

ð

7½dðx 2 x0 Þdðy 2 y0 Þda0 ¼ 2 A

ð

70 ½dðx 2 x0 Þdðy 2 y0 Þda0

A

þ ¼ 2 dðx 2 x0 Þdðy 2 y0 Þn0 ds0 :

ð2:3Þ

S

Here n is a unit normal vector to the interface and the prime on the gradient symbol denotes the gradient with respect to the primed variables. Although we have assumed that the area occupied by the marked fluid is finite so that S is a


90

closed contour, the contribution of most of the integral is zero and we can replace it by one over a part of the contour and drop the circle on the integral: ð 7H ¼ 2 dðx 2 x0 Þdðy 2 y0 Þn0 ds0 : ð2:4Þ S

By introducing local coordinates, tangent (s) and normal (n) to the front, we can write

dðx 2 x0 Þdðy 2 y0 Þ ¼ dðsÞdðnÞ; and evaluate the integral ð ð 2 dðx 2 x0 Þdðy 2 y0 Þn0 ds0 ¼ 2 dðs0 Þdðn0 Þn0 ds0 ¼ 2dðnÞn: S

ð2:5Þ

ð2:6Þ

S

This allows us to use a one-dimensional delta function of the normal variable, instead of the two-dimensional one in Eq. (2.3). Although we have used two dimensions in the discussion above, the same arguments apply for threedimensional fields as well. If the density of each phase is assumed to be constant, it can be written in terms of the constant densities and the Heaviside function:

rðx; y; tÞ ¼ ri Hðx; y; tÞ þ ro ð1 2 Hðx; y; tÞÞ:

ð2:7Þ

Here, ri is the density where H ¼ 1 and ro is the density where H ¼ 0. The gradient of the density is given by 7r ¼ ri 7H 2 ro 7H ¼ ðri 2 ro Þ7H ð ¼ Dr dðx 2 x0 Þdðy 2 y0 Þn0 ds0 ¼ DrdðnÞn;

ð2:8Þ

where we have put Dr ¼ ro 2 ri : In conservative form the Navier –Stokes, with the singular surface tension term added, are:

›ru þ 7·ruu ¼ 27P þ rf þ 7·mð7u þ 7T uÞ þ skdðnÞn: ›t

ð2:9Þ

This equation is valid for the whole flow field, even if the density field, r, and the viscosity field, m, change discontinuously. Here u is the velocity, P is the pressure, and f is a body force. Surface forces are added at the interface. k is the curvature for two-dimensional flows and twice the mean curvature for threedimensional flows and n is a unit vector normal to the front. Mass conservation is given by

›r þ 7·ru ¼ 0: ›t

ð2:10Þ


91

In most cases, the fluids are taken to be incompressible so that the density of a fluid particle remains constant: Dr ¼ 0: ð2:11Þ Dt This reduces the mass conservation equation to 7·u ¼ 0:

ð2:12Þ

Usually the viscosity in each fluid is also taken to be constant: Dm ¼ 0: ð2:13Þ Dt The ‘one-fluid’ equations are an exact rewrite of the Navier –Stokes equations for the fluid in each phase and the interface boundary conditions. To show the equivalence, it is simple to start with the ‘one-fluid’ formulation and show how the equations for each phase and the boundary conditions emerge. The velocity can be written as u ¼ H1 u1 þ H2 u2 ;

ð2:14Þ

where H2 ¼ 1 2 H1 : u1 is the velocity in fluid 1 and u2 in fluid 2 and u1 ¼ u2 at the interface. Similarly r ¼ H1 r1 þ H2 r2 ; m ¼ H1 m1 þ H2 m2 ; and P ¼ H1 P1 þ H2 P2 : Substituting these expressions into Eq. (2.9), using Eq. (2.1), that H1 H1 ¼ H1 ; H2 H2 ¼ H2 ; H1 H2 ¼ 0; and separating out the terms with H1 ; H2 ; and d, we obtain ›r1 u 1 þ 7·r1 u1 u1 ¼ 27P1 þ r1 f þ 7·m1 ð7u1 þ 7T u1 Þ ð2:15Þ ›t ›r2 u 2 þ 7·r2 u2 u2 ¼ 27P2 þ r2 f þ 7·m2 ð7u2 þ 7T u2 Þ ð2:16Þ ›t ½½2PI þ mð7u þ 7T uÞ·n ¼ skn:

ð2:17Þ

In the last equation, the brackets denote the jump across the interface. Equations (2.15) – (2.17) are, of course, the governing equations written separately for each phase and the usual statement of continuity of stresses at a fluid boundary. The governing equations as listed above assume that the only complication in multifluid flows is the presence of a moving phase boundary with a constant surface tension. While these equations can be used to describe many problems of practical interest, additional complications quickly emerge. The presence of a surfactant or contaminants at the interface between the fluids is perhaps the most common one, but other effects such as phase change are common too. We will address the state-of-the-art in computing more complex system later in this article. There is, however, one issue that needs to be addressed here. While the Navier– Stokes equations, with the appropriate interface conditions, govern the evolution of a system of two fluids separated by a sharp interface, the topology of

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the interface can change by processes that are not included in the continuum description. Topology changes are common in multiphase flow, such as when, for example, drops or bubbles break up or coalesce. Topology changes in multifluid flows can be divided into two broad classes: † Films that rupture. If a large drop approaches another drop or a flat surface, the fluid in between must be ‘squeezed’ out before the drops are sufficiently close so that the film becomes unstable to attractive forces and ruptures. † Threads that break. A long and thin cylinder of one-fluid will generally break by Rayleigh instability where one part of the cylinder becomes sufficiently thin so that surface tension ‘pinches’ it in two. The exact mechanisms of how threads snap and films break are still being actively investigated. There are, however, good reasons to believe that threads can become infinitely thin in a finite time and that their breaking is ‘almost’ described by the Navier– Stokes equations (Eggers, 1995). Films, on the other hand, are generally believed to rupture due to short range attractive forces, once they are a few hundred Angstrom thick. While it seems unlikely that films of viscous fluids would ever drain completely, there are recent evidences that suggest that inviscid films can become infinitely thin in a finite time, under certain circumstances (Hou et al., 1997). The assumption that films rupture due to attractive forces not included in the usual continuum description, has recently been challenged by Shikhmurzaev (2002) who argues that with the proper continuum description films can eventually disappear. While these ideas are still in the formative stage, they have the potential to have significant impact on how films are described computationally.

B. Numerical Implementation When a fluid interface is captured on a fixed grid, a marker function f is introduced such that f ¼ 1 in one-fluid and f ¼ 0 in the other fluid. The marker function is advected by the fluid, so once the fluid velocity is known, f can, in principle, be updated by integrating

›f þ u·7f ¼ 0 ›t

ð2:18Þ

in time. Here, f is essentially the Heaviside function H in Eq. (2.1), but in the numerical implementation f can have a smooth transition zone between one value and the other. We will therefore follow established convention and use f to refer


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to a marker function introduced for computational reasons. Integrating Eq. (2.18) for discontinuous data is, in spite of its apparent simplicity, one of the hardest problems in computational fluid dynamics. For low order advection schemes the discontinuity is rapidly smeared so the sharp interface will eventually disappear, and high order advection schemes produce unacceptable wiggles or oscillations in f. It is perhaps counter-intuitive, but advecting f accurately is harder than advecting a shock wave since a shock that is smoothed will attempt to regenerate itself, but there is no mechanism that will resharpen a discontinuity in f, once it has been smoothed. For low order advection, the reason for the excessive diffusion of f is that the fluxes between the cells are computed from the average value of f in each control volume and f will start to flow out of a cell long before the cell is full. The first method to successfully simulate the finite Reynolds number motion of free-surface and two-fluid flows was the marker-and-cell (MAC) method, where marker particles distributed uniformly in each fluid were used to identify each fluid. Different markers were used for each phase and the material properties were reconstructed from the marker particles. Separate surface markers were also sometimes introduced to facilitate the computation of the surface tension. While the historical importance of the MAC method for multiphase flow simulations cannot be overstated, it is obsolete by now. In current usage, the term ‘MAC method’ usually refers to a projection method using staggered grids. In the VOF method the marker particles are replaced by a marker function f that describes the location of each phase. To accurately advect f, the distribution of f in a cell (or the location of the interface) is reconstructed from the value of f in an interface cell and its neighbors. The location of the interface allows the fluxes to be computed more accurately and, in particular, prevent f from flowing out of a cell before it is full. The main difficulty with early VOF methods was maintaining the integrity of the boundary between the different fluids and the computation of the surface tension. In the method described by Hirt and Nichols (1981), the interface was approximated by straight line parallel to the y-axis for advection in the x-direction and parallel to the x-axis for y-advection. Generally this leads to deterioration of the interface (in the form of flotsam and jetsam) and later implementations have used arbitrarily oriented line segments (Ashgriz and Poo, 1991; Youngs, 1982, for example). Other improvements include methods to compute the surface tension by differentiation of the f function by Brackbill et al. (1992) and Lafaurie et al. (1994), and the use of sub-cells to improve the resolution of the interface by Chen et al. (1997). A review of the VOF method can be found in Scardovelli and Zaleski (1999). In the VOF method, the transition between one-fluid and the other takes place over about one grid cell. While it is clearly desirable to limit the thickness

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of the (unphysical) transition zone as much as possible, the development of several methods that smear the transition zone over a few grid cells suggests that such smoothing improves the isotropy and integrity of the interface. In the method of Unverdi and Tryggvason (1992a,b), the interface is tracked by connected marker points that are used to construct an indicator function to identify the different fluids and to compute surface tension. The method and its applications are discussed in Tryggvason et al. (2001). The level set method differs from the VOF in that instead of advecting a discontinuous marker function f, a continuous function (usually denoted by f) is advected with the flow. The interface is identified with f ¼ 0. To reconstruct the material properties of the flow (density and viscosity, for example), and to find the normal to the interface, a marker function is constructed from f f ¼ f ðfÞ:

ð2:19Þ

Unlike in the VOF method where f is nearly discontinuous, in the level set method f is given a smooth transition zone from one-fluid to the next. This is the same approach taken in the front tracking method of Unverdi and Tryggvason (1992a,b) and in the phase field method and clearly results in better behaved solutions than in the original VOF method. However, to be able to do the mapping f ¼ f ðfÞ in such a way that the thickness of the transition zone remains under control, f must have the same shape near the interface for all times. If f is simply advected by the fluid, this is not the case. Where the interface is stretched the gradient of f becomes steeper and if the interface is compressed the gradient becomes smaller. To deal with this problem, Sussman et al. (1994) introduced a reinitialization procedure where they modified f by solving

›f þ sgnðfÞðl7fl 2 1Þ ¼ 0: ›t

ð2:20Þ

at each time step. Here, sgn is the sign function. t is a pseudo-time and Eq. (2.20) is integrated to steady state at every time step, thus enforcing l7fl ¼ 1: This makes f a distance function and ensures that the slope near the interface is always the same. This step was critical in making level sets work for fluid dynamics simulations, but unfortunately introduced other problems such as loss of mass. While many of the early difficulties with level set methods have been overcome, the original developers apparently have not been satisfied with the performance of the method and they have attempted to improve the performance in various ways. Sussman et al. (1998) improved mass conservation, a hybrid VOF/level-set method was developed by Sussman and Puckett (2000), and Fedkiw et al. (1999) developed a ‘ghost fluid’ method based on assigning fictitious values to grid


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points on the other side of a fluid discontinuity. The level set method is reviewed by Osher and Fedkiw (2001) and by Sethian (2001). While the front tracking method and the level set methods can be described as a somewhat roundabout way to integrate Eq. (2.18) in time, in the CIP (see Yabe et al., 2001, for a review) Eq. (2.18) is integrated directly by using an advection method based on fitting the f profile using a cubic function. The success of the CIP method suggests that other advanced monotonic advection methods could be used as well, at least for short time evolution of the interface. In the phase field method, the governing equations are modified in such a way that the dynamics of the smoothed region between the different fluids is described in a thermodynamically consistent way. However, in actual implementations the thickness of the transition is much larger than it is in real systems and it is not clear whether keeping the thermodynamic conditions correct in an artificially thick interface has any advantages over methods that model the behavior in the transition zone in other ways. The phase field approach has found widespread use in simulation of solidification, but its use for fluid dynamic simulations is relatively limited, see Jacqmin (1999) and Jamet et al. (2001). The VOF method, the front tracking method of Unverdi and Tryggvason, the level-set method, the phase field method and the CIP method are usually implemented on regular structured staggered grids and the Navier—Stokes equations are usually solved by an explicit projection method. Ignoring minor differences between the exact implementation used by different authors, the fluid solvers are therefore essentially the same. Although the way the marker function identify the different fluids differs, all these methods have been used to generate impressive results, particularly for relatively short time scales. At long times and extreme values of the material parameters (low viscosity and large differences between the material properties) most of the methods can run into some difficulties, but the difficulties tend to be dependent on the exact problem being studied, the resolution used, and the details of the code used. It is probably true (and certainly the opinion of the authors of this review) that explicit tracking offers the potentially highest accuracy and greatest flexibility. However, tracking is slightly more complex than the use of a marker function and for many problems the extra complexity may not be needed. There is one area where there is some difference between methods that use a marker function and methods that track the interface using connected marker points. When thin films rupture, the interface must undergo topology changes. A marker function will fuse two-fluid blobs together once the thickness of the film between them is equal to the mesh size and the film will rupture. Tracking methods, on the other hand, will keep the fluids separate, no matter how thin the film becomes. While the default for marker function based methods is to always

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rupture thin films and for methods based on explicit tracking is to never rupture, both approaches can be modified to change the default behavior. Two drops identified by different marker functions would, for example, not coalesce automatically and the connectivity of the marker points can be changed to allow a thin film to rupture. While automatic coalescence can be very convenient in some cases, particularly if the topology change does not need to be treated accurately, this is also a serious weakness of such methods. Coalescence is usually strongly dependent on how quickly the fluid between the coalescing parts drains and simply connecting parts of the interface that are close may give the incorrect solution.

IV. Disperse Flows In many industrial and natural processes, multiphase flows consists of one phase in the form of well defined bubbles, drops, or solid particles dispersed in another continuous phase. Bubbly flows occur in boiling heat transfer, cloud cavitation, aeration and stirring of reactors in water purification and waste water treatment plants, bubble columns and centrifuges in chemical industry, cooling circuits of nuclear reactors, the exchange of gases and heat between the oceans and the atmosphere, and explosive volcanic eruptions, just to name a few examples. Similarly, drops are found in sprays used in atomization of liquid fuels, painting and coating, emulsions, and rain. Understanding the evolution and properties of dispersed flows is therefore of major technological as well as scientific interest. For engineering applications with a large number of bubbles, drops, or solid particles, computational modeling relies on equations that describe the average flow field. The two-fluid model, where separate equations are solved for the dispersed and the continuous phase, is the most common approach. Since no attempt is made to resolve the unsteady motion of individual particle, closure relations are necessary for the unresolved motion and the forces between the particle and the continuous phase. Closure relations are usually determined through a combination of dimensional arguments and correlation of experimental data. The situation is analogous to computations of turbulent flows using the Reynolds averaged Navier– Stokes equations where momentum transfer due to unsteady small-scale motion must be modeled. For details of two-fluid modeling see Drew (1983), Drew et al. (1993), and Zhang and Prosperetti (1994). For the turbulent motion of single phase flows, direct numerical simulations, where the unsteady Navier– Stokes equations are solved on fine enough grids to fully resolve all flow scales, have had a major impact on closure modeling. The goal of direct numerical simulations of multiphase flows is similar. In addition to


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information about how the drift Reynolds number, velocity fluctuations, and bubble dispersion change with the properties of the system, the computations should yield insight into how bubbles and drops interact, both with each other and the continuous phase. The simulations should show whether there is a predominant microstructure and/or interaction mode, and whether the flow forms structures that are much larger than the size of the dispersed particles. Information about the microstructure is essential for the construction of models of multiphase flows and can also help to identify what approximations can be made. Although the need for direct numerical simulations to help with the construction of reliable closure models has been recognized for a long time, it is only recently that major progress has been made. In the limit of high and low Reynolds numbers it is sometimes possible to simplify the flow description considerably by either ignoring inertia completely (Stokes flow) or by ignoring viscous effects and assuming a potential flow. For spherical particles, in both these limits, it is possible to reduce the governing equations to a system of coupled ordinary differential equations for the particle positions. For Stokes flow, Brady and collaborators (see Brady and Bossis, 1988, for a review) have investigated several problems, including the suspension of particles in shear flows. The simulations were used to examine the importance of Brownian motion compared to hydrodynamic effects and structure formation in shear flows, for example. Using the Lattice Boltzman method, Ladd (1993, 1997) has examined the sedimentation velocity of several thousand spherical particles in Stokes flow and how it depends on the size of the system. In the other limit, where viscosity is assumed to be small, Sangani and Didwania (1993) and Smereka (1993) simulated the motion of spherical bubbles in a periodic box and observed that the bubbles tended to form horizontal clusters, particularly when the variance of the bubble velocities was small. For both Stokes and potential flows, problems with deformable boundaries can be simulated using boundary integral techniques. For Stokes flow, boundary integral methods have been used to examine the dynamics of various suspension problems. For recent papers see, for example, the study by Li and Pozrikidis (2000) of the dynamics of two-dimensional drops in a channel and the simulation of a few three-dimensional drops in a channel by Zinchenko and Davis (2000). For finite Reynolds number dilute flows with very small particles, a number of investigators (including Squires and Eaton, 1990; Wang and Maxey, 1993; Truesdell and Elghobashi, 1994) have used the so-called point particle approximation where the dispersed phase is represented by points moving in an otherwise constant-density flow. The force on each point particle is specified by analytical models for very low and very high Reynolds numbers (Stokes flow and potential flow) and by empirical correlations for finite values. In some cases

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the particles are assumed to have no effect on the fluid flow, but in other cases the forces from the particles are added to the right-hand side of the Navier –Stokes equations. The fundamental assumption is that the length scales of the fluid motion are much larger than the size of the particles and that there are no direct particle – particle interactions. Furthermore, the wake is usually neglected completely. Recently, however, Pan and Banerjee (1997) and Maxey et al. (1997) have presented simulations where the particle size and its wake are accounted for by distributing the effect of the particle over a few grid points. For nondilute flows at intermediate Reynolds numbers it is necessary to solve the full unsteady Navier –Stokes equations. Such simulations for the unsteady motion of many bubbles or particles are relatively recent. Unverdi and Tryggvason (1992a,b) computed the interactions of 2 two- and threedimensional bubbles and Esmaeeli and Tryggvason (1996) followed the evolution of a few hundred two-dimensional bubbles. Esmaeeli and Tryggvason (1998, 1999) simulated the unsteady motion of several two- and three-dimensional bubbles. Mortazavi and Tryggvason (2000) examined the motion of a periodic row of drops in a channel and Mortazavi (1995) simulated several two- and three-dimensional drops in a pressure-driven channel flow. More recently, Bunner and Tryggvason (1999, 2002a,b) used a fully parallelized version of the method to examine three-dimensional systems with a much larger number of bubbles. The results of Bunner and Tryggvason will be discussed in more detail shortly. Other investigations include simulations of several two-dimensional bubbles in shear flow by Esmaeeli (1995) and a few studies of bubbles rising in a duct by Bunner (1999). For upflow, Bunner’s computations show concentration of bubbles near the bubble walls, in agreement with experimental observations. Other studies of the motion and interactions of many bubbles have been done by several Japanese authors. Early work, using the VOF method to compute the motion of a single two-dimensional bubble can be found in Tomiyama et al. (1993) and more recent work on bubble interactions, using both VOF and the Lattice Boltzman Method, is presented in Takada et al. (2000, 2001). Studies of the effect of bubbles on the drag in turbulent flows have been done by Kanai and Miyata (2001) and Kawamura and Kodama (2002). Major progress has also been made in the simulation of finite Reynolds number suspension of rigid particles. Feng et al. (1994, 1995) simulated the twodimensional, unsteady motion of one and two rigid particles, Hu (1996) computed the motion of a few hundred two-dimensional particles and fully threedimensional simulations of hundred particles were presented by Johnson and Tezduyar (1997). Recent papers include simulations of over 1000 spheres by Pan et al. (2002) and a study of the fluidization of 300 circular particles in plane Poiseuille flow by Choi and Joseph (2001).


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Homogeneous bubbly flow with many buoyant bubbles rising together in an initially quiescent fluid is perhaps one of the simplest example of dispersed flow. Such flows can be simulated using periodic domains where the bubbles in each period interact freely, but the configuration is repeated infinitely many times in each coordinate direction. In the simplest case, there is only one bubble per period so the configuration of the bubbles does not change as they rise. While such regular arrays are unlikely to be seen in an experiment, they provide a useful reference configuration for freely evolving array. As the number of bubbles in each period is increased, the regular array becomes unstable and the bubbles generally rise unsteadily, repeatedly undergoing close interactions with other bubbles. The behavior is, however, statistically steady and the average motion (averaged over long enough time) does not change. While the number of bubbles clearly influences the average motion for small enough number of bubbles per period, the hope is that once the size of the system is large enough, information obtained by averaging over each period will be representative of a truly homogeneous bubbly flow. Tryggvason and collaborators have examined the motion of nearly spherical bubbles at moderate Reynolds numbers in a number of papers and here we will briefly review some of their results. Esmaeeli and Tryggvason (1998) examined a case where the average rise in Reynolds number of the bubbles remained relatively small (1 – 2) and Esmaeeli and Tryggvason (1999) looked at another case where the Reynolds number is 20 – 30. In both cases, most of the simulations were limited to two-dimensional flows, although a few three-dimensional simulations with up to eight bubbles were included. Simulations of freely evolving arrays were compared with regular arrays and it was found that while freely evolving bubbles at low Reynolds numbers rise faster than a regular array (in agreement with Stokes flow results), at higher Reynolds numbers the trend is reversed and the freely moving bubbles rise slower. The time averages of the two-dimensional simulations were generally well converged but exhibited a dependency on the size of the system. This dependency was stronger for the low Reynolds number case than the moderate Reynolds number one. Although many of the qualitative aspects of the interactions of a few three-dimensional bubbles are captured by two-dimensional simulations, the much stronger interactions between two-dimensional bubbles can lead to quantitative differences. To examine a much larger number of three-dimensional bubbles, Bunner and Tryggvason (2002a,b) developed a parallel version of the method used by Esmaeeli and Tryggvason. Their largest simulations followed the motion of 216 three-dimensional buoyant bubbles per periodic domain for a relatively long time. The governing parameters were selected such that the average rise in Reynolds number was about 20 –30 (comparable to Esmaeeli and Tryggvason,

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1999, but not identical), depending on the void fraction, and deformations of the bubbles were small. Although the motion of the individual bubbles was unsteady, the simulations were carried out for a long enough time so the average behavior of the system were well-defined, as in the two-dimensional simulations of Esmaeeli and Tryggvason. Simulations with different number of bubbles were used to explore the dependency of the various average quantities on the size of the system. The average rise in Reynolds number and the Reynolds stresses were essentially fully converged for systems with 12 bubbles, but the average fluctuation of the bubble velocities required larger systems. Examination of the pair distribution function for the bubbles showed a preference for horizontal alignment of bubble pairs, independent of system size, but the distribution of bubbles remained nearly uniform. The results showed that there is an increased tendency for the bubbles to line up side-by-side as the rise in Reynolds number increases, suggesting a monotonic trend from the nearly no preference found by Ladd (1993) for Stokes flow, toward the strong layer formation seen in the potential flow simulations of Sangani and Didwania (1993) and Smereka (1993). To examine the usefulness of simplified models, the results were compared with analytical expressions for simple cell models in the Stokes flow and the potential flow limits. The simulations were also compared to a two-dimensional Stokes flow simulation. The results show that the rise in velocity at low Reynolds number is reasonably well predicted by Stokes flow based models. The bubble interaction mechanism is, however, quite different. At finite Reynolds numbers, two-bubble interactions take place by the ‘drafting, kissing, and tumbling’ mechanism of Fortes et al. (1987), where a bubble is drawn into the wake of a bubble in front. Once in the wake the bubble is shielded from the oncoming flow so that it accelerates and catches up with the bubble in front (‘drafting’). After colliding (‘kissing’) the bubbles ‘tumble’ and drift apart. This behavior is, of course, very different from either a Stokes flow where two bubbles do not change their relative orientation unless acted on by a third bubble, or the predictions of potential flow where a bubble is repelled from the wake of another one, not drawn into it. For moderate Reynolds numbers (about 20), the Reynolds stresses for a freely evolving twodimensional bubble array were comparable to Stokes flow while in threedimensional flow the results were comparable to predictions of potential flow cell models. Figures 1 and 2 show two examples from these simulations. To examine the effect of bubble deformation, Bunner and Tryggvason (2003) have done two sets of simulations using 27 bubbles per periodic domain. In one the bubbles are spherical, in the other the bubbles deform into ellipsoids of an aspect ratio of approximately 0.8. The nearly spherical bubbles quickly reach a well defined average rise velocity and remain nearly uniformly distributed in the computational domain. The deformable bubbles initially behave similarly, except


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Fig. 1. One frame from a simulation of the buoyant rise of 216 bubbles in a periodic domain.

that their velocity fluctuations are larger. After rising several bubble diameters, the nearly uniform distribution transitions to a completely different state where the bubbles accumulate in vertical streams, rising much faster than when they are uniformly distributed. This behavior can be explained by the dependency of the lift force that the bubbles experience on the deformation of the bubbles. Suppose a large number of bubbles come together for some reason. Since they look like a large bubble to the surrounding fluid, the group will rise faster than a single bubble, drawing fluid with them. A spherical bubble rising in the shear flow at the edge of this plume will experience a lift force directed out from the plume. The lift force on a deformable bubble, on the other hand, is directed into the plume as explained by Ervin and Tryggvason (1997). Spherical bubbles, temporarily crowded together, will therefore disperse but deformable bubbles will be drawn into the plume, further strengthening the plume. Bunner and Tryggvason (2003) only observed streaming for relatively high void fractions but speculated that the phenomenon could best be described as a nonlinear instability that required finite size perturbation in the bubble distribution to trigger it. At low void fraction the probability of sufficiently large perturbation was smaller and they suggested that their simulations for lower void fractions simply had not been conducted for long

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Fig. 2. A close up of the flow field around a few bubbles. From a simulation of the buoyant rise of 91 bubbles in a periodic domain.

enough time. To test this hypothesis, they examined the evolution of bubbles initially placed in a single column. As expected, the spherical bubbles immediately dispersed, but the deformable bubbles stayed in the column. Figure 3 shows two frames from the last simulation, where the spherical bubbles have dispersed but the deformable bubbles are crowded in a stream in the middle of the domain. The difference highlights the strong effect of deformation on the mechanism driving the interactions between the bubbles and on the collective motion of the bubbles. In addition to studies of bubbles with relatively modest rise Reynolds number in fully periodic domains, the method of Unverdi and Tryggvason has been used to examine a number of other bubble problems. Most of these simulations are, however, much more limited in scope than those reported in the papers by Esmaeeli and Tryggvason and Bunner and Tryggvason. It is well known that bubbles at high enough Reynolds number rise unsteadily, either wobbling as they rise or rising along a spiral path. Recent experimental studies by Ellingsen and Risso (2001) suggest that the wobbling mode may be a transitionary phase


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Fig. 3. Two frames from simulations of 27 bubbles. In the left frame the bubbles are nearly spherical but in the right hand frame the bubbles are ellipsoidal.

and that wobbly bubbles would eventually rise along spiral paths, if one waited long enough. Computationally it is found that two-dimensional bubbles in periodic domains start to wobble at much lower rise in Reynolds number than their three-dimensional counterparts. Esmaeeli et al. (1994) briefly examined bubbles that settle down into periodic wobbling and showed that the bubbles slow down significantly once they start to wobble. Go¨z et al. (2000) examined higher Reynolds numbers and found what looked like aperiodic motion at high enough Reynolds numbers. They suggested that the results indicated that real (threedimensional) deformable bubbles rising at high enough Reynolds number would exhibit chaotic motion. However, since air bubbles become spherical cap bubbles when their size (and rise Reynolds number) increases, such motion might not be observed under normal conditions. In bubble columns, continuous size distributions are generally observed but often the distribution is roughly bimodal, with many small and relatively few large bubbles. It may be expected that small bubbles are drawn into the wake of larger bubbles leading to drag reduction and thus an increase in the average rise velocity of such a group of bubbles. It could also be, however, that this is largely an effect of deformability and does occur for spherical bubbles only if their diameter ratio is sufficiently large. In order to isolate the effect of a bidispersed bubble size distribution, systems with an equal number of small and large

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spherical bubbles were simulated (Go¨z et al., 2001). The volume ratio of large to small bubbles was 2. The average rise velocity (of all bubbles) and the average turbulent kinetic energy of the liquid in dependence of the void fraction turned out to be very close to those of the corresponding monodisperse systems. There are differences, however, in the fluctuation velocities. The average vertical and horizontal fluctuation velocities over each size class are of similar magnitude as the corresponding values in the monodisperse case. The averages for the small bubbles therefore tend to be somewhat larger than the averages for the large bubbles, meaning that the small bubbles experience stronger fluctuations about their mean velocity. Due to the differential rise velocity of the large and small bubbles, the vertical fluctuation velocity averaged over all bubbles is, however, considerably larger in the bidisperse case than in the monodisperse case. For large differences in the bubble sizes considerable differences are observed. Preliminary report for systems where the large bubbles are eight times larger and deformable and rise along a spiral path, while the small bubbles are spherical, can be found in Go¨z et al. (2002).

V. Flows with Phase Change While major progress has been made in direct numerical simulations of ‘pure’ two-fluid flows, the incorporation of more complex physics is still at the early stages. Many simulations have been done of the heat and mass transfer of spherical drops, but relatively little is known about the effect of drops deformation on heat transfer, for example. Similarly, although a few authors have examined the effect of variable surface tension on the motion of a single drop or bubble, essentially nothing is known about the collective motion of drops with surfactants, the thermocapillary migration of many drops, or the effects of electric or magnetic fields on the collective motion of drops. In those cases the fluid flow is coupled to another field through forces at the fluid interface and extending current methods to such problems is relatively straight-forward. In addition to being of considerable technical importance in microscale and space applications, the few computations that have been done for such systems show that they also have a very rich and interesting dynamics. In a large number of engineering applications that involve multiphase flow, it is necessary to account for phase change, both between liquid and solid as well as liquid and vapor. For flows with phase changes—solidification and boiling—it is necessary to solve the energy equation in addition to the momentum and mass conservation equation. For the one-fluid approach the energy equation is

Computations of Multiphase Flows (neglecting internal heat generation due to friction): ð › ðcTÞ þ 7·cTu ¼ 7·k7T þ qdðx 2 xf ÞdA ›t A

105

ð4:1Þ

where c is the volumetric specific heat, k is the thermal conductivity and dðx 2 xf Þ is a three-dimensional delta function that is nonzero only at the interface where x ¼ xf : By integrating the energy equation across the front, we find that (assuming equal densities) 1 ›T ›T k V·n ¼ 2 k : ð4:2Þ L ›n l ›n v To solve the energy equation, it is necessary to specify the temperature at the phase boundary. We will assume a temperature equilibrium such that the temperature is continuous across the phase boundary, Tl ¼ Tg ¼ Tf :

ð4:3Þ

The temperature of the phase boundary can be found by assuming thermodynamic equilibrium. For solidification the Gibbs – Thompson Equations contain terms that lead to corrections due to curvature, local kinetics, and possibly other effects. This condition, taken from Alexiades and Solomon (1993), is given as: s v Tf ¼ Tm 1 2 k þ n ··· ð4:4Þ L h Here, s is the surface tension, h is the kinetic mobility, vn ¼ V·n; and k is the curvature. The surface tension and the kinetic mobility are generally anisotropic and are given by emperical functions (see Almgren, 1993, for example). For boiling, Juric and Tryggvason (1998) showed that the additional terms are small since length-scales resulting from the flow are considerably larger than those resulting from the thermodynamic conditions. It therefore is a reasonably good approximation simply to take the temperature at the phase boundary to be equal to the saturation temperature of the liquid at the system pressure Tf ¼ Tv :

ð4:5Þ

Although there are major similarities between boiling and solidification and most of the solution approach is the same, there are also some differences. The conversion of liquid into vapor or vice-versa is generally accompanied by significant change in volume. While this volume change is usually confined to the phase boundary, it must be accounted for. For solidification, however, it is usually permissible to neglect the volume change and treat both the melt and the solid, as well as the interface as incompressible. It is, on the other hand, necessary to account for the fact that the solid is rigid and in most cases stationary. In some cases it is possible to assume that the vapor in boiling problems and the solid in

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solidification problems are isothermal, but in general it is necessary to account for heat transfer in both phases.

A. Solidification Most materials used for man-made artifacts are processed as liquids at some stage. The way solidification takes place generally has major impact on the properties of the final product. The formation of microstructures, where some parts of the melt solidify with a different composition, is particularly important since the size and composition of the microstructure impact the hardness, ductility, and other bulk properties of the final product. Microstructures generally result from the unstable propagation of a solidification front into an undercooled melt. Although they are particularly common in binary alloys, where variable solute concentration can lead to localized constitutional undercooling, pure material can also form transient microstructure during solidification. Experimental investigations of the details of the solidification process can be difficult. The harsh thermal and chemical environments and limited optical access in directional solidification furnaces, for example, make it nearly impossible to obtain details of the transient temperature and solute distribution. Numerical simulations are a promising complement to experimental investigations and provide information that is hard to measure. Early computations of the large amplitude evolution include Ungar and Brown (1985) who used a boundary confirming finite element method to examine cellular solidification. Sethian and Strain (1992), using a level-set method, and Wheeler et al. (1993), using a phase field method, simulated the growth of dendrites. Simulations of the growth of two-dimensional dendrites in a pure material and in the absence of flow, have now become relatively routine (see, for example Udaykumar et al., 1999, for a recent reference and discussion of other work) and three-dimensional computations are starting to appear in the literature (Kobayashi, 1993; Schmidt, 1996; Karma and Rappel, 1997, 1998). Recent two-dimensional simulations of the solidification of binary alloys can be found in Warren and Boettinger (1995). For extensive reviews of the literature on computations of solidification see Wheeler et al. (1995), for example. A comprehensive review of methods for alloy solidification and morphological stability theory is provided by Coriell and McFadden (1993). An excellent theoretical introduction can be found in Alexiades and Solomon (1993). The simulations listed above assume no fluid flow. It is, however, reasonably well established experimentally that fluid flow can have significant impact on the growth of the microstructure (Glicksman et al., 1986; Lee et al.,


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1993; Bouissou et al., 1989). Simulations of the effect of fluid flow are, however, very recent. Tonhardt and Amberg (1998) simulated the effect of flow on a dendrite growing from a nucleation site on a wall and Beckermann et al. (1999) simulated the growth of single dendrite held fixed in uniform flow. Both used a phase field method, but whereas Tonhardt and Amberg modeled the solid as a fluid that was hundred times more viscous than the melt, Beckermann et al., imposed zero velocity in the solid. Juric (1998) and Shin and Juric (2000) developed a front-tracking method to follow the solidification front modeling the solid as a very viscous fluid, but in the front tracking method of Al-Rawahi and Tryggvason (2002) the velocity in the solid is exactly zero. These simulations are all for two-dimensional problems but Jeong et al. (2001) have presented a phase field method for fully threedimensional system and Al-Rawahi (2002) has done simulations of threedimensional systems with a front tracking method. In Fig. 4 one frame from a simulation of the growth of one three-dimensional dendrite, from Al-Rawahi (2002), is shown. In addition to the dendrite, the temperature and the velocity vectors in a plane cutting through the center of the domain are presented. A uniform inflow is specified on the left boundary, the top and bottom boundaries are periodic, and all gradients are set to zero at the outlet boundary. Initially a small spherical solid, perturbed slightly, is placed in

Fig. 4. One frame from a simulation of the growth of three-dimensional dendrite in a flow. From Al-Rawahi (2002).

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the center of the domain. The simulation is done using a domain that is a cube resolved by a 2563 grid. The nondimensional governing parameters and the anisotropy is selected in such a way that the dendrite arms grow along the coordinate directions. The definition of the nondimensional numbers used and their values are listed in Al-Rawahi (2002). The temperature of the incoming flow is equal to the undercooled temperature and as latent heat is released at the phase boundary, the flow sweeps it from the front to the back. This results in a thinner thermal boundary layer at the tip of the upstream growing arm and a relatively uniform temperature in the wake. The growth rate of the upstream arm is therefore enhanced and the growth of the downstream arm is reduced. Growth of side branches is also suppressed on the downstream side. Although the same qualitative trend is seen in simulations of two-dimensional systems, the wake structure is significantly different for two- and three-dimensional flow. In two dimensions the fluid must flow around the tip of the dendrites growing perpendicular to the flow direction, whereas in three dimensions the fluid can flow around the side of the side branches. This results in a much larger wake for the two-dimensional dendrite. In addition to increasing the growth rate of the dendrite growing into the flow, the flow reduces the radius of the tip, in agreement with the experimental observations of Lee et al. (1993) and Bouissou et al. (1989).

B. Boiling Boiling is one of the most efficient ways to remove heat from a solid surface. It is therefore commonly used in energy generation and refrigeration, for example. The large volume change and the high temperatures involved can make the consequences of design or operational errors catastrophic and accurate predictions are highly desirable. The change of phase from liquid to vapor and vice-versa usually takes place in a highly unsteady manner where the phase boundary is very convoluted. Direct numerical simulations therefore require the accurate incorporation of the unsteady phase boundary in addition to solutions of the Navier –Stokes and energy equation in each phase. Due to the complexity of phase change in liquid – vapor systems, most studies have been experimental and the results consist of correlations for specific conditions. Analytical studies are limited to spherical vapor bubbles (Lord Rayleigh, 1917; Plesset and Zwick, 1954; and Prosperetti and Plesset, 1978, for example), simple one-dimensional problems (see Eckert and Drake, 1972), and stability analysis (Prosperetti and Plesset, 1984). Numerical simulations have, until recently, relied on a number of simplifications. An example of such


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computation can be found in Lee and Nydahl (1989) who assume that the bubble remains hemispherical as it grows. More advanced computations started with Welch (1995), who simulated a fully deformable, two-dimensional bubble using moving triangular grids. He was, however, only able to follow the bubble for a relatively short time due to the distortion of the grids. Son and Dhir (1997) used a moving body fitted coordinate system to simulate film boiling for both twodimensional and axisymmetric flows, but were subject to similar limitations as Welch. The limitation to modest deformation of the phase boundary was overcome by Juric and Tryggvason (1998) who developed a front-tracking method for two-dimensional problems, and by Son and Dhir (1998) who used a level set function to follow the phase boundary in axisymmetric geometries. Son et al. (2002) have used this method to examine the coalescence of bubbles produced successively by a single nucleation site. Recently, Welch and Wilson (2000) developed a VOF method for boiling flows based on similar ideas and Shin and Juric (2002) presented a hybrid front-tracking/level-set method for three-dimensional simulations. Other recent simulations of boiling include axisymmetric computations of the growth and departure of a single bubble by Yoon et al. (2001) using a ‘mesh-free’ method. Boiling can take place under a variety of conditions. Like solidification it is usually initiated at a surface, although it can also be initiated at nuclei away from surfaces (homogeneous boiling). Boiling at a hot surface is usually classified as nucleation boiling if vapor bubbles form at well defined nucleation sites (small crevices that are slightly hotter than the rest of the surface) and the bubbles break away after they form. At higher heat transfer rates the hot surface is covered with a vapor film and the process is referred to as film boiling. Vapor is removed from the layer by the break-off of vapor bubbles, but vaporization at the liquid –vapor interface replenishes the layer. Generally, film boiling is undesirable since the vapor layer acts as an insulator, thus lowering the heat transfer rate and increasing the heater surface temperature. Fluid flow can, of course, have significant influence on the heat transfer and the vapor formation. If the liquid is initially stagnant, the process is called pool boiling, but if the liquid is flowing, it is referred to as flow boiling. Although boiling is usually discussed in the context of a liquid turning into vapor as heat is added, the reverse process, condensation, involves the same fundamental process except that heat is being removed. Similarly, boiling can take place when the system pressure is lowered but this situation is generally treated separately as cavitation. All simulations of boiling done so far assume one pure material that changes phase. Both boiling and condensation are, however, often modified by the presence of noncondensable gases mixed with the vapor. When the vapor pressure of the liquid vapor is only a small fraction of the total

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pressure, the local expansion of the liquid as it changes phase is dynamically insignificant, it is generally classified as evaporation, rather than boiling. Figure 5, from Esmaeeli and Tryggvason (2003) shows three frames from a two-dimensional simulation of film boiling. The top frame is at the earliest time. An initially quiescent liquid pool rests on a hot, horizontal surface, blanketed by a thin vapor film. As the liquid evaporates, the liquid/vapor interface becomes unstable and bubbles are periodically released from the layer. Gravitational acceleration is downward and the bubbles rise to the surface of the pool, break through the surface and release the vapor. Initially, both the vapor and the liquid are at saturation temperature, and the bottom wall is exposed to a constant temperature Tw which is higher than Tsat. The domain is periodic in the horizontal direction and is confined by a no-slip/no-through-flow wall at the bottom and an open boundary at the top. The domain is resolved by a 320 £ 160 grid, but the

Fig. 5. Simulation of two-dimensional film boiling at three times (time goes from top to bottom). A vapor layer blankets a hot wall and as the liquid evaporates, the liquid–vapor interface becomes unstable and bubbles are periodically released from the layer. The phase boundary and the velocity field are shown. From Esmaeeli and Tryggvason (2003).


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horizontal grid lines are unevenly spaced to resolve accurately the thin vapor film. As the simulation starts, the lower phase boundary undergoes a Rayleigh – Taylor instability while the upper interface barely moves. As the hot vapor rises, cold fluid is forced down towards the plate between the vapor plumes. The hot vapor plume evolves into a mushroom shape with a thin stem which eventually pinches off. As a bubble is formed, surface tension pulls back the stem towards the wall and a new thin film is formed. The process repeats itself as new bubbles are generated from the vapor layer. Initially, the system undergoes a transient motion where the size of the bubbles released from the vapor layer depends strongly on the initial perturbations of the layer. If the simulations are carried out long enough, however, the systems reaches a quasi-steady state independent of the initial conditions. The frames shown in Fig. 5 are at a time after the transient effects have disappeared. The heat transfer rate is usually given in terms of the Nusselt number, Nu, defined by Nu ¼ l=ðTw 2 Tsat Þ›T=›yy¼0 ; ð4:6Þ where l is the inviscidly most unstable wavelength for the vapor – liquid interface based on the parameters used here. In Fig. 6, Nu is plotted versus nondimensional time. The peaks in the Nusselt number corresponds to the time immediately before a bubble pinches off and the valleys correspond to times where the film thickness is relatively high. We have repeated this simulation using a deeper

Fig. 6. The Nu number versus time for the simulation shown in the previous figure. From Esmaeeli and Tryggvason (2003).

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liquid pool with essentially no change in the variation of the Nusselt number with time. Experimental correlations (see, for example, Berenson, 1961) are based on full three-dimensional flows. However, to evaluate the numerical results, we have computed the Nusselt number predicted by Berenson’s correlation to our numerical results. The Nu number based on this correlation, for the nondimensional numbers used here, is shown by a horizontal line in the figure. We note that Welch and Wilson (2000) found similar trend when they compared their two-dimensional results to the Berenson correlation.

VI. Conclusions Direct numerical simulations of multiphase flow have come a long way in the last half a decade or so. It is now possible to simulate accurately the evolution of finite Reynolds number disperse flows of several hundred bubbles, drops, and particles for sufficiently long time that reliable values can be obtained for various statistical quantities. Similarly, major progress has been achieved in the development of methods for more complex flows, including those where a liquid solidifies or evaporate. Simulations of large systems with boiling and solidification are therefore within reach. Much remains to be done, however, and it is probably true that the use of direct numerical simulations of multiphase flows for research and design is still in the embryonic state. The possibility of computing the evolution of complex multiphase flows—such as churn-turbulent bubbly flow undergoing boiling or the breakup of a jet into evaporating drops—will transform our understanding of multiphase flow. Currently, controls of most multiphase flow processes are fairly rudimentary and almost exclusively based on intuition and empirical observations. The savings realized if atomizers for spray generation, bubble injectors in bubble columns, and inserts into pipes to break up drops, just to name a few examples, could be improved by just a little bit would add up to a substantial amount of money. Reliable predictions would also reduce the design cost significantly for situations such as space vehicles and habitats where experimental investigations are expensive. And as the possibilities of manipulating flow at the very smallest scales by either stationary or free flowing MEMS devices becomes more realistic, the need to predict the effect of such manipulations becomes critical. While speculating about the long term impact of any new technology is a dangerous thing—it is easier to predict the near future and to identify what needs to be done to keep progress as fast as possible. Apart from the obvious prediction that computers will continue to become faster and more available, progress is going to take place mainly on two fairly broad fronts in the next few years.


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Development of numerical methods will focus on flows with complex physics. Although some progress has already been achieved for flows with variable surface tension, flows coupled to temperature and electric fields, and flows with phase change, much remains to be done before simulations of such systems become commonplace. In addition to the need to solve a large number of equations, coupled systems generally possess much larger range of length and time scales than simple two-fluid systems. Thus, the incorporation of implicit time-integrators for stiff systems and adaptive gridding will become even more important. It is also likely, as more and more complex problems are dealt with, that the differences between direct numerical simulations—where everything is resolved fully—and simulations where the smallest scales are modeled will become blurred. Simulations of atomization where the evolution of thin films are computed by ‘subgrid’ models and very small drops are included as point particles is a relatively obvious example of such simulations. Other examples include possible couplings with microscopic simulations of moving contact lines, kinetic effects at a solidification interface, and thin flames. Simulations of non-Newtonian fluids, where the microstructure has to be modeled in such a way that the molecular structure is accounted for in some way also falls under this category. The development of improved theoretical framework for the use of results generated by direct numerical simulations to improve understanding of multiphase flow and to help generate subgrid closure models will become critical. While considerable work has been done on modeling of multiphase flow, models such as the ‘two-fluid’ model do not, for example, provide a natural way to incorporate knowledge of universal small-scale behavior, except in the most rudimentary way. It is also likely that such models will have a range of complexity from correlations for simple problems to subgrid models for large eddy simulations. The availability of massive amount of detailed simulation data has increased greatly the urgency of further theoretical development of closure models at all levels. In addition to the development of more powerful numerical methods and more comprehensive theoretical framework to process the data, it will become important to attend to the ‘human’ aspect of the enterprise. The physical problems that we must deal with and the computational tools that are available are rapidly becoming very complex. The difficulty in developing fully parallelized software to solve the continuum equations (fluid flow, mass and heat transfer, etc.), where three-dimensional interfaces must be handled and the grids must be dynamically adapted, is putting such simulations beyond the reach of a single graduate student or even small research groups. It is simply going to be very difficult for a graduate student to learn everything that he or she needs to know and make significant new progress in four to five years. Lowering the

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‘knowledge barrier’ and ensuring that new investigators can enter the field of direct numerical simulations of multiphase flow may well become as important as improving the efficiency and accuracy of the numerical methods.

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Cracks and Fracture in Piezoelectrics YI-HENG CHENa and TIAN JIAN LUb a

School of Civil Engineering and Mechanics, Xi’an Jiao-Tong University, Xi’an 710049, China

b

Department of Engineering, University of Cambridge, Trumpington Street, Cambridge CB2 1PZ, UK

I. Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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II. Fundamental Formulations of Piezoelectricity. . . . . . . . . . . . . . . .

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III. Impermeable (Insulating) Crack . . . . . . . . . . . . . . . . . . . . . A. Impermeable Crack Model and Solutions . . . . . . . . . . . B. J-integral Analysis and Energy Release Rate . . . . . . . . C. Mechanical Strain Energy Release Rate . . . . . . . . . . . . D. Eigenfunction Expansion Forms and Weight Functions . E. Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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IV. Permeable (Conducting) Crack . . . . . . . . A. Conducting (Permeable) Crack Model B. Field Intensity Factors . . . . . . . . . . . . C. Energy Release Rate . . . . . . . . . . . . . D. Summary . . . . . . . . . . . . . . . . . . . . .

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V. Exact Electric Boundary Condition . . . . . . . . . . . . . . . . . . . . . . .

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VI. Non-linear Electric Boundary Condition. . . . . . . . . . . . . . . . . . . . A. Strip Electric Saturation Model (Generalized Dugdale Model) . B. Contact Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . C. Microstructure Near the Tip of an Impermeable Crack . . . . . .

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VII. Piezoelectric Bimaterial Systems . . . . . . . . . . . . . . . . . . . . . . . . .

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VIII. Other Developments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . A. Three-dimensional Cracks . . . . . . . . . . . . . . . . . . . . . . . . . . . B. Dynamic Fracture . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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IX. Concluding Remarks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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Appendix A . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .203 .. Appendix B . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .204 .. References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .


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Y.-H. Chen and T.J. Lu Abstract

Recent developments and current understanding on cracks and fracture in piezoelectric ceramic materials are presented. Focus is placed on the description and proper selection of electric boundary conditions along crack surfaces as well as their influence on piezoelectric fracture. Five different types of crack surface boundary conditions are addressed: (i) impermeable cracks with traction-free surfaces; (ii) permeable cracks with traction-free surfaces; (iii) cracks with exact electric boundary and traction-free surfaces; (iv) impermeable cracks with near-tip microstructural features; and (v) impermeable (or permeable) cracks with contacting surfaces. The first three conditions, namely, the impermeable crack (insulating crack), the permeable crack (conducting crack), and the exact electric boundary (which has the impermeable and permeable cracks as its two limiting cases), have been well studied in the literature and are commonly known as the linear models. The last two types, known as the non-linear models, are categorized as the mixed electric boundary condition, and are introduced to treat non-linear effects such as electric-field induced yielding and crack closure. Five different ways to formulate a fracture criterion for piezoelectric materials are presented and compared: (i) total energy release rate (or the J-integral); (ii) mechanical strain energy release rate (or the mechanical part of the J-integral); (iii) local and global energy release rates due to electrical yielding only; (iv) influence of mode-mixity on fracture toughness; and (v) maximum hoop stress. Contradicting views about the role of applied electric field in piezoelectric fracture are discussed in detail. Whilst most studies found that a positive electric field promotes crack propagation whereas a negative electric field impedes crack propagation, exactly the opposite trends have also been observed. Moreover, some recent investigations with indentation tests reveal that both positive and negative fields can promote crack growth if inelastic deformation is taken into account and/or fracture criterion based on mechanical strain energy release rate is used. In this review article, clarification of the above discrepancies is attempted in terms of different crack surface boundary conditions and different fracture criteria selected. Also discussed are interface crack problems in dissimilar piezoelectric materials and dynamic piezoelectric fracture. Both two- and three-dimensional solutions obtained with either analytical methods or numerical tools such as the method of finite elements are presented.

I. Introduction More than a century ago, in 1880, the Curie brothers discovered piezoelectric effect in natural crystals. In succession, the converse piezoelectric effect, i.e., changes in crystal dimensions upon the application of a voltage, was predicted by Lippmann from thermodynamic principles, and subsequently verified experimentally by the Curies. The first significant application of piezoelectric effect took place during World War I. A concerted effort was made to find a tool to locate submerged vessels to prevent ships from attacks by

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German submarines: piezoelectric crystal quartz was used to generate acoustic waves for this purpose. After World War II, von Hippel and associates (1946) in MIT, and Vul and Goldman (1947) in the former USSR, independently discovered barium titanate (BaTiO3). A few years later, Jaffe et al. (see, for example, Berlincourt et al., 1964) found that PbZrO3 –PbTiO3, now commonly known as lead zirconate titanate (PZT), has much better mechanical-electric coupling effect with a relatively high Curie temperature. Commercial applications of the piezoelectric effect in the 1960s and 1970s were largely concerned with acoustic instruments (Mason, 1981). During the last two decades, BaTiO3 and PZT ceramics have been widely used in smart structures and adaptive technologies as sensors, transducers, and, more recently, large displacement actuators. The actuator applications place stringent demands on the material due to stress concentration effects induced at large displacements, especially near edges and around material imperfections. Moreover, man-made functional materials such as piezoelectric composites and piezoelectric ceramic/polymer composites with enhanced electromechanical coupling characteristics have been developed, combining strong piezoelectric ceramics with compliant (possibly piezo-electrically active) polymers (Harrison et al., 1986; Haus and Newnham, 1986; Shaulov et al., 1989; Chan and Unsworth, 1989; Dunn and Taya, 1993; Sevostianov et al., 2001). Due to excellent piezoelectric and converse piezoelectric effects as well as good mechanical properties, these smart composites have become attractive candidates for use in transducers for underwater and biomedical imaging applications. Commonly used piezoelectric ceramics such as the PZT series are very brittle and hence susceptible to fracture during service. On one hand, commercial morphotropic PZT ceramics with typical grain sizes of 3– 10 mm often contain a large number of processing-induced pores or other types of defect, from which a macrocrack may be induced during service. On the other hand, because a high electric field is usually needed to generate large strains, actuation devices with appreciable deflections typically have layered structures with alternating ceramic layers and electrodes, resulting in further reliability issues. Microcracks at the domain level (, grain sizes) and macrocracks at the device level are often observed when these brittle materials are subjected to mechanical or electric loadings, or a combination of both. In particular, the high electric driving field needed to produce large displacements has been found to be responsible for mechanical and electrical degradation of actuators, with cracking being the most dominant mechanism (Deeg, 1980; Pohanka and Smith, 1988a,b; Newham and Ruschau, 1991). As this degradation significantly influences the mechanical as well as electrical performance of actuation devices, it is important to study piezoelectric fracture and to develop robust criteria for

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crack nucleation and propagation if smart and adaptive structures are to be implemented ubiquitously. The failure of a piezoelectric ceramic may be contributed to two main ingredients, which are often coupled. One is due to the formation and growth of microcracks subjected to a high monotonic, or cyclic electric field (Tobin and Pak, 1993; White et al., 1994; Cao and Evans, 1994; Sun and Park, 1995; Lynch et al., 1995; Yang and Suo, 1994; Wang and Singh, 1994, 1997; Park et al., 1997, 1998; Heyer et al., 1998; Schneider and Heyer, 1999; Xu et al., 2000; Tan et al., 2000). Here, the electroelastic field concentration at existing microcracks or other microdefects can lead to critical crack growth and subsequent mechanical failure or dielectric breakdown (McMeeking, 1989). The other failure origin is associated with the mechanical driving force due to actuation forces or other sources (Harrison et al., 1986; Winzer et al., 1989; Newham and Ruschau, 1991; Buijs and Martens, 1992; Furuta and Uchino, 1993; Fan and Qin, 1995; Lynch, 1996; Uchino, 1998). The fracture behavior of a linear piezoelectric ceramic subjected to combined mechanical-electric loadings is much more complicated than that associated with traditional ceramics where the use of Linear Elastic Fracture Mechanics (LEFM) suffices (see, for instance, Yamamoto et al., 1983; Cook et al., 1983; Okazaki, 1984; Pisarenko et al., 1985; Mehta and Virkar, 1990; Li et al., 1990; Dunn, 1994; Dunn and Taya, 1993, 1994; Wang and Singh, 1994, 1997). Mathematically, however, there is no unconquerable difficulty in solving crack and fracture problems of piezoelectric ceramics with linear constitutive relations. Nearly all mathematical methods customarily used in LEFM can be adopted in Linear Piezoelectric Fracture Mechanics (LPFM), including the complex potential theory (Lekhnitskii, 1963; Stroh, 1958, 1962), integral transformation, conformal mapping, and singular integral equations. Even for three-dimensional (3D) piezoelectric crack problems, the existing mathematical tools are sufficient (Sosa and Pak, 1990; Wang, 1992a,b; Wang, 1994; Wang and Huang, 1995a,b; Wang and Zheng, 1995; Chen and Shioya, 1999, 2000; Zhao et al., 1997a,b; Ding et al., 2000a,b). In fact, by adding an additional complex argument, the extension of the mathematical tools developed for LEFM to LPFM is relatively straightforward. Consequently, in this article, detailed mathematical manipulations will not be presented more than that is necessary. Those familiar with the mathematical treatment in LEFM should have little trouble with LPFM. The difficulty of establishing a successful LPFM lies rather in the physical understanding of piezoelectric fracture. As emphasized by Fulton and Gao (2001a), this difficulty stems from the fundamental discrepancy between theoretical predictions and empirical observations. For example, whilst


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experiments conducted on transversely isotropic piezoelectrics with cracks normal to the poling axis show an odd functional dependence of failure load on applied electric field, analyses based on LPFM predict an even dependence (a complete solution for a given crack configuration in 2D piezoelectric ceramics can always be obtained by using one of the established analytical or numerical techniques in LEFM). Consequently, a number of questions and sometimes confusions have been raised in the open literature. For example, how to treat the electric boundary condition along crack surfaces? Can fracture parameters widely used in LEFM, e.g., the J-integral and the stress intensity factors (SIFs), be used in LPFM? Is it necessary to introduce new fracture parameters in LPFM? What is the influence of electric loading on crack growth, i.e., does it impede or enhance crack initiation and propagation? In order to address the above issues and to provide a better physical understanding on piezoelectric fracture, a number of experiments have been performed with commonly used LEFM techniques: Vickers indentation, threepoint bending, and compact tension specimen. As in brittle materials, the indentation technique has been widely used to characterize material properties and fracture toughness of piezoelectric ceramics, because it is simple and can be performed on relatively small samples. With the Vickers indentation technique, Yamamoto et al. (1983), Okazaki (1984) and Pisarenko et al. (1985) observed the apparent anisotropy in fracture toughness of poled piezoelectric ceramics and its dependence on applied electric field. Later, Pohanka and Smith (1988a, b), Tobin and Pak (1993), Cao and Evans (1994), Sun and Park (1995), Wang and Singh (1994, 1997), Lynch et al. (1995), and Schneider and Heyer (1999) conducted similar experiments on a range of commercially available piezoelectric ceramics subjected to different combinations of mechanical-electric loadings. Despite these efforts, discrepancies between experimental measurements and theoretical predictions based on linear piezoelectric crack models prevail (McMeeking, 2001). It is now commonly accepted that the extension of fundamental fracture concepts for linear elastic materials to linear piezoelectric ceramics is not straightforward, even if the loading is purely mechanical or purely electric, hindered mainly by three obstacles. (a) The first obstacle is associated with the apparent anisotropic fracture toughness of piezoelectric ceramics due to poling. Pisarenko et al. (1985) found that the fracture toughness of a piezoelectric ceramic along two mutually perpendicular material planes differs by about 20%. Many indentation experiments showed that the critical length of a crack perpendicular to the poling direction is always much larger than that of a crack parallel to the poling direction. However, no such phenomenon has been observed in unpoled piezoelectric ceramics (Tobin and Pak, 1993).

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Possible mechanisms of piezoceramic microstructures affecting fracture toughness are discussed in Fulton and Gao (1997, 1998, 2001a,b), and Chen and Han (1999a,b). (b) The second obstacle in developing a fracture mechanics for piezoelectrics is perhaps the most significant, which is associated with the role of electric field in piezoelectric fracture. The electric field may either impede or enhance crack propagation, depending on both the electric boundary conditions along crack surfaces and the direction of poling in poled piezoelectric materials. As pointed out by Shindo et al. (2000), there is still confusion in the literature about the selection of proper electrical boundary conditions when studying piezoelectric crack problems. They added that the issue on how to impose the crack-surface electrical boundary condition is a controversial one. One commonly used boundary condition assumes that the normal component of the electric displacement along crack surfaces vanishes (Deeg, 1980). Under this condition the crack is impermeable to electric fields, i.e., its surfaces are chargefree and there is no electric displacement inside the crack gap. Another widely used boundary condition treats the crack as electrically permeable (Parton, 1976; McMeeking, 1989), requiring not only the normal component of the electric displacement but also the electric potential itself to be continuous across the crack, although the crack surfaces are no longer charge-free. More recently, McMeeking (2001) pointed out that existing theoretical treatments are all incomplete as far as electrostatic model is concerned, and the main deficiency is that almost all efforts in this area are based on the assumption that the permittivity is zero in the medium interior to the crack. (c) The third obstacle follows directly from the second, and is associated with the selection of fracture parameters when different electric boundary conditions are preferred. Although fracture is essentially a mechanical process, whether the traditional fracture parameters for brittle materials such as stress intensity factors (SIFs) and energy release rate (ERR) could be used to describe piezoelectric fracture remains to be verified. This is because the near-tip mechanical and electric fields are inherently coupled, and different selections of electric boundary conditions may lead to different distributions of near-tip stress field which, in turn, may lead to different fracture parameters (see, e.g., Pak, 1990a,b; Park and Sun, 1995a,b; Sun and Park, 1995; Heyer et al., 1998; Wang and Singh, 1994, 1997; Kumar and Singh, 1996a,b, 1997a – c, 1998; Ru, 1999a,b; Ru and Mao, 1999; Xu and Rajapakse, 2000a,b, 2001; Fulton and Gao, 2001a,b; McMeeking, 2001; Jiang and Sun, 2001; Rajapakse and Zeng, 2001). The observed anisotropic material behavior has been attributed to the difference in axis lengths in the a and c directions of a piezoelectric crystal in tetragonal phase below the Curie point, where a and c represent separately two perpendicular directions of the crystal. During the poling process


127

(usually performed under a very high electric field, about 1 MV/m or more), the crystal tries to switch the a and c axes in order to align the c axes along the poling direction. The difference in axis lengths and the constraints imposed by neighboring grains result in anisotropic residual stresses in the crystal. Unpoled PZT ceramics have isotropic properties, whilst the poling process always introduces anisotropy to the piezoelectric. The strain behavior of a poled PZT specimen under an applied electric field becomes anisotropic, depending upon the poling direction and the polarity of the electric field. If the polarity of the electric field is the same as that of poling (positive field), the specimen will normally expand in the poling direction and contrast in the perpendicular direction due to Poisson effect. Here, the additional domains tend to align with the electric field, and the dipoles in domains aligned with the electric field are stretched by the positive electric field. When the polarity of the applied electric field is opposite to that of poling (negative field), an effect opposite to that just described is observed (Wang and Singh, 1997). Using the indentation technique, Yamamoto et al. (1983), Cook et al. (1983), Okazaki (1984), Pisarenko et al. (1985), Tobin and Pak (1993), Sun and Park (1995), Wang and Singh (1994, 1997), and Schneider and Heyer (1999) have studied the anisotropic residual stress fields in poled piezoelectric materials and their effects on crack propagation. With regard to the selection of proper electric boundary conditions along crack surfaces, it is noted that the earliest attempt to analyze such problems was due to Parton (1976) and Polovinkina and Uliko (1978), although a little earlier Barnett and Lothe (1975) already studied dislocations and line charges in anisotropic piezoelectric insulators. Driven by the needs to develop modern smart and adaptive structures, significant advances in understanding how piezoelectric ceramics fracture under combined mechanical-electric loadings have been made in the last two decades (McHenry and Koepke, 1983; Pisarenko et al., 1985; Pak and Herrmann, 1986a,b; Zhou et al., 1986; Pohanka and Smith, 1988a,b; McMeeking, 1987, 1989, 1990, 1999, 2001; Pak, 1990a,b, 1992a,b; Tobin and Pak, 1993; Pak and Tobin, 1993; Shindo et al., 1990, 1996a,b, 1997a,b, 2000; Sosa and Pak, 1990; Sosa, 1991, 1992; Sosa and Khutoryansky, 1996, 1999; Suo et al., 1992; Suo, 1993; Benveniste, 1992; Wang, 1992a,b; Wang and Singh, 1994; Wang and Huang, 1995a,b; Wang and Zheng, 1995; Alshits et al., 1995; Zhang and Hack, 1992; Chen, 1993a,b, 1995; Chen and Lai, 1997; Yang and Suo, 1994; Park and Sun, 1995a,b; Gao, 1996; Gao et al., 1997a,b; Park et al., 1997, 1998; Zhang and Tong, 1996; Zhang, 1994a,b; Zhang et al., 1998; Zhao et al., 1997a,b; Kogan et al., 1996; Kumar and Singh, 1996a,b, 1997a– c, 1998; Li et al., 1990; Lee and Jiang, 1994; Li and Mataga, 1996a,b; Gao and Fan, 1998a,b, 1999a,b; Jiang et al., 1999;

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Lu and Chen, 1999; Wang and Han, 1999; Wang, 2000; Jiang and Sun, 2001; Yang, 2001). Relevant works by the present authors and associates in this area can be found in Chen and Tian (1999), Chen and Han (1999a,b), Han and Chen (1999), Tian and Chen (2000), Chen and Lu (2001), Ma and Chen (2001a –d), and Chen (2001). This article aims to review recent developments and current understanding on cracks and fracture in piezoelectric materials. The results will also be relevant to ferroelectric materials operating in the linear regime of the constitutive curve, except for a small region in the vicinity of a crack-tip (i.e., small-scale non-linearity). Focus will be placed on the description and proper selection of electric boundary conditions along crack surfaces as well as their influence on piezoelectric fracture. Five different types of crack surface boundary condition will be addressed. The first three conditions have been well studied in the literature, namely, the impermeable crack (insulating crack), the permeable crack (conducting crack), and the so-called exact electric boundary which has the impermeable crack and the permeable crack as its two limiting cases (Hao and Shen, 1994; Xu and Rajapakse, 2001). The last two types known as the non-linear electric boundary condition are introduced to treat non-linear effects in piezoelectric fracture. For example, inspired by the Dugdale model for ductile fracture, Gao and Barnett (1996), Gao et al. (1997a,b), and Fulton and Gao (1997, 1998, 2001a,b) proposed a strip saturation model to treat material non-linearity (e.g., electric yielding) ahead of an impermeable crack. The contact mechanical boundary condition was proposed by Ru (1999a,b), Ru and Mao (1999) and Kumar and Singh (1997c, 1998) to study the influence of electric-induced crack closure on near-tip stress distribution and dominant fracture parameters. Moreover, because most commercially available piezoelectric ceramics contain preexisting mircopores or microcracks, Chen and Han (1999a,b) and Ma and Chen (2001c) studied the effect of near-tip microcracks on the growth of a macrocrack. This review article comprises nine sections, and is organized as follows. In Section II, the fundamental formulations for linear piezoelectric materials are presented. Sections III– VI focus on the physical aspects of different electric boundary conditions and the associated fracture criteria for a plane crack. Major conclusions drawn in the past decade are presented, and some remarkably contradicting views are discussed and clarified in detail. Interface crack problems between two dissimilar piezoelectric materials are discussed in Section VII, whilst 3D crack solutions and dynamic fracture are presented in Section VIII. In Section IX, the main conclusions are summarized.


129

II. Fundamental Formulations of Piezoelectricity To proceed, the derivation of fundamental formulations in Linear Piezoelectric Fracture Mechanics is briefly described in this section. The governing equations and constitutive relations of linear piezoelectric ceramics can be established by combining the principles of elasticity, electrostatics and thermodynamics (Landau and Lifshitz, 1960; Berlincourt et al., 1964; Tieresten, 1969; Parton, 1976; Deeg, 1980; Mikhailov and Parton 1990; Parton and Kudryvtsev, 1988; Pohanka and Smith, 1988a,b; Pak and Herrmann, 1986a,b; Pak, 1990a,b; Ikeda, 1990; Jona and Shirane, 1993). The internal energy density P stored in any linear elastic dielectric can be written as

Pðs; DÞ ¼

1 2

sij sij þ 12 Ei Di ;

ð2:1aÞ

or, equivalently dP ¼ sij dsij þ Ei dDi :

ð2:1bÞ

Here, sij is the stress tensor, Di is the electric displacement vector, Ei is the electric field vector, and sij is the strain tensor, with sij ¼

1 2

ðui;j þ uj;i Þ;

ð2:2Þ

where ui is the displacement. The internal energy P is the thermodynamic potential with respect to the charges on the dielectric. However, if one chooses to derive the governing equations with Ei instead of Di as an independent variable (Ei is related to the electric potential, f, and Di to the electric charge, q), another thermodynamic potential with respect to the electric potential needs to be introduced. This can be achieved by defining an electromechanical enthalpy density (Landau and Lifshitz, 1960; Deeg, 1980; Pak, 1990a,b), as

or

Lðs; EÞ ¼ P 2 Di Ei ;

ð2:3aÞ

dL ¼ dP 2 Di dEi :

ð2:3bÞ

The second term on the right-hand side of Eq. (2.3a) is needed because in varying the energy while keeping the electric potential constant the external source (a large reservoir of charges at constant potential) has to do work by 2Di Ei (Landau and Lifshitz, 1960). Therefore, L is the correct thermodynamic potential when the mechanical displacement and the electric potential are taken to be the independent variables. The governing equations and the associated boundary conditions for a piezoelectric material can now be derived by taking L to be the Lagrangian


130

density and employing Hamilton’s principle, as ð ð ð d L dV 2 ð fi dui 2 qb dfÞdV 2 ðTi dui 2 qs dfÞdS ¼ 0; V

V

ð2:4aÞ

S

where fi is the body force, qb is the body charge, Ti is the applied surface traction, qs is the applied surface charge, V is the volume occupied by the material, and S is the boundary of the material. The electric field vector is related to f by Ei ¼ 2

›f : ›x i

ð2:4bÞ

For linear elastic piezoelectric materials, the electric enthalpy density is (Landau and Lifshitz, 1960):

Lðsij ; Ei Þ ¼

1 2

Cijkl sij skl 2

1 2

1ij Ei Ej 2 eikl skl Ei ;

ð2:5Þ

where Cijkl are the elastic moduli measured in a constant electric field, 1ij are the dielectric constants measured at constant strain, and eikl are the piezoelectric constants. Substitution of Eq. (2.5) into Eqs. (2.4a) and (2.4b) results in ) sij;j þ fi ¼ 0 ; ðgoverning field equationsÞ; ð2:6Þ Di;i ¼ qb ) sij nj ¼ Ti ðboundary conditionsÞ; ð2:7Þ Di ni ¼ 2qs and 9 ›L > ¼ Cijkl skl 2 ekij Ek > > = ›sij > > ›L ; Di ¼ 2 ¼ eikl skl þ 1ik Ek > › Ei

sij ¼

ðconstitutive equationsÞ;

ð2:8Þ

where ni is the outward unit normal vector to S. There are natural crystals, such as quartz, that exhibit piezoelectricity, but man-made piezoelectric ceramics such as barium titanate (BaTiO3) and lead zirconate titanate (PZT) exhibit much stronger piezoelectric coupling. Piezoelectric effects can be induced in these ceramics through a process called poling under a very strong DC electric field (, 1 MV/m) so that permanent dipole moments can be aligned (Pak, 1992a,b). Upon poling, an initially isotropic nonpiezoelectric ceramic becomes an anisotropic piezoelectric. The poled ceramic normally exhibits transversely isotropic elastic behavior with hexagonal symmetry of class 6mm, with the poling direction perpendicular to the isotropic plane (Pak, 1992a,b; Sosa, 1991, 1992). For convenience, the material constants


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defined in Eqs. (2.5) and (2.8), i.e., the elastic moduli Cijkl ; dielectric constants 1ij ; and piezoelectric constants eikl are listed in Appendix A for two commercially available piezoelectric ceramics, PZT-4 and PZT-5H (Berlincourt et al., 1964; Deeg, 1980; Pak, 1990a,b, 1992a,b; Sosa, 1992; Park and Sun, 1995b). Equations (2.6) –(2.8) are established for general 3D problems. For twodimensional (2D) problems in a plane-strain transversely isotropic piezoelectric ceramic, the constitutive Eq. (2.8) reduces to 0 1 2 30 1 2 3 111 s11 a11 a12 a13 b11 b21 ! B C 6 7B C 6 7 D1 B 122 C ¼ 6 a12 a22 a23 7B s22 C þ 6 b12 b22 7 ; ð2:9aÞ @ A 4 5@ A 4 5 D2 2112 a13 a23 a33 s12 b13 b23 0 1 " # s11 " # ! ! E1 b11 b12 b13 B D1 d11 d12 C B C ¼2 ; ð2:9bÞ @ s22 A þ E2 b21 b22 b23 d12 d22 D2 s12 where aij ; bij ; and dij are known as the reduced material constants (Sosa, 1991, 1992), which are related to the material constants of Eq. (2.8). When the poling direction is perpendicular or parallel to the plane crack, the following constants vanish: a13 ¼ a23 ¼ b11 ¼ b12 ¼ b33 ¼ d12 ¼ 0: For a plane PZT-4 ceramic the reduced material constants take values as given in Appendix A (Sosa, 1992). Using the Lekhnitskii (1963) complex potential theorem, Sosa (1991, 1992) derived the following relations between the mechanical and electric quantities and three complex potentials:

s11 ¼ 2Re

3 X

m2k w0k ðzk Þ;

ð2:10Þ

k¼1

s22 ¼ 2Re

3 X

w0k ðzk Þ;

ð2:11Þ

mk w0k ðzk Þ;

ð2:12Þ

lk mk w0k ðzk Þ;

ð2:13Þ

k¼1

s12 ¼ 22Re

3 X k¼1

D1 ¼ 2Re

3 X k¼1

D2 ¼ 22Re

3 X

lk w0k ðzk Þ;

ð2:14Þ

k¼1

E1 ¼ 2Re

3 X k¼1

ðb11 þ d11 lk Þw0k ðzk Þ;

ð2:15Þ


132

E2 ¼ 22Re

3 X

ðb21 m2k þ b22 þ d22 lk Þw0k ðzk Þ:

ð2:16Þ

k¼1

Here, the integral number k in Lekhnitskii’s complex arguments zk varies from 1 to 3 rather than from 1 to 2 for traditional anisotropic plane elastic problems. These three distinct arguments correspond to three distinct complex potentials, denoted separately by Fk ðzk Þ ¼ w0k ðzk Þ with k ¼ 1,2,3. The complex constants mk (k ¼ 1,2,3) represent the three characteristic roots with positive imaginary parts, which depend on the reduced material constants as well as on the poling direction. The complex constants lk (k ¼ 1,2,3) are related directly to the reduced material constants (Sosa, 1991). Unlike 2D problems, for a generalized plane problem where all stress and strain components in a 3D coordinate system remain non-zero in general but are dependent on two Cartesian coordinates x1 and x2 only. Ting (1986) and Suo et al. (1992) used Stroh’s complex potential theorem (1958, 1962) with four distinct complex arguments and presented a compact form representation derived directly from Eqs. (2.6) and (2.8): u ¼ ½uk ; fT ¼ af ðzÞ;

z ¼ x1 þ px2 ;

½Q þ pðR þ RT Þ þ p2 Ta ¼ 0:

ð2:17Þ ð2:18Þ

Here, the complex constant p and 4-component vector a remain to be determined, f ðzÞ is an arbitrary function of the complex argument z, the superscript T indicates transposition, and Q, R and T are 4 £ 4 matrices defined as: " # " # Ci1k1 e1i1 Ci1k2 e2i1 Q¼ ; R¼ ; eT1k1 2111 eT1k2 2112 ð2:19Þ " # Ci2k2 e2i2 T¼ ; eT2k2 2122 where Q and T are symmetric. To express stresses and displacements in terms of the general solution f(z), it is convenient to introduce the generalized stress concept, with the electric displacements taken as special stress components: ( ) ( ) sj2 sj1 T 0 ¼ ðR þ pTÞaf ðzÞ; ¼ ðQ þ pRÞaf 0 ðzÞ; D2 D1 ð2:20Þ ð j ¼ 1; 2; 3Þ;


133

or, in a compact form, as ½w ¼ bf ðzÞ;

1 b ¼ ðRT þ pTÞa ¼ 2 ðQ þ pRÞa: p

ð2:21Þ

The non-trivial solution condition of the fourth order eigen-value problem (2.18) leads to four pairs of complex conjugate characteristic roots with positive imaginary parts, pa (a ¼ 1,2,3,4) and the associated vectors, aa and ba : Let paþ4 ¼ p a ;

baþ4 ¼ b a ;

aaþ4 ¼ a a ;

ða ¼ 1; 2; 3; 4Þ;

ð2:22Þ

where the overbar denotes complex conjugate. Suo et al. (1992) proved that the general solution for physical quantities such as u, w, si2, si1, Ti in an anisotropic piezoelectric can be expressed in terms of the four-dimensional Stroh formalism (1958, 1962), as u ¼ 2Re½AfðzÞ;

ð2:23aÞ

½w ¼ 2Re½BfðzÞ;

ð2:23bÞ 0

½si2 ¼ ½wi;1 ¼ 2Re½Bf ðzÞ;

ð2:23cÞ

½si1 ¼ 2½wi;2 ¼ 2Re½Lf 0 ðzÞ;

ð2:23dÞ

Ti ¼ 22Re½BfðzÞ;

ð2:23eÞ

where according to the generalized stress and displacement concepts, u ¼ ½u1 u2 u3 fT ; T

½si1 ¼ ½s11 s21 s31 D1 ;

ð2:24Þ T

½si2 ¼ ½s12 s22 s32 D2 ;

fðzÞ ¼ ½ f1 ðz1 Þ f2 ðz2 Þ f3 ðz3 Þ f4 ðz4 ÞT ; A ¼ ½a1 a2 a3 a4 ;

ð2:25Þ ð2:26Þ

B ¼ ½b1 b2 b3 b4 ;

ð2:27Þ

L ¼ ½2b1 m1 2 b2 m2 2 b3 m3 2 b4 m4 :

ð2:28Þ

In these equations, A, B and L each denote a 4 £ 4 matrix that is dependent upon material constants. Ting (1986) and Deng and Meguid (1998) found that the matrices A and B have the following important orthogonal relations: ¼I TB þB TA AT B þ BT A ¼ A

A T ¼ BBT þ B B T ¼ 0 AAT þ A

B T ¼ I BAT þ BA T ¼ ABT þ A

¼A TB þ B T A ¼ 0: AT B þ BT A

ð2:29Þ

Furthermore, since A and B are in general non-singular, Suo et al. (1992) proposed two additional matrices: Y ¼ iAB21 ;

ð2:30Þ

H ¼ Y þ Y;

ð2:31Þ

134


pffiffiffiffi HT ¼ H) where i ¼ 21; Y and H are 4 £ 4 Hermitian matrices (i.e., YT ¼ Y; that are independent of the normalization for A, and H is positive-definite (Suo et al., 1992). More formulations specific for 3D problems will be discussed in Section VIII.

III. Impermeable (Insulating) Crack A. Impermeable Crack Model and Solutions With reference to Fig. 1, consider a plane crack of length 2a in an infinitely large piezoelectric ceramic. Let the uniform mechanical-electric loading at 1 1 1 remote be denoted by s1 22 ; s12 ; D2 ; where D2 represents the electric displacement component perpendicular to the crack plane. (Alternatively, the remote electric loading can also be specified by the electric field component E21 .)

Fig. 1. A typical crack embedded in a plane piezoelectric material.


135

Note that in Fig. 1 the poling direction of the piezoelectric material is oriented by an angle b with respect to the x2-axis. The two extreme cases where the poling direction is either perpendicular or parallel to the crack surfaces are represented by b ¼ 0 and 908, respectively. Although the earliest model of the electric boundary condition along crack surfaces is a conducting one (Parton, 1976), in the last decade some researchers did not consider this to be realistic. Deeg (1980) is perhaps the first to introduce the charge-free condition along crack surfaces, i.e., the impermeable model (also known as the insulating crack). Parton and Kudryvtsev (1988), Pak (1990a,b, 1992a,b) and Sosa (1991, 1992) subsequently used the impermeable crack model to analyze piezoelectric crack problems. Suo et al. (1992) in particular argued that there is an electric potential drop across a plane piezoelectric crack. With the existence of this drop accounted for, Park and Sun (1995a,b) and Sun and Park (1995) concluded that the impermeable crack model is more realistic when the cracked specimen is immersed in silicon oil (Jiang and Sun, 2001). Mathematically, the impermeable crack model dictates that both surfaces of a plane crack are charge-free, or, equivalently, the normal component D2 of the electric displacement vector D distributed along crack surfaces vanishes (Fig. 1): 2 Dþ 2 ðxÞ ¼ D2 ðxÞ ¼ 0

x [ ð2a; aÞ:

ð3:1Þ

For mechanical quantities, the Griffith crack model customarily used in LEFM has been adopted in LPFM, with traction-free conditions specified: 2 sþ 12 ðxÞ ¼ s12 ðxÞ ¼ 0 2 sþ 22 ðxÞ ¼ s22 ðxÞ ¼ 0

x [ ð2a; aÞ:

ð3:2Þ

From lengthy but straightforward manipulations Pak (1992a,b) and Sosa (1991, 1992) found that both the stress and electric fields near the crack-tip (Fig. 1) have the traditional inverse square root singularity irrespective of crack orientation or poling direction. The crack-tip SIFs and the electric displacement intensity factor (EDIF) are related to remote electric-mechanical loading by pffiffiffiffi KI ¼ s1 ð3:3aÞ 22 pa; pffiffiffiffi ð3:3bÞ KII ¼ s1 12 pa; p ffiffiffiffi ð3:3cÞ Ke ¼ D1 2 pa: The model-I and mode-II SIFs, KI and KII, govern the near-tip singular stress fields, whereas the EDIF, Ke ; governs the near-tip singular electric field. With KIII reserved to denote the mode-III (anti-plane shear) SIF for a Griffith crack, Ke is commonly known as the mode-IV intensity factor (Suo et al., 1992).

136


For convenience, it has been proposed to denote both SIFs and EDIF by field intensity factors. For an impermeable crack in an infinite plane piezoelectric material as shown in Fig. 1, Eq. (3.3) reveals that the SIFs are independent of remote electric loading whereas the EDIF is independent of remote mechanical loading, although from Eqs. (2.10) – (2.16) it is seen that the presence of an electric field disturbs the near-tip stress fields significantly. (It is also interesting to note that these field intensity factors are independent of material symmetry.) Using the transversely isotropic PZT-4 ceramic with material constants provided by Berlincourt et al. (1964), Sosa (1992) studied the angular distributions of neartip stress fields as perturbed by the near-tip electric field. Park and Sun (1995b) subsequently used the complex Fourier transformation to study the characteristics of tangential stresses near the crack-tip as functions of angle u for PZT-4 subjected to either purely electric or purely mechanical loading (Fig. 2), or a combination of both (Fig. 3). From the maximum hoop stress criterion widely used in LEFM, Kumar and Singh (1997a,b) found that remote electric loading has a significant effect on the location angle of maximum hoop stress, along which crack extension is expected to occur. In other words, when an electric loading is applied, even though it is parallel to the poling direction, crack kinking may still occur due to the electric disturbance of near-tip stress fields (Sosa, 1992; Park and Sun, 1995b).

Fig. 2. Distribution of hoop stress at the tip of an impermeable crack subjected to a purely mechanical or a purely electric loading (Park and Sun, 1995b).


137

Fig. 3. Distribution of hoop stress at the tip of an impermeable crack subjected to a combined mechanical and electric loading (Park and Sun, 1995b).

As fracture is a mechanical process from the phenomenological point of view, some researchers have proposed to use the SIFs as a fracture criterion in the same way as they are used in LEFM, without considering the effect of EDIF. In other words, this approach implies that fracture toughness KIC is a purely mechanical quantity independent of electric field. However, although this criterion can account for the anisotropic nature of fracture in piezoelectric ceramics under purely mechanical loading (McHenry and Koepke, 1983; Pisarenko et al., 1985), it cannot explain the experimentally observed influence of electric field upon piezoelectric fracture (Tobin and Pak, 1993; Pak and Tobin, 1993). For example, McHenry and Koepke (1983) measured crack propagation velocities under electric loading and observed that the electric field increases crack speed, with crack propagation deviating from its original direction under a sufficiently strong electric field. Tobin and Pak (1993) performed Vickers indentation tests in poled PZT-8 with an applied electric field of 0.47 MV/m and found that its apparent fracture toughness (KIC) may either decrease or increase, depending on the direction of the applied electric field. Consequently, SIFs should not be used as a fracture criterion for an insulating crack, since they do not reflect the influence of electric field on fracture. That is, with a fixed remote mechanical loading, the fracture behavior of an impermeable crack will be different if the electric loading is altered, although the SIFs at the crack tips remain unchanged.


138

B. J-Integral Analysis and Energy Release Rate As a fundamental fracture parameter for LEFM, the J-integral is equivalent to the crack-tip energy release rate (ERR) per unit crack extension (also known as the total potential energy release rate abbreviated by TPERR). Naturally, because the EER is consisted of both mechanical and electric parts, it has been suggested that the ERR (or J ) be used, instead of the mechanical SIFs, as a fracture criterion for piezoelectric materials (Suo et al., 1992). Based on the conservation laws in piezoelectrics, Pak and Herrmann (1986a,b) established the formulation of the Jintegral, as J¼

þ ›up ›f sij sij dx2 2 ni sip ds 2 Di Ei dx2 þ ni Di ds ›x 1 ›x 1 C

ð3:4Þ

i;j;p ¼ 1;2; where C is a closed contour surrounding one of the crack tips only, see Fig. 1. The J-integral is related to the SIFs by (Suo et al., 1992) 0

KII

1

B C C J ¼ 14 KII KI Ke Hp B @ KI A;

ð3:5Þ

Ke where H p is a 3 £ 3 matrix related to piezoelectric material constants as: 21

Hp ¼ 2ReðiAp Bp Þ; 2

p 1 p2 p3

ð3:6aÞ

3

6 7 7 Ap ¼ 6 4 q 1 q2 q3 5 ;

ð3:6bÞ

r1 r2 r3 2

2m1 2m2 2m3

6 Bp ¼ 6 4 1

1

3

7 1 7 5:

ð3:6cÞ

2l1 2l2 2l3 Whether or not the crack of Fig. 1 is perpendicular to the poling direction, it is evident that the coupling terms arising from the SIFs and EDIF in Eq. (3.5) contribute non-trivially to the J-integral. Park and Sun (1995b) reported detailed


139

data for the ERR of a Mode-I crack in PZT-4 when the crack is perpendicular to the poling direction as well as the applied electric field, with GI ¼

pa 2 22 1 1 ½1:48 £ 10211 ðs1 22 Þ þ 5:34 £ 10 s22 D2 2 2 2 8:56 £ 107 ðD1 2 Þ ðN=mÞ;

ð3:6dÞ

or GI ¼

pa 2 210 1 1 ½2:76 £ 10211 ðs1 s22 E2 22 Þ þ 1:23 £ 10 2 2 8:56 £ 1029 ðE21 Þ2 ðN=mÞ;

ð3:6eÞ

2 2 1 1 where s1 22 ; D2 ; and E2 have the units of N/m , C/m , and V/m, respectively. However, although the ERR correctly represents the inherent coupling between mechanical and electric quantities, a new problem arises from the 1 negative last term in Eqs. (3.6d) or (3.6e). Depending upon the ratio of s1 22 to D2 1 1 (or s22 to E2 ), the ERR may be either positive, zero or negative. In other words, for a fixed mechanical loading, use of the ERR-based fracture criterion implies that the presence of an electric field always impedes or even arrests crack growth (Deeg, 1980; Pak, 1990a,b), irrespective of whether the electric field is coincident with the poling direction (the positive electric field) or the opposite (the negative electric field). Moreover, Eqs. (3.6d) and (3.6e) reveal that, in the absence of any mechanical loading, it is impossible for a purely electric loading to drive the crack because the last term in Eqs. (3.6d) and (3.6e) is always negative. This is in contradiction with all available experimental observations. For example, McHenry and Koepke (1983) observed that the presence of an electric field increases crack speed, and crack propagation would deviate from its original direction if the electric field is sufficiently strong. Cao and Evans (1994) reported electric-field-induced crack growth in PLZT and PZT ceramics in both poled and unpoled states. In particular, Tobin and Pak (1993) and Pak and Tobin (1993) presented optical micrographs to show the effect of applied electric fields on indentation crack patterns in poled PZT-8 specimens (the material properties of PZT-8 can be found in Appendix A3). These patterns show that cracks perpendicular to the poling direction are significantly longer than cracks parallel to the poling direction. When an electric field is applied during indentation, further elongation of cracks perpendicular to the poling direction was observed whereas relatively little crack growth was observed parallel to the poling direction (see Table I). The data presented in Table I reveal the role of applied electric field in poled piezoelectric ceramics with indentation cracks. In the absence of an electric field, the crack length perpendicular and parallel to the

140


Table I CRACK LENGTHS VERSUS ELECTRIC FIELD STRENGTH FOR PERPENDICULAR AND PARALLEL CRACKS IN POLED AND UNPOLED PIEZOELECTRIC CERAMICS Specimen/electric field/orientation Unpoled/0.0(105 V/m)/perpendicular Unpoled/0.0(105 V/m)/parallel Poled/0.0(105 V/m)/perpendicular Poled/0.0(105 V/m)/parallel Poled/ þ 4.7(105 V/m)/perpendicular Poled/ þ 4.7(105 V/m)/parallel Poled/ 2 4.7(105 V/m)/perpendicular

Crack length (mm)

Standard deviation

25.20 24.15 31.80 16.33 53.73 17.83 21.50

2.21 1.73 4.29 2.45 6.56 4.07 4.82

poling direction is 31.80 and 16.33 mm, respectively. Upon applying a positive electric field (E2 ¼ 0.47 MV/m), these become 53.73 and 17.83 mm, respectively, implying that the positive electric field promotes crack growth in the direction perpendicular to poling whilst it has little influence on crack growth in the transverse direction. When a negative electric field (E2 ¼ 2 0.47 MV/m) is applied, the crack length perpendicular to the poling direction decreases to 21.50 mm, suggesting that crack growth in the direction perpendicular to poling is hindered by a negative electric field. Similar results are reported in Sun and Park (1995). Figure 4 shows the indentation cracking pattern in a poled PZT-8 specimen observed by Tobin and Pak (1993) with an applied electric field of

Fig. 4. Vickers indentation induced cracking pattern (mag. ¼ 500 £ ) in poled PZT-8 with an applied field of 0.47 MV/m (Tobin and Pak, 1993).


141

E2 ¼ ^ 0.47 MV/m. Microstructural examinations of the polished specimen surface reveal a large number of distributed pores with an average pore size in the range of 3 – 8 mm, resulting from the consolidation process used in the manufacture of the material. Okazaki (1984) suggested that residual stresses in the near surface material due to polishing (or processing) may lead to the dependence of fracture toughness on load or crack length. Tobin and Pak (1993) evaluated this possibility by indenting PZT-8 with different mechanical loads ranging from 0.3 to 0.7 kg. Figure 5 plots the effect of crack size on fracture toughness of poled PZT-8 measured by Tobin and Pak (1993). It is seen from Fig. 5 that the fracture toughness of poled PZT-8 is relatively independent of crack size. Thus, it appears that residual stresses are not expected to play a significant role in this material. Furthermore, Tobin and Pak (1993) found that a positive DC electric field enhances the anisotropy of fracture toughness whilst a negative DC electric field impedes the anisotropy. These results cannot be explained by the ERR criterion such as Eq. (3.6d) or (3.6e). It is now commonly accepted that poled PZT and unpoled (or depoled) PZT have significantly different fracture behaviors. The results of Tobin and Pak (1993) show that the poling of PZT-8 introduces significant anisotropy in crack growth rates and fracture toughness, e.g., the fracture toughness for cracks parallel to the poling direction is much larger than that for cracks perpendicular to the poling direction, whereas depoled PZT-8 has no such anisotropy (a poled specimen can be depoled by air annealing above the Curie point, i.e., 300 8C for PZT-8, see Appendix A3). However, all the experimental studies prior to Park and Sun (1995b) did not explicitly mention the type of electric boundary condition on crack surfaces.

Fig. 5. Effect of crack size on fracture toughness of poled PZT-8 (Tobin and Pak, 1993).

142


For an insulating crack, Pak (1990a,b) applied the ERR fracture criterion to anti-plane (Mode III) piezoelectric fracture problems. At a fixed mechanical load, the electric load is found to either enhance or inhibit crack growth, depending on the magnitude, direction, and type of the electric load. In fact, a decade earlier, Deeg (1980) already showed mathematically that the growth of an insulating crack could be arrested by applying a suitable electric load, although this work was never published in the open literature. Using the J-integral as a fracture criterion, Pak (1992a,b) made extensive use of Deeg’s results to reiterate and expand upon some of the salient features of electro-elastic coupling effects on piezoelectric fracture, and concluded that for an impermeable crack the results obtained by applying the ERR criterion are significantly different from those predicted by using the SIFs criterion. Wang and Singh (1994, 1997) used the Vickers indentation technique to study crack propagation in PZT EC-65 ceramics subjected to a combination of mechanical and electric loads. The material coefficients for PZT EC-65 are listed in Appendix B. They found that a positive electric field inhibits crack propagation, whereas a negative field enhances crack propagation. Such effects become more profound with increasing electric field strength and decreasing mechanical loading. These experimental observations are apparently not consistent with the results of Tobin and Pak (1993) and Park and Sun (1995b). On one hand, we note that the PZT EC-65 ceramic used by Wang and Singh (1994) is much softer than the PZT-8 ceramic used by Tobin and Pak (1993). For example, the compliance of PZT EC-65 (18.3 £ 10212 m2/N; see Appendix B1) is approximately twice the compliance of PZT-8 (1/11.8 £ 10210 ¼ 8.4 £ 10212 m2/N; see Appendix A3). On the other hand, we note that the indentation force 0.5 2 1.2 kg used by Wang and Singh (1997) is significantly larger than the force 0.3 2 0.7 kg used by Tobin and Pak (1993). Later, in Section VI, we will discuss the non-negligible influence of inelastic deformation in indentation tests (Jiang and Sun, 2001). Based on the criterion of maximum hoop stress, Kumar and Singh (1996a,b, 1997a,b) studied a Mode-I impermeable crack perpendicular to the poling direction by using the finite element method (FEM), and compared their numerical results with the experimental data of Wang and Singh (1994, 1997) for both PZT-EC-64 and PZT-EC-65 materials. The results reproduced here in Figs. 6(a) and (b) and 7 show that the critical crack length is reduced by a positive electric field and increased by a negative electric field, which are contrary to the predictions based on either the SIFs criterion (Sosa, 1992) or the ERR criterion (Pak, 1992a). The above discussions clearly indicate that the fracture behavior of a piezoelectric ceramic under electric and/or mechanical loading is still ambiguous.


143

Fig. 6. Critical crack length plotted as a function of electric field for PZT-EC-64 and PZT-EC-65 subjected to: (a) positive electric loading; (b) negative electric loading (Wang and Singh, 1994, 1997).

In particular, it is uncertain whether an applied electric field would impede or enhance crack propagation. Nevertheless, it becomes clear, at least, that either the approximately impermeable crack model is not realistic or new fracture parameters other than the SIFs or ERR need to be introduced. These topics will be explored in detail below. C. Mechanical Strain Energy Release Rate Assuming that the impermeable crack model is adequate, Park and Sun (1995a,b) and Sun and Park (1995) proposed a new fracture criterion based on


144

Fig. 7. Critical crack length plotted as a function of electric field for PZT-EC-64 and PZT-EC-65 subjected to a combined mechanical and electric loading (Wang and Singh, 1994, 1997).

the mechanical strain energy release rate (MSERR). They argued that, since fracture is a mechanical process, it is more logical to exclude the electric energy and only use the mechanical strain energy released during crack extension as the fracture criterion. The MSERR is defined by the mechanical part of the crack closure integral, as: GM I ¼ Lim

1 ðd s ðxÞDu2 ðd 2 xÞdx ðMode IÞ; 2d 0 22

ð3:7Þ

GM II ¼ Lim

1 ðd s ðxÞDu1 ðd 2 xÞdx ðMode IIÞ: 2d 0 12

ð3:8Þ

d!0

d!0

For a Mode-I crack, it follows from Eq. (3.7) that GM I ¼

1 4

p p p ðH21 KI KII þ H22 KI2 þ H23 KI Ke Þ;

ð3:9Þ

where Hp21, Hp22 and Hp23 are elements of the matrix H p defined in Eq. (3.6). Using a PZT-4 compact tension specimen with its poling direction perpendicular to the crack, Park and Sun (1995a) found experimentally that electric discharging through air between the upper and lower electrodes of the specimen occurs when the electric field exceeds 0.5 MV/m (about half of the electric poling field strength). In order to prevent such electric discharging and therefore enforce an insulating crack condition (3.1), the compact tension specimen was immersed in a tub filled with silicone oil. It is then found that the


145

MSERR for an insulating crack perpendicular to the poling direction of PZT-4 is given by (Park and Sun, 1995a) GM I ¼

pa 2 22 1 1 ½1:48 £ 10211 ðs1 22 Þ þ 2:67 £ 10 s22 D2 Þ ðN=mÞ; 2

ð3:10Þ

GM I ¼

pa 2 210 1 1 ½2:12 £ 10211 ðs1 s22 E2 Þ ðN=mÞ; 22 Þ þ 2:67 £ 10 2

ð3:11Þ

or

1 1 where s1 22 ; D2 ; and D2 have the same units as those in Eq. (3.6). Although the mechanical and electric quantities are still coupled in the MSERR, it is seen from Eqs. (3.10) and (3.11) that the negative terms appearing in Eqs. (3.6d) and (3.6e) now vanish, and the MSERR is linearly proportional to 1 the remote electric loading D1 2 (or E2 ) at a given non-zero mechanical loading, 1 i.e., s22 ¼ const – 0: Using the MSERR as a fracture criterion, Park and Sun (1995a,b) predicted that a positive electric field decreases the critical stress at which fracture occurs and hence enhances crack propagation, whereas a negative electric field increases the fracture load and hence impedes crack propagation (see Fig. 8 for a center crack and Fig. 9 for 2 mm off-center crack). These predictions are consistent with the Vickers indentation results of Tobin and Pak (1993) on PZT-8 specimens. As previously discussed, Tobin and Pak (1993) also found that the apparent fracture toughness of a crack perpendicular to the poling direction is reduced by a negative electric field and increased by a positive electric field.

Fig. 8. Fracture load as a function of electric field strength for a center crack in PZT-4 (Park and Sun, 1995a,b).

146


Fig. 9. Fracture load as a function of electric field strength for a 2 mm off-center crack in PZT-4 (Park and Sun, 1995a,b).

Although Tobin and Pak (1993) made extensive use of the results of Pak (1990a,b) and Sosa (1992), they did not explicitly mention the type of crack surface electric boundary conditions pertinent to their indentation fracture specimens. We note that their results agree well with the experimental measurements of Park and Sun (1995a,b) on compact tension specimens, but not with those of Wang and Singh (1994, 1997) and Kumar and Singh (1996a,b). As mentioned earlier, the PZT EC-65 material used by Wang and Singh (1994, 1997) is much softer than PZT-4, PZT-5H and PZT-8, which may contribute to the experimental discrepancy. To further examine the results of Park and Sun (1995a,b), Kumar and Singh (1996b) used two commercial finite element packages, ANSYS and ABAQUS , to calculate the MSERR for a center crack in a finite PZT-4 plate. Their numerical results on the MSERR exhibit different trends from those obtained by Park and Sun (1995a,b). More importantly, Kumar and Singh (1996b) made some critical comments on the work of Park and Sun (1995a, b), as they found several numerical errors in Park and Sun (1995a,b). In succession, Balke et al. (1998) corrected these errors, but the conclusions reached by Park and Sun (1995a,b) still hold, suggesting that the empirically defined ‘MSERR’ is perhaps a better fracture mechanical parameter than the total potential energy release rate (TPERR). Consequently, MSERR appears to be a viable concept for studying the fracture toughness of a piezoelectric and its dependence on electric field with compact tension (Park and Sun, 1995a,b), indentation (Sun and Park, 1995) or other types of specimens. More recently,


147

Jiang and Sun (1999) employed the MSERR to successfully characterize the fatigue behavior of piezoelectric ceramics. If the applied negative electric loading is sufficiently high, crack closure may occur even under Mode-I mechanical loading (Kumar and Singh, 1996b, 1997c). This prediction coincides with the indentation results of Cao and Evans (1994). Under such conditions, the conventional traction-free condition and the MSERR fracture criterion are no longer valid (Kumar and Singh, 1997c). This topic was re-examined in more detail by Ru (1999a,b). Analytical studies on piezoelectric cracks and fracture have been carried out by many researchers (see, e.g., Deeg, 1980; McHenry and Hoepke, 1983; Pak, 1990a,b, 1992; Suo et al., 1992; Sosa, 1990, 1992, 1994; Zhang and Hack, 1992; Wang and Singh, 1994, 1997; Kumar and Singh, 1996a,b, 1997a –c, 1998). It is shown that crack growth can be either inhibited or enhanced, depending on the combinations of electric and mechanical loadings and nature of crack surface boundary conditions. For certain ratios of the electrical-to-mechanical loading, it has been predicted that crack arrest is possible. However, contradicting results from these theoretical studies have been obtained, depending on which fracture criterion (SIFs or ERR) or boundary condition is used. In sharp contrast with the extensive theoretical studies, experimental results obtained under simultaneous mechanical and electrical loadings are rather limited. The earlier experimental work most widely quoted was performed by McHenry and Koepke (1983) who examined the effect of electric fields on subcritical crack growth in PZTs by using a double torsion technique. They found that a high electric field (either AC or DC) applied perpendicular to the crack plane enhances crack propagation in unpoled PZTs. For poled PZTs, their study did not show quantitatively whether the applied electric field enhances or inhibits crack propagation. Instead, it was reported that the application of an electric field perpendicular to the crack plane always turns the crack in a direction opposite to the poling direction. Using prefabricated cracks, Cao and Evans (1994) reported electric-field-induced fatigue crack growth in piezoelectric ceramics, whilst White et al. (1994) reported the fracture behavior of cyclically loaded PZT without any pre-existing cracks. Wang and Singh (1994, 1997) systematically investigated the fracturing behavior of piezoelectric ceramics under simultaneous mechanical and electrical loadings. Following the work of Okazaki (1984), Pisarenko et al. (1985), and Mehta and Virkar (1990), they used the Vickers indentation technique to study the correlations between electric field strength and crack length. These correlations enable a quantitative estimate of the influence of electrical and mechanical loadings on crack propagation. The typical value of the coercive field selected by Wang and Singh (1997) is 1.2 MV/m for a soft PZT and 2.4 MV/m for a hard PZT. They also applied fields


148

Fig. 10. Schematic of a crack perpendicular (’) or parallel (// ) to the poling direction in a PZT specimen under (a) a positive; (b) a negative electric field (Wang and Singh, 1997).

much smaller than these to avoid de-poling of the sample. For example, fields between 0.5 and 0.8 MV/m were applied in the direction opposite to the poling whereas fields between 0.5 and 1.0 MV/m were applied along the poling direction. Figure 10(a) and (b) illustrate schematically the electric-field polarity and crack orientation for both positive and negative electric fields (Wang and Singh, 1997). In the presence of an electric field, cracks in poled PZT specimens are found to propagate anisotropically in a manner similar to that observed without the electric field. For both positive and negative polarities, cracks perpendicular to the electric field propagate more than those parallel to the electric field, although the degree of this anisotropy is dependent upon electric field strength and polarity (see Tables II and III reproduced from the work of Wang and Singh, 1997). On the other hand, cracks emanating from the corners of

STATISTICAL ANALYSIS

Direction

Perpendicular Parallel

OF

Table II CRACK LENGTH VARIATION UNDER POSITIVE ELECTRIC FIELD FIXED INDENTATION LOAD OF 1.2 kg Crack length (mm)

c1 (E ¼ 0 V/mm)

c2 (E ¼ þ1000 V/mm)

158.2 ^ 8.5 90.8 ^ 4.3

147.3 ^ 10.4 78.2 ^ 6.7

FOR A

95% confidence interval

219.9 , c2 2 c1 , 22.0 218.0 , c2 2 c1 , 27.3


STATISTICAL ANALYSIS

Direction

Perpendicular Parallel

OF

149

Table III CRACK LENGTH VARIATION UNDER NEGATIVE ELECTRIC FIELD FOR A FIXED INDENTATION LOAD OF 1.2 kg Crack length (mm)

c1 (E ¼ 0 V/mm)

c2 (E ¼ 2800 V/mm)

138.8 ^ 5.2 90.4 ^ 7.8

163.0 ^ 9.4 89.9 ^ 3.6

95% confidence interval

17.1 , c2 2 c1 , 31.3 26.7 , c2 2 c1 , 25.8

an indentation propagate more or less along a straight line, with no turning or skewing observed, as shown in Fig. 11(a) and (b) (Wang and Singh, 1997). It should be emphasized that Wang and Singh’s work (1997) was also based on the impermeable crack model. Wang and Singh (1997) observed decreasing crack lengths with increasing positive electric field applied in the direction either perpendicular or parallel to the crack plane (Tables II and III). In comparison, the crack length increases with increasing negative electric field applied normal to the crack, whilst it remains more or less unchanged with increasing negative electric field parallel to the crack. These results are not only in contradiction with those of Park and Sun (1995a,b), but also difficult to explain intuitively based on piezoelectric effect and simple mechanical strain considerations. Wang and Singh (1997) observed

Fig. 11. SEM images showing (a) anisotropic crack growth under indentation and electric field (E ¼ þ 0.5 MV/m); (b) no turning or skewing during crack extension (Wang and Singh, 1997).

150


that a positive (negative) electric field induces tensile (compressive) strain in the direction of the electric field, implying that a crack may propagate less under tensile strain and more under compressive strain. Cheng et al. (1992) pointed out that grinding and polishing during sample preparation may cause domain reorientation at the surface of a piezoelectric material. However, Wang and Singh (1997) eliminated such influences in their experiments. To explain the mechanisms underneath the above experimental phenomena, Wang and Singh (1997) argued that the local electric field in a piezoelectric sample is distorted by the introduction of indentation cracks. As shown in Fig. 12, the electric field is shielded by a crack, and the flux lines have to go around the crack in order to keep the total electric flux constant, leading to an enhancement of the electric field at the tip of the crack. It is known that crack propagation perpendicular to an applied electric field is reduced relative to that in the absence of the field, while cracks parallel to the field show little or no change in length compared to that without the electric field. Wang and Singh (1997) concluded that the enhancement of electric field at the tip of a crack causes an increase in strain near the tip, resulting in a compressive stress at the tip as the material near the tip attempts to expand but is constrained by the surrounding material. Figure 13(a) and (b) illuminate the influence of applied electric field on the stress state in front of a crack-tip (Wang and Singh, 1997). This mechanism, which considers the change in stress states between regions behind and in front of a crack-tip, as schematically shown in Fig. 13, may be used to explain the experimental results listed in Tables II and III. When an indentation crack is introduced in a piezoelectric subjected to a positive electric field (Fig. 13(a)), the material in front of the crack will continue to

Fig. 12. Shielding of an electric field around an indentation crack (Wang and Singh, 1997).


151

Fig. 13. Illustration of two different effects of electric field shielding on stress state in front of a crack-tip: (a) positive field; (b) negative field (Wang and Singh, 1997).

elongate, whereas the material in the wake of the crack will tend to relax back to its original position because the material in this region is less active in terms of the piezoelectric effect. Consequently, the material behind the tip is constraining the elongation of the material in front of the tip, resulting in a crack closing force at the crack-tip which offsets the crack opening force from the indentation and leads to a reduction in crack length under the positive electric field (Wang and Singh, 1997). In the case of a negative applied electric field (Fig. 13(b)), the material behind the crack-tip will tend to expand back to its original position once the electric field is shielded, while the material in front of the crack-tip continues to contract, which develops an additional crack opening force in the vicinity of the crack-tip and enhances crack extension (Wang and Singh, 1997). With a constant applied electric field, the effect of varying mechanical loading on crack propagation may also lead to different experimental observations regarding the electric field effect. By systematically increasing the indentation load from 0.2 to 1.4 kg, Wang and Singh (1997) observed largest crack lengths for the perpendicular cases under a negative electric field and smallest crack lengths for the parallel cases under a positive electric field; the crack lengths observed for the perpendicular cases under a positive electric field and for the parallel cases under a negative electric field are intermediate. It becomes clear now that, for an insulating crack, different selections of fracture parameters (e.g., SIFs, ERR, and MSERR) can lead to different and mutually contradicting predictions. This point will be elaborated in more detail below. With reference to Eq. (3.4), it is seen that the J-integral can be divided into two distinct parts. The first, called the mechanical part, is associated only with mechanical quantities, whereas the second, called the electric part, is associated

152


only with electric quantities, although both quantities are coupled in the constitutive equation (2.8). Consequently, for an insulating crack, whether the SIFs (or the J-integral) can be used as a fracture criterion depends on the magnitudes and signs of the two types of energy. Pak (1990) found that, for a Mode-III insulating crack, the mechanical part of the ERR, i.e., the first term in Eq. (3.4), is always positive, which represents the energy released when the crack grows by a unit length. However, the electric part, i.e., the second term in Eq. (3.4) is always negative, representing energy absorbance during crack growth. In other words, the mechanical energy and electric energy in the J-integral formulation Eq. (3.4) play opposite roles in crack propagation. Figure 14(a) –(c) present the calculated values of J-integral (or G) for a Mode-I impermeable crack oriented at angle a with respect to the axis perpendicular to poling direction (Liu and Sheng, 2001). Again, it is seen that the mechanical part is positive and the electric part is negative, consistent with the results for a Mode-III insulating crack (Pak, 1990a, b). Furthermore, the electric energy has, in general, a magnitude much larger than its mechanical counterpart. Since fracture is a mechanical process at the macroscopic level, some researchers have challenged the physical meaning of a negative ERR and questioned how the electric energy is ‘absorbed’ during crack extension. Park and Sun (1995a,b) insist that neither the SIFs nor the J-integral (ERR) can be used to describe the stability and growth of an impermeable crack in linear piezoelectric ceramics. Instead, as the MSERR is always positive and represents the energy due to crack opening, they propose to use the MSERR as a new crack driving force in LPFM: that is, a new fracture criterion based on the MSERR is applicable when the crack surfaces are charge-free. This charge-free condition can be ensured by putting a crack specimen in a tub filled with silicone oil (Park and Sun, 1995a). However, since the value of GM I evaluated from Eq. (3.9) vanishes altogether in the absence of remote mechanical loading, as a fracture criterion the MSERR has a major shortcoming since it cannot explain why a purely DC electric field could cause the formation of cracks in BaTiO3 (McMeeking, 1989) or the extension of existing cracks in PZT ceramics (Cao and Evans, 1994; Tan et al., 2000; Xu et al., 2000). The experimental results obtained by Xu et al. (2000) under a 2.5 MV/m DC or AC electric field with no mechanical loading are shown in Fig. 15(a) – (d). Many researchers, including the present authors, emphasized that the ERR and MSERR are macroscopic concepts that are established by idealizing the poled or unpoled ferroelectric ceramics as uniform homogenous materials. In the vicinity of a crack-tip, a non-linear process zone (albeit ill-defined) exists which contains heterogeneities induced from mismatch between the pores (or particles) and matrix phases (Tobin and Pak, 1993; Schneider and Heyer, 1999; Tan et al.,


153

Fig. 14. Comparison between a permeable crack and an impermeable crack (Liu and Sheng, 2001): (a) mechanical; (b) electrical; (c) total energy release rate.

2000; Xu et al., 2000). As a result, the non-linear properties induced from the ferroelectric and ferroelastic switching zone (Yang and Suo, 1994; Zhu and Yang, 1997; Heyer et al., 1998) become relevant. It has been anticipated that a non-linear analysis would provide more realistic results. Gao and Barnett (1996)

154


Fig. 15. Microcracking induced by a purely DC electric field (Xu et al., 2000).

proposed a modified Dugdale (1960) model for an insulating crack. Based on the concept of local and global energy release rates for an electrically yielded crack in a piezoelectric ceramic, Gao and Barnett (1996) and Gao et al. (1997) established a “strip saturation” model to account for the non-linear behavior in the near-tip process zone. This model and others based on dielectric non-linearity at the crack-tip (Gao et al., 1997; Ru, 1999a,b; Zhu and Yang, 1997) appear to support the experimental results of Park and Sun (1995a). Figure 16 shows the typical profile of near-tip switching zone in a poled piezoelectric ceramic (Fulton and Gao, 2001b). Suo (1993) defined the increase in SIFs and electric field intensity factor to qualitatively describe the domain switching effect in the process zone. Han and Chen (1999) presented a simple mathematical tool to treat interacting inhomogeneities in a plane piezoelectric ceramic, whilst Chen and Han (1999a,b) studied microcrack shielding problems in PZT ceramics. Their results on multiple crack interactions are shown in Figs. 17 and 18 for crack-tip SIFs and MSERR, respectively. Although the cracks are assumed impermeable, it is seen that, in general, neither the SIFs nor the MSERR vanishes even under purely electric loading (i.e., KI1 ¼ KII1 ¼ 0). This is attributed to the fact that the neartip electric field and hence the SIFs and MSERR of a crack are significantly disturbed by neighboring cracks. These theoretical results have important practical implications. Inhomogeneities cannot be avoided in any ferroelectric


155

Fig. 16. Profile of a near-tip switching zone under: (a) a purely mechanical loading; (b) a combined mechanical and electrical loading (Fulton and Gao, 2001b).

ceramic: see, for instance, Fig. 4 from Tobin and Pak (1993) or Fig. 11(a) and (b) from Wang and Singh (1997), where a number of pores with size of a few micrometers are observed. Therefore, even based on the empirical MSERR concept for impermeable cracks, a purely electric loading is expected to induce microdefect interactions and subsequent microcrack growth. In other words, MSERR may still play the role as the crack driving force under purely electric loadings if the existence of near-tip microcracks is accounted for.

Fig. 17. SIFs due to crack interaction as influenced by electric loading (Han and Chen, 1999).

156


Fig. 18. MSERR due to crack interaction as influenced by electric loading (Han and Chen, 1999).

D. Eigenfunction Expansion Forms and Weight Functions The discussion hitherto shows that, under certain conditions, the impermeable crack can be a useful model to study piezoelectric fracture. In practice, the important task is to evaluate the crack-tip SIFs and EDIF, from which both the ERR and MSERR can be calculated and then crack stability or critical crack length be estimated. As the Williams eigenfunction expansion form (EEF) has proven to be a powerful tool for analyzing fracture in brittle materials, Sosa and Pak (1990) performed a 3D eigenfunction analysis for an impermeable crack in piezoelectric materials. Similarly, as the Bueckner work conjugate integral (BWCI) and the resulting weight function method (WFM) have been well addressed in LEFM, Ma and Chen (2001a) used the compact formulations of Stroh’s complex potential theorem (Suo et al., 1992) to find the explicit function expressions of EEF for a semi-infinite impermeable crack in an anisotropic piezoelectric material. The work of Ma and Chen (2001a) is based on the fundamental results of Chen (1985), Chen and Hasebe (1995), and Chen and Ma (2000) for traditional non-piezoelectric materials, and it is found that the pseudo-orthogonal properties of EEF established for traditional materials (Chen, 1985; Chen and Hasebe, 1995; Chen and Ma, 2000) are still valid in piezoelectric materials. Based on Bett’s reciprocal theorem (Parton and Kudryvtsev, 1988), the extension of Bueckner’s (1973) work conjugate integral to treat a semi-infinite


157

Fig. 19. A semi-infinite crack in a plane piezoelectric material (Ma and Chen, 2001a).

insulating crack in a plane piezoelectric material leads to (see Fig. 19): B¼

ð G

ðuai s bij 2 ubi s aij Þnj ds þ

ð G

ðfa Dbk 2 fb Dak Þnk ds

ð3:12Þ

ði ¼ 1; 2; 3; j; k ¼ 1; 2Þ; where the superscripts a and b refer to two stress – displacement states in the anisotropic piezoelectric solid, respectively. By using the generalized stress and generalized displacement concepts, Eq. (3.12) can be rewritten as B¼

ð G

ðuai sbij 2 ubi saij Þnj ds

ði ¼ 1; 2; 3; 4; j ¼ 1; 2;

ð3:13Þ

u4 ¼ f; s41 ¼ D1 ; s42 ¼ D2 Þ; or, equivalently B¼

ð G

ðuai dTib 2 ubi dTia Þ

ði ¼ 1; 2; 3; 4Þ:

ð3:14Þ

Following Suo’s formulation (1992), Eq. (3.14) can be expressed in a compact matrix form, as B¼

ð

T

½AfðzÞ þ AfðzÞb d½BfðzÞ þ BfðzÞa G

2

ð3:15Þ

ð

aT

b

½AfðzÞ þ AfðzÞ d½BfðzÞ þ BfðzÞ ; G


158

where the matrices A and B are defined in Eq. (2.27), while the vector fðzÞ refers to the EEF for the crack of Fig. 19. Its formulation will be given below. By assuming that the integral contour G in Eq. (3.15) is piecewise smooth in the x1 2 x2 plane (Fig. 19) and integrating by part, one obtains ð T T f a B ¼ ½f b ðAT B þ BT AÞdf a þ f b ðAT B þ BT AÞd G ð3:16Þ ð T T f a : T B þ B T AÞdf a þ f b ðA TB þB T AÞd ½fb ðA þ G

Then, from the orthogonal properties of the matrices A and B as shown in Eq. (2.29), Eq. (3.16) simplifies to ð ð T T T B¼ ½f b df a þ fb dfa ¼ 2Re ½f b df a : ð3:17Þ G

G

This clearly shows the path-independent nature of the Bueckner-integral in anisotropic piezoelectric solids, i.e., BG ¼ Bg ;

ð3:18Þ

where G and g are two closed contours surrounding the crack-tip (Fig. 19). Without going into details, the eigenfunction expansion forms for the crack in Fig. 19 are given by f a ðzÞ ¼ ½ f1a ðz1 Þ; f2a ðz2 Þ; f3a ðz3 Þ; f4a ðz4 ÞT l=2

l=2

l=2

ð3:19Þ

l=2

¼ ½Al z1 ; Bl z2 ; Cl z3 ; Dl z4 T ; f b ðzÞ ¼ ½ f1b ðz1 Þ; f2b ðz2 Þ; f3b ðz3 Þ; f4b ðz4 ÞT k=2

k=2

k=2

ð3:20Þ

k=2

¼ ½Ak z1 ; Bk z2 ; Ck z3 ; Dk z4 T ;

where l/2 and k/2 are the eigenvalues. Ma and Chen (2001a) derived the following pseudo-orthogonal properties of the EEF for piezoelectric materials: ( 22pl Im½ETl Ek l þ k ¼ 0; B¼ ð3:21Þ 0 l þ k – 0; where the vector El is related to the complex coefficients in Eq. (3.19) by El ¼ ½Al ; Bl ; Cl ; Di : Equation (3.21) proves to be a powerful tool with which some important results have been established. For example, by taking the two states a and b associated with K ¼ ½KII ; KI ; KIII ; Ke to be

s bij ¼

›s aij ; ›x 1

ubi ¼

›uai ; ›x 1

fb ¼

›f a ; ›x 1

Dbj ¼

›Daj : ›x 1

ð3:22Þ


159

Ma and Chen (2001a) proved that the Bueckner-integral is just twice the J-integral even though mechanical and electric quantities are coupled: B ¼ 2 12 KT ImðB2T B21 ÞK ¼

1 2

KT HK ¼ 2J:

ð3:23Þ

Moreover, if the two states a and b selected are such that

s bij ¼ saij þ fb ¼

›s aij ; ›x 1

›fa ; ›x 1

ubi ¼

›uai ; ›x 1

Dbj ¼ Daj þ

›Daj ; ›x 1

ð3:24Þ

then the Bueckner-integral is just twice the M-integral (Chen and Tian, 1999). Obviously, due to the pseudo-orthogonal property (3.21), only the term in the EEF (3.19) corresponding to l ¼ 1 contributes to the Bueckner-integral (3.17). Just as in the case of brittle materials, the J-integral and M-integral can be considered as two special cases of the Bueckner-integral in piezoelectric materials. The pseudo-orthogonal property (3.21) can also be used to find certain useful weight functions for calculating field intensity factors like the SIFs and the EDIF. For an insulating crack, such weight functions are (Ma and Chen, 2001a): 2 !

KII

3

7 6 7 6 ð "6 KI 7 7 ¼ p1ffiffiffiffi 6 d1 d 2 d3 d4 6 uai s bij nj ds; 7 2p G 6 KIII 7 5 4 Ke

ð3:25Þ

where ual is a complementary state which represents the higher order singular term (Bueckner, 1973; Chen, 1985) and does not yield any stress or electric displacement on the outside closed contour G (Ma and Chen, 2001a), while s bij is a mechanical-electric state for a given crack configuration with external boundary G. The elements dj in the vector ½d1 ; d2 ; d3 ; d4 are specially introduced, which can be either unit or zero depending on which field intensity factor is to be distinctly calculated. Therefore, for any given geometric configuration of a general plane anisotropic piezoelectric solid containing a crack subjected to an arbitrary combination of mechanical and electric loadings, Eq. (3.25), together with the complementary potentials proposed by Ma and Chen (2001a), enables one to calculate the Mode I, Mode II, and Mode III SIFs as well as the EDIF at the crack-tip distinctly by selecting the corresponding value of dj to be unit or zero without any special treatment of the near-tip singular region.

160

Y.-H. Chen and T.J. Lu E. Summary

(1) The impermeable crack presents an approximate model to treat the electric boundary on crack surfaces, provided that the charge-free condition as well as the traction-free condition on the surfaces are satisfied. An impermeable crack may be realized by immersing crack specimens into an insulating fluid, e.g., silicone oil (Park and Sun, 1995a, b; Sun and Park, 1995; Jiang and Sun, 2001). Physically, this is reasonable because there are several orders of magnitude difference between the values of dielectric constants of silicon oil and PZT ceramics. (2) Either the SIFs or ERR ( ¼ J) can be used as a fracture criterion for an impermeable crack, especially when the ratio of electric to mechanical loading is relatively small (Kumar and Singh, 1996b). On one hand, under the charge-free condition for an insulating crack the SIFs are only related to the remote mechanical loading, whereas the EDIF is only related to the remote electric loading. On the other hand, the ERR ( ¼ J) comprises two types of energy: mechanical energy and electric energy. Unlike the energy release role of the mechanical energy, the electric energy generally plays the role of energy absorbance for an insulating crack. (3) Under certain conditions, the MSERR proposed by Park and Sun (1995a, b) appears to be a useful fracture criterion for impermeable cracks. A key concept of the MSERR fracture criterion is that fracture is a mechanical process from phenomenological viewpoint. (4) With the MSERR criterion, a positive electric field enhances crack growth while a negative electric field impedes crack growth, provided that the charge-free condition as well as the traction-free condition on the crack surfaces are always satisfied. That is, a sufficiently large mechanical loading should always be in place, otherwise the crack surfaces may either contain distributed charges (Hao and Shen, 1994; Xu and Rajapakse, 2001) or in contact with each other (Kumar and Singh, 1997c, 1998; Ru, 1999a,b). (5) The MSERR criterion with the impermeable crack model cannot be used to describe crack initiation and growth induced by a purely electric loading. The contradicting results in the literature regarding impermeable cracks may be attributed to two reasons. Either cracking is actually of the multiple mirocracking type (Han and Chen, 1999) or the charge-free and traction-free conditions have been violated (McMeeking, 1989; Cao and Evans, 1994; Ru, 1999a,b). Furthermore, Narita and Shindo (1998, 1999) studied a crack in a finite piezoelectric strip, and found that the dominant parameters at the


161

crack-tip are significantly influenced by the external boundaries of the strip. In this case the remote electric loading will contribute to the mechanical SIF due to the interaction between the crack and strip boundaries. (6) Crack kinking may occur due to the inherent anisotropy of poled PZT ceramics, especially when the applied electric field or the crack is neither perpendicular nor parallel to the poling direction. When the charge-free condition is satisfied, the crack kinking angle may be predicted by using the maximum hoop stress criterion (Nemat-Nasser, 1994; Kumar and Singh, 1997a,b). The angular distributions of the near-tip stresses in a double edge-notch specimen obtained by Kumar and Singh (1996a) with the method of finite elements are close to those calculated analytically by Sosa (1992) for an infinite PZT-4 and Pak (1992a,b) for an infinite PZT-5H. (7) Although the insulating crack model has been studied for more than 10 years, controversial results have been obtained due to its idealization. For example, the FEM results of Kumar and Singh (1996a,b, 1997a,b, 1998) show that a positive electric loading decreases the SIF and a negative loading increases the SIF, which appear to agree well with experimental data of Wang and Singh (1994, 1997). This means that even for a purely insulating crack their results are contrary to those of Park and Sun (1995a,b), due mainly to different selections of fracture parameters. Models that assume permeable cracks as well as crack-tip non-linearity are obviously needed for further clarification. For example, the influence of crack-tip non-linear zone, i.e., the ferroelectric and ferroelastic switching zone (Yang and Suo, 1994; Suo, 1993; Heyer et al., 1998), the strip saturation model (Gao and Barnett, 1996; Gao et al., 1997a,b), and the change of the mechanical boundary condition (the traction-free condition may fail under strong electric fields) remain to be properly studied.

IV. Permeable (Conducting) Crack Parton (1976) was apparently the first to introduce the concept of conducting (permeable) cracks. For cracks sufficiently sharp, Mikhailov and Parton (1990) asserted that the air gap within the crack is conducting under a sufficiently high electric field (e.g., over 0.5 MV/m). McMeeking (1987, 1989, 1990, 1999, 2001) applied the conducting crack model to treat breakdown problems in

162


dielectric materials such as unpoled BaTiO3 as well as PZT ceramics. The breakdown problems are motivated by experimental observations that, at very high levels of the electrostatic field (near breakdown levels), the Mode-I stress intensity factor KI for cracks of a few microns in length can approach the fracture toughness of common ceramics. For such problems, only the converse piezoelectric properties associated with electric-field-induced stresses need to be accounted for, but not the piezoelectric properties (i.e., stress-induced charges). When calculating the SIFs at the tip of a conducting crack in a plane isotropic linear elastic homogeneous dielectric subjected to a purely electric loading, McMeeking (1987, 1989, 1990) found that there exist forces caused by the actions of the electric field on charges induced in the crack, and concluded that a purely electric loading alone can induce KI at the tip of a conducting crack in a dielectric such as BaTiO3. His results provide a good explanation of dielectric breakdown at very high electric field levels. Moreover, based on the invariant Gintegral of Cherepanov (1979), McMeeking (1990) performed the J-integral analysis of electrically induced mechanical stresses for plane cracks in elastic dielectrics. He then used this integral to calculate the mechanical SIF at the tip of a conducting crack in the presence of an electric field. This can be used to determine the mechanical stresses that would be present when a dielectric containing such cracks experiences insulator failure at very high electric field levels. Piezoelectric materials as poled ferroelectric materials are a special type of dielectrics. The significance of McMeeking’s results (1987, 1989, 1990) is the possible extension of the conducting crack model from dielectrics to poled ferroelectrics. He calculated the crack-tip energy release rate for a piezoelectric compact tension specimen and then attempted to establish a fracture mechanics for brittle piezoelectric materials. It is anticipated that, by using the conducting crack model, albeit still approximate, some of the confusions raised in the previous section for insulating cracks could be clarified. Zhang and Hack (1992), Dunn (1994), and Hao and Shen (1994) recognized that, although the magnitude of the normal electric displacement on crack surfaces is in general very small, the electric displacement and the corresponding electric potential are continuous across the crack. Sosa (1991, 1992) initially studied insulating cracks but later focused on permeable cracks (Sosa and Khutoryansky, 1996), recognizing that the assumption of electric impermeability at the boundary of a cavity may lead to erroneous conclusions, which becomes particularly relevant for sharp cracks or very slender elliptical cavities. The conducting crack model has gained popularity in recent years. Hao and Shen, (1994), Gao and Fan (1999a,b), Shindo et al. (1996a,b, 1997a,b, 2000), Ru and Mao (1999), Xu and Rajapakse (1999, 2000a,b, 2001) obtained a series of


163

analytical solutions for different crack configurations by using the conducting crack model. It has been shown that the so-called exact boundary condition accounting for the piezoelectric-vacuum interface can always be reduced to the conducting crack, as long as the crack is not parallel to the poling direction (Xu and Rajapakse, 2001). On the other hand, the exact boundary condition reduces to an insulating crack when the crack is parallel to the poling direction (Xu and Rajapakse, 2001). White et al. (1994) and Heyer et al. (1998) carried out experimental measurements to check the validity of the conducting crack model. The experiments were performed under conditions (dry air or vacuum) different from that (silicone oil) used by Park and Sun (1995a), Sun and Park (1995), and Jiang and Sun (2001). Recently, Rajapakse and Zeng (2001) studied the toughening of conducting cracks due to domain switching. Except for a few cases where the cracks are parallel to the poling direction, in most of these experiments the cracks are chosen to be perpendicular to the poling direction. Very high electric fields (much larger than 0.5 MV/m) approaching the breakdown level and relatively low electric fields (much smaller than 0.5 MV/m) were both used in these experiments.

A. Conducting (Permeable) Crack Model With reference to Fig. 1, the permeable electric boundary condition on crack surfaces assumes that not only the normal electric displacement but also the electric potential are continuous across the crack gap (Parton, 1976), i.e., 2 Dþ 2 ðx1 Þ ¼ D2 ðx1 Þ;

ð4:1aÞ

fþ ðx1 Þ ¼ f2 ðx1 Þ:

ð4:1bÞ

The most significant electric quantity to be determined on a crack surface is the distributed normal electric displacement D02 ðx1 Þ: Unlike the impermeable crack model, this quantity is unknown for a conducting crack: it represents the change when the electric field strides across the crack gap. Chen and Lai (1997) found that the electroelastic field inside a plane inhomogeneity is uniform under uniform far-field loading, i.e., D02 ðx1 Þ ¼ constant: Moreover, Eq. (4.1b) is equivalent to the continuity of tangential electric field on crack surfaces: E1þ ðx1 Þ ¼ E12 ðx1 Þ:

ð4:2Þ

Using the mapping technique in complex potential analysis and accounting for the traction-free condition (3.2), Gao and Fan (1999a,b) and Xu and Rajapakse


164

(2001) independently found that the normal electric displacement on crack surfaces is related to uniform far-field loadings by Im D02 ¼

3 X

1 kk ð2Ak1 s1 22 þ Ak2 s12 Þ

þ D1 2 ;

k¼1

Im

3 X

ð4:3Þ

kk Ak3

k¼1

where kk and Ak1 ; Ak2 ; Ak3 (k ¼ 1,2,3) are material constants defined in Eq. (3.6b). Xu and Rajapakse (2001) added that, as the complex coefficients in the 0 complex potentials are related to the electric quantity D1 2 2 D2 rather than the 1 remote electric displacement D2 itself, Eq. (4.3) suggests that for a conducting crack the remote electric loading (i.e., D1 2 in Fig. 1) has no influence on the complex potential solutions. This implies that, for a conducting crack, the remote mechanical loading plays a much more significant role than the remote electric loading does. Consequently, traditional mechanical SIFs may be used as a fracture criterion for conducting cracks. Indeed, Eq. (4.3) shows that, in contrast to vanishing D02 for impermeable cracks, the assumption of permeability leads to non-zero crack surface electric displacements. Both far-field mechanical and electric loadings may contribute to D02, although, according to Chen and Lai 0 (1997), the value of D1 2 2 D2 remains unchanged when the electric loading is varied at fixed mechanical loading.

B. Field Intensity Factors For a conducting crack as shown in Fig. 1, the field intensity factors are given by (Gao and Fan, 1999a; Xu and Rajapakse, 2001) pffiffiffiffi KI ¼ s1 ð4:4aÞ 22 pa; p ffiffiffiffi ð4:4bÞ KII ¼ s1 12 pa; p ffiffiffiffi 0 Ke ¼ ðD1 ð4:4cÞ 2 2 D2 Þ pa: By comparing Eqs. (4.4a) –(4.4c) with Eqs. (3.3a)– (3.3c), it is seen that whilst the SIFs of a conducting crack are identical to those for an impermeable crack, the EDIF is different from that of Eq. (3.3c) due to the existence of D02 : Furthermore, from Eqs. (4.3) and (4.4c), it is seen that the EDIF depends on 0 1 1 0 D1 2 2 D2 but not on D2 itself. Also note that D2 2 D2 as formulated by Eq. (4.3) depends only on remote mechanical loadings. Upon setting the remote electric 0 loading D1 2 to zero in Eqs. (4.3) and (4.4c), it is seen that D2 on crack surfaces


165

does not vanish. In other words, remote mechanical loadings as preferred by s1 22 0 and s1 12 in Fig. 1 do contribute to the electric quantity D2 and hence the EDIF. This special piezoelectric effect induced from remote mechanical loadings implies that mechanical quantities such as the SIFs may play a more dominant role for conducting cracks. Electric quantities such as the EDIF can be formulated directly from mechanical quantities as seen in Eqs. (4.3) and (4.4c). Because the EDIF of Eq. (4.4c) is different from that of Eq. (3.3c), the angular distributions of near-tip stresses will also be different, resulting in different kinking angles for a mixed-mode crack. Xu and Rajapakse (2001) presented the hoop stress at the tip of a conducting crack, as pffiffi 3 X a 1 1 0 3=2 suu ¼ pffiffiffi Re ½Ak1 s1 : ð4:5Þ 22 2 Ak2 s12 2 Ak3 ðD2 2 D2 Þðcos u þ mk sin uÞ 2r k¼1 0 1 Again, it is seen that the hoop stress depends upon D1 2 2 D2 but not on D2 ; that is, the hoop stress at the tip of a conducting crack-tip depends only on mechanical loadings and is independent of any electric loading. This provides another theoretical evidence that the traditional SIFs may be used as the fracture criterion for a conducting crack. On the other hand, based on the maximum hoop stress criterion adopted by Kumar and Singh (1996a,b, 1997a,b, 1998), the kinking of a permeable crack is found to occur at a different angle from that for an impermeable crack, when both cracks are subjected to the same combination of electromechanical loadings. In the absence of remote mechanical loading, D02 equals D1 2 and hence the EDIF vanishes; consequently, the hoop stress vanishes for a permeable crack but not for an impermeable crack. On the other hand, in the absence of remote electric loading, the remote mechanical loading gives rise to non-vanishing D02 and hence the EDIF does not vanish. In this limiting case, the hoop stress in Eq. (4.5) does not vanish but is still different from that for an impermeable crack.

C. Energy Release Rate As an approximate model, the permeable crack appears to have gained more popularity than the impermeable crack in recent years. Unlike Park and Sun (1995a,b) who put fracture specimens in an insulating oil, McMeeking (1989) considered a crack-like flaw in a dielectric material filled with dry air or vacuum. Let the dielectric permittivity of the dielectric be denoted by 1m and that of the medium inside the flaw by 1f : An elliptical flaw with aspect ratio b=a p 1 has a normalized electric field concentration approximately equal to ð1 þ


166

b=aÞ=ðb=a þ 1f =1m Þ (McMeeking, 1989). For a crack-like flaw in barium titanate (BaTiO3) or PZT-8 containing vacuum or dry air (Tobin and Pak, 1993; Park and Sun, 1995a), 1f =1m < 1023 (see, e.g., Appendix A3, where 1T33 =10 ¼ 1000). Consequently, to an excellent approximation, such flaws may be taken as electrically permeable. Xu and Rajapakse (2001) calculated the energy release rate (ERR) for a conducting crack (Fig. 1). Regardless of the poling direction of the piezoelectric ceramic, the ERR or the J-integral can be derived by using a crack closure integral. With the right crack-tip (Fig. 1) extending by a small amount da, the ERR of the system is G ¼ GM þ GE ¼ Lim

da!0

1 ðda {si2 ðx; 0Þui ðda 2 x; ^pÞ 2da 0 þ D2 ðx; 0Þfðda 2 x; ^pÞ}dx;

ð4:6Þ

where (Xu and Rajapakse, 2001) GM ¼

3 X pa 2 1 2 1 1 Im { 2 qk Ak1 ðs1 22 Þ þ pk Ak2 ðs12 Þ þ ðqk Ak2 2 pk Ak1 Þs12 s22 2 k¼1 1 1 0 þ ðpk Ak3 s1 12 þ qk Ak3 s22 ÞðD2 2 D2 Þ};

3 pa X 1 1 0 G ¼ Im {ð2sk Ak1 s1 22 þ sk Ak2 s12 ÞðD2 2 D2 Þ 2 k¼1

ð4:7aÞ

E

ð4:7bÞ

0 2 þ sk Ak3 ðD1 2 2 D2 Þ }:

It should be emphasized that Eq. (4.6) accounts for both mechanical and electric quantities for a conducting crack, whereas the MSERR proposed by Park and Sun (1995a,b) for an insulating crack corresponds to mechanical quantities only in the closure integral (cf. Section III.C). From Eqs. (4.7a) and (4.7b) it is seen that both GM and GE depend only on s1 22 1 0 1 1 and s1 12 ; since the terms involving D2 2 D2 are related to s22 and s12 by Eq. (4.3) regardless of the electric loading D1 2 itself. Consequently, for a conducting crack, from the energy balance point of view, mechanical loadings always play a dominant role. For comparison with an impermeable crack, Fig. 14(a) – (c) present the numerical results for GM and GE of a permeable crack (Liu and Sheng, 2001; Chen, 2001). It is seen that for a permeable crack GM is always positive and has a value much larger than GE : In other words, unlike an impermeable crack, the permeable crack model results in G < GM : Chen and Lu (2001) analyzed multiple strongly interacting permeable and impermeable cracks; their results for two interacting cracks are plotted in Fig. 20(a) –(c). Again, it is seen that G < GM


167

Fig. 20. (a) Total; (b) mechanical; (c) electrical energy release rate at the tip of crack 2 plotted as a function of its orientation angle relative to crack 1 in PZT-4 (Chen and Lu 2001).

and GE < 0: These results show that traditional fracture criteria such as KI ¼ KIC or G ¼ GIC are directly applicable in piezoelectric fracture, provided that the conducting crack assumption is valid.

168


McMeeking (1999) proposed a mechanism of piezoelectric fracture consistent with the standard features of elasticity and dielectricity. A Griffith-type energy balance is used to account for the influence of electric field and mechanical loading on fracture. Finite element calculations on energy release rates (the Jintegral) are carried out for a PZT-4 compact tension specimen. The poling direction was assumed to be perpendicular to the crack in the specimen (Fig. 21), with two thin electrodes placed on its upper and lower halves, respectively. Elementary considerations show that, for homogeneous materials, the J-integral is path-independent as long as the integration contour begins and ends at the crack-tip and encloses the tip, encompassing the singularity there (Fig. 22). Numerical evaluation of J is performed by using the domain integral technique

Fig. 21. A compact tension specimen subjected to a combined mechanical and electric loading (McMeeking, 1999).

Fig. 22. Crack-tip enclosing contour for J-integral calculation.


169

(Li et al., 1985), after the finite element solution is obtained. In order to avoid mesh dependence, the finite element mesh and domain integral for J are such that values within 3% of exact solutions for J could be obtained for a number of test problems. Crack-tip energy release rate for a piezoelectric compact tension specimen in the presence of an applied electric field and a mechanical load is plotted in Fig. 23 (McMeeking, 1999), which is normalized by the crack-tip ERR at the same mechanical load but without the applied electric field. The results in Fig. 23 are shown as a function of the applied electric field, with each curve representing a different level of mechanical load F. It is seen that an applied electric field, either positive or negative, reduces the ERR, although at a fixed level of the mechanical load, a positive electric field has a somewhat bigger effect than a negative field (Fig. 23). It is also clear from Fig. 23 that a negative ERR can be induced with a large electric field applied in conjunction with a moderate mechanical load. A negative ERR means (as mentioned in previous sections) that, when the crack propagates, energy is required instead of it being made available by the process (McMeeking, 1999). Numerical results of the crack-tip ERR are also plotted in Fig. 24 for a compact tension specimen made of materials having the same material parameter ratios as PZT-4 (McMeeking, 1999). The results, shown in this way, indicate that

Fig. 23. Crack-tip ERR for a piezoelectric compact tension specimen in the presence of an applied electric field and a mechanical load normalized by the ERR at the same mechanical load but without the applied electric field for selected different levels of mechanical load (McMeeking, 1999).

170


Fig. 24. Crack-tip ERR for a piezoelectric compact tension specimen in the presence of an applied electric field and a mechanical load normalized by the ERR at the same mechanical load but without the applied electric field for selected ratios of electric field to mechanical load (McMeeking, 1999).

at a given ratio of electric field to mechanical load, the electric field reduces the ERR by a bigger fraction when the mechanical load is high. The trend shown in Fig. 24 is consistent with the perception that a large crack opening due to a high mechanical load causes the greatest perturbation of the applied electric field and therefore leads to the most significant fractional change to the ERR. McMeeking (1999) found this significant, because there is a separate curve for each value of the applied load in Fig. 24, due to the non-linearity of the boundary condition on crack surfaces. The non-linear boundary condition can be written as 210 fðxÞ þ dðxÞDn ðxÞ ¼ 0; where dðxÞ is the crack opening displacement. One difficulty nevertheless remains, as experiments show (Tobin and Pak, 1993; Park and Sun, 1995a,b) that a positive electric field transverse to the crack encourages crack growth whereas a negative field has the opposite effect. McMeeking (1999) noticed that the assessment from his results leaves out the possibility that the applied electric field affects the fracture toughness as well as the crack-tip ERR. From this point of view, the experimental phenomena may be explained by the decreasing material fracture toughness when an electric field transverse to the crack is applied. On a phenomenological basis, such a model only requires a fracture toughness that is a function of the mode-mixity KE =KI present at the crack-tip. Here, KE is the electric field intensity factor which can be


171

Fig. 25. Crack-tip mode-mixity for a piezoelectric compact tension specimen in the presence of an applied electric field and a mechanical load for selected ratios of electric field to mechanical load; mode-mixity is defined as the electric field intensity factor (EFIF) divided by the Mode-I stress intensity factor (McMeeking, 1999).

readily calculated once KI and Ke (the EDIF) are determined. Numerical results of KE/KI obtained by McMeeking (1999) are plotted in Fig. 25. Such an approach has gained credence in the study of mode-mixity effect on purely mechanically driven fracture, where it has been well established that the effective fracture toughness depends strongly on the mode-mixity KII/KI (Hutchinson and Suo, 1992). To assess the possible dependence of fracture toughness on mode-mixity KE =KI in the case of PZT-4, McMeeking (1999) reconsidered the data reported by Park and Sun (1995a,b). However, McMeeking (1999) found that his numerical calculations (Fig. 25) did not correspond exactly to the experiment measurements of Park and Sun (1995a,b). The difference lies in the fact that the cracks in Park and Sun’s test specimens were cut by using a 0.46 mm thick diamond wheel, except for the last 1 mm of the crack which was cut with a razor blade. Consequently, the cracks in their experiments are not cleavage cracks of zero gap width (assumed in McMeeking’s numerical analysis), but rather like notches with finite width. Since the crack gap width is known (from numerical calculations) to have a major influence on the crack-tip ERR, the effect of a finite gap width on experimental results must be assessed. McMeeking (1999) concluded that in the case of small-scale process zones where mechanical

172


and electric non-linearities are confined to the immediate vicinity of the crack-tip, a conventional fracture mechanics characterized by the total ERR is viable. That is, the fracture toughness of a piezoelectric in terms of the total ERR can be viewed as a function of the mode-mixity KE =KI : The experimental and numerical results of Heyer et al. (1998) for a conducting crack appear to confirm the modemixity concept. However, like the insulating crack approximation, the conducting crack model also has its limitations. Specifically, it fails to explain the experimentally observed crack growth under a purely electric field (Tobin and Pak, 1993; Cao and Evans, 1994; Heyer et al., 1998). Again, more advanced models incorporating non-linearity have been proposed. Heyer et al. (1998) used the increase in the SIFs and electric field intensity factor as defined in Suo (1993) to qualitatively describe the domain switching effect in the process zone for a permeable crack. Their results are confirmed by their own experimental data obtained with four-point bending specimens, although the cracks are parallel rather than perpendicular to the poling direction. The poled piezoelectric ceramic used by Heyer et al. (1998) is PZT-PIC 151 (PI-Ceramic GmbH, Lederhose, Germany), with typical grain sizes of 5 –10 mm and material constants as listed in Appendix A. Rajapakse and Zeng (2001) used the strip saturation model (Gao and Barnett, 1996; Gao et al., 1997a,b) and the domain switching effect to study a conducting crack. Similarly, as in the case of an insulating crack, controversial results regarding the fracture criterion also exist for a conducting crack. Whilst some researchers reported that the SIF is always reduced by the presence of an electric loading, either positive or negative, others found that a positive electric field (but not the negative field) decreases the SIF. Using the J-integral as the fracture criterion, Suo et al. (1991, 1992, 1993) found that an electric field applied perpendicular to crack surfaces produces a positive driving force for a conducting crack and a negative driving force for an insulating crack. Whereas the latter statement is in agreement with Pak (1992a,b), Park and Sun (1995b) and Zhang et al. (1998), the former appears to be in contradiction with Eq. (4.7). McMeeking (1989) Zhang and Tong (1996) and Zhang et al. (1998) found that the permeable crack condition may underestimate the effect of the electric field on crack growth, whereas the impermeable model may overestimate the effect. The elliptical crack model as used by a number of researchers (e.g., Xu and Rajapakse, 2001) only accounts for the effect of initial crack opening on crack growth. For a slit crack (as the limiting case of an elliptical crack) the electric boundary may be very sensitive to crack opening caused by the applied loading, since the dielectric constant of PZT is much higher than that of dry air or vacuum.


173

D. Summary For a conducting crack in an infinite plane piezoelectric ceramic subjected to remote mechanical-electric loadings, the following conclusions can be summarized: (1) There exists a constantly distributed electric displacement D02 on crack surfaces, induced by both the applied electric and mechanical fields. However, the difference between the far electric displacement and D02, 0 i.e., D1 2 2 D2 ; is only dependent upon the remote mechanical loading. (2) Both analytical and experimental results reveal that the SIFs can be used as a fracture criterion to predict the stability and growth of a conducting crack. (3) The hoop stress at the tip of a conducting crack is independent of remote electric loading. As a result, crack kinking is governed by mechanical parameters only. (4) The electric intensity factor is induced from the applied mechanical loading rather than the electric loading. In the absence of electric loading, i.e., D1 2 ¼ 0; the positive piezoelectric effect of the material induces the distributed D02 : (5) The ERR or J-integral of a conducting crack plays a role similar to that in LEFM. In contrast to the case of an insulating crack, the mechanical part of the ERR always has an overwhelming dominance over its electric counterpart. However, the role the electric field plays for a conducting crack exhibits a feature opposite to that for an insulating crack. For example, the experimental results of Wang and Singh (1994, 1997) with specimens in dry air show that a positive electric loading impedes crack growth and a negative electric loading enhances crack growth, whereas the experimental results of Park and Sun (1995a) with specimens immersed in silicon oil for insulating cracks show exactly the opposite trends. (6) For experiments carried out with sharp cracks (slit cracks) in dry air or vacuum, the conducting crack approximation appears to be valid, because the dielectric permittivity of a typical piezoelectric ceramic is several orders of magnitude larger than the permittivity of dry air or vacuum. However, when a crack specimen is immersed in an insulating oil (Park and Sun, 1995a; Sun and Park, 1995), the crack surfaces should no longer be taken as conducting. This provides a good explanation for the apparently contradicting experimental results of Park and Sun (1995a) and Wang and Singh (1997) or Heyer et al. (1998).


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(7) Like the insulating crack approximation, the conducting crack is a linear model and hence cannot be used to explain the experimentally observed crack growth induced by a purely electric loading. Only when certain nonlinear features of the material are taken into account will the electric induced cracking be predicted. For instance, to study the non-linear features associated with the near-tip process zone, McMeeking (1989) made use of the non-linear constitutive relations of dielectrics, and Rajapakse and Zeng (2001) studied the toughening of conducting cracks due to domain switching. (8) Motivated by the well-known mixed mode fracture in LEFM, the concept of mode-mixity (ratio of electric field intensity to stress intensity) may be suitable for describing the fracture toughness of poled piezoelectric ceramics. Although Heyer et al. (1998), McMeeking (1999) and many others reported experimental and numerical results to support this concept, substantial further investigations are needed before a mixed-mode LPFM could be established.

V. Exact Electric Boundary Condition To explain why there exist controversial results in the literature, the socalled exact electric boundary condition has been proposed. With vacuum or air inside a crack, this condition takes into account the connection condition across the interface between vacuum (or air) and piezoelectric ceramic (Hao and Shen, 1994; Sosa and Khutoryansky, 1996; Gao and Fan, 1999a,b; Xu and Rajapakse, 2001; Chen and Lu, 2001). The dielectric permittivity of vacuum denoted by 1v is generally taken to be 8.85 £ 10212 C2/N m2 (Xu and Rajapakse, 2001), about 1000 times smaller than that of commonly used dielectrics (McMeeking, 1989; also see Tobin and Pak, 1993 for PZT-8). With reference to Fig. 1, assume the interior of the crack is filled with either vacuum or dry air. The exact electric boundary is formulated as (Hao and Shen, 1994): 2 2 þ D02 ðuþ ð5:1Þ 2 2 u2 Þ ¼ 1v ðf 2 f Þ: The electric displacement D02 is given by Xu and Rajapakse (2001) as Im

3 X

1 1 0 sk ½Lk1 s1 22 2 Lk2 s12 2 Lk3 ðD2 2 D2 Þ

k¼1

D02 ¼ 21v Im

3 X k¼1

; qk ½Lk1 s1 22

2

Lk2 s1 12

2

Lk3 ðD1 2

2

D02 Þ

ð5:2Þ


175

where sk and qk (k ¼ 1,2,3) are material constants. If 1v ¼ 0; Eq. (5.1) reduces to the insulating crack model (3.1) with charge-free surfaces, whereas if there 2 is no gap between the crack surfaces, i.e., ðuþ 2 2 u2 Þ ¼ 0; Eq. (5.1) reduces to the conducting crack model (4.1). The exact electric boundary condition (5.1) therefore represents a more realistic and general model to study cracks and fracture in piezoelectric materials. Equation (5.2) is a second-order algebraic equation for the unknown D02. Although it has two different roots in general, only one is realistic, leading to a positive mechanical part of the ERR (Xu and Rajapakse, 2001). For a plane crack perpendicular to the poling direction of PZT-4 (b ¼ 0, Fig. 1), Xu and Rajapakse (2001) used the exact boundary condition and found that D02 is always negative. For example, under a constant mechanical loading s1 22 ¼ 1 MPa; they obtained 24 that D02 ¼ 2 5.173 £ 1025 for D1 ¼ 2:0 £ 10 ; D02 ¼ 2 2.460 £ 1024 for 2 24 0 1 1 D2 ¼ 0; and D2 ¼ 2 4.403 £ 10 for D2 ¼ 22:0 £ 1024 : Here, the value of D02 under a purely mechanical loading ðD1 2 ¼ 0Þ represents a typical mechanicalelectric transformation (positive piezoelectric effect versus converse piezoelectric effect, i.e., electric-mechanical transformation). Similar results are obtained for PZT-5H. Indeed, Eq. (5.2) shows that, in contrast to a vanishing D02 for impermeable cracks, the use of exact electric boundary condition would result in non-zero crack surface electric displacements for conducting cracks, because both far field mechanical and electric loadings may contribute to D02. With the exact electric boundary condition (5.1), the crack-tip field intensity factors (SIFs and EDIF) are still given by Eq. (4.4), except that D02 is now determined according to Eq. (5.2). Consequently, like the insulating or conducting crack model, the exact electric boundary condition cannot explain the electric-field-induce crack formation and growth, since the SIFs are independent of the far electric field. Figure 26(a) – (c) show the EDIF as a function of remote electric field when the crack is oriented at 08, 308, and 908 relative to the poling direction, corresponding to the case of a perpendicular, inclined, and parallel crack, respectively (Xu and Rajapakse, 2001). It is seen from Fig. 26(a) – (c) that the results for a permeable crack and an exact electric boundary crack coincide well with each other for the cases of b ¼ 08 and 308, whereas the results for an impermeable crack agrees well with those for an exact electric boundary crack in the case of b ¼ 908. Similar results are shown in Fig. 27(a) – (c), where the trends exhibited by the mechanical part of the ERR are plotted as a function of applied electric field (Xu and Rajapakse, 2001). From these results it may be concluded that the exact electric boundary crack reduces to a permeable crack when the poling direction and the far electric field are not parallel to the crack, and to an impermeable crack if the poling direction as well as the far electric field are parallel to the crack. The first

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Fig. 26. EDIF plotted as a function of applied electric field for impermeable, permeable, and exact electric boundary cracks, with poling direction taken as b ¼ 08, 308, 908, respectively, and s1 22 ¼ 0:6 MPa (Xu and Rajapakse, 2001).

conclusion appears to be realistic since several researchers (Hao and Shen, 1994; Dunn, 1994; Sosa and Khutoryansky, 1996; McMeeking, 1989; Gao and Fan, 1999; Xu and Rajapakse, 2001) have already pointed out that the exact electric boundary condition can always be reduced to a permeable crack when the applied electric field is high and the crack is perpendicular to the applied electric field. The second conclusion is nevertheless in sharp contrast with the experimental results of Heyer et al. (1998) who used a permeable crack model and aligned the crack parallel to both the poling direction and the applied electric field. Furthermore, when b ! 908, a sharp change occurs by switching from permeable to impermeable electric boundary condition. At present, no other experimental or analytical results support such conclusions. The only possibility is that the three electric boundary conditions


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Fig. 27. Mechanical part of ERR plotted as a function of applied electric field for impermeable, permeable, and exact electric boundary cracks, with poling direction taken as b ¼ 08, 308, 458, 608, 908, respectively, and s1 22 ¼ 0:6 MPa (Xu and Rajapakse, 2001).

studied so far—impermeable, permeable and exact—are perhaps all too idealistic. Several non-linear models have therefore been proposed, taking into account either the non-linear process zone near the crack-tip or crack surface contact. For example, based on experimental evidence (Cao and Evans, 1994), Kumar and Singh (1998) and Ru (1999a,b) emphasized that the crack faces in contact with each other will significantly affect the fracture behavior of piezoelectric materials. This topic will be discussed in detail below. VI. Non-linear Electric Boundary Condition To remove the notable discrepancy between predictions from linear piezoelectric models and experiment measurements, a great deal of theoretical

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and experimental studies have been devoted to insulating cracks as well as conducting cracks, and yet controversial results are often reported. Doubts are subsequently raised about the range of validity of linear constitutive relations (2.8) or (2.9a) and (2.9b), and it has been suggested that the confusion may be clarified by introducing non-linear features, especially in the near-tip process zone. On the other hand, because the energy integral for a crack in a linear piezoelectric is proportional to the square of the applied electric field, the energy release rate (ERR) is independent of the field direction. Recall that similar results exist for elastic cracks in plane and anti-plane deformations, where a Mode-I crack under compressive stressing does not grow even if its driving force (ERR) is positive and increases with increasing compressive loading. Here, because the crack surfaces are in contact, the traction-free model, based on which the energy integral is calculated, is no longer valid. Consequently, in addition to the electric boundary condition with the traditional traction-free model (3.2), the effect of mechanical boundary condition on crack growth in piezoelectric materials is also important (Kumar and Singh, 1998). Furthermore, as in conventional ceramics (Hutchinson, 1987; Ortiz, 1987), microcracking in the process zone near the tip of a macrocrack should also be considered since a number of micropores and/or microcracks have been observed experimentally (see, e.g., Fig. 11(a) and (b)). All these topics, i.e., the near-tip Dugdale model, the contact crack model, and the microcrack shielding model for piezoelectric ceramics will be reviewed below.

A. Strip Electric Saturation Model (Generalized Dugdale Model) Gao et al. (1997a,b) attempted to close the gap between theory and experiments by considering the effects of electric non-linearity near a cracktip. Ferroelectric crystals are known to exhibit strong electrical non-linearities at large electric field strength levels (Jona and Shirane, 1993). The electrical polarization of these crystals is accomplished by ionic movement, which is expected to become more sluggish as the electric field strength is increased. Some of the non-linear features have already been modeled by Yang and Suo (1994), Lynch et al. (1995), Hao et al. (1996), Park and Shrout (1997), Shin and Lee (2000), and Popa and Calderon-Moreno (2001). Under small-scale yielding conditions, both mechanical and electric fields exhibit crack-like singular behaviors at a global length scale. The ERR at this scale can be calculated based on a linear piezoelectric crack without electrical yielding. However, the concept of the MSERR proposed by Park and Sun (1995a,b) has been questioned by these authors. It is argued that, on physical grounds, all mechanical forces are of


179

electromagnetic origin and energy transfer between electric and mechanical fields is the essence of electromechanical devices, and hence there is no fundamental reason to separate a physical process into an electric part and a mechanical part. Electrical yielding occurs before mechanical yielding if the solid is electrically more ductile. At a length scale smaller than the electrical non-linear zone but larger than the mechanical non-linear zone, the mechanical part of the near-tip field is still singular near the crack-tip while the electric part varies smoothly due to yielding. It is at this length scale that the analysis of the so-called local ERRs for an electrically yielding crack can be performed. Fracture is in general a multi-scale physical process involving length scales spanning many orders of magnitude, as illustrated schematically in Fig. 28 (Gao et al., 1997a,b). Gao (1993, 1996) has explored this multi-scale viewpoint of fracture when studying dynamic fracture, arguing that both crack velocity and limiting wave speed are related to length scales. Similarly, for an electromechanical fracture process, the crack driving force may be either small or large at the global length scale, but it is the local energy release rate, which is directly related to the energy available for crack growth. The length scale selected for such analysis needs only to be sufficiently small to capture the essential features of significance. For fracture under combined electromechanical loadings, this length scale appears to be that associated with electrical yielding (Gao et al., 1997a,b). The selection of a proper length scale may be of general importance for modeling fracture under coupled fields (mechanical, electrical, magnetical, etc.),

Fig. 28. Schematic of multi-scale singularity fields in piezoelectric fracture (Gao et al., 1997a).

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but it does not imply that the selection of electric boundary conditions on crack surfaces is no longer important. This is because both near-tip microstructures and electrical continuity across the crack (even though not near the crack-tip) affect the fracture process significantly. For example, Chen and Han (1999a,b) studied near-tip microcracking in piezoelectric ceramics, and found that the role of applied electric loading is affected by near-tip microstructures. On one hand, for certain microstructures, a positive electrical loading promotes crack growth whereas a negative electrical loading impedes crack growth. On the other hand, the opposite is found to hold for different microstructures. They added that the influence of near-tip microstructures on electric quantities is much larger than that on mechanical quantities. This suggests that the contradicting results of Park and Sun (1995a,b) in comparison with those of Wang and Singh (1994, 1997) may be attributed to different experimental conditions, manifesting as either different electric boundary conditions or different microstructures near the cracktip. Consequently, unlike conventional ceramics (or piezoelectric ceramics) subjected to a purely mechanical loading, the Vickers indentation, uniaxial tension, and bending tests under combined electromechanical loadings may lead to different results. Systematic further research should therefore be directed to address the effects of both microstructures and electric boundary conditions. Gao et al. (1997a,b) used an impermeable crack as a first step to study the effect of electrical yielding on the ERR. The key step was to propose a strip saturation model in which electrical polarization reaches a saturation limit along a line segment in front of a finite crack in an infinite piezoelectric medium. It is argued that plastic yielding is difficult to nucleate in brittle piezoelectric ceramics such as lead zirconate titanate and barium titanate. Consequently, the crack-tip electric saturation zone in these materials is much larger than the plastic yielding zone and hence plays a more pivotal role in near-tip non-linearity. Under such conditions, the effect of plastic yielding can be neglected. The strip saturation model is built upon the classical model of Dugdale (1960) for plastic yielding near a crack-tip in a thin metal sheet. Such an extension is nevertheless not straightforward, because electromechanical coupling problems are much more complicated than plastic yielding. A large number of material parameters need to be taken into account, requiring often the use of complex matrices for numerical evaluations. For convenience, Gao et al. (1997) assume that the crack-tip field is electrically yielded but mechanically singular. For poled ferroelectric ceramics, there are typically five independent elastic constants, three independent piezoelectric constants and two independent dielectric constants. With focus placed on a qualitative understanding of the interaction between electric and mechanical loadings, Gao et al. (1997) solved


181

the strip saturation model by using a simplified electroelasticity formulation with drastically reduced number of material constants and degrees of freedom. In this formulation, only three independent material constants, representing separately elastic, piezoelectric and dielectric material properties, and two degrees of freedom associated with elastic displacement and electric potential are considered (see, e.g., Fig. 1): 0

s11

0

1

1

B C B B C B Bp B s22 C B C B B C B B C B Bp B s33 C B C B C ¼ MB B B C B B0 B s23 C B C B B C B B0 Bs C B B 13 C @ A @

s12

0

p

p

0 0

1

p

0 0

p

1

0 0

0

0

1 0

0

0

0 1

0

0

0 0

0

10

s11

0

1

0

CB B C CB B C B0 0 CB s22 C CB B C CB B C CB B C 0 CB s33 C B0 CB B C CB C 2 eB CB B C 0 CB 2s23 C B0 CB B C CB B C C B1 C B 0 CB 2s13 C B A@ @ A p

0

2s12

21

0

1

C C 21 C0 1 C C E1 C 1 CB C CB C CB E2 C; C@ A 0 C C E C 3 0 C C A

0 0 1 0 0

0 ð6:1aÞ

1

0

0

D1

1

0

0

B B C B B C B D 2 C ¼ eB 0 @ @ A D3

0

0

0 1

0

0

1 0

1

0 0

21 21

s11 C B C B C B s 22 0 1B C C 1 0 B C B B CB s33 C B C CB 0 CB C þ 1B 0 @ C AB B 2s23 C C B 0 0 B C B 2s C B 13 C A @

0

0

10

E1

1

1

CB C CB C 0 CB E2 C; A@ A

0

1

E3

2s12 ð6:1bÞ where the symbol p is used to denote those elements eliminated by Gao et al. (1997a,b). The following global ERR (or apparent ERR under small-scale yielding) is obtained: J¼

pa 2 2 ½s 2 ðM1 þ e2 ÞE1 2M 1

ðfor cracks perpendicular to poling axisÞ; ð6:2aÞ

J¼

pa 2 s ^ 1 4M

ðfor cracks parallel to poling axisÞ;

ð6:2bÞ

^ ¼ M þ e2 =1; 1^ ¼ e1; and 1^ ¼ 21=1 whilst the real constants M, e and where M 1 are coefficients defined in Eq. (6.1).


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After introducing an electrically yielded segment in front of an insulated crack-tip, Gao et al. (1997a,b) obtained the local ERR, as ! pa e2 JL ¼ 1þ ðeE1 þ s1 Þ2 ðfor cracks perpendicular to poling axisÞ: 2M M1 ð6:3Þ Obviously, the local ERR, JL, differs from the global ERR, J, in a fundamental way. Using a fracture criterion based on the global ERR (6.2a), one would predict that an electric field inhibits crack growth, which is not consistent with experimental results (Tobin and Pak, 1993; Cao and Evans, 1994; Park and Sun, 1995a,b). On the other hand, use of the local ERR (6.3) would lead to the prediction that fracture is promoted by a positively electric field and inhibited by a negatively electric field. This is in broad agreement with existing experimental observations. For cracks parallel to the poling axis, Gao et al. (1997a,b) concluded that there is no need to distinguish between the local and global ERRs because J ¼ JL : This result suggests that the electric field has no influence on the ERR, again in agreement with experimental observations for cracks parallel to the poling axis. The following comments appear to be in place: (1) Gao et al. (1997a,b) proposed a fracture criterion based on the ERR concept, i.e., the apparent ERR reaching a critical value Jcr: J¼

pa 2 ½s 2 ðMe þ e2 ÞE2 ¼ Jcr ; 2M f

from which the failure stress sf is obtained as qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi sf ¼ s 2f0 þ ðM1 þ e2 ÞE2 ;

ð6:4aÞ

ð6:4bÞ

where sf0 denotes the critical fracture stress in the absence of any electric field. Note that sf depends on the magnitude of the applied electric field but not on its sign. Numerical calculations by Park and Sun (1995a,b) and Kumar and Singh (1996) for practical material properties and crack geometries show similar results. However, at present there is no experimental data to support this criterion. (2) The Park-Sun fracture criterion based on the MSERR leads to JM ¼

pa 2 ½s þ es1 E1 ¼ Jcr ; 2M 1

qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi sf ¼ s 2f0 þ e2 E2 =4 2 eE=2:

ð6:5aÞ

ð6:5bÞ


183

Experimentally, Park and Sun (1995a,b) found that the failure load is insensitive to changes in the electric field, i.e., the first term in the square root of Eq. (6.5b) dominates over the second term, yielding

sf ø sf0 2 eE=2:

ð6:6Þ

Park and Sun (1995a,b) found that this criterion agrees approximately with the measured critical load of cracks perpendicular to the poling direction for both simple tension and three-point bending PZT-4 specimens. (3) Using a fracture criterion based on the local ERR of an electrically yielded crack, Gao et al. (1997a,b) obtained " # pa e2 JL ¼ 1þ ðsf þ eEÞ2 ¼ Jcr ; ð6:7aÞ 2M M1 or

sf ¼ sf0 2 eE:

ð6:7bÞ

In general, this criterion agrees well with experimental observations. In Vickers indentation tests under a fixed indenting force, cracks perpendicular to the poling direction are observed to grow longer than those in the case without an electric field. By comparing Eq. (6.7b) with Eq. (6.6), it is interesting to note that the local ERR criterion differs from the MSERR criterion only by a factor 2 in the slope of the sf 2 E curve. Piezoelectric ceramics are mechanically brittle and it is difficult to measure experimentally the sf 2 E slope to within a factor of 2 accuracy. Therefore, the local ERR concept (Gao et al., 1997a,b) appears to support the empirical MSERR concept (Park and Sun, 1995a,b), although, as Gao et al. (1997a,b) emphasized, fracture is a multiple scale process and there is no fundamental reason to separate a physical process into electric and mechanical parts. Of great interest is the comparison between local and global ERRs under small-scale yielding in the absence of mechanical loading. From Eqs. (6.2a) and (6.3) for cracks perpendicular to the poling axis, it follows that pa 2 J¼2 ð1 þ e2 =MÞE1 ðapparent ERRÞ; ð6:8aÞ 2 ! pae2 e2 1þ E2 JL ¼ ðlocal ERRÞ; ð6:8bÞ 2M M1 1 As expected, the apparent ERR is negative and hence represents an energy absorbance process whereas the local ERR is positive, representing an energy release process. For cracks parallel to the poling axis, in the absence of mechanical loading, both the local and apparent ERRs vanish.

184


Equation (6.8a) is obviously not consistent with existing experimental observations. Cao and Evans (1994) and Lynch et al. (1995) have demonstrated experimentally that cracks perpendicular to the poling axis can grow stably under a cyclic electric field applied in the poling direction without any mechanical loading. Under the same conditions (without mechanical loading) they found that cracks parallel to the poling axis show no significant growth, which is in agreement with the prediction of Eq. (6.2b). Although Lynch et al. (1995) proposed mechanisms different from those of Gao et al. (1997a,b), it is noticed that the local ERR does provide a simple fatigue fracture criterion consistent with experiments. For simplicity Gao et al. (1997a,b) eliminated a number of elements in the constitutive equation (6.1), which have the same order of magnitude as the remaining constants M, e, and 1. Fulton and Gao (1997, 1998, 1999, 2001a,b) extended the work of Gao et al. (1997a,b) to treat the effect of local polarization switching on piezoelectric fracture and proposed a model based on microstructural features, taking account of all non-zero constants in the constitutive equation. The polarization switching and saturation typical of ferroelectrics is simulated by a collection of discrete electric dipoles superimposed on a medium satisfying the linear piezoelectric constitutive law. This model leads to a local crack driving force consistent with experimental observations. Fulton and Gao (2001a,b) found that the new fracture criterion can be applied to cracks at any orientation under arbitrary loading conditions, and it requires no additional empirical characterization of the material. Following the work of Gao et al. (1997a,b), Wang (2000) presented a complete solution by using the complex potential technique in which all elements in the constitutive equation are taken into account. All these studies demonstrate that the non-linear fracture mechanics of piezoelectric ceramics is only associated with electric yielding, which is different from conventional understanding in non-linear fracture mechanics of structural materials where mechanical yielding leads to different crack-tip singularities, e.g., the HRR singularity. A few attempts have recently been made to explain the experimental results by accounting for the permeability of air that fills open cracks or flaws. However, Fulton and Gao (2001a) show that the presence of air does not alter the character of their analytical predictions, and the approximation of a perfectly insulating medium is valid except in the case of an almost perfectly sharp crack. To further study the non-linear behavior such as domain switching, let us briefly review crack shielding problems (Hutchinson, 1987; Ortiz, 1987). It has been well established that dislocations (or microcracks) near a crack-tip can provide significant shielding on the local ERR from the remote applied ERR (global ERR). From the mathematical analogy between a dislocation and an


185

electric dipole (Barnett and Lothe, 1975), it is known that the local ERR will differ from the apparent ERR if there are electric dipoles near the crack-tip. Chen and Han (1999a,b) presented numerical results on near-tip microcracking in PZT-4 ceramics subjected to either a purely mechanical loading or a combined electromechanical loading. They found that the remote J-integral is divided into two distinct parts: one is the local ERR and the other is contributed by near-tip microcracks. We will discuss this topic in more detail in the forthcoming section. The near-tip dipoles play a similar role as that of near-tip microcracks, even if the electromechanical coupling for the former is significantly different from the latter. The piezoelectric effect of ferroelectric ceramics stems from the ensemble behavior of microscopic electric dipoles under an applied electric field. For convenience, assume that an insulated crack in PZT-4 is subjected to a critical Mode-I stress intensity Kcr, with no voltage applied. Based on McMeeking’s (1999) analysis of Park and Sun’s (1995b) experiments, a reasonable estimate of pffiffiffi this parameter is Kcr ¼ 1 MPa m: The size of the fracture process zone is then approximately given by 1 Kcr 2 M ry ¼ ; ð6:9Þ 2 smax where smax is the limit of the linear response of a cracked PZT-4. In the absence of plastic deformation by dislocation motion, this limit can be interpreted as the cohesive strength, about 1/30 of the elastic modulus. Fulton and Gao (1997, 1998, 2001a,b) found that r M y < 40 nm: Furthermore, based on a saturation limit of 1 MV/m, they estimated that the length scale of electrical non-linearity is r Ey < 90 mm; which is much larger than ryM : This finding prompted Fulton and Gao (2001a) to focus on electric saturation yielding in the near-tip process zone, with the effect of plastic yielding neglected. Zhang and Jiang (1995) attempted to clarify the situation from a different angle. They emphasized that experimental investigations suggest microcracking may be the main cause for commonly encountered degradation of ferroelectric actuators, namely electric fatigue. It is widely accepted that stress concentration is responsible for the onset of cracking. The formation of ferroelectric twins, i.e., 908 domains, causes severe stress concentration at intersections of domain walls with grain boundaries, due to the incompatibility of lattice distortions with grain boundary constraints. Zhang and Jiang (1995) analyzed the asymptotic behavior of stress and electric fields near these intersections. The asymptotic analysis is carried out within the framework of electrostatics for deformable continua assuming that the electromechanical state in each of the ferroelectric domains is slightly distorted from one of the natural states of the crystal. Furthermore, they

186


developed a piecewise-linear model and found that the concentrations of stress and electric fields exhibit a r 2l power-law singularity, with r representing the radial coordinate originating at the intersecting point and l denoting a constant in the range [0, 1]. The severity of the field concentration depends upon crystal orientation and the orientations of domain walls with respect to the grain boundary with which they intersect. Considering that microcracks may nucleate and grow during cooling or poling, Zhang and Jiang (1995) also studied the interaction of ferroelectirc twins with pre-existing microcracks, both intergranular and transgranular. It has been established that this interaction makes the singularity substantially stronger than the conventional r 21/2 singularity and hence promotes crack growth.

B. Contact Model Following the experimental findings of Cao and Evens (1994), Kumar and Singh (1997c, 1998) studied the influence of applied electric fields and mechanical boundary conditions on crack-tip stress distributions in piezoelectric materials. Using the finite element method they found that, under a negative electric field, the assumption of traction-free crack surfaces as the mechanical boundary condition is invalid since crack closure may occur. Ru (1999a) studied the electric-field-induce crack closure in linear piezoelectric media, and presented general conditions for crack closure in terms of remote loading parameters, which delimit the applicability of the conventional traction-free model for both insulating and conducting cracks. After making detailed manipulations and obtaining exact solutions for a closed crack in a plane piezoelectric ceramic, Ru (1999a) concluded that, in a poled ferroelectric, crack closure occurs for a conducting crack parallel to the poling axis when subjected to an electric field in the poling direction, and for an insulating crack perpendicular to the poling axis when subjected to an electric field opposite to the poling direction. A crack closure model that assumes continuous normal traction/displacement and vanishing tangential traction across the closed crack is then proposed. Of great significance is that the numerical results of Ru (1999a) for a closed crack predict a non-zero mode-I SIF induced from electric-field loading. This result is in sharp contrast with the conventional traction-free crack model, which predicts that an electric field cannot produce any SIF at cracks in linear piezoelectric materials. Ru’s work therefore sheds further light on the physical understanding regarding piezoelectric cracking induced by a purely electric loading. It is noticed that Cheng et al. (2000) performed an investigation on PZT ceramics subjected to compression– compression loading, and also found that crack closure occurs.


187

C. Microstructure Near the Tip of an Impermeable Crack Gao et al. (1997) emphasized that non-linear modeling of a piezoelectric crack is important. On one hand, it is common to measure fracture strength of brittle ceramics under three-point bending without a pre-existing crack in the sample (Han and Wang, 1999; Han and Chen, 1999). In this case, fracture may initiate from more than one type of microflaws. Conducting species inside a microcrack may cause a fundamental change in its behavior and microslits may be electrically more permeable than a macrocrack crack-like flaw (McMeeking, 1999, 2001). On the other hand, as shown in Figs. 4 and 11(a) and (b), microcracks or other defects are often found before an indentation test is conducted (Tobin and Pak, 1993; Sun and Park, 1995; Wang and Singh, 1994, 1997; Schneider and Heyer, 1999). Unlike microcrack shielding problems in conventional ceramics such as Al2O3 and SiC studied by Ortiz (1987) and Hutchinson (1987), where stress-induced microcracking near the tip of a macrocrack was considered, piezoelectric ceramics always contain processinginduced microdefects such as pores and cracks whose size, location and orientation are seldom dependent on the near-tip stress field of a macrocrack. The presence of these macrostructures near the tip of an impermeable macrocrack may significantly disturb both near-tip stress and electric fields, which in turn will affect the role of the applied electric field (positive or negative). Chen and Han (1999a,b) studied the interacting between parallel microcracks and an impermeable macrocrack. The poling direction was taken as perpendicular to both the macrocrack and microcracks. With two elementary solutions and by using the pseudo-traction-electric displacement method (PTED), this interacting problem was reduced to a system of Fredholm integrals, which were then solved numerically with the Chebyshev integration scheme. The ERR or J-integral analysis was used by Chen and Han (1999a,b) only as a consistency check rather than a fracture criterion. Based on the MSERR criterion of Park and Sun (1995a, b), they found that the location and orientation of microcracks significantly affect the role of applied electric field on macrocrack growth. The main results of Chen and Han (1999a,b) are shown in Fig. 29(a) – (c) where numerical results for the M1 are plotted as functions of applied normalized MSERR denoted by GMt I =GI electric loading. Here, the location angle f of a single microcrack takes the value of 108, 808 and 1608, respectively, whereas the normalized distance dn ¼ ðd 2 aÞ=a is taken to be 0.3, a and d being separately the half-length of the microcrack and the distance between the macrocrack tip and the microcrack center. In Fig. 29(a) –(c), the solid lines refer to the case where no microcrack is present and the dashed lines refer to the case of a single microcrack interacting

188


Fig. 29. Mechanical strain energy release rate (MSERR) plotted as a function of remote electric field as influenced by near-tip microcracking: (a) f ¼ 108; (b) f ¼ 808; (c) f ¼ 1608.

with a macrocrack. It is seen that, although the MSERR is proportional to the applied electric field Ke1 =KI1 as found by Park and Sun (1995a,b), the effect of Ke1 on the macrocrack tip MSERR is dependent upon the location angle f of the microcrack. At a fixed value of Ke1 =KI1 ; the MSERR at the macrocrack tip is


189

increased by the existence of a microcrack if its location angle f is relatively small (say, w ¼ 108, Fig. 29(a)) and decreased if f is relatively large (Fig. 29(b) and (c)). This result presents an evidence that, besides the domain switching in the near-tip process zone, microcracking provides another source for the localenergy release rate (Fulton and Gao, 2001a,b). Consequently, based on the results of Chen and Han (1999a,b), it may be concluded that microcracks in the near-tip process zone not only reduce the effective elastic moduli and release the residual stresses (the two sources of shielding effect studied by Hutchinson, 1987), but also disturb the near-tip electric field. The latter provides another source of shielding. Finally, it is worth noting that in the recent work of McMeeking (2001) the remanent polarization vector P r and the remanent strain srkl of polarization were added into the constitutive equations of piezoelectric ceramics. This was motivated by the fact that theoretical treatments, such as an ad hoc neglect of the electric contribution to the ERR (Park and Sun, 1995a,b) or the apparent dissipation of energy by the saturation of the electric polarization (Gao et al., 1997; Fulton and Gao, 1997), are hard to justify. In addition, charge separation in the material and discharges within the crack during fracture have not been considered but would seem to be relevant phenomena. McMeeking (2001) introduced a more complicated constitutive equation instead of Eq. (2.8), as

sij ¼ Cijkl ðskl 2 srkl Þ 2 ekij Ek ; Di ¼ eijk ðsjk 2 srjk Þ þ 1ij Ej þ Pri ;

ð6:10Þ

which connects the electric displacement, stress, electric field and strain (Landau and Lifshitz, 1960; Jaffe et al., 1971; Eringen and Maugin, 1990). McMeeking (2001) explored some new aspects of a crack-tip fracture mechanics by combining the effects of mechanical and electrical loading with strain and polarization behavior for dielectrics with a special emphasis on piezoelectrics. It is found that it is important to determine the current crack configuration so that an accurate estimate can be made for the ERR due to electric effects. In previous sections, we have shown that under certain conditions the applied electric field in the absence of stress loading has no effect on crack propagation. McMeeking’s study (2001) reveals that discharge or charge separation plays an important role in determining how the electric field influences crack growth such that if substantial discharge/charge separation occurs, the prime influence of the electric field is eliminated. A strip specimen subjected to fixed grip conditions with uniform electrode charge density was analyzed by McMeeking (2001) as a special case to illustrate general points. It is asserted that in the case of small-scale process zones in which mechanical and electrical non-linearities are confined very close to the crack-tip, a conventional

190


fracture mechanics controlled by the TERR should work with a parameter used to account, as in mode-mixity, for the degree of crack-tip polarization intensity.

VII. Piezoelectric Bimaterial Systems As a natural extension from homogeneous piezoelectric materials, interface cracks between dissimilar piezoelectric ceramics have received increasing attention (Herbert, 1982; Harrison et al., 1986; Zhang and Jiang, 1995; Chung and Ting, 1996; Hao et al., 1996; Meguid and Zhong, 1997; Deng and Meguid, 1998; Ding et al., 1996, 1997, 1999; Qu and Li, 1991; Qin and Mai, 1998, 1999, 2000; Qin and Yu, 1997; Qin and Zhang, 2000; Ru et al., 1998; Ru, 2000; Pan and Yuan, 2000; Tian and Chen, 2000; Ma and Chen, 2001b; Narita and Shindo, 1998a,b, 1999a,b; Liu et al., 1999, 2001; Gao and Wang, 2000, 2001; Gao et al., 2001; Kwon and Lee, 2000; Shen et al., 2000a,b; Li et al., 2000; Wang, 2000a,b; Xu and Rajapakse, 2000a,b; Herrmann et al., 2001; Zhou et al., 2001; Wang and Noda, 2001). Research in this area is motivated by strong engineering demands to design new composite materials with mechanical-electric coupling properties such as polymer/piezoelectric composites, BaTiO3/PZT composites, metal/PZT bimaterials, and dissimilar PZT materials. Haus and Newnham (1986) performed an experimental and theoretical study of PZT/polymer composites for hydrophone applications. Shaulov et al. (1989) and Chan and Unsworth (1989) studied PZT/polymer composites. Sevostianov et al. (2001) examined the effective mechanical and piezoelectric properties of piezocomposites including both polymer/PZT composites and BaTiO3/PZT. Their results constitute a theoretical framework for the design of piezocomposites with prescribed overall properties. Most theoretical studies on interface cracks in dissimilar piezoelectric materials built upon previous analyses of interface cracks in convenient dissimilar isotropic or anisotropic materials. Kuo and Barnett (1991) studied the stress singularities of interface cracks in bonded piezoelectric half-spaces. Qin and Yu (1997) studied an arbitrary-oriented plane crack terminating at the interface between dissimilar piezoelectric materials. Liu et al. (1999) studied a screw dislocation interacting with a piezoelectric bimaterial interface. Deng and Meguid (1998) studied a conducting rigid inclusion at the interface of two dissimilar piezoelectric materials. Kwon and Lee (2000) studied a moving interfacial crack between piezoelectric ceramic and elastic layers. Ma and Chen (2001b) presented a Williams eigenfunction expansion form for a semi-infinite interface crack in piezoelectric bimaterials and introduced the Bueckner


191

conjugate integrals and related weight functions. Both oscillating singularity model and contact model near the tip of an interface crack have been studied. For example, Herrmann et al. (2000, 2001) introduced a near-tip contact model for interface cracks in piezoelectric bimaterials. Besides these, Tian and Chen (2000) studied the interaction between a microcrack and an interface macrocrack in metal/piezoelectric bimaterials. Both insulating and conducting crack models have been developed for piezoelectric bimaterial systems. Mathematically, an interface crack in dissimilar piezoelectric materials exhibit oscillatory singularity at the crack-tip, like its counterpart in dissimilar elastic anisotropic materials (Hutchinson and Suo, 1992; Beom and Atluri, 1995; Ma and Chen, 2001e; Liu, 2003). Ma and Chen (2001b) established the eigen-function expansion form (EEF) for a semi-infinite interface crack by using the Stroh theorem (1958, 1962) and the compact formulations (Suo et al., 1992). The crack configuration considered is shown in Fig. 30, where the upper and lower half-planes are occupied by two dissimilar piezoelectric materials labeled by #1 and #2, respectively. From Eqs. (2.17) – (2.31) and the charge- and traction-free conditions on the crack surfaces, they found that the Williams type EEF is given by 1 1 X X 1 i1 k n 21 f 01 ðzÞ ¼ pffiffiffiffiffi B21 ðI þ i b ÞYðz ; z Þ a z þ B ðI þ a Þ ibn z n ; ð7:1aÞ n 1 1 2 2pz n¼0 n¼0

f 02 ðzÞ ¼

f 1 ðzÞ ¼

1 1 X X 1 i1 k pffiffiffiffiffi B21 an z n þ B21 ibn z n ; ð7:1bÞ 2 ðI 2 ibÞYðz ; z Þ 2 ðI 2 aÞ 2 2pz n¼0 n¼0 1 1 X X 1 z nþ1 pffiffiffiffiffi B21 ; Xðzi1 ; zk ; nÞan z nþ1 þ B21 ibn 1 ðI þ ibÞ 1 ðI þ aÞ nþ1 2 2pz n¼0 n¼0

ð7:2aÞ 1 1 X X 1 z nþ1 f 2 ðzÞ ¼ pffiffiffiffiffi B21 : Xðzi1 ; zk ; nÞan z nþ1 þ B21 ibn 2 ðI 2 ibÞ 2 ðI 2 aÞ nþ1 2 2pz n¼0 n¼0

ð7:2bÞ Here, the subscripts 1 and 2 refer to the upper and lower half-planes, respectively, an and bn are complex coefficients vectors, while a and b are the generalized Dundurs parameters defined by (Dundurs and Markenscoff, 1993; Beom and Atlury, 1995, 1996):

a ¼ ðL1 2 L2 ÞðL1 þ L2 Þ21 ;

ð7:3aÞ

21 b ¼ ðL21 1 þ L2 ÞðM1 2 M2 Þ;

ð7:3bÞ


192

Fig. 30. An interface crack between two dissimilar piezoelectric ceramics (Ma and Chen, 2001b).

where L1, L2, M1, and M2 are known matrices defined by Eqs. (2.28) and (2.31) in the upper and lower piezoelectric materials, respectively. The two real parameters 1 and k in Eqs. (7.1a), (7.1b), (7.2a) and (7.2b) are separately the oscillatory and singularity indices, given by 1 tanh21 h; p 1 k ¼ tan21 v; p

1¼

ð7:4aÞ ð7:4bÞ

where

h¼

v¼

(

$

(

$

)1=2 1=2 2 1 2 14 trðb Þ ; 4 trðb Þ 2kbk 1 4

2

%2

2

%2

trðb Þ 2kbk

1=2

ð7:5aÞ

)1=2 2

þ 14 trðb Þ

:

ð7:5bÞ

The matrix function Yðzi1 ; zk Þ in Eqs. (7.1a), (7.1b), and (7.2a), (7.2b) can be written explicitly as Yðzi1 ; zk Þ ¼ D1 zi1 þ D2 z2i1 þ D3 zk þ D4 z2k ; where

! 1 1 iv2 i 3 2 2 b2 b ; v I2b þ D1 ¼ 2 h2 þ v2 h h ! 1 1 iv2 i 3 2 2 bþ b ; v I2b 2 D2 ¼ 2 h2 þ v2 h h

ð7:6Þ

ð7:7aÞ ð7:7bÞ


193

! 1 1 h2 1 3 2 2 b2 b ; D3 ¼ h Iþb 2 2 h2 þ v2 v v

ð7:7cÞ

! 1 1 h2 1 3 2 2 D4 ¼ bþ b : h Iþb þ 2 h2 þ v2 v v

ð7:7dÞ

The singular stress-electric field along the bounded interface near the crack-tip is then given by 1 k T2 ðx1 Þ ¼ pffiffiffiffiffiffiffi Yðxi1 1 ; x1 Þgðx1 Þ: 2px1

ð7:8Þ

The vector denoting the field intensity factors is given by (Beom and Atluri, 1995) pffiffiffiffiffi K ¼ limþ 2pxYðx12i1 ; x2k 1 ÞT2 ðxÞ;

ð7:9Þ

x!0

where K ¼ ½K1 ; K2 ; K3 ; Ke T : Since Yðx12i1 ; x2k 1 Þ and T2 ðx1 Þ are real, K is also real. The intensity factor vector K may be considered as an extension of the elastic version proposed by Wu (1990) and Qu and Li (1991), and provides a unique characterization of the crack-tip state for piezoelectric bimaterials. For 2D problems, the three field intensity factors: the SIFs, K1 and K2, and the EDIF, Ke, are inherently coupled at the tip of an interface crack. This feature may be used to describe microcracking induced by a purely electric loading at the matrix-particle interface. On the other hand, the coefficient vector b0 in Eq. (7.1) represents stresses acting parallel to the crack surface (i.e., s11 and D1), which are referred to as the generalized T-stress following Rice (1988). The stresses s11 and D1 are uniform but different in each of the two materials. Following the procedures outlined in Ting (1986), Deng and Meguid (1998) and Ma and Chen (2001b,e) obtained the following orthogonal property relations valid in both material #1 and material #2: 8 T k ¼ I Tk B k þ B Tk A > Ak Bk þ BTk Ak ¼ A > > > > < Bk ATk þ B Tk ¼ Ak BTk þ A k B Tk ¼ I kA > A AT þ A Tk ¼ Bk BTk þ B k B kA Tk ¼ 0 > k k > > > : T k ¼ A Tk Bk þ B Tk Ak ¼ 0 Ak Bk þ BTk A

ðk ¼ 1; 2Þ:

ð7:10Þ


194

Using the Bueckner-integral (1973), Ma and Chen (2001b) obtained the pseudo orthogonal properties of the EEF (7.1a), (7.1b) or (7.2a), (7.2b), as ( ) 2 ð X bT a B-integral ¼ 2Re ½f k df k k¼1

Gk

8 1 0 2 a ImðW1 Þan > > < 4 m ¼ 2pb0m W2 bn > > : 0

nþmþ1¼0

ð7:11Þ

m þ n þ 1 ¼ 21 ; other cases

where W1 ¼

DT1 ða^1 þ b^1 ÞD2 DT2 ða1 þ b1 ÞD1 DT3 ða^1 þ b^1 ÞD4 þ þ m þ 12 þ i1 m þ 12 2 i1 m þ 12 þ k þ

W2 ¼

DT4 ða1 þ b1 ÞD3 ; m þ 12 2 k

ð7:12aÞ

21 T 2T 21 ðI þ aÞT ImðB2T 1 B1 ÞðI þ aÞ þ ðI 2 aÞ ImðB2 B2 ÞðI 2 aÞ ; ð7:12bÞ mþ1 21 b^1 ¼ ðI þ ibÞT B2T 1 B1 ðI þ ibÞ;

ð7:12cÞ

21 a^1 ¼ ðI 2 ibÞT B2T 1 B1 ðI 2 ibÞ:

ð7:12dÞ

Equation (7.11) is very effective at finding new weight functions, from which the crack-tip dominant parameters, i.e., the elements of vector K in Eq. (7.9) can be calculated distinctively, without any special treatment of the near-tip singular region. Following Comninou (1977, 1978), Herrmann and Loboda (2000) and Herrmann et al. (2001) considered various contact zone models near the tip of a permeable interface crack in piezoelectric bimaterials. By introducing an artificial frictionless contact zone and by assuming electrically permeable crack surfaces, they reduced the problem to a combined Dirichlet-Riemann boundary value problem and obtained an exact analytical solution. The stresses, electrical displacements and derivatives of the displacement and electric potential jumps along the material interface are all given in a clear analytical form. The SIFs and ERRs at the singular points are subsequently found, from which the electrical intensity factor is determined. The relationships between the fracture parameters of various interface crack models are obtained. Particularly, the contact zone length denoted by l0, the corresponding SIF denoted by k02 as well as the SIFs of the classical


195

oscillating model denoted by K1, K2 are expressed in terms of the SIFs of an artificial contact zone model denoted by k1, k2 as follows K1 ¼

K2 ¼

1 _ pffiffi _ _ pffiffi _ ½ðt 1 ak1 2 mt 2 k2 Þcosðb þ cÞ þ ðt 2 ak1 2 mt 3 k2 Þsinðb þ cÞ; 11 ð7:13aÞ 1 _ pffiffi _ _ pffiffi _ ½ðt ak1 2 mt 3 k2 Þcosðb þ cÞ 2 ðt 1 ak1 2 mt 2 k2 Þsinðb þ cÞ; m11 1 ð7:13bÞ _

_

_

where a, b, c, t 1 ; t 2 ; t 3 ; m, 11 are real coefficients defined in Herrmann and Loboda (2000), Herrmann et al. (2000) and Herrmann et al. (2001). Figure 31(a) – (c) show the functional dependence of l0 on the remote tensile to shear stress ratio t=s: Here, the labels I and II refer to two different material combinations: CTS-19 (upper half-plane)/PZT-4 (lower half-plane) and Cadmium Sulfide (upper half-plane)/Barium Sodium Niobate (lower half-plane). The corresponding material constants are listed in Appendix B2. Figure 31(a) – (c) illuminate the variation of l0 (in a logarithmic scale) for relatively small values of t=s and for moderate values of t=s; respectively. When the load ratio is relatively small (Fig. 31(a) and (b)), the contact zone length l0 is very small as in conventional bimaterials (Comninou, 1977, 1978; Rice, 1988). For example, when t=s ¼ 290; l0 ¼ 1=e5 ¼ 0:0067: For large values of t=s (Fig. 31(c)), l0 gradually increases to a limiting value of 0.30525 for the CTS-19/PZT4 bimaterial and 0.30537 for the Cadmium Sulfide/Barium Sodium Niobate bimaterial. These results are very similar to those derived by Comninou (1978) in classical interface crack problems under a shear field. Although interface cracks in dissimilar elastic materials have been studied extensively, much less is known of interface cracks in dissimilar piezoelectric materials, and experimental data is very limited. More systematic studies are needed for a better understanding of interfacial fracture under combined mechanical-electric loadings.

VIII. Other Developments A. Three-Dimensional Cracks Penny-shaped cracks in 3D infinite piezoelectric materials have been analyzed by many researchers (Du and Wang, 1992; Du et al., 1994, 1995; Wang, 1992a,b;

196


Fig. 31. Contact zone length plotted as a function of load ratio t=s for (a) 0 # 2t=s # 9; (b) 10 # 2t=s # 90; (c) 100 # 2t=s # 9000 (Herrmann and Loboda, 2000).

Wang, 1994; Wang and Huang, 1995a,b; Wang and Zheng, 1995; Park, 1994; Chopra, 1995; Kogan et al., 1996; Ding et al., 1996, 1997, 1999, 2000a,b; Zhao et al., 1997a,b; Akamatsu and Tanuma, 1997; Chen and Shioya, 1999, 2000; Jiang et al., 1999; Colla et al., 1999; Pan and Yuan, 2000; Pan and Tonon, 2000; Jiang and Sun, 2001), and both axis-symmetrical and non-symmetrical loading


197

conditions have been considered. Although existing experimental data on electric induced crack formation and growth are mainly associated with 3D surface cracks, e.g., Vickers indentation cracking (Yamamoto et al., 1983; Tobin and Pak, 1993; Cao and Evans, 1994; White et al., 1994; Sun and Park, 1995; Lynch, 1996; Ramamurty et al., 1999; Xu et al., 2000; Tan et al., 2000; Jiang and Sun, 2001) or 3D interface cracks, the study of 3D cracks in piezoelectric ceramics has so far mainly focused either on mathematical solutions or on experimental data (Tobin and Pak, 1993). Direct comparisons between theoretical predictions and experimental measurements for 3D cracks are scarce, with notable exceptions of Sridhar et al. (1999) and Jiang and Sun (2001) who studied indentation cracking in piezoelectric ceramics. Both insulating and conducting crack models have been extended to treat 3D piezoelectric cracks. However, as in the case of 2D cracks, contradicting results about the role of applied electric field as well as fracture criterion selection remain to be clarified or verified. As previously mentioned, mathematically, 3D crack problems in LPFM pose no special difficulty even though mechanical and electric quantities are coupled. This is mainly due to the inherent correspondence between elasticity and piezoelectricity (Milton and Movchan, 1995; Chen, 1993a,b, 1995; Chen and Lai, 1997). For example, the integral transformations (Sneddon and Lowengrub, 1968) provide a powerful tool for this purpose in a way similar to that in LEFM. Du and Wang (1992) and Wang (1992a,b) are probably the first to study 3D defects in piezoelectrics. Wang and associates (1994, 1995a – d) subsequently solved a penny-shaped insulating crack and presented a general solution for 3D piezoelectric cracks. Based on the 3D complex potential theory of Fabrikant (1989, 1991), Fabrikant et al. (1994), and Ding et al. (1996, 1997, 1999, 2000a,b) focused on 3D conducting cracks. Hill (1997) proposed new 3D piezoelectric boundary elements in his PhD thesis. Shindo et al. (1997b) studied the bending of a symmetric piezothermoelastic laminated plate containing a through crack by using the finite element method. Whilst a few general-purpose, commercially available finite element codes such as ANSYS (1995, 1996) and ABAQUS (1995) may be effective in providing approximate solutions for such problems, care must be taken to ensure numerical accuracy as this is not always attainable even with a fine mesh. For example, Kumar and Singh (1996b) found that sometimes ANSYS (1995) does not generate any useful results and gives error messages if the applied electric field is relatively high, whereas the use of ABAQUS (1995) has no such problems. On the other hand, Heyer et al. (1998) used a more recent version of ANSYS (1996) to calculate a four-point bending fracture specimen and have not encountered any numerical difficulty. Physically, as in the case of a 2D crack, the main obstacle is associated with how to treat the electric and mechanical boundary conditions for a 3D crack.

198


Different selections of such boundary conditions have lead to different predictions. For example, a number of researchers found that, for a pennyshaped conducting crack, the presence of an electric field (either positive or negative) always impedes crack growth, because the crack opening always decreases with increasing electric loading. This result is not consistent with the experimental results of Park and Sun (1995a,b) and Wang and Singh (1994, 1996) for 2D cracks. As the growth of Vickers indentation cracks in piezoelectric ceramics is significantly affected by the application of an electric field (Tobin and Pak, 1993; Cao and Evans, 1994), Giannakopoulos and Suresh (1999) developed a theory for axisymmetric indentation in an attempt to explain the experimental results of Ramamurty et al. (1999). Many researchers (Okazaki, 1984; Tobin and Pak, 1993; Cao and Evans, 1994; Sun and Park 1995; Schneider and Heyer, 1999; analyzed the elasto-electric field due to a rigid conical punch on a transversely isotropic piezoelectric half-space. Jiang and Sun (2001) recently presented a 3D analysis of indentation cracking in piezoelectrics. They noted the wedge effect caused by inelastic deformation on the role of the applied electric field. In order to model the wedge effect as a free surface, an approximate analytic solution for a half penny-shaped crack in a piezoelectric half-space is obtained by modifying the solution for a full penny-shaped crack. The solution in conjunction with the MSERR is used to quantitatively account for the effect of electric field on crack growth under varying indentation forces. Following Sun and Park (1995) and Park and Sun (1995a,b), the electric boundary condition used by Jiang and Sun (2001) was impermeable, e.g., by immersing the specimen in silicon oil rather than in dry air or vacuum. For convenience the crack induced by Vickers indentation (Fig. 32(a)) was modeled as a semicircular surface crack in a half-space as shown in Fig. 32(b). Thus, the only difference between this problem and the penny-shaped crack problem studied by Wang (1994) is the additional traction-free boundary condition on the free surface, i.e., the x– y plane in Fig. 32(b). Furthermore, Jiang and Sun (2001) assumed that the wedge is piezoelectric and the inelastic region can be represented by a one-dimensional piezoelectric rod element. Upon the application of a positive electric field, the representative piezoelectric rod would elongate and in turn the wedge force develops. In contrast, if a negative field is applied to the rod, 1808 domain switching would take place since the voltage on the crack surface is very high and the length of the rod is very small. Since it is difficult to predict when and how a complete domain switching would occur in the wedge, Jiang and Sun (2001) introduced a simple empirical model, subjected to the provisos that: (i) domain switching in the wedge region


199

Fig. 32. (a) Vickers indentation induced surface crack (Sun and Park, 1995; Jiang and Sun, 2001); (b) typical indentation-induced crack beneath the surface (Sun and Park, 1995).

occurs when a negative electric field is applied, which is based on the observation that the electric field across the crack surface exceeds the coercive field; (ii) the piezoelectric constant e33 in Eq. (2.8) or in Appendix A1 decreases after domain switching since the 1808 domain switching may not be fully completed (Lynch, 1996; Schaufele and Hardtle, 1996). Using the MSERR criterion, Jiang and Sun (2001) calculated the critical crack length under two mechanical loadings P ¼ 9.8 and 49 N. Their numerical results plotted in Fig. 33(a) and (b) are in general consistent with the experimental data of Sun and Park (1995). It is concluded that, through coupling with the tensile residual stress at the crack front, a positive electric field would produce a higher driving force GM I (the MSERR, Park and Sun, 1995a,b) for crack growth (Fig. 33(a)). In contrast, when a negative electric field is applied, due to the wedging effect, tensile stresses at the crack front increase and the crack could still grow (Fig. 33(b)), though to a much less extent in comparison with Fig. 33(a). Here, due to the existence of inelastic deformation induced from a larger indenter force of P ¼ 49 N, the negative field

200


Fig. 33. Crack length plotted as a function of electric field strength for (a) P ¼ 9:8 N; (b) P ¼ 49 N (Jiang and Sun, 2001).

no longer impedes crack growth. Consequently, different selections of the Vickers indentation force P can significantly influence the effect of applied electric field on crack growth. More research on 3D cracks is obviously warranted, provided that all the remaining controversial results in 2D problems are clarified. For example, substantial 3D finite element simulations are needed to study Vickers indentation cracking under either positive or negative electric field, so as to obtain more information about the electric-field-induce crack closure. For this purpose,


201

3D weight functions need to be developed for a surface crack in linear piezoelectric materials, in a way similar to those established by Rice (1989) and Bueckner (1989) for LEFM. Further investigation is also needed to quantify the inelastic effect as well as wedge effect on piezoelectric crack growth in Vickers tests.

B. Dynamic Fracture As in the case of LEFM, static fracture and dynamic fracture in piezoelectric ceramics have been studied individually although, physically, both research directions are based on the energy balance concepts. There has been a number of investigations in the literature, which aim to provide analytical solutions of different crack configurations in an infinite piezoelectric medium subjected to dynamic loading (see, e.g., Furuta and Uchino, 1993; Norris, 1994; Shindo and Ozawa, 1990; Shindo et al., 1996a,b; Khutoryansky and Sosa, 1995; Li and Mataga, 1996a,b; Wang et al., 1998; Yu and Qin, 1996a,b; Yu and Chen, 1998; Chen and Karihaloo, 1999; Sosa and Khutoryansky, 1999, 2001; Daros and Antes, 2000). Mathematically, dynamic crack problems in infinite piezoelectric media can always be solved with existing tools such as the Wiener-Hopf technique and integral translation techniques (Sneddon and Lowengrub, 1968). However, it is very difficult to summarize the main conclusions derived so far on dynamic piezoelectric fracture, simply because statistic piezoelectric fracture is yet to be fully understood. The study on dynamic crack propagation in piezoelectric media will be more fruitful only after similar problems in static fracture have been completely solved.

IX. Concluding Remarks Over the last two decades, at least five different ways have been proposed to establish a possible fracture criterion for piezoelectric ceramics: (i) total energy release rate or the J-integral (Pak, 1990a,b; McMeeking, 1989); (ii) mechanical strain energy release rate (Park and Sun, 1995a,b) or the mechanical part of the J-integral (Chen and Lu, 2001); (iii) local and global energy release rates due to electrical yielding only (Gao, 1996; Gao and Barnett, 1996; Gao et al., 1997a, b); (iv) dependence of fracture toughness on mode-mixity (Heyer et al., 1998; McMeeking, 1999); (v) maximum hoop stress (Kumar and Singh, 1997a –c).

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However, at present no universal fracture criterion exists which can be used to fully explain piezoelectric fracture under arbitrary electromechanical loadings. On one hand, the fracture toughness of a poled piezoelectric ceramic exhibits an apparent anisotropy: under identical mechanical and electric loading conditions, a crack parallel to the poling direction propagates much less than a crack perpendicular to the poling direction does. No such anisotropy has been found for unpoled piezoelectric ceramics. On the other hand, there exist contradicting views on how an applied electric field may affect the growth of a crack perpendicular to the poling direction. Supported by a number of experimental results and analytical predictions, the overwhelming view is that a positive electric field promotes crack propagation whereas a negative electric field impedes crack propagation. However, Wang and Singh (1994, 1997) have found exactly the opposite trends in their experiments, i.e., a positive electric field impedes crack propagation and a negative electric field enhances crack propagation. We believe that the above discrepancy may be caused by different electricmechanical boundary conditions on crack surfaces selected by different investigators. There exist at least five different boundary conditions: (i) an impermeable crack with traction-free surfaces; (ii) a permeable crack with traction-free surfaces; (iii) a crack with exact electric boundary and traction-free surfaces; (iv) an impermeable crack with near-tip microstructure; (v) an impermeable or permeable crack with contacting surfaces. Again, similar to the selection of piezoelectric fracture criteria, there is no universal crack surface condition that can be applied to describe all the experimental phenomena. A slit crack or very sharp crack can be modeled as a permeable crack with traction-free surfaces (McMeeking, 1989, 1990, 1999), and its fracturing behavior can be characterized by using the SIFs as a fracture criterion. The impermeable crack model is valid when cracked specimens are immersed in silicon oil (Park and Sun, 1995a; Sun and Park, 1995; Gao et al., 1997a,b), for which the MSERR (Park and Sun, 1995a,b) or the local ERR (Gao et al., 1997; Fulton and Gao, 1997) can be used as fracture criterion. The contradicting experimental data and analytical results reported in the literature require a better physical understanding of piezoelectric fracture in order to apply proper (or new) boundary condition and fracture criterion for each specific case. Three-dimensional finite element calculations with different selections of mechanical-electric boundary conditions along crack surfaces are needed to study cracks induced by Vickers indentation, and hence to clarify the contradicting results reported by Tobin and Pak (1993) and Wang and Singh (1994, 1997). Naturally, several powerful techniques developed for conventional brittle materials such as the weight function method (Rice, 1989; Bueckner,


203

1989) need to be extended to treat mechanical-electric coupling problems in poled piezoelectric ceramics when using finite element codes such as ANSYS and ABAQUS . These techniques may be used to determine dominant fracture parameters, and to check the validity of various fracture criteria.

Acknowledgements The authors would like to thank the National Science Foundation of China through Grant no. 10072046 and the Royal Society K.C. Wong Award for partially supporting this work. TJL would also like to thank the US Office of Naval Research MURI Program (project no. 340-6206-2) for partial financial support.

Appendix A A1. Material properties of PZT-4 ceramic (Deeg, 1980; Park and Sun, 1995b) c11 ¼ 13:9 £ 1010 c12 ¼ 7:78 £ 1010 ; c13 ¼ 7:43 £ 1010 ; c33 ¼ 11:3 £ 1010 ; c44 ¼ 2:56 £ 1010 (all in unit of N/m2); e31 ¼ 26:98; e33 ¼ 13:84; e15 ¼ 13:44 (all in unit of C/m2); 111 ¼ 6:00 £ 1029 ; 133 ¼ 5:47 £ 1029 (all in unit of C/Vm) Reduced material constants of PZT-4 (Sosa, 1992): a11 ¼ 8:205 £ 1022 ; a12 ¼ 23:144 £ 10212 ; a22 ¼ 7:495 £ 10212 ; a33 ¼ 19:3 £ 10212 ; ðall in unit of m2 N 21 Þ; b21 ¼ 216:62 £ 1023 ; b22 ¼ 23:96 £ 1023 ; b13 ¼ 39:4 £ 1023 (all in unit of m2 C21); d11 ¼ 7:66 £ 107 ; d22 ¼ 9:82 £ 107 VN 21 (all in unit of VN21).

A2. Material properties of PZT-5H ceramic (Deeg, 1980; Pak, 1992a,b) c11 ¼ 12:6 £ 1010 N/m2 ; c12 ¼ 5:5 £ 1010 ; c13 ¼ 5:3 £ 1010 ; c33 ¼ 11:7 £ 1010 ; c44 ¼ 3:53 £ 1010 ; e31 ¼ 26:5 C=m2 ; e33 ¼ 23:3; e15 ¼ 17:0; 111 ¼ 151 £ 10210 C=Vm; 133 ¼ 130 £ 10210 C=Vm:


204

A3. ELECTROMECHANICAL PROPERTIES OF POLED PZT-8 (TOBIN AND PAK, 1993)

Density Curie temperature Relative dielectric constants, 1T33 =10 Piezoelectric charge constants, d33 d31 d15 Open circuit elastic constants, cD 11 cD 33

7600 kg/m3 300 8C 1000 225 £ 10212 m/V 297 £ 10212 m/V 330 £ 10212 m/V 9.9 £ 1010 N/m2 11.8 £ 1010 N/m2

A4. Material properties of PZT-PIC 151 (Heyer, et al., 1998) C11 ¼ 1:1 £ 1011 Pa; C12 ¼ 6:3 £ 1010 Pa; C13 ¼ 6:4 £ 1010 Pa; C33 ¼ 1:0 £ 1011 Pa; C44 ¼ 2:0 £ 1010 Pa; C66 ¼ 2:2 £ 1010 Pa; 111 ¼ 111010 ; 133 ¼ 85210 ; 10 ¼ 8:85 £ 10212 C=Vm; e31 ¼ 9:6 C=m2 ; e33 ¼ 215:1 C=m2 ; e15 ¼ 212 C=m2 :

Appendix B B1. MATERIAL CONSTANTS OF SOFT PZT EC-65 (WANG AND SINGH, 1997)

Young’s modulus (GPa) Compliance C33 ( £ 10212 m2/N) Curie temperature Mechanical Q factor Dielectric constant at 1 kHz Coupling factor Piezoelectric constants ( £ 10212 m/V) d33 d31 d15 Piezoelectric constants ( £ 1023 Vm/N) g33 g31 g15

66 18.3 350 8C 100 1725 0.72 380 2173 584 25.0 211.5 38.2


205

B2. MATERIAL CONSTANTS FOR CTS-19, CADMIUM SULFIDE, AND BARIUM SODIUM NIOBATE (HERRMANN AND LOBODA, 2000)

Piezoelectrics

CTS-19

C11 £ 10210 (N/m2) 210

C33 £ 10

2

(N/m )

Cadmium sulfide

Barium sodium niobate

11.22

9.07

23.9

10.6

9.38

13.5

5.1

5.0

2

6.22

2

111 £ 1010 (C/Vm)

2.49 23.4 9.45 15.1 72.57

1.5 20.24 20.21 0.44 0.77

6.5 20.4 2.8 4.3 18.99

133 £ 1010 (C/Vm)

82.74

0.81

2.74

210

C13 £ 10

(N/m )

210

C44 £ 10 (N/m ) e31 (C/m2) e15 (C/m2) e33 (C/m2)

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Environmental Mechanics Research in China LI JIACHUN, LIU QINGQUAN, and ZHOU JIFU Department of Engineering Sciences, Institute of Mechanics, Chinese Academy of Sciences, Beijing 100080, China

I. Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . II. Environmental Problems in China . . . . . . . . . . A. Water Resources and Hydrology Hazards . . B. Soil Erosion and Desert Invasion . . . . . . . . C. Other Issues . . . . . . . . . . . . . . . . . . . . . . .

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IV. Case Study . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . A. Sediment Transport and Sea Water Intrusion in the Yangtze River Estuary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . B. Runoff Generation and Soil Erosion on the Loess Plateau . . C. Terrestrial Processes in Arid Areas, the Northwest China . .

Abstract By recalling mankind’s path during past 50 years in the present article, we mainly highlight the significance of environmental issues today. In particular, two major factors leading to environment deterioration in China such as water resources and coal burning are stressed on. Present-day environmental issues are obviously interdisciplinary, of multiple scales and multi-composition in nature. Therefore, a process-based approach for environment research is absolutely necessary. A series of sub-processes, either physical, chemical or biological, are subsequently analyzed in order to establish reasonable parameterization E-mail address: [email protected] (L. Jiachun). ADVANCES IN APPLIED MECHANICS, VOL. 39 ISSN 0065-2156

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scheme and credible comprehensive model. And we are now in a position to answer questions still open to us, improve existing somewhat empirical engineering approaches and enhance quantitative accuracy in prediction. To illustrate this process-based research approach, three typical examples associated with the Yangtze River Estuary, Loess Plateau and Tenggli Desert environments have been dealt with, respectively. A theoretical model of vertical flow field accounting for runoff and tide interaction has been established to delineate salinity and sediment motion which are responsible for the formation of mouth bar at the outlet and the ecological evolution there. A kinematic wave theory combined with the revised Green – Ampt infiltration formula is applied to the prediction of runoff generation and erosion in three types of erosion region on the Loess Plateau. Three approaches describing water motion in SPAC system in arid areas at different levels have been improved by introducing vegetation sub-models. However, we have found that the formation of a dry sandy layer and biological crust skin are additional primary causes leading to deterioration of water supply and succession of ecological system.

I. Introduction The 20th century has seen the emerging need for environmental fluid mechanics and the need has been increasing with time after the World War II. The grave pollution events such as the stifling smoke in hazy winter of London were astonishingly alarming to rouse attention in the 1950s. As pointed out in the book ‘Silent Spring’ by R. Carson in the 1960s, the revelation of severe pesticide pollution as a primary cause set a milestone for the world to recognize the threat from environment degradation. The United Nations Conference on Human Environment (UNCHE) was convened in 1972 to promote the resolution of these emerging problems. Consequently, the United Nations Environment Program (UNEP) was established in Nairobi, Kenya, followed by signing up of a series of international treaties. Under these circumstances, the representatives of both developed and developing countries gathered at the United Nations Conference on Environment and Development (UNCED) held at Rio de Janeiro, Brazil to explore the compromise between environment protection and economic development in the world. Just as G. H. Brundtlands suggested in the well-known book ‘Our Common Future,’ a new concept of sustainable development has become a consensus. It means that such a kind of developmental mode must meet the demand of the current generation without doing any harm to the coming generations (Fuwa, 1995; Li and Wu, 1998). The environmental problem can perhaps be traced back as early as to the time of sparsely populated and primarily agrarian world long past, but the impacts on the communities were then negligible. With the world population rapidly growing and urban centers expanding, the adverse effects have gradually become vital and urgent (Reible, 1999).

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Nowadays, the environment issues in contemporary society are noted to have the following characteristics and salient features: (1) Comprehensive environmental irregularities occur more frequently. As a result, the so-called environment problems should be investigated and understood in a broader sense. They include worldwide climate anomalies such as the cold summer in 1980, warm winter in 1989, dry weather in 1984, 1986 and 1987. In addition, such natural disasters as abrupt variations in temperature, rainfall, wind force and water level generally have inflicted heavy tolls upon the living world and properties. Toxic industrial discharges and daily life contaminants released to the air, water and soil bodies constitute the primary cause of environmental pollution. Degradation of the ecological environment arises usually by overgrazing and deforestation. Such ecosystem unbalances are known to lead to worsening consequences that are responsible for desertland formation, pasture degeneration and loss of biological diversity. In particular, the aforementioned episodes tend to occur interactively rather as solitary cases. For example, the pollution by phosphorus compounds tends to give rise to overgrowth of algae, thus resulting in breaking out of red tide. The global warming tends to induce flooding of coastal cities owing to the rise of sea surface level (Bigg, 1996; Singh, 1996). (2) The loss in environmental events becomes increasingly more severe. The principal characters of a contemporary society are industrialization and urbanization, the speed of which is much accelerated in today’s developing countries. In case an accident should take place suddenly without warning, the loss of properties and lives could be overwhelming. Such cases now seem to be the rule rather than merely incidental. This is clearly exhibited by the long list of the natural hazards such as flood and drought, hurricane and tornado, storm surge and tsunami, volcano eruption and earthquake, landslide and debris flows, snow avalanche and dust storm, etc. It is estimated in a report issued by the Adversary Committee on International Decade of Natural Disaster Reduction (INDNR) that natural hazards have claimed more than 2.8 million lives, together with adversely affecting 820 million people during 1979 – 1990. On the most severe occasions in Bangladesh and China, one single disaster is already seen to have taken more than one quarter million lives, plus the accompanied property damage evaluated at 25 –100 billion USD. (3) The influences of adverse human activities can be significant. Some of the environmental problems may be due to certain natural causes. Nevertheless, the adverse human activities now have become comparable with or even surpassing the roles of natural factors. The major impact of this category is evidently shown by the total effects of burning of fossil fuels on the local and regional air quality and on its role as a dominant contributor to the green house gases emitted into

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the atmosphere. Aerosols affect climate by scattering and absorbing solar radiation. At the same time, they form condensation nuclei that are critically responsible for cloud formation and precipitation. The natural sources of aerosols are volcano debris released in volcano eruption and the dissolved salt contained in the sea water spray thrown into the air during wave breaking and left to ride in air updrafts. The routes of human activities in altering aerosol number density and size distribution are through industrial emission and biotic marine fluxes of Dimethyl Sulphide (DMS) flux into the air. As is known and reported, the combustion of fossil fuel and the burning associated with deforestation may add 87 Tg sulfur, about 1.5 times of the amount by natural emission, annually into the atmosphere. The daily phenomena of low visibility range, increased cloud cover, and especially the acid rain falling over North America and European continents amply afford the evidence of sulfate aerosol accumulation. The land use and cover change (LUCC) such as from forest to arable land and from savannah to desert in habitable regions is also enormous. The effects of LUCC are not only restricted to altering surface albedo, moisture and thermal characters, but also through affecting global carbon cycling. It is reported that over the last 200 years, the contributions from fossil fuel combustion and land use conversion are approximately comparable. In addition to its central role in affecting global change, many facets of human welfare are closely linked to LUCC, including biological diversity and food production (Bernard, 1981; Jager, 1988; Bigg, 1996). Looking forward to the future at the turn of the century, the world population will most likely continue to grow on this globe already populated with more than 6 billion of people, especially in Africa, India and Southeast Asia. Furthermore, the magnifying requirement of material civilization will certainly result in consumption of more energy and resources. It seems that the pressure of environment degradation will not be alleviated shortly and hence the environmental issue will certainly remain a challenging problem in the 21st century (Fuwu, 1995; Li and Wu, 1998; Reible, 1999).

II. Environmental Problems in China As a developing country with an area of 9.60 million km2 and population of 1.295 billion, China is facing great challenge in environmental issues to maintain the current speed for economic booming. In most western part of China far from east coastal line are mountains and plateaus. The Tibet plateau prevents vapor over the India Ocean from penetrating into inner land. Consequently, water is always a central issue in most provinces in China. Except for natural factors,


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humanity’s activities, especially dramatic explosion of population and inappropriate cultivation of land further deteriorate the situation. In the following, we shall address in detail China’s principal environmental issues, in particular, those associated with water resources:

A. Water Resources and Hydrology Hazards It is estimated that the total amount of water resources available is 2800 billion cubic meters annually, ranking the sixth in the world. However, the water supply per capita is 2162 m3, merely one fourth of the world’s average level. Moreover, they are not evenly distributed either seasonally or geographically. The situation that contaminated water tends to be released without processing further aggravates its emergency. More than tens of big cities, in particular, the metropolises are commonly confronted with shortage of water. Some of the great rivers such as the Yellow River even dries out downstream more frequently in the 1990s, which may last as long as half a year. Excessive use of water without planning in upstream provinces accounts for the cause. Except for management in water use, the government has lately decided to initiate ‘China’s South Water to North Transfer’ project including three routes in the near future in order to mitigate the severe crisis. The middle route project (MRP) with a free flow trunk as long as 1240 km shall annually transfer 14.5 billion m3 of water to highly cultivated and water-deficient Huang-Huai-Hai plain. Similarly, the Tarimu river in the south Xinjiang province formed a green corridor in the desert in the last century. However, the ecological environment downstream have been severely threatened by river shrinkage of 320 km since 1950. Recent transfer of 0.7 billion m3 of water from the Bosten lake into the Tarimu river refills the disappearing Teitema lake, saving patches of dying popular diversifolia. In contrast, the great rivers are not always calm as have been reflected by more frequent catastrophic flooding in the 1990s. The disaster in 1998, comparable to those in 1931 and 1954, was the most severe one, resulting in total property loss of 25 billion USD (Table I). Consecutive torrential rains due to anomalous activity of atmospheric circulation and reclamation of land around lakes in the Jingjiang region were responsible for why high level water failed to retreat during July and August of 1998. Densely populated and well-developed flood diversion areas brought about more difficulties for us to make decision. Besides, land subsidence, debris flow, saltwater intrusion, sediment transport are all among the same category (Singh, 1996).

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Table I Comparison of the Yangtze River Flood in 1954 and 1998 Hankou

1954 1998

Yichang

Flux (m3/s)

Water level (m ASL)

Flux (m3/s)

Water level (m ASL)

70,700 72,300

29.73 29.43

63,600 66,800

55.73 54.50

B. Soil Erosion and Desert Invasion Owing to unfavorable climatic conditions in the Northwest China, arid and semi-arid areas occupy 3.78 million km2 (39% of total area). Desert scatters along the north border of China and forms a belt as long as 4000 km with the width of 600 km. Their total area amounts to 0.71 million km2. The largest one is the Takramagan desert with an area of 0.33 million km2. Anthropogenic activities along with adverse weather conditions accelerate the speed of desert invasion, the annual rate of which turns out to be 2.46 thousand km2. Total area has amounted to 2.62 million km2 (27%) (Table II). Soil erosion carrying away 5 billion ton fertile surface soil annually by wind or water is the major cause leading to soil erosion. Take the Loess Plateau as an example, the eroded area is 0.43 million km2. The most serious situation has an erosion module as large as 30,000 ton/km2. As for downstream, the direct consequences are waterway blockage and water pollution. Furthermore, wind erosion brings about more frequent sandstorms in the north China in the last decade. During recent economy boost in the west China, we should pay prior attention to these problems in order to render this area more beautiful than before. C. Other Issues An additional influential issue in this respect is the coal-dominated energy structure, namely, coal (70%), petroleum (22%), hydropower (5%), natural gas Table II China’s Land Use Indices

Area (million of km2) Proportion (%)

Arid and semi-arid

Deserted land

Desert

Erosion

3.78 39.3

2.62 27.2

0.71 7.39

3.60 37.5


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Table III China’s Major Population and Energy Production Indices

1953 1964 1982 1990 2000

Population (billion)

GDP per capita (USD)

Coal (billion T)

Oil (billion T)

Gas (billion m3)

Electricity (G kWh)

0.62 0.72 1.03 1.16 1.29

– – 196 313 842

0.11 0.20 0.62 1.00 1.00

– – 0.10 0.15 0.16

– – – 1.53 3.00

12 73 379 623 1209

(2%) and others. It is reported that China’s GDP has surpassed 10,000 USD and GDP per capita has been enhanced from 114 USD in 1970 to 842 USD in 2000. We may expect that the living standard will stride forward in foreseeable future. Although the growth rate of population has been slowed down to less than 1%, we still have to make great effort to combat pollution during the period of economy boost. The government has taken measures to accelerate the construction of carbon-free hydro- and nuclear power station. Total capacity of hydropower and nuclear power plants amounts to 72.8 and 2.1 GW, respectively, by the end of 2000. The production of natural gas with lower carbon emission is also doubled from 15.3 to 30 billion m3 during the last 10 years (Table III). Therefore, the pattern of energy consumption is unable to change significantly in a short period though, the implement of energy conservation and efficiency enhancement along with the above-mentioned policy have certainly contributed to reducing global carbon dioxide and local sulfur oxides emissions (Jager, 1988; Zhang, 1994).

III. Process-Based Research Approach Conventionally, the task of an environmental engineer is to copy with the foregoing episodes, aiming at minimizing any potential adverse effects and abating human’s losses of traditional industrial and agricultural activities. Sometimes it is simply understood as finding some solutions to manipulate pollution events. Any efforts in modeling are regarded as useless for ‘being wise after the event’. For a long time, collecting field data became a major job for the researchers in this field to establish an empirical formula. Nevertheless, this mode of investigation has often given way to process-based approaches. Since the former tends to obtain right answer for the wrong reasons rather than right reason, in the mean while, the latter verifying each process component as well as the final output and avoiding parameter optimization is more credible and preferred for

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scientists and engineers in environmental field nowadays. It has been recognized that this kind of research approach, namely process-oriented modeling can fulfill the task of demonstrating the past event, predicting the future evolution, assessing the impact of an action and evaluate the effectiveness of human’s intervention. In particular, the experiments on a large scale or for a long-term are always expensive or even beyond people’s capability, the process-based model is the unique selection (Reible, 1999; Parsons and Abrahams, 1992). Basically, any environmental problem is aroused by fluid-carried mass or energy transport. Even earthquake is indirectly relevant to mantle convection, which drives tectonic motion, magma plume and volcano eruption. However, the flows in nature seem to be considerably different from typical industrial flows with external forcing such as boundary layer, channel flows, pipe flows, mixing layer, wakes and jets. In general, buoyancy due to density difference is dominant over or at least equally important as shear stress in them. On the other hand, how the mass and energy are redistributed by convection-accompanied diffusion is also crucial. Moreover, it is commonly found that many physical, chemical or even biological phenomena are involved in more complex environmental issues. Hence, the analysis of them underlying the environmental problem is a preliminary step for process-oriented research. Therefore, we will describe those processes peculiar to environmental fluid flows in nature at first (Lumley et al., 1996; Li, 2001a).

A. Sub-process Analysis 1. Flows in Nature a. Atmosphere Boundary Layer Atmospheric boundary layer (ABL) driven by outer free atmosphere and sensitively affected by ground indicates the lowest part of troposphere where human beings are inhabitating. Its importance is evidently attributed to the fact that the turbulent flows in ABL play a decisive role in climate variation and air pollution. According to atmospheric stability, ABL is usually classified into three categories: unstable convective boundary layer (CBL), neutral boundary layer and stable boundary layer (SBL). CBL in atmosphere is a multi-layered structure with the height h of about 1– 2 km and consists of surface layer (, 0.1 h), mixing layer (0.1 – 0.8 h) and entrainment layer (0.8 –1.2 h). The surface layer is characteristic of velocity shear and constant momentum flux. The mixed layer exhibits strong mixing feature of its buoyancy-driven turbulence and the highest


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layer is influenced by entrainment effects, capping version and stable atmosphere aloft. In contrast, SBL ranging from 10 to hundreds of meters demonstrates small fluctuation and strong vertical inhomogeneity. In the 1960s, one of the most significant advances is the Monin –Obukhov theory and people endeavored to elucidate the mean behaviors such as fluxes and profiles. The most popular one accounting for stability effects on the average wind speed u is the Businger – Dyer formula verified by observations in Wagnara, Australia: u z u ¼ p ln 2 c M ð3:1Þ z0 k where up is the friction velocity, k the Karman constant, z0 the roughness and cM a function of 6, which is of different form for stable and unstable circumstances and

6¼

z ; L

L¼

q ðt=rÞ3=2 ; kgðH=rCp Þ

ð3:2Þ

in which L denotes the Monin – Obukhov length measuring the relative importance of wind shear and buoyancy. The statistical behaviors and coherent structure of turbulence were obtained by large eddy simulation (Deardorff, 1970; Moeng, 1984), in which large scale eddies are responsible for most of the energy transfer whereas those smaller than grid size can only be modeled. As for CBL, the research was directed to counter gradient transfer, entrainment nearby the tropopause and the role played by convective vortices and the asymmetry in updraught and downdraught flows has been identified. In contrast, the investigation of SBL relatively lags behind because of the difficulties in the interaction between internal waves and turbulence, variable height of SBL and low turbulence intensity measurement (Li, 1993; Garratt and Taylor, 1996). Owing to the damping induced by vegetation, the velocity profile exhibits inflexion nearby canopy and a shear layer is formed there. The mixing layer analogue brought forth lately by Finnigan (2001) has further deepened our understanding of canopy turbulence. The emphasis of the future study should be laid on cloud capped ABL, inhomogeneous underlying surface effects and their applications in global and regional circulation models (Garratt and Taylor, 1996). b. Bottom Boundary Layer The bottom boundary layer means the layer inside which the flow is substantially influenced by the bed and is generated by moving water or a plate back and forth in the laboratory. In coastal engineering, such kind of unsteady boundary layer is formed under water surface wave or tidal flows adjacent to

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the bottom (Nielsen, 1992; Li et al., 1999b). The thickness of this oscillatory boundary layer is estimated as: pffiffiffiffi ds ¼ nT ð3:3Þ where n is the kinetic viscosity, and T a typical period. Generally speaking, a wave-induced boundary can be as thick as a few millimeters, whereas tideinduced boundaries can extend to the whole water body owing to its longer period. Hence, wave-induced high stresses at the seabed could initiate sediment to re-suspend, while tidal flows could carry away suspended sediment. Hence, it is considerably significant to establish the quantitative links between the flow fields and the rates of morphological changes. The velocity profile of a laminar oscillatory boundary layer of circular frequency v in z direction is given by: uðz; tÞ 2 u1 ðtÞ ¼ Av expð2ð1 þ iÞz=ds Þeivt

ð3:4Þ

where A indicates the amplitude. The transition to turbulence is generally dependent on the Reynolds number in terms of the oscillatory boundary layer thickness: rffiffiffiffiffi 2v Re ¼ A ð3:5Þ n and bed roughness. The flow can be classified as laminar, disturbed laminar, intermittent turbulent and full-developed turbulent states. For smooth bed, the flow is found to remain laminar for Re , 100, to become turbulent for Re . 3500, and to appear partly laminar and partly turbulent for Re lying in between (Nino and Garcia, 1996; Li et al., 1999b). If we consider motion adjacent to solid or free surface or around small structure in waves, the viscous wave theory should be applied. Previous in-depth research (Lamb, 1945; Phillips, 1977) has given dissipation effects in waves due to viscosity. If the viscosity is assumed small, the difference between ideal and viscous waves with wavenumber k merely lies in that the amplitude decays exponentially at the rate of: d ¼ 2n k 2

ð3:6Þ

In practical applications, we need to examine the combined wave –current interaction boundary layer and also consider the effects of coastal line and topography. c. Buoyancy-Induced Convection As we know, density variations due to in-homogeneity in temperature and physico-chemical composition in various natural environment can give rise to buoyancy-induced flows, such as CBL, upwelling in the ocean, thermohaline


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circulation, and mantle convection (Turner, 2000). With the characteristic velocity left unknown, the non-dimensional parameter for buoyancy-induced convection is the Grashof number defined as: Gr ¼

agDTL3 n2

ð3:7Þ

where a is the bulk expansivity, g the gravity acceleration, n the kinetic viscosity and L a characteristic length. For environmental flows at low speed with small density difference, we tend to assume the Boussinesq approximation, that is, the fluid is assumed incompressible except for that in the buoyancy term. The thermal plume over a heat source such as chimney or cooling water exhaust thus penetrates upward into stably stratified fluid, entrains ambient fluid and finally descends. The previous work has quantitatively answered the question concerning the rise height, flow inversion and mixing process. More attention should be paid to convection in porous media and with phase change (Lumley et al., 1996).

d. Flows in Rotating System To investigate geophysical flows on this rotating planet with angular velocity V, the effects of the Coriolis force is measured by the ratio of the inertial and Coriolis forces, the so-called Rossby number: Ro ¼

U fL

ð3:8Þ

where f ¼ 2V sin f is the Coriolis parameter at the latitude f, U and L are the characteristic velocity and length, respectively. Since the air in the atmosphere generally moves faster than the water in the ocean, the rotating effects should be accounted for those flows with length scale order of 1000 km (for the atmosphere) or with length scale order of 100 km (for the ocean). Typical examples are geostrophic flows, Taylor’s column, Ekman effect and western strengthening of wind-driven circulation. If we restrict ourselves to a narrow latitude zone, we can adopt the f-plane or the b-plane approximation. A unique feature of rotating systems is their conservation of potential vorticity. Therefore, a Beta gyre appears during the movement of tropical cyclones. Such internal asymmetric structure is responsible for anomalous trajectory (Li and Kwok, 1997). Since the vertical velocity component is much smaller than the horizontal ones, the large-scale motion essentially is quasi-two-dimensional. Without vortex stretching, we have adverse energy cascade and normal entropy cascade. For this reason, geophysical fluid is a natural laboratory of two-dimensional turbulence.

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In addition, people find that the vortices within the critical distance tend to merge (Hopfinger, 1988; Lumley et al., 1996; Turner, 2000).

2. Transport of Mass and Energy In nature, there are varieties of minute particulates of different sizes such as smoke, dust, volcano debris, salt, fog, sediment, pollen and virus, of which those with diameter between 1 and 100 nm constitutes aerosol and suspensions in liquid. For investigating such systems, a non-Newtonian model (for densely concentrated sediment, mud), or a multiphase model, or passive scalar model may be applied. Here, by the so-called passive scalar model, it is meant that the concentration of dispersate is so rare that the motion of the dispersive medium is not affected, that is, there is no coupling between dispersate and dispersive medium. Instead, we shall be rather concerned with the diffusion, settling and flocculation of dispersate in dispersive media.

a. Diffusion In laminar flows, the diffusivity can be derived based on kinetic theory. On the other hand, the Taylor theory for turbulent flows shows that the displacement variance in short duration is proportional to the time square and that in long-term to the time interval. Consequently, the turbulent diffusivity is finally proportional to the velocity variance and Lagrange integral time scale (Brenner and Stone, 2000). As far as suspending liquid is concerned, the migration of minute particles with radius R at temperature T in fluid is caused by the Brownian motion, and the corresponding diffusivity may be written as (where k is the Planck constant): D¼

kT 6pmR

ð3:9Þ

Recent research on passive scalar indicates the differences in statistical behaviors between velocity and passive scalar fields: pdf of passive scalar is no longer Gaussian even for Gaussian velocity field; the passive scalar of grid turbulence exhibits a scale law even at low Reynolds numbers; passive scalar demonstrates more intermittent than the velocity and inhomogeneity appears in an inertial and dissipative range. Therefore, this area requires more exploration (Warhaft, 2000) to clarify for better understanding. In addition, we should emphasize on a key point that the turbulence structure in large scale plays a dominant role in the transport of momentum, mass and energy.


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b. Deposition Dispersate particles deposit in the gravity field and thus become separated from the dispersive medium. For small particles moving in dilute suspension at low Reynolds numbers, the Stokes formula is generally used to calculate the viscous drag and particle settling speed. For spherical droplets or bubbles in gas or water, the drag is smaller, versus their solid counterpart, owing to the motion of the internal fluid brought by the outer fluid, yielding a somewhat larger settling speed. Further, for larger Reynolds numbers and dispersate deformation, the settling speed can be calculated based on the following expression (Levich, 1962): sffiffiffiffiffiffiffiffiffi 2r 0 gV U¼ Cd rS

ð3:10Þ

in which V is the volume of a liquid droplet or gas bubble, S its frontal area, Cd stands for the drag coefficient, r and r 0 are the densities of particles and liquid (or gas), respectively. In addition, surfactants in liquid may further alter the deposition process. The aforementioned process is essential to studies of sediment motion, dry deposition, and wet scavenge in formulating acid rain models.

c. Flocculation It is possible that dispersate particles may move to distances so close that the intermolecular Van der Waals force plays an active role in leading to flocculation. However, the repulsive force due to discharge at the particle surface is a stable factor to prevent them to come too close. Otherwise, flash flocculation would take place instead. We may obtain the temporal evolution of the number density in flocculation due to the Brownian motion as: n¼

n0 ; 1 þ t=T

T ¼ ð8pDRn0 Þ21

ð3:11Þ

where R is the flocculation radius (equal to the diameter of the particle), D the diffusivity by the Brownian motion, T the characteristic time of flocculation. Flocculation is found to occur during the transport of aerosols in atmosphere and of sediment in estuaries. In the presence of prevailing flow fields, flocculation in shear flows is called the gradient flocculation. In this connection, dissolution, emulsification, etc. seem to need further investigations. Absorption, adsorption, and similar mass exchange at fluid interfaces should also deserve attention in environmental engineering.

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3. Other Physical, Chemical and Biological Processes a. Radiation Solar radiation is the energy source for living creatures on the earth to grow and the major driving force of atmospheric and oceanic motion as well. The average flux at the outer edge of atmosphere on average is defined as the solar constant with minor variation: I0 ¼ 1:97 Kar=cm2 min ¼ 1374 w=m2 which is updated based on the recent satellite measurements. Varying with the solar activity and distance between the sun and the earth, the solar radiation actually is represented as: I¼

I0 ðsin f sind þ cosf cosd cos vÞ r2

ð3:12Þ

where r is the relative distance between the sun and the earth, f the latitude, d the declination, and v the temporal angle. Penetrating through the atmosphere, the direct radiation is estimated by transparency or turbidity and cloud quantity S. There are gas molecules, aerosol and water droplets in the atmosphere. The Rayleigh scattering occurs around the gas molecules with the radius smaller than electromagnetic wave length, while Mie scattering occurs around aerosols with diameter equivalent to electromagnetic wavelength. The scattering coefficient dependent on wavelength is inhomogeneous in different directions. In this way, the total radiation on the ground Q turns out to be the sum of the direct radiation S and the scattered radiation q: Q¼Sþq

ð3:13Þ

Depending on the color, moisture and roughness of the underlying surface, the ground reflects the incident waves, namely, albedo a. On the other hand, the long wave reflection from the ground is partly returned back to the ground, namely, the adverse radiation. The difference between them is known as effective radiation E. Based on the radiation balance at the ground, the net radiation R is derived as: R ¼ ð1 2 aÞQ 2 E;

ð3:14Þ

which is responsible for where the sensible, latent and ground fluxes come from (Rosenberg, 1974; Ten Berge, 1990; Dingman, 1994). In water quality models, the solar radiation penetration should also be taken into consideration to understand the growth and metabolism of phytoplankton (Hamilton and Schladov, 1997).


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b. Chemical Reaction Some typical chemical reactions in the environmental sciences and technology that should be concerned with are listed as follows (Fuwa, 1995; Lei et al., 1998; Reible, 1999): Reactions in ozone layer: While ozone is uniformly distributed in the troposphere, it drastically decreases in the stratosphere due to diffusion and dissociation. Ultraviolet radiation depletes ozone through the reactions by chlorine atom from chloroflurocarbons (CFC) released to the atmosphere as follows: Cl þ O3 ! ClO þ O2

ClO þ O ! Cl þ O2

ð3:15Þ

Since atomic chlorine may be continuously generated by the reaction of chlorine monoxide with atomic oxygen in stratosphere, the foregoing chain reaction may indeed be self-maintaining. This explains why single atomic chlorine in the atmosphere may remain for such a long time to break tens of thousand of ozone molecules. The reactions in the stratosphere seem to be very complicated; hydrogen, free radical of hydroxide, methane and oxides of nitrogen take part in the process directly or indirectly. Very probably, the inhomogeneous reaction of gas phase at the surface of sulfate aerosol should be considered to elucidate the formation of ozone hole over the Antarctic continent. Mechanism of acid rain formation: The oxides of sulfur or nitrogen in the atmosphere are the principal elements to acidify rain. The reactions to produce acidified materials are as follows: SO2 þ OH þ M þ O2 þ H2 O ! H2 SO4 þ M þ HO2

ð3:16Þ

where M is a third matter. The chain reaction may be continued because the free radical of OH is generated again in the reactions. As for the reaction of nitric acid formation, it turns out that: OH þ NO2 þ M ! HNO3 þ M

ð3:17Þ

of which the reaction rate is one order higher than that for sulfuric acid formation. As a result, the lifetime of oxides of nitrogen is only one day, much shorter than that of oxides of sulfur. For this reason, it is only for long distance transport of pollutants that we need to consider effects of chemical reactions, dry deposition and wet scavenge (Lei et al., 1998). c. Biological Processes For green plants, the metabolism of water, mineral and organic matters are going on regularly. Water absorbed through roots is evaporated through

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leaf stoma. The metabolism of organic matters is fulfilled by photosynthesis: light; chlorophyll

6CO2 þ 12H2 O ! C6 H12 O6 þ 6H2 O þ 6O2

ð3:18Þ

which turns the inorganic compounds into primary sugar to form complex sugar, starch, cellulose, etc. and stores the photo energy in the form of dry material in the plant. It should be noticed that plants are capable of adapting to the ambient environment. For instance, the evaporation diminishes due to lack of water in dry season, a process that can be represented with parameterization (Rosenberg, 1974). In modeling eutrophication of lakes and oceans, many factors should be considered such as the growth of zooplankton, metabolism of phytoplankton, cycling of nutritive salt, dissolved oxygen, conservation of total phosphorus and nitrogen. In considering long-term climate models, the geophysical biochemical cycling of dioxide oxygen is of significance (Hamilton and Schladov, 1997). d. Ecological Succession As is well known, all living creature experiences a process from a preliminary to an advanced one, or from a simpler to more complicated ones. When resources are enough to maintain a favorable condition, species evolve and ecological system tends to mature. Generally speaking, such natural evolution takes hundreds of million or billion years. Nevertheless, the alteration of species in short term (a few or tens years) known as succession occurs by external humanity’s disturbance such as storm, wild fire, overgrazing and deforestation. The researchers should pay due attention to this kind of environmental variation (Bernard, 1981; Luo and Shen, 1994).

B. Parameterization The foregoing environmental problems cover a wide range of the spatial scales reducing from global, synoptic, meso-watershed to turbulence, and the temporal scales from century to seconds (Beniston, 1998). It is because the external forcing such as periodic variation of solar magnetic field and sunspot, motions of the sun, moon and earth is intrinsically of multiple scale in nature. For instance, the tilt of the earth axis with an average of 23.58 varies between 22.18 and 24.58 with a period of 40,000 years. And it precesses at a period of 20,000 years. In addition, the orbit eccentricity also changes with a period of 105,000 years (Bigg, 1996; Dingman, 1994; Watt, 1998). These variations in the solar orbit account for the fact that the ice sheet expands and recedes with a period of 100 kyr modulated by


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cycles of 20 or 40 kyr due to radiation variation. The nonlinear interaction between external forcing with different periods leads to manifesting of multiple scale phenomena. The parameterization in environmental problems has two implications. On the one hand, the sub-process analysis should be ended up with a quantitative description by supplementing new variables, new terms or even new equations and by altering boundary conditions reflecting the constituting sub-processes. On the other hand, we may have subgrid model to represent the influences arising from unresolved scales on the resolved scales, which is commonly seen in the research of turbulence and geophysical fluid mechanics. We consider it as a significant task in solving environmental problems (Smagorinsky, 1963; Galperin and Orszag, 1993; Lesieu and Metais, 1996; Zhou, 1998; Li and Xie, 1998; Li, 2001b).

C. Integrated Models We note that while incorporating the last two procedures, an integrated model can be constructed. For establishing such a comprehensive model, the main points should cover factors as follows: firstly, the model should be simple and concise enough to achieve computational efficiency as well as to reflect its validity. Secondly, the relationship between different physical elements and factors involved should be correctly established. In this regard, the aforementioned sub-processes are indispensable elements in modeling the phenomenon found in nature. Therefore, each of them should be thoroughly investigated and verified by analysis, computation and even by laboratory experiments. Of course, we are able to borrow the mature results available in different fields. With such a comprehensive model established, more challenging undertakings such as answering controversial issues, revealing implied mechanism and improving prediction can be fulfilled. IV. Case Study A. Sediment Transport and Sea Water Intrusion in the Yangtze River Estuary Scientists in the hydrodynamics field in China have focused their investigations on estuarine environmental problems in recent decades. On the one hand, many environmental events such as water pollution, toxic algae bloom, sea water intrusion, and waterway blockage have more frequently occurred than

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ever before in over-populated areas. On the other hand, the complexity involved in interactions between various underlying influential factors seems to be really overwhelming. As a case study, the Yangtze River estuary, the most important waterway in China, is indeed typical. In the Yangtze River Estuary, there exists a very complex topography of islands and shoals spreading over the multi-forked river exit to the open sea. The quite large Chongming Island separates the North and South Branches of the Yangtze, while the Changxing and Hengsha Islands further divide the South Branch into the North and South Waterways. Again, the South Waterway bifurcates into the North and South Channels (see Fig. 1). Unfortunately, Typhoon 8310 in 1983 left with a legacy of a vast blockage of the South Channel with a great amount of sedimentation. People had to give up the South Channel, resorting instead of dredging the North Channel for a navigation passage since then. However, the minimum water depth in this channel was only about 7 m near the mouth bar formed by sediment deposition. Consequently, a waterway deepening project has been planned to construct two levees, with three groins attached at each of them, along the sides of the North Channel (the layout of the first phase hydrostructures is shown in Fig. 1) in order to maintain a proper flow rate of water in the navigation waterway and to facilitate sediment sluice (Zhou et al., 2000).

Fig. 1. General Bathymetry of the North and South Channel in the Yangtze River Estuary. The thin solid lines are bathymetric isobars with a 5-m increment. In the domain, there are four gauging stations: Hengsha (HS), Middle of North Channel (MNC), Jiuduandong (JDD) and Zhongjun (ZJ), which are marked by circled stars. The two thick solid lines in the center of the domain are the levees of the first phase project with groins N1, N2, N3 and S1, S2, S3 attached.


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Associated with this project, engineers and scientists are much concerned with the following problems: What are the major impacts on the environment that are feasible to arise from the project, for instance, will the sea water interface by sea water intrusion shift downstream or upstream? Will there be any alteration in sediment transport and turbidity maximum, which could have direct influence on the growth of phytoplankton and zooplankton? What will be the annual and diurnal variations in salinity and sediment concentration in the Yangtze River Estuary? Whether or not the waterway may maintain a certain depth at affordable maintenance cost in dredging. (Luo and Shen, 1994). We plan to delineate our recent fundamental investigations in this section to explore these challenging problems in the area of estuarine environment.

1. Ruling Factors for Estuarine Flow and Mass Transport a. Runoff and Tide Interaction Runoff and tide are two major dynamic factors that can substantially affect estuarine flow and mass transport. In the Yangtze River Estuary, both runoff discharge and tide flooding volume are enormous. The annual total runoff recently recorded amounts to 925 billion m3 with the mean discharge as high as 29,300 m3/s. The maximal discharge of 92,600 m3/s and the minimal one of 6020 m3/s were reported in history. It belongs to a weakly or intermediately mixing estuary with mean tidal level of 2.66 m observed at Zhongjun tide station. In the meantime, the tidal flood discharge may reach 264,000 m3/s, nine times the mean runoff (Zhou et al., 1999b). For better understanding of sediment transport processes in estuaries, we should first examine the vertical structure of estuarine flows in detail as due to interaction of tide and runoff (Zhou et al., 1999a). In estuarine regions, tide forces the fluid to oscillate back-and-forth, thus resulting in an essentially periodic flow field. For instance, there are four different phases or segments in one tidal cycle: namely, low water rapid (LWR) with maximal ebb speed; high water rapid (HWR) with maximal flood speed; low water slack (LWS) when the flow turns from ebb to flood; and high water slack (HWS) when the reverse occurs. For the sake of convenience, we define uzmax and Umax as the top velocity and the depthintegrated mean velocity, respectively, at LWR; uzmin and Umin as the minimum and depth-mean minimum velocities at HWR, respectively. Around the instants of rapid water stage, the current runs in a manner very much similar to the unidirectional flow in rivers, owing to long tidal period, lasting usually 12 or 24 h. While around the slack water period, flow reversal occurs. The flow is evidently characterized by the fact that the current in the lower layer is generally opposite

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to that in the upper layer. Nevertheless, runoff superposed on tides may produce a seaward net flux throughout. Therefore, we have uzmax . 2 uzmin and Umax . Umin generally. To facilitate analysis, we decompose the runoff and tidal components as follows: U ¼ Uc þ Uw

ð4:1Þ

uðzÞ ¼ uc ðzÞ þ uw ðzÞ;

ð4:2Þ

where U is depth-averaged mean velocity, uðzÞ the longitudinal current component at vertical coordinates z, Uw, Uc denote the tide and runoff components of U, respectively; and likewise for the uw ðzÞ; uc ðzÞ components of uðzÞ: Obviously, uw ðzÞ can be derived from uzw, the tide component of the speed at the free surface, by applying an oscillatory boundary layer theory. Similarly, uc ðzÞ can be expressed in terms of uzc ; the runoff component of the speed at the free surface, by using the logarithmic law for unidirectional flow velocity, which is accurate enough for engineering purpose. For further analysis, we define uzc ¼ uzw ¼

1 2

½uzmax þ uzmin ;

ð4:3Þ

½uzmax 2 uzmin cos f

ð4:4Þ

1 2

where f denotes the tidal phase. The above variables are functions of x, t or x, z, t. For simplicity, the symbols x and t will be omitted. According to the logarithmic law of velocity profile, we have pffiffi ! n g z umax ¼ Umax 1 þ ð4:5Þ 1=6 k hmax pffiffi ! n g z umin ¼ Umin 1 þ ð4:6Þ 1=6 k hmin where k is the von Karman constant, n is the Manning roughness coefficient, and hmax, hmin are the water depths corresponding to Umax at LWR and Umin at HWR, respectively. Hence, the vertical two-dimensional structure of estuarine flow can be delineated by oscillatory boundary layer theory and runoff-tide decomposition approach as stated above. In this way, uðzÞ; vertical distribution of horizontal velocity component, can be deduced from the time series of U, depth-averaged velocity. Furthermore, wðzÞ; distribution of vertical velocity component, can be obtained with the help of continuity equation.


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We begin our analysis by investigating the relationship between uc ðzÞ and uzc : Integrating the logarithmic formula uc ðzÞ 1 ¼ ln z þ const upc k

ð4:7Þ

over the vertical stretch yields Uc 1 1 ¼ ln h 2 þ const upc k k

ð4:8Þ

in which upc is the friction velocity of the runoff component of estuarine flow and h the water depth. Substitution of h for variable z in Eq. (4.7) leads to uzc 1 ¼ ln h þ const upc k

ð4:9Þ

Subtracting Eq. (4.7) from Eq. (4.9), we have uzc 2 uc ðzÞ 1 h ¼ ln up c k z

ð4:10Þ

Subtracting Eq. (4.9) from Eq. (4.8), we have uzc ¼ Uc þ

up c k

ð4:11Þ

By using the Manning formula we have: pffiffi upc ¼ n gUc =h1=6

ð4:12Þ

Solving the simultaneous Equations (4.10) – (4.12), we obtain the relationship between uc ðzÞ and uzc which reads 0 B B uc ðzÞ ¼ uzc B B1 2 @

1 C 1 hC C ln zC A k h1=6 1 þ pffiffi n g

ð4:13Þ

We consider next the relationship between uw ðzÞ and uzw : As the reference frame is fixed at the bottom, the velocity at the water surface oscillating with a simple harmonic mode as uzw ¼

1 2

ðuzmax 2 uzmin Þcos f ¼ A cos f

ð4:14Þ

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is found to induce all the lower horizontal layers of fluid to move back and forth, due to the viscous effects, with a velocity distribution (Nielsen, 1992): p p z z uw ðzÞ ¼ A cos f 2 A exp 2 2f cos d d

ð4:15Þ

pffiffiffiffiffiffi in which d ¼ 2n /v is the so-called Stokes length, n the kinetic viscosity, v the oscillating circular frequency, A the amplitude, p a constant which is 1 for laminar flow and may be 1/3 for turbulent flow, whereas the phase f is unknown but should satisfy Uw ¼

1 ðh u ðzÞdz h 0 w

ð4:16Þ

where Uw ¼ U 2 Uc and Uc ¼

uzc pffiffi n g 1þ k h1=6

ð4:17Þ

ð4:18Þ

which is obtained by solving Eqs. (4.11) and (4.12) simultaneously. Integrating Eq. (4.16), with uw ðzÞ substituted by Eq. (4.15), we have qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi Uw ¼ ðA þ aÞ2 þ b2 sinðf þ cÞ ð4:19Þ Hence, Uw ffi 2c f ¼ arcsin pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ðA þ aÞ2 þ b2

ð4:20Þ

in which c ¼ arctanððA þ aÞ=bÞ; and

a¼

Ad 2q {e ðq þ 1Þ½ðq 2 1Þcos q 2 ðq þ 1Þsin q þ 1}; 2ph

ð4:21Þ

b¼

Ad 2q {e ðq þ 1Þ½ðq þ 1Þcos q þ ðq 2 1Þsin q 2 1}; 2ph

ð4:22Þ

where q ¼ ðh=dÞ p : In summary, we now have established a relation between uðzÞ and U by means of Eqs. (4.13), (4.15), (4.18) and (4.19) and also w(z), in association with the interaction between runoff and tide.


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b. Sediment Flocculation due to Salinity Sediment particles in estuaries are generally very fine. For instance, the mean diameter of the suspended particles is 0.0086 mm in the Yangtze River Estuary. A group of fine particles of this kind tends to flocculate through physical and chemical processes at the particle surface. According to Chien and Wan (1986), particles smaller than 0.01 mm in diameter display a conspicuous flocculation effect. The finer the particles, the stronger the physical and chemical effects and hence the more easily flocculation would occur. Once sediment flocculates, the particles settle down in groups rather than individually. Thus, the settling velocity, which is dependent on the grain diameter for an individual particle, may substantially increase by thousands of times. In salty turbid water environment, sediment concentration and salinity in sea water are important factors that can prompt sediment to flocculate. Figure 2 (Chien and Wan, 1986) shows that the settling velocity increases abruptly with salinity in sea water when salinity is within a certain range, which is also dependent on the sediment concentration. Therefore, mixing process in estuary is bound to exert crucial impact on sediment transport there. Generally speaking, there are three types of mixing processes corresponding to different kinds of estuaries: sharply stratified estuaries or salt-wedge estuaries, partially stratified estuaries, and well-mixed estuaries. Quantitatively, they are

Fig. 2. Variation of settling velocity with salinity at different sediment concentration (Chien and Wan, 1986).

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classified by the mixing index N, defined as the ratio of runoff volume to flood volume within one tidal cycle. Conventionally, we have the following criteria: N $ 0:7

sharply stratified

0:1 # N , 0:7 partially stratified N , 0:1

well mixed

In a sharply stratified estuary, runoff discharge as compared to the tidal range is so large that fresh water runs seawards along the upper layer and salt water intrudes upstream from the lower layer. Mixing only takes place in the neighborhood of the interface between the two layers by turbulent entrainment. In regard to the quite extensive literature on this subject, the studies have mainly focused on the interfacial mixing mechanism and dependence of location and geometry of the arrested salt-wedge on runoff and tidal discharge (Grubert, 1989, 1990; Kurup et al., 1998). In contrast, the runoff discharge as compared to tidal range is rather small in well-mixed estuaries, hence the tidal force can potentially produce strong turbulence, resulting in a vertically uniform salinity profile. The dynamic process can be examined by depth-integrated models. As for the partially stratified estuary, such as the Yangtze River Estuary, the mixing index, being 0.18 on average, varies with seasons and the salinity distribution displays distinct three-dimensional characteristics. To describe this kind of mixing process, it is indispensable to have a better understanding of vertical structure of estuarine flows. c. Resuspension in Turbulent Flows The mechanism of sediment resuspension in turbulent flows is a challenging problem. It is essential to specify a well-founded resuspension flux formulation in sediment transport modeling with the flux directly serving as the bottom condition in a vertical two-dimensional or three-dimensional numerical model. At the same time, this flux is implicitly represented in the source term in a depthaveraged two-dimensional model. The mechanism of sediment resuspension has been explored for several decades. The intermittence of sediment inception has provided a resemblance to turbulence. Early in the 1960s, Sutherland (1967) put forward an idea that a mechanism to entrain sediment grains into suspension would correspond to flow ejections associated with coherent structures in the near-bed region of a turbulent boundary layer. This idea has been verified more recently by empirical evidence. In fact, the sediment-laden boils observed at the free surface of rivers exhibit the effects of burst events on sediment transport in alluvial channels (Jackson, 1976). Also, different indoor experiments have been conducted to examine


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the entrainment mechanism of particles into suspension. Particularly, Nino and Garcia (1996) studied particle –turbulence interactions in near-wall region with natural sediment particles and proved the strong dependence of sediment inception and entrainment on turbulent bursting events. This led them to draw the following conclusions. Sediment particles tend to accumulate in the low-speed regions and form wall streaks accompanied by the presence of counter-rotating quasi-streamwise vortices. The wall streaks extend about 1000 – 2000 wall units in the streamwise direction, and have a transverse wavelength of about 100 wall units. Particles are picked up from the bed by flow ejection events occurring downstream of the shear layers, and particle ejection angles are in the range 10 –208, very similar to the angle of inclination of the shear layers, typically being 148. The number of particles reaching the outer regions of the wall layer increases with the dimensionless bed shear stress. Particles would be only rarely deposited by the action of sweep events, but rather they appear to be falling back toward the bed as they lose correlation with the turbulent structures that lifted them from the bed and kept them in suspension for some time. These studies, however, delineate only qualitatively that the mechanism of turbulent bursting is supposed to be responsible for sediment inception and suspension. More recently, Cao (1997) has established a quantitative formula for sediment entrainment flux based on the turbulent bursting mechanism. His formula relates the flux to the temporal and spatial scales of bursting events so as to take the form: sffiffiffi S t0 t0 En ¼ PC0 d 21 ð4:23Þ g tc rs 2 r pffiffiffi in which C0 is the volumetric concentration of bed material, P ¼ l Sg=nTBþ ; l < 0.02 the averaged area of all bursts per unit bed area, TBþ < 100 the nondimensional bursting period, S ¼ ðrs 2 rÞ=r; rs the sediment density, d the diameter of sediment grain, tc the critical incipient stress, and t 0 the skin friction of bed shear stress. In estuaries, the bulk density of bed material is usually smaller when fine particles start to settle down. It then gradually grows with time due to a compact process, during which the incipient condition alters with bed mixture’s bulk density. To take this factor into account, we introduce Tang – Cunben’s formula (Tang, 1963) " # 1 gb 10 k 3:2ðgs 2 gÞd þ ð4:24Þ tc ¼ 77:5 d gb0 where g ¼ rg; gs ¼ rs g; gb0 ¼ 1.6 g/cm3 is the bulk density of compact bed material; k ¼ 2.9 £ 1024 g/cm; gb is the bulk density of bed material, which is not a constant but varies with tidal phase.

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Li Jiachun et al. 2. Model and its Validation

For long and narrow estuaries, a vertical two-dimensional model is applicable. Here, we are ready to propose a quasi-two-dimensional vertical model instead of the conventional one. This model computes vertical two-dimensional flow by profiling one-dimensional current obtained from the Saint Venant Equations via the vertical structure of estuarine flow as discussed in Section IV.A.1.a. Therefore, the model is advantageous in requiring less CPU time and affording more apparent physical implications. In addition, it can be easily extended to three-dimensional circumstances (Zhou and Li, 1999; Zhou and Li, 2000). a. Governing Equations The one-dimensional continuum and momentum equations for shallow water are

›z ›Uh þ ¼ 0; ›t ›x

ð4:25Þ

›U ›U ›z gh ›r lUlU þU ¼ 2g 2 2g 2 ; ›t ›x ›x r ›x C h

ð4:26Þ

where t is the time; x, z the horizontal (seawards) and the vertical (upwards) coordinates, respectively; z the water level; h the depth; U the vertical mean velocity; g the gravitational acceleration; C the Chezy coefficient; and r the depthaveraged value of salt water density r, which is defined by the state equation:

r ¼ r0 þ hs;

ð4:27Þ

in which r0 is the density of fresh water; s the salinity; h a constant, which approximates to 0.0007 provided s is measured in parts per thousand (ppt). To further derive the vertical velocity component, we introduce the vertical twodimensional continuity equation

›u ›w þ ¼ 0; ›x ›z

ð4:28Þ

where u and w, respectively, denote the x- and z-component of the flow velocity. Then, the salinity and sediment concentration can be obtained via the respective vertical two-dimensional advection –diffusion equation: ›s ›us ›ws › ›s › ›s þ þ ¼ Dx Dz þ ; ð4:29Þ ›x ›z ›t ›x ›z ›x ›z ›c ›uc ›ðw 2 vf Þc › ›c › ›c þ þ ¼ N N þ : ð4:30Þ ›x x ›x ›z z ›z ›t ›x ›z


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in which c is the sediment concentration; vf the settling velocity of flocculated sediment particles, Dx and Dz the horizontal and vertical turbulent diffusivities for salinity, Nx and Nz horizontal and vertical turbulent diffusivities for sediment. It should be mentioned that the main point in solving Eq. (4.30) is how to specify the bed boundary condition more reasonably. As we have argued in the previous paragraph, an entrainment function En is used for bottom sediment flux, i.e., 2Nz

›c ¼ En ; ›z

ð4:31Þ

where En is a function of either incipient velocity or incipient stress. For unsteady sediment transport, incipient stress is preferred. Here, we adopt the turbulent bursting-based formula as cited in Section IV.A.1.c. b. Solution Methods and Procedures The methods and steps to solve the foregoing equations are as follows. First, we solve Eqs. (4.25) – (4.27) simultaneously, yielding the time series of mean speed U and water level z in a tidal cycle. Resorting to the theory we have proposed, u(z) is obtained via Eqs. (4.3), (4.4), (4.13), (4.15) and (4.17) –(4.20). Then, w(z) is directly derived by continuity equation (4.28). And the patterns in salinity and sediment concentration are ultimately produced by integrating the advection– diffusion equations (4.29) and (4.30) in sequence. In order to solve Eqs. (4.25) and (4.26) numerically, we choose the Preissmann implicit finite difference scheme, which is advantageous in its stable nature and less CPU time demand. Equations (4.29) and (4.30) are discretized by an up-wind difference scheme and the alternative direction implicit (ADI) method. However, iterations are always necessary due to the s being unknown in the state equation. c. Validation of the Model Observational database in the North Channel waterway of the Yangtze River Estuary is used for the model validation. Hourly tide levels were acquired on September 3– 5, 1989 at four gauging stations at Hengsha, Middle of North Channel, 18 km downstream of Hengsha, and at Jiuduandong, 40 km downstream of Hengsha (see Fig. 1). The salinity, sediment concentration and horizontal velocities were measured at SN2, SN3 and SN4, 8, 12 and 16 km downstream of Hengsha station, respectively. Taking Hengsha station as the upstream boundary and Jiuduandong as the downstream boundary, the simulated spatial domain is meshed by 20 grids with horizontal step Dx ¼ 2 km and vertically divided into 10 layers. The time step for temporal integration from 6:00 a.m. on Sep. 4 to 6:00 a.m. on Sep. 5 is Dt ¼ 60 s. The roughness coefficient is

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assumed to be n ¼ 0:01: With zðx; 0Þ ¼ 0 and Uðx; z; 0Þ ¼ 0 as the initial conditions, the simulated time series of water level at the Middle of North Channel station is compared very well with the measurement as shown in Fig. 3 (Zhou and Li, 2000). The calculated flow field in the vertical plane around HWR and LWR is shown in Fig. 4 (Zhou and Li, 2000), demonstrating that the horizontal velocity component is much greater than the vertical one. Comparing the simulated horizontal velocity component with the corresponding data in-situ in Fig. 5, we find that the calculated flow field represents the real patterns of estuarine flows approximately. For the sake of validating the salinity and sediment concentration prediction, the downstream boundary is replaced by that at SN4 owing to lack of salinity and sediment concentration data at Jiuduandong station. The water level at SN4 obtained from the computation for water level validation is taken as the boundary conditions. The salinity at Hengsha as the upstream boundary can be assumed to be zero because the Hengsha station is situated approximately at the limit section which divides salty and fresh water (Wang et al., 1995). According to Kuang (1993), the horizontal diffusivity assumes the value Nx ¼ Dx ¼ 500 m2/s, and the vertical diffusivity has the form 8 > < kup z 1 2 z ; z=h , 0:5; h Nz ¼ Dz ¼ ð4:32Þ > : kup h; z=h $ 0:5:

Fig. 3. The water level at the Middle of North Channel station. The time duration is from 6:00 a.m. (corresponding to t ¼ 0 h) on Sep. 4 to 6:00 a.m. (corresponding to t ¼ 24 h) on Sep. 5, 1989.


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Fig. 4. The simulated flow vector field. The solid line is the bed profile. (a) Around high water rapid (HWR); (b) around low water rapid (LWR).

Here up is the friction velocity of the mean flow. Transition from the initial salinity field sðx; z; 0Þ ¼ 0 to a stable state takes a few tidal cycles. Figures 5– 7 (Zhou and Li, 2000; Zhou et al., 1999a) present some comparison of the calculated time series of velocity, salinity and sediment concentration at the site of SN3 with the measured ones at six different heights of 1.0h, 0.8h, 0.6h, 0.4h, 0.2h and 0.0h. The agreements are satisfactory, i.e., well within permitted errors for engineering practice.

3. Variation of Salinity Generally speaking, mixing processes of sea water in estuaries are very complicated due to the interactions of different dynamic factors and irregular

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Li Jiachun et al. Fig. 5. Comparison of the calculated (line) and observed (dots) velocity time series. The time duration is from 6:00 a.m. (corresponding to t ¼ 0 h) on Sep. 4 to 6:00 a.m. (corresponding to t ¼ 24 h) on Sep. 5, 1989.

Environmental Mechanics Research in China Fig. 6. Comparison of the calculated (line) and observed (dots) salinity time series. The time duration is from 6:00 a.m. (corresponding to t ¼ 0 h) on Sep. 4 to 6:00 a.m. (corresponding to t ¼ 24 h) on Sep. 5, 1989 (reproduced from Zhou et al. (1999a) with permission by Scientia Sinica).

247

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Li Jiachun et al. Fig. 7. Comparison of the calculated (line) and observed (dots) sediment concentration time series. The time duration is from 6:00 a.m. (corresponding to t ¼ 0 h) to 6:00 p.m. (corresponding to t ¼ 12 h) on Sep. 4, 1989.


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Table IV Cases for Studying Salinity and Sediment Concentration Case label

a b c d e

Presumed conditions

Calculated results

te (m)

uc (m/s)

N

2.0 1.0 0.5 0.5 0.5

0.1 0.1 0.1 0.2 0.3

0.13 0.18 0.22 0.40 0.57

Ds/Dx (km21) 0.48 0.61 0.69 1.09 1.50

topography. For simplicity, a plane bottom is assumed to facilitate analyzing the influences of runoff and tides on the mixing processes. The adopted parameters are referred to those of the North Channel waterway of the Yangtze River Estuary. For example, the depth is taken as 10 m below theoretical datum plane. The hydrological factors are listed in Table IV. In this table, te denotes the amplitude of tide level, half of the tidal range, uc stands for the runoff component of velocity at the upstream boundary, and N is the mixing index, expressing the relative strength of runoff and tides. a. Longitudinal Distribution of Mean Salinity Some representative variations of salinity with runoff and tides have been obtained by case studies (see Fig. 8). Comparisons of cases a, b, c and of cases c, d, e reveal the variations of salinity with tides and runoff, respectively. Figure 8 suggests that the longitudinal mean salinity distribution is dependent on tidal range, phase angle and runoff discharge. The gradient reaches its smallest for the most upstream sea water intrusion at HWS and its greatest for the most downstream sea water intrusion at LWS, and it is between these two values at all the other moments. At a given phase, the gradient decreases with increasing tidal range or hydrodynamic tidal force, if runoff discharge remains unchanged. When te ¼ 2.0 m, uc ¼ 0.1 m/s, the longitudinal salinity profile is approximately linear. If tidal range remains the same, an increase in the runoff discharge results in a large salinity gradient, with the longitudinal profile deviating from a linear one. In fact, we may as well introduce a single variable, namely the mixing index, to explain the variation of longitudinal salinity distribution at a given phase of the tides and runoff. For case a to e, the degree of mixing becomes gradually weaker with increasing mixing index Table IV while the salinity gradient grows correspondingly. When the index takes the value 0.13, the longitudinal salinity profile becomes approximately a linear one.

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Fig. 8. Variation of the longitudinal distribution of the mean salinity with runoff and tides. Square: at HWS; Dot: at HWR; Upper-Triangle: LWR; Down-Triangle: LWS. (a) case a; (b) case b; (c) case c; (d) case d; (e) case e (reproduced from Zhou et al. (1999a) with permission by Scientia Sinica).

b. Variations of the Salt Front The ‘front’ concept was early used in meteorology to represent an interface between two air masses with different temperatures. This concept has been introduced to oceanography to describe the interface between two water masses with different temperatures or densities at which the gradient of certain hydrological quantities (such as salinity, density, etc.) is large. Nearby the front, the salinity favors flocculation of sediment particles, and the circulation caused by density difference tends to capture the suspended sediment. All these effects result in the formation of ‘turbidity maximum’ in a certain zone of estuary.


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The salt front intensity can be quantified by the longitudinal salinity gradient Ds=Dx: In Table IV, a list of the mean salt front intensities at HWS and LWS is given. The figures in the table show that Ds=Dx increases with N, namely, the salt front for weakly mixing case becomes more intense with runoff discharge increasing or tidal range decreasing. Thus, it can be inferred that the salt front intensity in flood seasons is greater than that in dry seasons. On the other hand, the salt front intensity during ebbing is greater than that during flooding of tide, owing to the seaward movement of the salt front forced by runoff on ebb and the upstream movement of the salt front forced by tide on flood. These statements are demonstrated in Fig. 8 (Zhou et al., 1999a). Figure 9 (Zhou et al., 1999a) plots in Ds=Dx against N, approximately exhibiting a linear increase in salt front intensities with mixing indices, with different slopes at different phases. The slope turns from its minimum at HWS to its maximum at LWS. Field measurements indicate that a salt front with the greatest longitudinal salinity gradient exists in an estuary all the year round. Its intensity and position are phase-, runoff- and tide-dependent. This is implied by the data in Fig. 8. At the HWS instant, the greatest salinity gradients of cases a, b, c, d and e are 0.70, 0.94, 1.00, 1.37 and 1.78 km21, respectively. The fronts are located at some 20, 25, 28, 32 and 36 km downstream from the Hengsha station, with the position displacement reaching 16 km approximately.

Fig. 9. Relationship of salt front intensity and mixing index (reproduced from Zhou et al. (1999a) with permission by Scientia Sinica).

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Li Jiachun et al. c. Salinity Distributions in the Vertical Plane

Salinity contours in the vertical plane (see Fig. 10, Zhou et al., 1999a) describe the patterns of salinity distribution both in the longitudinal and vertical directions, exhibiting different characteristics related to different combinations of the hydrological factors involved, including certain periodic variations for a given combination of hydrological factors. In Fig. 10(a) –(d) are plotted the contours of case b (te ¼ 1.0 m, uc ¼ 0.1 m/s) at slack water and rapid water phases, which once again depict the characteristics of the longitudinal mean salinity distribution shown in Fig. 8, e.g., the salt front intruding the most upstream at HWS under the tide effects, and being pushed seawards to the most downstream end at LWS by the runoff force. In addition, a rudimentary salt-wedge is seen from the figures as a result of the density flow. On the other hand, in the vicinity of the upstream end, which may lie beyond the uppermost reach by the sea water, and the downstream end, where there is little fresh water, the salinity distributes uniformly in the vertical direction. Therefore, the vertical salinity gradient experiences a small-tolarge-to-small variation, suggesting uniform vertical salinity profiles at the two ends and non-uniform ones in the intermediate range. Figure 10(d) – (f) shows of the salinity contours for different tidal ranges, 0.5, 1.0 and 2.0 m, with the same runoff discharge. It can be seen that the salt front approaches more upstream with increasing tidal force when runoff discharge remains unchanged, with a 5-km shift of the front locations. Likewise, a comparison of the salinity contours for different uc values, 0.1, 0.2 and 0.3 m/s with a given tidal range, is made between Fig. 10(f)– (h), suggesting that an increase of runoff discharge constrains the upstream intrusion of the salt water and makes the salt front move seawards, with a 15 km shift of the front. Therefore, effects of runoff and tides make the salt front oscillate downstream and upstream, respectively, with the former effect being more apparent.

4. Turbidity Maximum Zone The turbidity maximum zone, in which the sediment concentration is relatively higher, is a natural phenomenon almost found in every estuary. Studies on ecological environment in estuaries have shown that the turbidity maximum plays an important role in the growth of phytoplankton and zooplankton (Gu et al., 1995; Xu et al., 1995). In the turbidity maximum, suspended sediment reduces the light transmittance of water, affecting the photosynthesis and, hence, the chlorophyll content of water. In addition, the content and distribution

Environmental Mechanics Research in China Fig. 10. Variations of salinity contours with time, runoff and tides. (a) Case b at HWS; (b) case b at HWR; (c) case b at LWR; (d) case b at LWS; (e) case a at LWS; (f) case c at LWS; (g) case d at LWS; (h) case e at LWS (reproduced from Zhou et al. (1999a) with permission by Scientia Sinica). 253

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of nutrient substances are bound to change with sediment transport due to the strong adsorption capacity of sediment particles to the nutritious salt and heavy metal elements (Shen and Shi, 1999). It is also of significance to study turbidity maximum in order to elucidate the formation of the mouth bar. We have examined the variation of sediment concentration with runoff and tides by case studies as discussed in the foregoing section, aiming at revealing the mechanism why turbidity maximum occurs. Figure 11 (Zhou and Li, 2000) shows distributions of concentration around high slack water phase. A common characteristic can be found in each case: a high concentration zone exists in the middle of the simulated domain. As a matter of fact, vertical sediment transport can be accounted for by the balance of resuspension and settling. The former depends on the interaction of bed and current, and the latter represents the effect of salinity. In the upstream domain, resuspension is dominant due to slow settling of sediment grains. This causes the concentration to increase. With increasing salinity downstream, the settling velocity vf increases correspondingly. At the same time, the mean flow velocity is reduced. Therefore, the settling effect becomes the dominant one, resulting in a decrease in the sediment concentration. In this low-to-high-to-low process of concentration, a midway high concentration zone exists naturally. This special zone is generally called ‘turbidity maximum’. It is around this area that a mouth bar forms easily. Figure 11 also shows the variation of turbidity maximum position with the relative strength of runoff and tides. From case b to e, the mixing index, N, changes from small to large. Correspondingly, the turbidity maximum shifts from upstream to a downstream area. This is simply because the smaller the N, the stronger the tides, the farther the sea water intrudes upstream. Meanwhile, the salt water mixes with the fresh water more uniformly. As a result, vf increases immediately along the whole water column, and the concentration decreases. Secondly, our calculated results indicate that the concentration distribution is phase dependent. The turbidity maximum position varies in a tidal cycle. It is most upstream at HWS and most downstream at LWS, shifting 8 km or so. This is also attributed to the non-uniform salinity distribution. With the upstream intrusion of flood tides, salinity increases and then vf increases as well due to the flocculation effect. Hence, the turbidity maximum moves upstream till HWS. On the contrary, it moves seawards till LWS. It seems that the shifting distance within one tidal cycle is independent of the mixing index N. But the averaged position within a tidal cycle varies with N. The larger the N, the more downstream the turbidity maximum locates. The maximal shifting distance of mean position of the turbidity maximum approximates to 20 km.

Environmental Mechanics Research in China Fig. 11. Contours of sediment concentration corresponding to different mixing indices. (a) Case b at HWS; (b) case c at HWS; (c) case d at HWS; (d) case e at HWS.

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Fig. 12. Sediment concentration contours of case d at HWS (c¯e ¼ 2.0 kg/m3). The turbidity maximum feature is not noticeable if the sediment concentration at the entrance is large enough.

Finally, the concentration distribution is also adjusted by the entrance sediment condition even if the hydraulic factors remain the same. Figure 12 (Zhou and Li, 2000) displays some concentration contours of case d when c e ; the depth-averaged sediment concentration at the entrance, is equal to 2.0 kg/m3. It is found that the concentration decreases seawards while variations of the turbidity maximum features are not noticeable. In this case, suspended load qs continues to decrease seawards (see Fig. 13, Zhou and Li, 2000, dqs =dx is always negative) owing to the dominant settling effect over resuspension, and hence deposition occurs everywhere. When c e is small, dqs =dx changes from positive to negative, suggesting that erosion occurs upstream followed by downstream deposition.

B. Runoff Generation and Soil Erosion on the Loess Plateau The Loess Plateau in the central China with an area of 430,000 km2 is noted for its ancient culture and serious water and soil loss. About 287,600 m2 of area in this region has an annual erosion rate greater than 1000 ton/km2. Moreover, the eroded soil totally amounts to more than 2.2 £ 109 tons, among which about 1.6 £ 109 ton, on average, is delivered into the Yellow River, rendering the river one of the highest sediment-laden


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Fig. 13. Influence of sediment supply at the entrance on the curve of suspended load vs distance. Square: c¯e ¼ 0.0 kg/m3; dot: c¯e ¼ 0.5 kg/m3; upper-triangle: c¯e ¼ 2.0 kg/m3.

rivers in the world. The serious soil erosion of long period leads to the fragment geomorphy in this region. Especially, the recent violent human activities considerably aggravate the situation and have brought about a series of consequences such as lean soil, desert land formation, arable land diminish and production reduction. The eroded soil input into the Yellow River and carried downstream to block water channel is responsible for frequent flooding disasters in the middle and lower reaches. Furthermore, the sediment with adsorbed toxic elements from pesticide and fertilizer in the cultivated land may cause water pollution by concentration effect. Therefore, it is an urgent and challenging task to prevent ecological environment deterioration in Western China due to soil erosion. During the recent decades, there have been extensive investigations on soil erosion in the Loess plateau area. A general qualitative description of this issue such as erosion type, erosion intensity, erosion geomorphy and their regional distribution has been acquired. Many empirical and semi-empirical relationships have been established to estimate runoff and soil erosion quantitatively. However, most of the previous researches aiming at a certain specific erosion type or region only covered very limited parameter range. As a matter of fact, surface feature, topography and soil characteristics of the Loess Plateau along with rainfall conditions are substantially varied in different regions. The research

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in a specific region without quantitative comparison and mechanism exploration is unable to reflect overall roles of various factors in the process of erosion. Accordingly, the research of soil erosion at present tends to shift from the empirical approach to the process-based dynamic description such as WEPP in USA and ANSWERS in Europe. In recent years, some process-based models are also developed in China (Chen et al., 1996; Qi and Huang, 1997). Tang and Chen (1997) have ever presented a more theoretical model for predicting transportation capacity of overland flows. But comparatively, the dynamic study of erosion process on the Loess Plateau is relatively weak. The present investigation is mainly concerned with the physical mechanism of overland flows and soil erosion on the Loess Plateau. Since the water erosion caused by rainfall is a rather complicated process affected by both water and soil interactively, three research methods including theoretical analysis, laboratory experiment and mathematical model were used in the research. As a first step, the objective of the current research therefore is to examine the fundamental laws of overland flows, runoff generation and rill erosion in several typical regions on the Loess Plateau, which will certainly be helpful for gaining an insight into these processes on the Loess Plateau in the Northwest China.

1. Formation of Runoff on the Slope Hillslope runoff erosion is such a process that sheet flow generated during rainfall scours the soil surface. The water flow produces eroding force and plays an active role in the whole process. It is of primary importance to first understand the runoff generation characteristics. The motive power led to erosion is the acting force of surface flow, while the erosion-resisting capacity is dependent on the stability of soil body. That is to say, the soil loss on the hillslope occurs when the scouring capability of surface runoff exceeds the erosion-resisting capacity. The whole process can be divided into three stages. Firstly, when the rainfall intensity is greater than the soil infiltration rate and the surface ponding capability, the excess rain flows down the hillslope under the action of gravity and thus forms the surface sheet flow. Then, when the scouring ability is greater than the erosion-resisting capacity of soil, the scour of soil particles is initiated. And finally, the scoured soil is transported downstream by overland flow. Therefore, the soil erosion is mostly dependent on the overland flow caused by rainfall. To sum up, when the rainfall intensity exceeds soil infiltration rate, water begins to accumulate in the depressed pool or vacancy. It is not until beyond the surface ponding ability that water flows down the slope under the action


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of gravity and form sheet flow. That is just the beginning of runoff generation. Obviously, precipitation, infiltration and overland flow formation are the most fundamental hydrodynamic processes in soil erosion study.

a. Precipitation on the Loess Plateau The Loess Plateau belongs to the semi-humid and semi-arid region, the annual precipitation is in the scope of 400 – 800 mm. The annual precipitation in whole Loess Plateau area, on average, is about 500 mm, and its seasonal distribution in a year is very non-uniform. More than 70% of precipitation occurs from July to September. Moreover, these rainfalls are mostly rainstorms with shorter duration and indefinite location. The precipitation of one storm is often more than 10% of total annual amount. What is unique for the erosion on the Loess Plateau is that it always occurs during rainstorm. The rainfall number giving rise to apparent erosion does not exceed 10% of the total ones. On average, three rainfalls each year may cause serious erosion, only occupying about 4% of total rainfall number. As roughly estimated, 80% of the total erosion quantity is yielded in 1.3 times of rainfalls while 50% of the total erosion quantity in 0.6 times of rainfalls (Wang and Jiao, 1996). Thus we would rather lay emphasis on the erosion of the Loess Plateau under rainstorm conditions. So far, there is still no standard rainfall frequency and intensity available in the Loess Plateau area. According to Wang and Jiaos (1996) research, the rainstorms in this region can be divided into three types: uniform rainstorm, a single-peak rainstorm and multi-peaks rainstorm. The single-peak rainstorm is assumed as a single-peak rain process of short duration with high peak, while multi-peaks rainstorm is assumed as multi-peaks rain process of long duration with relatively low peak in which one peak value is equal to three to five times of the others.

b. Infiltration Process Soil infiltration is a precursory process for runoff formation. While its influencing factors include cumulative infiltration quantity, infiltration rate, saturate conductivity (or infiltration coefficient), saturate volumetric water content, i.e., efficient porosity, and initial volumetric water content, etc. So a reasonable description of soil infiltration process is considerably essential for overland flow modeling. The famous Green – Ampt model with explicit physical meaning was put forward in 1911, then the models of Horton (1940) and Philip (1957) in succession. So far the Green –Ampt model is still commonly used

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owing to its simplicity and credible applications. In particular, the improvements by Mein and Larson and Chu have extended its applicability to the circumstances of both steady and unsteady rainfall. Although the Horton model is widely applied in China, the parameters involved do not bear apparent physical meaning as the Green – Ampt model. So we select the revised Green –Ampt model (Mein and Larson, 1973; Chu, 1978) to describe the soil infiltration process. The governing equations of soil infiltration may be written as: dI ¼ K½1 þ ðus 2 ui ÞS=I dt I I ¼ Kt 2 Sðus 2 ui Þln 1 þ Sðus 2 ui Þ

i¼

ð4:33Þ

where I is the cumulative infiltration quantity (m), i the infiltration rate (m/s), K the saturate conductivity of soil (or infiltration coefficient) (m/s), us the saturate volumetric water content, i.e., the efficient porosity(%), ui the initial volumetric water content (%), and S the soil suction (m). The classic Green –Ampt formula is merely applied to ponding water infiltration process on dry soil. Mein and Larson (1973) generalized it to the infiltration process during rainfall. Suppose p is the steady rainfall intensity. In initial stage of rainfall, all the rain infiltrates into soil. The moment when infiltration rate i equals p ponding occurs. The cumulative infiltration, Ip ; may be derived by the Green –Ampt model: Ip ¼

ðus 2 ui ÞS ; p=K 2 1

ð4:34Þ

from which the ponding moment tp ¼ Ip =p is yielded. Thus the infiltration rate in the whole process can be expressed as i¼p

t # tp

i ¼ K½1 þ ðus 2 ui ÞS=I t . tp

;

ð4:35Þ

where I is the cumulative infiltration depth after the ponding time (including Ip ). For the sake of convenience, the formula may be revised as K½t 2 ðtp 2 ts Þ ¼ I 2 Sðus 2 ui Þln 1 þ

I ; Sðus 2 ui Þ

ð4:36Þ


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in which ts , tp represents the time needed for reaching the same accumulative infiltration Ip (or i ¼ p) supposing ponding occurs initially. It is determined by Ip Kts ¼ Ip 2 Sðus 2 ui Þln 1 þ ð4:37Þ Sðus 2 ui Þ The main idea of revision is to assume the entire process as ponding infiltration from the very beginning. Therefore, the whole curve should be translated to the left by tp – ts and translate back to the right by tp – ts. In this manner, we may continue to use the Green –Ampt model after ponding by introducing a pseudo-time concept. However, the steady rainfall is unable to meet the requirements in actual applications. Chu (1978) generalized the GA-ML model again to the unsteady rainfall process. The main procedure is to sort the ground surface conditions into four kinds for each time step: (1) (2) (3) (4)

no ponding at the beginning, no ponding at the end no ponding at the beginning, ponding at the end ponding at the beginning, ponding at the end ponding at the beginning, no ponding at the end

At the beginning of each time interval, the total rainfall quantity, the total infiltration quantity and the total excess quantity are known. Whether the ponding occurs at the end of the interval could be distinguished by two indices, where cu is used for determining whether ponding occurs at the beginning of the interval, while cp is used for determining whether ponding occurs at the end of the interval: cu cu cu cp cp cp

¼ Pðtn Þ 2 Rðtn21 Þ 2 KSM=ði 2 KÞ . 0 ponding at the end of the interval, , 0 still no ponding ¼ Pðtn Þ 2 Iðtn Þ 2 Rðtn21 Þ . 0 still ponding at the end of the interval, , 0 ponding disappears,

where M represents us 2 ui : Provided that i , K; there is no ponding all the time and we need not use these two indices. c. Kinematic Wave Theory Because of the very thin depth of overland flow and the complicated boundary conditions, it is a tough task to describe the movement of this kind of

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flow appropriately. Usually, one-dimensional shallow water equation (Saint Venant equation) is used in modeling of overland flow (Emmett, 1978).

›v ›h ›h þv þ ¼ qp ›t ›x ›t

ð4:38Þ

›v ›v ›h v þv þg þ qp þ gðSf 2 S0 Þ ¼ 0 h ›t ›x ›x

ð4:39Þ

h

in which t is the time, x the distance, h and v represent the runoff depth and velocity, qp means rain excess, S0 the slope, Sf the energy slope, and g the gravity acceleration. Nevertheless, Saint Venant equation always needs more parameters. As for runoff, most researchers prefer to describe overland flows by the so-called kinematic wave theory. Actually, it is a simplified form of the one-dimensional Saint Venant model. Woolhiser and Ligget (1967) have ever analyzed the onedimensional unsteady overland flow. They found that when the kinematic wave number K . 20ðK ¼ S0 L=h0 F02 Þ; (in which S0 is the slope, L the runoff length, h0 the depth of the runoff at the distance L, F0 is the Froude number) and F0 . 0.5, the kinematic wave model can describe overland flow quite well. Actually the kinematic wave number is certainly much greater than 10 on the Loess Plateau as Shen (1996) reported. Therefore, the kinematic wave approximation turns out to be a suitable mathematical description of the runoff generation process on the Loess Plateau. Till now there have been some revisions in the kinematic wave model (Ponce et al., 1978; Govindaraju, 1988). If the streamwise variation of water depth is considered by introducing an additional pressure gradient term, it becomes diffusion wave model. The main assumption of the kinematic wave model is that the gravity component along the slope is equal to the resistance force ðSf ¼ S0 ¼ sin uÞ and thus a concise relationship between discharge and water depth is established. Considering that the slope gradients on the Loess Plateau are generally precipitous, Eqs. (4.38) and (4.39) could be simplified as (Chen and Liu, 2001):

›h ›q þ ¼ p sin u 2 i ›t ›x 1 1=2 q ¼ h5=3 S0 n where x is the coordinate alone the flow direction (m), t the time (s), h the water depth (m), q the unit discharge (m2/s), p indicates the rainfall intensity (m/s), S0 ¼ sin u the slope gradient and u the inclination angle of slope (in degree), and n the Manning roughness coefficient.


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Fig. 14. Variation of the runoff generation process on a permeable slope. The comparison of the calculated (line) and experimental (dots) unit discharge at the outlet (experimental data are from Lima, 1992).

The laboratory experimental data of Lima (1992) are used to verify the performance of the model. This experiment was proceeded in a flume with length 1 m, width 0.5 m and slope S0 ¼ 0.1. The rainfall intensity was 0.03741 mm/s together with other soil parameters as follows K ¼ 1:67 £ 1026 m=s;

us ¼ 0:506;

ui ¼ 0:0107;

S ¼ 0:02 m

Figure 14 is the comparison of calculated result and the experiments of the unit discharge at the outlet. The simulated diagram shows good agreement between them except during the very short initial period. It is especially encouraging to find the consistency between the theory and the experiment when runoff is about to stop. The results indicate that the model presented in this paper can efficiently simulate the runoff generation on the slope. Moreover, we have analyzed various factors affecting the major parameters as well as the process of rainfall – infiltration –runoff generation (Chen et al., 2001a). The results are shown in Fig. 15, and we can obtain the following principal conclusions: (1) Each dynamic parameter (unit discharge, water depth, velocity, shear stress) grows as the rainfall intensity is increasing; (2) Each dynamic parameter increases as the infiltration rate is diminishing; (3) Each dynamic parameter rises as the slope becomes longer; (4) Each dynamic parameter decreases as the volumetric soil water deficiency (the difference between the saturated water content and the initial water content) becomes larger;

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Fig. 15. Variation of runoff generation with various major affecting factors. (a) –(d) Unit discharge (m3/s) vs time. ((a) For different initial volumetric water contents. (b) For different saturate conductivity of soil. (c) For different lengths of slope surface. (d) For different slope gradients). (e) Unit discharge and water depth vs slope degree. (f) Velocity and sheer stress vs slope degrees.

(5) The water depth and the discharge rate of overland flow decrease with the increase of the slope gradient. (6) The slope gradient plays both positive and negative roles on the flow velocity and shear stress of overland flow. The flow velocity and shear stress increases at first and then began to decrease when the slope gradient reaches their respective critical values. Although the corresponding critical


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slopes are not equal, both of them are estimated within the range of about 40– 508. This conclusion also accords with our theoretical analysis (Liu et al., 2001) well.

2. Runoff Generation Characteristics in Typical Erosion Regions a. Typical Erosion Regions Generally speaking, the soil erosion intensity is gradually increasing from the south to the north in the Loess Plateau area (Chen et al., 1988; Chen, 1996). According to Liu’s research (1965), this area could be partitioned into three belts correspondingly, namely clay loess belt, loess belt and silt loess belt. We choose one typical region in each belt and then simulate runoff generation in these typical regions in order to analyze the characteristics of runoff generation under different rain types. These three regions are marked as region N, M and S, respectively (see Fig. 16, Chen et al., 2001a). Region N is located at the Yellow River valley with extremely intensive erosion in the northeast of Shanxi province just adjacent to the border of Sanxi province. The soil there belongs to silt loess, its grain is relatively coarse

Fig. 16. The sketch of the Loess Plateau and the selected typical regions.

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and the clay content is quite low. Region M denotes the area around Gu Yuan– Ping Liang – Xi Feng with intermediate intensity of erosion in the central Shanxi. The soil there belongs to loess and its property is between clay and silt loess. Region S is at the Weinan plain with slight erosion intensity. The soil there belongs to clay loess, its grain is relatively fine and the clay content is high. The modeled slopes are all assumed flat and barren without any vegetation cover. The calculated slope length is 30 m in each case study. We take the average values of the slope distribution provided by Jiang (1997) as slope gradient. In the long run, the slope inclination of regions N, M, S are selected as 288, 22.68, 14.38, respectively. The basic characteristics of soil have great effects on the runoff generation and soil erosion processes. According to the previous research (Jiang, 1997; Yang, 1996; Tang and Chen, 1994; Qi and Huang, 1997), we have determined the characteristic parameters of soil in the three typical regions in Table V. b. Determination of Rainfall Process Choosing uniform rainstorm, three single-peak rainstorms (Type A) and two multi-peaks (Type B) rainstorms, we have designed rainfall patterns for all circumstances. The whole duration of Type A rainstorm is 1 h and the total rainfall is 60 mm while 6 h and 120 mm for Type B rainstorm (with two peaks). The peak of type A1 rainstorm is in the middle of the duration, whereas the peak of Type A2 at the beginning and the peak of Type A3 rainstorm at the end. The higher peak of Type B1 rainfall is before the lower peak and the Type B2 rainfall is just the opposite. These diagrams are shown in Fig. 17 (Chen et al., 2001a). c. Runoff Generation Characteristics By using the runoff generation model mentioned above, we have calculated the runoff generation processes of six rainfall types, respectively (Chen et al.,

Table V Soil Characteristic Parameters Parameter

N region

M region

S region

Infiltration coefficient K (mm/min) Porosity us (%) Initial moisture weight percentage (%) Soil suction S (m) Roughness coefficient n

0.75 57 14.4 £ 50% 0.06 0.03

1.08 55 21.0 £ 70% 0.10 0.03

0.92 50 20.7 £ 70% 0.15 0.03


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Fig. 17. The types of rainfall process. Based on the analysis on rainfall characteristics on the Loess Plateau, six types of rainstorms are designed for calculation: a uniform rainstorm, three types of single-peak rainstorms (Type A) and two sorts of multi-peaks (Type B) rainstorms.

2001b), that is, a uniform rainstorm, three Type A rainstorms and two Type B rainstorms. The calculated results of the runoff generation ratio in every region and for every rain type are listed in Table VI. The runoff generation ratio is mostly above 50% for Type A rainfall, or even near 70% in some areas. In contrast, it is around 30% for Type B rainfall or near 50% at most. This result is consistent with the conclusion by Wang and Jiao (1996). We can find from the table that the runoff generation rate is increasing from the south to the north. In the context of runoff generation, we can predict that the erosion force in the northern region is stronger than that in the southern region under the same rainfall conditions. Furthermore, the comparison of different rain types shows that the runoff generation ratio of

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Table VI Runoff Generation Ratio of Every Typical Region Under Different Rainfalls Region Rain type

N region

M region

S region

Uniform A1 A2 A3 B1 B2

0.666 0.680 0.664 0.678 0.497 0.502

0.518 0.556 0.509 0.559 0.301 0.332

0.485 0.529 0.474 0.537 0.281 0.322

Type A rainstorm is always larger than that of Type B rainfall. And the later the peak of rainfall intensity appears, the larger the runoff generation ratio. Thus, the rainfall with later peak exhibits more powerful erosion capability than that with earlier peak. The runoff generation characteristics in different erosion intensity regions, including the unit width discharge q, water depth h, flow velocity u, shear stress t varying with time at the slope outlet for each rain type have also been studied as shown in Figs. 18 and 19 (Chen et al., 2001a), respectively. By carefully analyzing the runoff generation process in the Loess Plateau region under different types of rainfall conditions, we have obtained the following main conclusions: Under the same rainfall conditions, the runoff generation ratio in the Loess Plateau region tends to increase from the south to the north. At the same time, the hydraulic parameters such as the unit discharge, flow velocity, shear stress as major influencing erosion factors generally agree with this tendency. The runoff generation ratio of Type A rainstorm with shorter duration and higher intensity is larger than that of Type B rainstorm with longer duration and relatively lower intensity. The peak values of flow parameters for Type A rainfall are higher than Type B rainfall, whereas it lasts for shorter duration correspondingly. The effects of the two types of the rainfall manifest themselves differently in different regions. Generally speaking, the later the peak of rainfall, the higher the peak values of runoff generation parameters and the higher the erosion capability of sheet flows. 3. Dynamic Overland Flow Hydraulics a. Foundation of Dynamic Model While the surface of a hill slope appears undulant in the transverse direction, the overland flow may direct to lower area, and form concentration flow route,


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Fig. 18. Runoff generation characteristics of Type A1 rainfall in different erosion intensity regions, including the unit width discharge q, water depth h, flow velocity u, shear stress t varying with time at the slope outlet. Type A1: single-peak rainstorm. nor: region N, mid: region M, sou: region S.

which ultimately results in rill formation. Because of the fragment geomorphy in the Loess Plateau area, the reasonable simulation of this concentrated flow route on the surface landform of hillslope is of primary importance for prediction of runoff generation and soil erosion in this area. Obviously, a one-dimensional dynamic model is inadequate to properly simulate the flow routing caused by the landform. On the other hand, it is also infeasible to simulate the process of flow concentration by a two-dimensional full Saint Venant equation system because of the thin flow depth and the complex geomorphy. One potential numerical approach is to introduce twodimension grids on the slope surface, and the flow in a cell is simulated by

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Fig. 19. Runoff generation characteristics of B1 Type rainfall in different erosion intensity regions, including the unit width discharge q, water depth h, flow velocity u, shear stress t varying with time at the slope outlet. Type B1: multi-peaks rainstorm. nor: region N, mid: region M, sou: region S.

one-dimensional dynamic model. The different flowline routine is specified in every grid and the inflow discharge into and the outflow discharge from the adjacent grids is calculated. Based on this concept Scoging (1992) developed a distribution mathematical model of hillslope overland flow by determining the direction of flow according to the heights of the four corners of computational elements or cells. The flowline vector was represented as originating from the highest corner of a cell, with its magnitude and direction determined by the relative heights of four corners of the cell. The side intersecting the vector was considered the outlet of the cell for flow, assuming that all the water of


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the element flows out from this side. So every element exports water to the adjacent element, at the same time every element receives the water flow from the four adjacent elements. Using a one-dimensional difference equation obtained between the downstream cell and its several upstream neighboring cells, the process of the run-off and the concentrated water flow route was described for the whole area. Nevertheless, this method possesses an apparent drawback in flow routing. Actually, water in a cell tends to flow out through two lower side boundaries instead of one side boundary. Based on the analogue approach to previous investigation, we have developed a dynamic model that can handle the runoff concentrating process more adequately. As in Fig. 20, the slope is partitioned into rectangle grids. This so-called rectangle is the projection of the grids to the horizontal plane. Assuming the element plane as a bilinear element, represented as: z ¼ f ðx; yÞ ¼ ðax þ bÞðcy þ dÞ

ð4:41Þ

where z is the height of the element, x and y are the abscissa and ordinate, respectively. Grad f ðx; yÞ ¼

›f ›f iþ j ›x ›y

ð4:42Þ

Taking the slope gradient of the center point of the element as the slope gradient of the whole element, the representative slope gradient u can be obtained by:

Fig. 20. Sketch of two-dimensional grids and cell flow direction. (a) Two-dimensional grids (The projection of grids to the horizon plane forms rectangles). (b) Cell flow direction and four corner marks.

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sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ðz1 þ z2 2 z3 2 z4 Þ2 ðz1 2 z2 2 z3 þ z4 Þ2 tgu ¼ lGrad f ðx; yÞl ¼ þ ð4:43Þ 4dx2 4dy2 and the flowing vector direction angle g looks ›f ›f dx ðz1 2 z2 2 z3 þ z4 Þ ¼ tgg ¼ dy ðz1 þ z2 2 z3 2 z4 Þ ›y ›x

ð4:44Þ

where z1, z2, z3 and z4 are the heights of four corners, respectively; dx and dy values are the projected grid lengths on the horizontal plane in the x and y directions, respectively. Assuming that the water flow direction is controlled completely by the landform of hillslope, and the water interaction may not affect its flow direction, the water in every element is considered flowing out along the flow direction angle into the two adjacent grids. So the flow route may usually intersect with two sides of the cell. If the unit discharge is q, then the unit discharge along the x and y directions are: qx ¼ q cos g;

qy ¼ q sin g

Consequently, we can derive the governing equations of overland flow as follows: 8 ›h ›qy ›qx > > < ›t þ ›x þ ›y ¼ p cos u 2 i ð4:45Þ > > : q ¼ uh ¼ 1 h5=3 S1=2 ; q ¼ q cos g; q ¼ q sin g x y 0 n

b. Laboratory Experiments The process of runoff generation and flow concentration on a hillslope with rugged landform was investigated experimentally in the Chinese National Key Laboratory of Soil Erosion and Dryland Farming in Loess Plateau, the Northwestern Institute of Water and Soil Conservation of Chinese Science Academy, China. The drop-former type rainfall simulator was employed to generate artificial rainfall. Raindrops were formed at 16 m above the ground, falling down at a fixed speed near the surface of the test plot. The rainfall intensity is adjustable in the range of 15 –200 mm/h. The soil used in this experiment was the local loess of Yangling in Shanxi province of China. The soil was packed into a 320 cm long, 100 cm wide and 30 cm deep wooden-box with holes on the bottom, and controlled to reach a bulk


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density of 1.3 g/cm3. In the experiment, the slope gradient of the soil box was secured at 108. The initial surface landform is shown in Fig. 21 exhibiting three main grooves on the slope surface. The rainfall intensity adopted in this experiment was 1.6 mm/min, and the rainfall duration was 1 h. The runoff volume at the test plot outlet was measured, and the flow velocities at several specific points were also measured using stained method at different times from the beginning of runoff formation. c. Model Validation Using the proposed model, the process of runoff generation and flow route concentration on the test plot were simulated (Fig. 22). The model parameters used in simulation are shown in Table VII. Figure 23(a) shows the distribution of simulated flow depths on the entire test slope. In the figure, the position with larger values of water depth is on the right where the water starts to concentrate into a route. Compared to the experimental

Fig. 21. Initial geomorphic contours of the slope surface (Unit: mm). The test wooden-box is 320 cm long, 100 cm wide and 30 cm deep.

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Li Jiachun et al. Table VII Compute Parameters

Porosity of soil Initial water content Permeability coefficient Soil suction Roughness coefficient

50.27% 22.62% 0.1 mm/min 0.15 m 0.03

Fig. 22. Comparison of simulated and experimental results of runoff discharge. Line: simulation, dots: experiment.

result in Fig. 23(b), the agreement between simulated result and experimental observation is excellent, implying that the present model is capable of simulating the runoff generation and concentrated flow route on the hillslope.

4. Rill Erosion Estimation Rill erosion is a principal form of soil erosion caused by overland flow. Generally speaking, overland flow brings forth sheet erosion at first, then start to cause rill erosion with increasing flux. The rill erosion is commonly seen in the Loess Plateau area as rainstorm occurs on the steep slope there. Previous research


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Fig. 23. Comparison of simulated and observed flowline routine concentration on the slope as shown in Fig. 12(a) Simulated result of water depth contours (unit: mm). (b) Observation of flowlines on the slope surface.

has shown that the rills on the hillslope of Loess Plateau are usually as wide as 5 –20 cm with the depth less than 20 cm (Liu et al., 1998). After the rill occurs on the hillslope, the sheet flow will convert into concentrated flowline, and its depth, velocity and erosion capability all rapidly increase. Lots of research (Zhang, 1991; Zheng and Kang, 1998) demonstrated that the rill erosion intensity would be severe, even scores of times of the interrill erosion, and the sheet erosion can be ignored compared with the rill erosion. So understanding rill erosion mechanism is of primary significance for prediction of soil erosion. a. Experimental Research Generally, the formation of rill is irregular, and its length, width, depth and form are changing during erosion process. For non-uniform soil, rugged hillslope surface and unsteady flow, the generation and development of rill are more stochastic. The situation brings much difficulty in rill erosion study. In order to better understand the rill erosion law, a series of rill erosion experiments with soil

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Fig. 24. The sketch of experimental equipment. (a) Flume A: Soil box that is used to pack soil. In experiment, a 10 cm long, 5 cm wide and 2 cm deep rill from the outlet is artificially made along the center line of flume A. (b) Flume B: Trestle flume. Its slope gradient can be adjusted according to experimental requirement. During experiment, the flume A was fixed in the flume B.

flume was carried out in the Chinese National Key Laboratory of Soil Erosion and Dryland Farming in Loess Plateau, the Northwest Institute of Water and Soil Conservation of Chinese Science Academy, China. The experimental soil flume is shown in Fig. 24. During experiments, the soil was firstly packed into flume A 20 cm thickness, and then adjusted to reach actual bulk density of 1.3 g/cm3. A 10 cm long, 5 cm wide and 2 cm deep groove from the outlet is artificially made along the center line of flume A. Then we fixed the flume A in the flume B and adjusted the flume B to required slope gradient. The water flow is supplied at the top inlet of flume instead of artificial rainfall. This experimental method can easily manipulate the flow discharge (runoff rate) well, and assure the water flow in the groove directly to form rill erosion at the very beginning of experiment. Thus, we can directly obtain the relation of erosion rate and runoff by neglecting splash erosion. In addition, relatively larger amount of rill flow discharge may be obtained in a small scale test. The soil used in experiments is the local loess of Yangling in Shanxi province of China. Before the experiments, the soil was sprinkled in water so as to approximately saturate the surface soil. Five slopes of 5, 10, 15, 20, 258 and five flow discharges of 0.01 £ 1023, 0.025 £ 1023, 0.05 £ 1023, 0.075 £ 1023, 0.1 £ 1023 m3/s for each slope were designed in experiments. The rill erosion rate, rill flow velocity, depth and width were measured carefully. Part of experimental results demonstrating the relations among sediment transport rate, rill flow discharge and slope gradient are shown in Fig. 25.


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Fig. 25. The experimental results of rill erosion. (a) Sediment transport rate vs rill flow discharge in five different slope gradients (5, 10, 15, 20, 258). (b) Sediment transport rate vs rill flow discharge in five different discharge grades (0.01 £ 1023, 0.025 £ 1023, 0.05 £ 1023, 0.075 £ 1023, 0.1 £ 1023 m3/s).

Although the results fluctuate greatly owing to the inherent randomness of rill erosion, the rule that rill erosion rate varies with slope gradient and rill flow discharge is comparatively clear. Total erosion quantity increases as the slope gradient and the discharge of rill flow. Lots of research show that the width of rill has relatively good correlation with runoff discharge Q and slope gradient S. Supposing that there exists exponential relationship, we obtain the following formula by using multi-variable linear regression. B ¼ 0:596Q0:316 S20:23

ð4:46Þ

where B is the rill width (m), Q the discharge of rill flow (m3/s), and S the slope gradient. The correlation coefficient in the above formula is r ¼ 0.674. This formula is close to the relationship obtained by Zhang (1999) on Loess hillslope. We found that the wet perimeter of rill flow has better relation with runoff discharge Q and slope S. If the rill width B is replaced by wet perimeter p, then p ¼ 1:512Q0:371 S20:245

ð4:47Þ

It is obviously a better relationship compared with formula (4.46) with correlation coefficient r ¼ 0:9011.

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Li Jiachun et al. b. Rill Erosion Model

Similar to that in open channel, the sediment transport in the rill flow should also satisfy the mass conservation law. Supposing the sediment transport process in the rill is steady, the following equation can be utilized to express the rill erosion rate (Huang et al., 1996): dqs ¼ Dr þ Di dx

ð4:48Þ

where qs is the erosion rate per unit width, Dr the rill erosion rate, and Di the interrill erosion rate. The sediment coming from interrill is supposed to be totally transported through rill flow. This differential equation is also adopted in other physics-based erosion model, such as WEPP. Supposing the rill erosion rate is proportional to the difference of saturated sediment transport capacity and real sediment transport rate qs in rill flows, we have Dr ¼ aðTc 2 qs Þ

ð4:49Þ

Obviously, Tc represents the transport capacity of rill flows. a is a coefficient, its reciprocal, 1=a with a length dimension, implying the distance that sediment concentration of rill flow reestablishes from zero to the maximum capacity, i.e., saturated sediment transport. The expression of a should be determined by further study. Physically, two aspects should be pointed out in formula (4.49): one is that the erosion capability of rill flow has a limit value relating to its transport capacity, which is not more than aTc ; the other is that the erosion capacity of rill flow may decrease gradually as the sediment concentration increases. When the sediment concentration in rill flow is saturated, the rill erosion stops. The interrill erosion is negligibly small compared with rill erosion, so Eq. (4.48) is simplified in the following form: dqs ¼ aðTc 2 qs Þ dx

ð4:50Þ

Supposing that Tc is constant, the soil is homogeneous, and the sediment transport rate is zero at the inlet, we can solve Eq. (4.50) analytically: qs ¼ Tc ð1 2 e2ax Þ;

ð4:51Þ

which can be considered as a basic formula describing rill erosion. Then, the coefficient a turns out to be

a¼2

lnð1 2 qs =Tc Þ x

ð4:52Þ


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Since there is no formula of sediment transport rate especially for sheet flow and rill flow till now, the sediment transport rate in river dynamics is still widely used. Some scholars used experimental data of sheet flow and rill flow verified their validity, but unable to obtain consistent result between majority of them. Foster and Meyer (1977) and Alonso et al. (1981) recommended Yalin’s formula, Moore and Burch (1986) suggested Yang’s formula, Lu et al. (1989) considered Engelund and Fredsoe’s as the best and Govers (1992) thought Low’s was better. According to the experimental data, we find that the results given by Yalin are more reasonable for the soil of Loess Plateau. Yalin’s formula can be expressed as qs ¼ GY 0:5 ðY 2 Yc Þdðgdðs 2 1ÞÞ0:5 rs 0:635 lnð1 þ asÞ G¼ 12 Yc as 2:45 Y as ¼ 0:4 Yc0:5 21 Yc s

ð4:53Þ ð4:54Þ ð4:55Þ

where Y is the dimensionless shear stress, Y ¼ t=½ðrs 2 rÞgd ; t is flow shear stress, N/m2. Yc is the dimensionless critical shear stress force, Yc ¼ tc =½ðrs 2 rÞgd ; tc is the critical shear stress, N/m2. r is the density of fluid, kg/ m3, rs the density of sediment, kg/m3, s ¼ rs =ðrs 2 rÞ: d is the diameter of sediment, m, g gravitational acceleration, m/s2, R hydrodynamic radius, m. S0 ( ¼ sin u) represents the slope gradient, u is the angle of a hillslope. Generally, the soil is mainly separated by rill flow in the form of single particle or aggregation of some particles to be similar to the process of sediment transport in open channel flow. So we choose Dou’s (1999) formula to describe the critical shear stress tc : 2 " #5=2 " pffiffiffiffiffi #3 0 1=3 d r 2 r g 1 þ gh d d=d 5 s 0 0 2 43:6 tc ¼ k r gd þ ð4:56Þ r dp g0p d where k is a non-dimensional parameter, k ¼ 0.128, d0 the characteristic diameter (mm), dp the referenced diameter, and is taken as 10 mm, h the water depth, 10 the parameter of adherence force, for ordinary sediment 10 ¼ 1.75 cm3/s2, d the thickness parameter of water film, d ¼ 2.31 £ 1025 cm, g0 the dry density of sediment (N/m3), g0p is the stable dry density of sediment (N/m3). According to our research, g0p can be determined by the following formula:

g0p ¼ 0:68gs ðd=d0 Þn n ¼ 0:080 þ 0:014ðd=d25 Þ

ð4:57Þ

in which d0 ¼ 1 mm, d25 is the grain diameter, than which 25% of the total particles by weight is finer.

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Formula (4.56) can relatively express the sediment incipient motion law of various diameters well, such as coarse sediment, fine sediment, cohesive sediment, and light sediment. For fine sediment-like soil, this formula specially considers two forces, the cohesive force between particles and the additional pressure of water film. In addition, the slope gradients on the Loess Plateau are usually large, so the influence of slope on soil particle’s incipient motion should be taken into account. In accordance with Zhang et al. (1989), we have the critical shear stress on a slope:

d0 tc ¼ k r dp 2

1=3 "

pffiffiffiffiffi ## " rs 2 r g0 5=2 10 þ g cos u hd d=d 3:6 g cos ud þ r g0p d ð4:58Þ

Using Eqs. (4.52), (4.53) and (4.58), a and its governing law can be determined by means of our experimental data on rill erosion. Let La ¼ 1=a; the experimental results show that the value of L has a trend to increase with the increase of discharge of rill flow and slope gradient, but the correlation is not very satisfactory. Considering the influence of various factors, such as the effective shear stress of the rill flow t 2 tc ; the hydraulic radius R, the velocity u, the diameter of soil particles d, the density of soil particles under water ðrs 2 rÞg; and the slope gradient S0, the relation can be acquired by using multi-variable regression of experimental data under the assumption of an exponential relation. 0:148 La t 2 tc u 21:022 1:550 pffiffiffiffi ¼ 15165:8 S0 R ðrs 2 rÞgd gd

ð4:59Þ

where the correlation coefficient r ¼ 0.9314. The predicted value given by the above formula is compared with experiments in Fig. 26. The rill erosion model is validated by a series of indoor artificial rainfall experiments which were accomplished by Li Zhanbin at the Chinese National Key Laboratory of Soil Erosion and Dryland Farming in Loess Plateau, Northwestern Institute of Water and Soil Conservation of Chinese Science Academy. The test plot used in the experiments is 320 cm long and 100 cm wide. According to lots of experiments, there are three rills to distribute evenly in a 1 m width hillslope, and the influx coefficient of rill flow is supposed to be 0.8. Two sets of experimental results are shown in Fig. 27. The agreement between the predicted results and the experimental data shows that the present rill erosion model may effectively simulate rill erosion process and predict erosion rate on the Loess Plateau.


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Fig. 26. Comparison of calculated (line) and experimental results (dots) of La value. La has a length dimension, it represents the distance that sediment concentration of rill flow re-establishes from 0 to maximum value, i.e., saturated sediment carrying capacity.

c. Rill Erosion Characters The rill erosion process is simulated with the present model according to variable slope gradient, slope length and rain intensity. Figure 28 shows the simulated results of rill erosion rate varying with time under different slope gradients, slope lengths and rain intensities.

Fig. 27. Comparisons of simulated and artificial rainfall experimental results of rill erosion rate. (a) Experimental condition: rain intensity: 1.0 mm/min, slope gradient: 208 (b) Experimental condition: rain intensity: 1.71 mm/min, slope gradient: 308.

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Fig. 28. Simulated results of rill erosion rate with different slopes, rill lengths, slopes and rain intensities. (a) Rill erosion rate in seven different slope gradients (5, 10, 15, 20, 25, 30, 358). (b) Rill erosion rate in six different slope lengths (5, 10, 15, 20, 30, 50 m). (c) Rill erosion rate in six different rain intensities (0.5, 0.75, 1.0, 1.5, 2.0, 2.5 mm/min).

Analyzing the simulated results, we may come to the following main conclusions: The rill erosion rate is proportional to the length of slope. Because the infiltration rate decreases and runoff increases gradually, the sediment transport rate increases rapidly at the beginning and then slows down gradually till to a stable value. Correspondingly, the accumulative erosion quantity increases drastically at first and then increases linearly. The longer the slope is, the more


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obvious the trend is. The fact shows that the main factor influencing rill erosion is runoff discharge as different slope lengths reflect the different runoff discharges at the outlet. The sediment transport rate and the accumulative erosion quantity both increases with slope gradient. Generally speaking, the runoff discharge grows as slope gradient increases. However, it directly leads to the increase of shear stress. As a result, rill erosion rate may increase as slope gradient increases. The influence of rain intensity on the rill erosion is similar to the slope gradient. The erosion rate and accumulative erosion quantity increases with rain intensity almost linearly. It is accounted for by larger runoff discharge during intensive raining. In addition, we find that the influence of rain intensity variation on the erosion rate in slight rainfall is more obvious than that in heavy rainfall.

C. Terrestrial Processes in Arid Areas, the Northwest China It is reported by UNEP that desert land formation occurs at the rate of approximately 6 million hectares annually owing to global climate change and human activities. Meanwhile it has also become a bottleneck in the economic development of Northwest China. For example, Xinjiang as the largest province in China is rich in petroleum, gas and metal. Although local precipitation seems little, there is enough sunshine and water supply from rain in mountain area and melted snow. Therefore, we have cultivated a large expanse of oasis though many environment problems such as shrinking of rivers, drying of lakes and invasion of desert have become more severe than ever (Wu, 1992). An additional example is the ‘San Bei’ windbreak aiming to protect the railway from being buried by advancing desert. In the 1950s, a few kinds of drought-enduring plants surprisingly survived using cellular straw technique at the southern border of the Tenggeli Desert (Figs. 29 and 30). During the infant stage, they grew normally, thus forming a windbreak belt to hinder the movement of the desert (Li and Ouyang, 1996). Nevertheless, these plants started to wither due to the increase of water requirement and the formation of biological crust skin. Hence, the study of energy and water cycling at the interface of atmosphere and land is certainly beneficial to improving environment quality and developing regional economics. Of course, the objective of terrestrial process study is twofold. A better understanding of momentum, energy and mass exchange between land and atmosphere may provide reasonable parameterization scheme for meteorologists in AGCM as well.

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Fig. 29. Artemisia ordosica—a typical drought-enduring plant which was fitted by the cellular straw technique in the southern border of the Tenggeli Desert in order to prevent desert invasion and protect the Bao-Lan Railway since 1950s.

Fig. 30. Location of windbreak, where black line is the BaoLan rail way, grey line is the Yellow river.


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1. SPAC System Models We have defined soil –plant –atmosphere as a Continuum (SPAC) to facilitate us to apply classical fluid mechanics or heat and mass transfer theories available in this field. There have been a number of models used for describing air, water and heat transport in the SPAC system so that micrometeorological states in vegetation and soil or parameterization scheme can be acquired. Generally speaking, these models may be classified as the following three categories: a. Force-Restore Method Basically, the force-restore model is of hydrologic nature. With planetary parameters measured at the anemometer height at one site, the force-restore model aims to predict water and heat motion in soil at different sites (Lin and Sun, 1986). Deardorff (1978) has compared different models among simple schemes and proved that the force-restore method is most satisfactory. Sun and Lu (1992), Liu et al. (1996) and Sun et al. (1998) extended the theory to an area partly covered by vegetation. Liu and Sun have made a critical review on ecological aspects of water cycle by the same approach in 1999 (Liu and Sun, 1999). By considering the diurnal and seasonal variation in temperature and different heat diffusivity, the soil is divided into three layers, i.e., surface layer d, diurnal layer depth d1 (5 –10 cm) and annual layer of depth d2 (1 –2 m):

a

dTg 2Gð0; tÞ 2p ¼ 2 ðT 2 TÞ cd1 dt t g dT 2Gð0; tÞ ¼ dt cd2

where Tg and T are the temperature at the surface layer and depth-averaged temperature, respectively, Gð0; tÞ the heat flux into the soil directly derived from the energy balance at the ground surface, t the diurnal period, c the volumetric heat capacity of soil and a a weighted factor. Based on the conservation principle, we may derive water cycling equations by dividing soil into shallow and deep layers with thickness d 1 and d 2 in a similar way: du E d 1 1 ¼ Ib 2 b sb þ ðIc 2 U1 Þsc 2 q12 dt rw ð4:61Þ d u 2 d 2 ¼ q12 2 q23 2 U2 sc dt where u1 and u2 are the volumetric moisture contents in the upper and lower layers, U1 and U2 are the water uptake from upper and lower layers through roots, Eb is

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the evaporation at bare soil, Ib and Ic are infiltration at bare or vegetated land occupying sb and sc of total area. Finally, q12 and q23 are the water transfer between layers. In particular, we have noticed that real evaporation may differ from potential evaporation due to various factors such as: the effects of border in 20 m, the species and height of plants (usually conifer evaporated less than deciduous trees), the period of growth (coverage is larger during elongation and evaporation more in blossom and milk period). Generally, actual evaporation is estimated by potential evaporation multiplied by evapotranspiration coefficient: Em ¼ K c E0

ð4:62Þ

where E0 ¼

DRn þ gEa Dþg

ð4:63Þ

in which D is saturated vapor pressure, g dry – wet constant, Rn denotes net radiation, Ea means drying force of air. Water is insufficient to supply plants growing in arid areas so that the evaporation is reduced correspondingly. As a matter of fact, that is one kind of physiological response to an unfavorable environment. Insufficient supply of water leads to lack of water in the plant body so as to close their stomata. The task for us is to determine two parameters: critical water capacity uk (about 75% of field capacity) under which the evaporation is greatly reduced and the assimilation process almost ceases; the water capacity for zero evaporation up ; which is even lower than wither point. Then, the evapotranspiration coefficient may be written as (Li et al., 1997): 8 9 u . uk ; > > > > < = K 0c ¼ uk $ u . up ; ð4:64Þ > > > > : ; q # up

b. Turbulent Modeling As a matter of fact, people have recognized that the transfer of momentum, energy and mass is essentially turbulent. However, the force-restore model has entirely neglected the atmospheric motion, thus merely predicting the evolution of physical quantities in the soil and fluxes at the ground surface. The atmospheric states are heavily dependent on the observation in the surface layer. As early as in 1960s’, Tan and Ling (1961) firstly applied turbulent models in the turbulence research in canopy. By regarding branches and leaves


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of forest or crops as a sink of momentum and assuming the mixing length proportional to the height z, they successfully yielded wind profile with an inflexion at the top of canopy, which is in good agreement with observations. Mellor and Yamada (1974) developed a hierarchy of secondorder models to simulate unsteady atmospheric boundary layer with varying thermal stratification. In regard to the coupling with soil layer, earlier crude model only used empirical relation of aerodynamic and water vapor resistance. Deardorff (1978) further proposed a concept of surface wetness function to represent this kind of interaction between air and soil. In 1980s’, Naot and Mahrer (1982) and Ten Berge (1990) have established a coupling model for heat and water transfer in the lower atmosphere and the upper soil over bare or vegetation-covered field. Based on the previous investigations, we have constructed a comprehensive coupling model (Xie et al., 1998; Li et al., 1999a) both for AGCM parameterization in climate prediction and agricultural production enhancement or environment protection in micrometeorology with revised source and sink terms. Then, the Reynolds Averaged Navier –Stokes equations including rotation and stratification effects for horizontally homogeneous atmospheric flows look like: ›u › tx ¼ f ðv 2 vg Þ 2 2 Cd AðzÞulul ›z r ›t ›v › ty ¼ 2f ðu 2 ug Þ 2 2 Cd AðzÞvlvl ›z r ›t ! ›T › H b ðzÞ ¼2 þ 2AðzÞðTl 2 TÞ=r ›z rC p ›t ð4:65Þ ›q › E ¼2 þ 2AðzÞðql 2 q Þ=ðrs ðzÞ þ rb ðzÞÞ ›z r ›t ty ›v ›e t ›u g H › ›e ¼ x þ þ KM þ T rCp ›z ›t r ›z r ›z ›z 2

ðCeÞ3=2 þ Cd AðzÞðu2 þ v 2 Þ3=2 lM

where u, v, T, q are the mean velocity components, potential temperature and specific humidity, respectively. The symbol overbar indicates the Reynolds Averaged Quantities. ug and vg denote the components of geostrophic wind, while f ¼ 2V sin f is the Coriolis parameter with V ¼ 7.27 £ 1025/s, f the local latitude. r is the density of atmosphere, CP the heat capacity, g the acceleration of gravity. e the turbulent kinetic energy, tx and ty are shear stress

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components, H and E the sensible and latent heat transfer fluxes, respectively. For flows in ABL, the Boussinesq approximation is usually assumed, meaning atmosphere is regarded as incompressible except in the buoyancy term. And people tend to use K – e model to close RANS equations, in which the turbulent diffusivity for momentum, energy and mass can be proportional to the mixing length written as: KMM;H;E ¼ lM;H;E ðCeÞ1=2

ð4:66Þ

where the mixing length are dependent on the stratification states and their expressions are omitted here for brevity. Cd is the aerodynamic drag, rb and rs are the impedance corresponding to leaf boundary layer and stomata, respectively, AðzÞ denotes leaf area index, which turns out to zero for bare land. Similarly, we may derive the equations for heat and water in soil: ›T › ›T ¼ l ›z ›t ›z ›u › ›pðu; TÞ › ¼ Kðu; TÞ rw 2 rw g Kðu; TÞ 2 Sðz; TÞ ›z ›z ›t ›z where T and u are temperature and volumetric water content in soil, l the thermal conductivity, C the volumetric heat capacity, rw the density of water, p the pressure potential and Kðu; TÞ the hydraulic conductivity. Sðz; tÞ is the water absorption function for roots (Molz, 1981) written as: E ðtÞLðzÞf ðuÞ Sðz; tÞ ¼ ðLcr LðzÞf ðuÞdz

ð4:68Þ

0

where Ec ðtÞ is the flux of transpiration, LðzÞ the root density distribution function, f ðuÞ a function associated with water absorption resistance of root implying reduction of water absorption as soil moisture is less than field capacity until wilting point. With seven variables, we specify the corresponding conditions as vanished normal derivatives at upper and lower boundaries, respectively. In addition, we require equilibrium in energy transfer at the interface between atmosphere and land to determine the temperature at the ground, thus leading to: Rn þ H þ LE ¼ G

ð4:69Þ

where Rn is the net radiation, H and E are sensible and latent heat flux and finally G is the ground heat flux. No doubt, the iteration is unavoidable in the computation for satisfying the last condition.


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c. Large Eddy Simulation During the last two decades, people have commonly recognized the role of large eddy simulation (LES) as a powerful research tool, which is most suitable for unsteady three-dimensional complex turbulent flows in industry and natural environment. The main point in LES is that the large-scale motion is resolved while the small scale motion is modeled or, in geophysical terminology, parameterized. On the one hand, LES obtains three-dimensional unsteady instantaneous flow fields to yield any statistical quantities instead of the Reynolds averaged ones, which relies on great many empirical constants. On the other hand, LES seems more perspective to simulate the complex flows in the foreseeable future due to its moderate CPU time and storage memory demands. Initial applications are mainly focused on the geophysical fluid flows, in particular, the convective atmospheric boundary layer (Deardorff, 1970). Recently, the scope of research has extended to additional geophysical as well as industrial flows which exhibit complicated features such as stable boundary layer, turbulence in plant canopy, compressible flows, acoustics and combustion (Galperin and Orszag, 1993; Piomelli 1999; Li, 2001b). The investigation of turbulence over and in plant canopy is of significance due to demand in understanding terrestrial processes. Previous research was based on the models such as biosphere – atmosphere transfer scheme (BATS) and simple bioshere model (SiB). Turbulent modeling and LES for planetary boundary layer was introduced in the last decades (Deardorff, 1970; Moeng, 1984). With the same vegetation model as in the previous research, we present a new subgrid model for turbulence within and over vegetative canopy in this paragraph. The earliest subgrid scale (SGS) model of dissipative eddy viscosity is due to Smagorinsky (1963). As we know, SGS stress tensor can be decomposed into isotropic and an-isotropic components. The former determines the rate of global SGS dissipation, namely, the net energy flux from the resolved to the unresolved scales while the latter determines the mean shear stress and therefore the mean velocity profile (Baggett, 1997). Modeling dissipation and stress are two different tasks fulfilled by different models. It has been shown that modeling the former is absolutely of primarily significance. It seems that LES is able to reproduce characteristics of moderately complex turbulent flows such as mean velocity and rms velocity fluctuation even if they are mostly based on the variants of the Smagoringsky mode. To describe the energy cascade drain, Smagorinsky (1963) assumed that SGS stress may be represented as:

tij ¼ 22nT S ij

ð4:70Þ

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where 1 S ij ¼ 2

›u j ›u i þ ›x j ›x i

! ð4:71Þ

is the strain rate, nT the eddy viscosity defined as the product of the mixing length of grid size D and the velocity difference at this length scale: nT ¼ ðCs DÞ2 lSl

ð4:72Þ

¼ ð2S ij S ij Þ1=2 lSl

ð4:73Þ

and

The determination of the Smagoringsky constant Cs ranging from 0.18 to 0.23, even reduced to 0.10 for most practical flows is crucial for correct estimation of dissipation. The presence of shear near the wall or in transitional flows diminishes the dissipation. Since the Smagoringsky model is valid only for convective ABL far above the underlying surface, whereas structure –function model becomes more suitable by considering overestimation of dissipation due to intermittence and underdevelopment of small-scale motion in shear turbulence (Lesieu and Metais, 1996). Then, we assume the viscosity coefficient as:

nM ¼ bnM1 þ ð1 2 bÞnM2 23=2

nM2 ¼ 0:105CK Fðx; DÞ ¼

DFðx; DÞ1=2

3 1X ½luðxÞ 2 uðx þ Dxi ei Þl2 þ luðxÞ 2 uðx 2 Dxi ei Þl2 6 i¼1

ð4:74Þ ð4:75Þ ð4:76Þ

where nM1 and nM2 are those for convection or shear dominant region, b a ratio ranging from 0 to 1 representing situation in the vicinity of or far from canopy top. F is the local structure-function of filtered velocity field of the second order (Li and Xie, 1998).

2. Observations Despite of expensive investment, field experiments are always indispensable for model validation in the atmosphere – land process research. The most influential international projects in this field are First ISLSCP Field Experiment (FIFE), Hydrological Atmosphere Experiment (HAPEX) and Heihe River Basin Field Experiment (HEIFE) under the framework of World Climate Research Program (WCRP) and International Biosphere – Geosphere Program (IGBP)


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(Ye and Lin, 1995). In FIFE, synchronized observations of varying terrestrial process due to grazing and wildfire were conducted within an area of 15 £ 15 km2 at the middle of Kansas State, USA by satellite, airplane and ground-based equipment. HAPEX-MOBILHY over a field of 100 £ 100 km2 in the southwest of French, where scattered were mixture crops or forest (mainly pine) was implemented by an observatory network equipped with automatic meteorology and hydrology apparatus. However, HEFEI was firstly initiated by famous Chinese meteorologists D.Z. Ye in May 1986 at a WCRP meeting in Geneva. Chinese and Japanese scientists were jointly performing 5 years’ observations in the middle reach of the Heihe River along Hexi corridor, Gansu province with an area of 70 £ 90 km2. In particular, people have implemented intensive observation at the border of desert, gobi and oasis in order to better understand the ABL structure, radiation, water budget and turbulent flux in arid areas. Objectives of the largescale projects mainly aim at global and regional climate research as well as meteorology study. Since 1950s, Chinese Academy of Sciences started to establish field experiment observatories. Up to now there have been tens of observatories available, representing typical ecological systems such as crops, forest, grassland, meadow, desert, glacier and marine biology. They have become centers of longterm observation and scientific research and a window open to the world community (CAS, 1988). In regard to atmosphere –land process study, we have a constant collaboration with Shapotou Desert Research Station, Yucheng Experimental Station and Aksu water Budget Experimental Research Station by joint projects. For instance, Shapotou Station located at 378270 N and 1048570 E belongs to the semi-arid area of northwest inner land with a hot, dry and windy climate. The average temperature in this area is 9.68C and the unevenly distributed precipitation, mainly in July to September, is 186.2 mm on average. Due to low relative humidity (usually 40%, the lowest is 10%, especially in spring) and strong winds (averaging 5 m/s over 200 days), sand transport is very severe. There is little vegetation coverage because of deep underground water table and poor water-holding ability of sandy soil (usual water capacity is 2 –3%). One or two experiments have been conducted nearly every year. Apart from routine weather and hydrology observations by local engineers, we have performed different kinds of measurements by a MAOS-1 automatic meteorology observation system, including net pyrradiometer by TBB-1 sensor, heat flux into ground by CN-3 thermal flux plate, ground and soil temperature by HBW-2 and HBW-2B sensors. To explore boundary layer structure, tower-mounted instruments at different heights are used: for instance, specific moisture by HTF-2 ventilated psychrometer sensor and wind speed cup or sonic anemometers. In dense vegetation layer, we would rather use more sensitive wind

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sensor VF-1 with magnetic bear and EC9-b wind direction sensor with high frequency response. As for water content in soil, we usually measure it by weighting augered soil and evapotranspiration by Lysimeter for low moisture cases (Kaimal and Finnigan, 1994; Li and Ouyang, 1996; Yao et al., 2001). 3. Discussion on Water and Air Motions in SPAC System a. Water Requirement of Plants Numerical simulation has been conducted for the fields with bare soil or covered with Caragana korshinskii or Artimisia ordisica during 20 –27th, September, 1993. The soil there consists of fine sand containing little organic matter, calcium carbonate and nutrient. Its field capacity of water is 3.34 –3.96% and wither capacity 0.7%. We have compared the simulated soil temperature and water content in agreement with measurement. The variations of evapotranspiration over Caragana korshiskii and Artemisia ordosica have been shown in Figs. 31 and 32 (Li et al., 1997). We find that the evapotranspiration by Caragana korshinskii is larger, implying that Artemisica ordosica with little water requirement among drought-enduring plants is preferred in order to save water.

Fig. 31. Variation of evapotranspiraton from Caragana korshiskii with time solid line indicates numerical data; dash line indicates observation data at Shapotou during Sepp. 20–27, 1993.


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Fig. 32. Variation of evapotranspiraton from Artemisia ordosica with time solid line indicates numerical data; dash line indicates observation data at Shapotou during Sepp. 20–27, 1993.

Quantitatively, the total amount of evapotranspiration for Caragana korshinskii is around 1.4 mm/d, almost double of that for bare soil. The evapotranspiration for Artemisia ordosica is in between. Usually, evapotranspiration may reach maximum at 15:00 p.m. and then reduce to minimum late in the night. A drizzle is responsible for negative evapotranspiration during the seventh night on September 25, 1993. b. Roles of Dry Layer and Biological Crust Skin As stated in the previous paragraph, 68.7% of precipitation occurs in June to September. Annually, 50 days on average are rainy with 10 days larger than 5 mm and seldom continuous precipitation. We may find that there are two factors in moisture distribution in soil to restrict water supply and plant growth. Figs. 33 and 34 (Li et al., 1997) provide volumetric capacity of water in the soil by numerical simulation and observations. Obviously, a dry sand layer with moisture only around 1% can be identified at the depth of 6 –7 cm and the most humid layer occurs at the depth of 50 cm or so. The reason for the phenomenon is that little precipitation and fierce evaporation in arid areas prevent water from penetrating into the deep layer. If precipitation is less than 8 mm, it is even not enough to wet upper dry sand and therefore regarded as ineffective. Moreover,

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Fig. 33. Variation of soil moisture at the surface, depths of 50 and 120 cm with time. Solid line indicates numerical simulation; blank circles, triangles and squares denote measurements at different depths at Shapotou during Sep. 20–27, 1993.

Fig. 34. Vertical profile of soil moisture. Solid line indicates numerical simulation; blank circles denote measurements at different depths at Shapotou during Sep. 20–27, 1993.


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deeper underground water may not reach soil of shallow layer. On the other hand, there has formed a biological crust skin in the elder plant region due to dust deposition and rotted fading material accumulation in microorganism environment. Consisting of 12.6– 28.7% of silt and 1.78– 3.88% of clay grains and covered with moss, the sponge-like biological crust skin usually may contain much more water than sand. Despite its capability of erosion resistance, the crust skin reduces water infiltration into deep layer. Observation shows that soil water content in the elder plant region with thicker crust skin of 1.5– 2.0 cm is less than that with thinner crust skin in the younger plant region. The forgoing described climate and soil water distribution factors account for the continuous reduction of moisture in deep soil layer of 100 –200 cm in elder plant region, which leads to the overspread of shallow root plants and the degeneration of deep root trees. Generally, rare vegetation flourishes and dense vegetation fades. Therefore, the optimal distance between individual plants is proved 1.5 m instead of 1 m (Li and Wu, 1998; Yao et al., 2001).

c. Turbulent Structure Within and Over Vegetation Canopy We have simulated a region of 192 m £ 192 m £ 64 m with 96 £ 96 £ 32 grids, and integrated 6400 time steps of 0.1 s by LES. The computations are conducted on SGI Origin 2000 parallel supercomputer at LASG, CAS for 11 h for each case. Convective and weakly convective ABL with the Monin– Obukhov length 2 40 m, 2 700 m and LAI 2, 5 have been calculated as examples (Li and Xie, 1999). Mean horizontal velocity profile, Reynolds stress and turbulent kinetic energy have been simulated and well compared with both observation and simulation results (Shaw and Schumann, 1992; Shaw et al., 1988; Patton, 1997) (see Figs. 35 –38, Li, 2001a). This model also exhibits smaller fraction of SGS energy than Patton’s (1997). Such organized structure of ramp pattern in temperature scalar field has been observed for strong convective ABL by Gao et al. (1989). It means that a sharp sloping frontal between relatively warm, humid air being expelled from the forest and cooler, drier air being swept into the canopy from aloft. This kind of turbulent structure caused by strong velocity shear is responsible for the exchange of warm and cold air in ejection and down sweeping process. In the simulation of turbulence within and above the vegetation canopy, we have also identified such structure for strong convective situation by this new model. About 15% of the tree height over the canopy, we may find a slant front of width 3 –6 m with drastic temperature variation separating warm and cold regions (see Figs. 39 –40, Li, 2001a).

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Fig. 35. Comparison of longitudinal velocity profile within and over a forest with LAI ¼ 2.

Fig. 36. Comparison of longitudinal velocity profile within and over a forest with LAI ¼ 5.


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Fig. 37. Comparison of Normalized Reynolds stress within and over a forest by LES, where blank squares denote numerical simulation for resolved and unresolved components; blank circles denote numerical simulation by Shaw et al. (1988).

V. Concluding Remarks As we have previously stated in the present article that although people have come to adequately recognize the significance of environment protection and sustainable development, China is still confronting twofold pressure. One is from swift economic development at the annual rate of 7 –8% in GDP and endeavor to complete the transition to a developed country in tens of years. Another is from delayed growth peak in population despite of successful family planning policy during past decades. Major sources to generate environment hazards in today’s China are found to be unfavorable distribution in water resources and coal burning. Certainly, radical alleviation of these problems can substantially ameliorate China’s living surroundings in future. Resolution of these environmental issues that are obviously interdisciplinary, of multiple scales and multi-composition in nature needs joint efforts of scientists from relevant communities of scientific branches. The central tasks of scientists in environmental fluid mechanics circle should be at first focused on clarifying

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Fig. 38. Comparison of Turbulent kinetic energy within and over a forest by LES, where blank squares denote numerical simulation for resolved and unresolved components; solid crosses denote numerical simulation by Patton (1997). The present results with smaller proportion of subgrid energy are better than Patton’s results.

dynamic processes to better understand how natural flows in atmosphere, hydrosphere, geosphere and biosphere accompanied with momentum, energy and mass transfer occur, instead of direct prediction or forecast of an environment event from the very beginning. Therefore, the proper objective of their research aims at qualitatively describing environment episode or designing typical experiments on small scale in order to reveal the mechanism implied in them. Then, we should manage to answer these questions, which remain open to us or still in controversy. Through this kind of process-based research, we are now in a position to improve existing somewhat empirical engineering approaches and enhance quantitative accuracy in prediction. Finally, the numerical simulation based on thus established models, which have been validated and proved credible by comparison with observation, becomes such a powerful tool for prediction, assessment and manipulation of environmental events in order to partly replace expensive long-term and large scale field experiments. The ultimate goal may very likely be to provide a scientific foundation for policy-makers in decision from macroscopic and strategic point of view. To illustrate this process-based research approach, three typical examples associated with the Yangtze River Estuary, Loess Plateau and Tenggli Desert


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Fig. 39. Velocity field and thermal contours nearby a forest canopy, where a temperature ramp structure can be clearly identified for a strongly convective boundary layer (L ¼ 240 m).

Fig. 40. A typical ramp structure of temperature over a forest canopy for a strongly convective atmospheric boundary layer.

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environments have been dealt with, respectively. A theoretical model of vertical flow field accounting for runoff and tide interaction has been established to delineate salinity and sediment motion in the estuarine region. We have simulated diurnal and seasonal variations of salt front and turbidity maximum, which are responsible for the formation of mouth bar at the outlet of a river and the evolution of ecological environment there. A kinematic wave theory combined with the revised Green – Ampt infiltration formula is applied to the prediction of runoff generation and erosion in three types of erosion region on the Loess Plateau in excellent accord with observational reality. A new rule of runoff routing we put forward is able to properly recover two-dimensional overland flows on rugged slope. Moreover, the influential factors of dominant rill erosion compared to inter-rill erosion can be used as valuable references in returning reclaimed sloping land into vegetation covered field as an effective measure of erosion prevention. Three approaches describing water motion in SPAC system at different levels have been improved by introducing vegetation sub-models, which have ability to reflect momentum, moisture and heat transfer between atmosphere and land in arid areas. As drought-enduring plants grow up, the water requirement is necessarily rising. However, we have found that the formation of a dry sandy layer and biological crust skin are additional primary causes leading to deterioration of water supply and succession of ecological system. To enlarge distance between individual plant, to select water-saving trees and to artificially break crust skin therefore seem to be suitable measures for windbreak to survive for a longer time. Facing such a tough task in a country with vast territory and more than one billion of population, we certainly need sustainable efforts in research and in harness of natural environment. As a result, we ultimately achieve sustainable development in economy and society. At the same time, we should also take an active part in the study of those problems relevant to global environments and industrial processes.

Acknowledgments We wish to thank Professor Z. M. Zheng, who initiated the research in environmental fluid mechanics at the Institute of Mechanics, CAS as early as in 1980s. Parts of computation and experiments were accomplished by Drs. Z. T. Xie, L. Chen and T. Wang when they were working at this institute or as Ph.D. students. Professor D. L. Yao et al. have undertaken systematic observation associated with land – atmosphere interaction. We acknowledge Professor C-H Moeng, NCAR, Boulder, Colorado for generous permission of using their LES code. Our sincere gratitude is especially extended to Professor T. Y. Wu for


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his encouragement and advice during the preparation of this article. The financial support of the National Natural Science Foundation of China under the grant No. 19832060, 10002023 and 10176032 is highly appreciated. We are also grateful to the support of Chinese Academy of Sciences and long-term collaboration with the Shapotou Desert Research Station, Yucheng Experimental Station and Acsu Water Budget Experimental Research Station, CAS in observation.

Appendix: Acronyms ABL BATS CAS CBL CFC DMS FIFE HAPEX HEIFE HWR HWS IGBP INDNR ISLSCP LAI LES LUCC LWR LWS MRP NCAR SBL SGS SiB SPAC UNCED UNCHE UNEP WCRP

Atmosphere Boundary Layer (p230) Biosphere – Atmosphere Transfer Scheme Chinese Academy of Sciences (p297) Convective Boundary Layer ChloroFluroCarbons (p237) DiMethyl Sulphide First ISLSCP Field Experiment (p296) Hydrological Atmospheric Pilot EXperiment program EEIhe river basin Field Experiment High Water Rapid (p241) High Water Slack International Geosphere –Biosphere Program International Decade of Natural Disaster Reduction (p225) International Satellite Land Surface Climatology Project Leaf Area Index (p301) Large Eddy Simulation (p295) Land Use and Cover Change Low Water Rapid Low Water Slack Middle Route Project (p227) National Center for Atmospheric Research Stable Boundary Layer Subgrid Scale model Simple Biosphere model Soil– Plant –Atmosphere Continuum (p291) United Nation Conference on Environment and Development United Nation Conference on Human Environment (p224) United Nation Environment Program World Climate Research Program

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Author Index

Numbers in italics refer to pages on which the complete references are cited

Balke, H., 124, 126, 146, 153, 161, 163, 172, 173, 174, 176, 197, 201, 204, 205, 208 Banerjee, S., 98, 118 Barnett, D. M., 127, 128, 132, 133, 134, 135, 138, 142, 147, 153, 154, 156, 161, 172, 172, 185, 190, 191, 201, 205, 208, 209, 213 Bartman, A. B., 45, 76 Bazhekov, I. B., 88, 119 Beckermann, C., 107, 114 Belitsky, G. R., 52, 75 Beniston, M., 232, 302 Benveniste, Y., 127, 205 Beom, H. G., 191, 193, 205 Berenson, P. J., 112, 114 Bergersen, B., 53, 76 Berlincourt, D. A., 123, 129, 131, 136, 205 Bernard, J. N., 220, 232, 302 Bigg, G. R., 219, 220, 232, 302 Birkhoff, G., 64, 76 Blake, J. R., 84, 114, 118 Boal, D., 53, 76 Boettinger, W. J., 106, 120 Bogomolov, V. A., 52, 76 Boris, J. P., 88, 118 Borisov, A. V., 16, 76 Bossis, G., 84, 115 Bouissou, Ph., 106, 108, 114 Boulton-Stone, J. M., 84, 114 Brackbill, J. U., 93, 114 Bradford, J. M., 278, 303 Bradley, S. G., 86, 114

A Abo-Shaer, J. R., 26, 75 Acrivos, A., 85, 118, 120, 224, 227, 228, 304 Adler, M., 43, 75 Ahmad, N. A., 106, 120 Airault, H., 45, 75 Akamatsu, K., 195, 205 Al-Rawahi, N., 88, 94, 107, 114, 120 Alguero, M., 186, 206 Almgren, R., 105, 114 Alonso, C. V., 279, 302 Alshits, V. I., 127, 205 Amberg, G., 107, 120 Ananth, R., 106, 108, 118 Anderson, D. M., 83, 114 Anglin, J. R., 26, 75 Antes, H., 201, 206 Aranson, S. Kh., 52, 75 Aref, H., 5, 17, 19, 22, 29, 68, 74, 75, 78, 85, 120 Ashgriz, N., 93, 114 Atluri, S. N., 191, 193, 205

B Bahr, H. A., 124, 126, 153, 161, 163, 172, 173, 174, 176, 197, 201, 204, 208 Bai, R., 98, 118 Baker, G. R., 84, 114

307

308

Author Index

Brady, J. F., 84, 115 Braun, R. J., 106, 120 Brenner, M. P., 228, 302 Brevig, P., 84, 120 Brown, R. A., 107, 120 Bueckner, H. F., 156, 159, 194, 201, 202, 205 Buijs, M., 124, 205 Bunner, B., 88, 94, 98, 99, 115, 120 Burch, G. J., 279, 304 Butler, T. D., 87, 117 Butts, D. A., 26, 76

C Cabral, H. E., 22, 76 Cai, Q. G., 265, 302 Calderon-Moreno, J. M., 178, 211 Calogero, F., 20, 76 Campbell, L. J., 26, 45, 76, 77 Cao, H. C., 124, 125, 139, 147, 152, 160, 172, 177, 182, 184, 186, 197, 198, 205 Cao, M. S., 190, 215 Cao, W., 124, 127, 209 Cao, Z. X., 241, 302 Carman, G. P., 124, 127, 211 Cassol, E. A., 279, 304 Chahine, G. L., 85, 115 Chan, H. L. W., 123, 190, 205 Chan, R. K.-C., 86, 115 Chang, E. J., 98, 118 Chapman, R. B., 86, 115 Chen, B., 190, 195, 197, 207 Chen, G. X., 258, 266, 302, 305 Chen, H. D., 88, 119 Chen, H., 127, 210 Chen, L., 262, 263, 265, 266, 302, 304 Chen, S., 93, 115 Chen, T., 127, 163, 164, 197, 206 Chen, W. Q., 124, 195, 206 Chen, Y.-H., 126, 128, 154, 156, 159, 160, 166, 174, 180, 185, 187, 189, 190, 191, 193, 194, 201, 206, 208, 210, 213 Chen, Y. Z., 265, 302, Chen, Y., 254, 305 Chen, Z. T., 201, 206, 214 Cheng, B. L., 186, 206 Cheng, S., 150, 206 Cherepanov, G. P., 162, 206 Chi, Y., 190, 195, 197, 207 Chien, N., 239, 302 Choi, H. G., 98, 115

Chu, T., 260, 261, 302 Chung, F. R. K., 53, 76 Chung, M. Y., 190, 206 Chushko, V. M., 124, 125, 127, 137, 147, 211 Cokelet, E. D., 84, 118 Colla, E. V., 195, 206 Collier, L., 124, 125, 178, 184, 210 Comninou, M., 194, 195, 206 Cook, J. L., 87, 117 Cook, R. F., 124, 127, 206 Cook, W. R., 189, 209 Coriell, S. R., 106, 115, 116 Coutris, N., 95, 117 Coxeter, H. S. M., 10, 76 Crowdy, D., 4, 76 Crowe, C., 83, 115 Crum, M. M., 46, 76 Curran, D. R., 123, 129, 131, 136, 205

D Daly, B. J., 86, 115 Dandy, D. S., 87, 115 Dantzig, J. A., 107, 117 Daros, C. H., 201, 206 Davis, R. H., 85, 97, 120 de Josselin de Jong, , 85, 117 DeGregoria, A. J., 85, 115 Deardorff, J. W., 225, 285, 289, 302 Delhaye, J. M., 95, 117 Den Hartog, G., 295, 305 Deng, W., 133, 190, 193, 207 Derr, L., 2, 76 Dhir, V. K., 109, 119 Didwania, A. K., 84, 119 Diepers, H. J., 107, 114 Ding, H. J., 124, 190, 195, 197, 206, 207 Dingman, S. L., 231, 232, 302 Domm, U., 71, 76 Domon, W., 127, 162, 197, 213 Donnelly, R. J., 27, 76 Dou, G. R., 279, 302 Drake, R. M., 108, 115 Drescher, J., 124, 126, 146, 153, 161, 163, 172, 173, 174, 176, 197, 201, 204, 205, 208 Drew, D. A., 97, 115 Dritschel, D. G., 2, 22, 76, 78 Du, S. Y., 190, 195, 197, 201, 207, 210, 214 Du, Y., 293, 295, 306 Dugdale, D. S., 154, 180, 207 Dundurs, J., 191, 207

Author Index Dunn, M. L., 124, 207 Dunn, M., 123, 124, 162, 176, 207 Duraiswami, R., 85, 115 Durkin, D., 29, 76 E Eaton, J. K., 97, 119 Eckert, E. R. G., 108, 115 Eggers, J., 92, 116 Elghobashi, S., 97, 120 Ellingsen, K., 102, 116 Emmett, W. W., 262, 302 Epstein, M., 190, 212 Eringen, A. C., 189, 207 Ervin, E. A., 102, 116 Ervin, E., 104, 116 Eshelby, J. D., 4, 76 Esmaeeli, A., 88, 94, 98, 99, 104, 116, 120 Evans, A. G., 124, 125, 139, 147, 152, 160, 172, 177, 182, 184, 186, 197, 198, 205 F Fabrikant, V. I., 197, 207 Fadda, D., 93, 115 Fajans, J., 29, 76 Fan, H., 124, 207 Fan, J. X., 127, 162, 163, 164, 174, 180, 183, 187, 208 Fan, T. Y., 190, 210 Fang, D. N., 127, 190, 195, 209, 210 Fatemi, E., 94, 119 Fedkiw, R. P., 95, 118 Feng, J., 88, 98, 116 Finnigan, J. J., 293, 303 Finnigan, J., 225, 302 Fisher, J., 64, 76 Floryan, J. M., 83, 116 Foote, G. B., 86, 116 Fortes, A., 100, 116 Foster, G. R., 279, 302 Frank, F. C., 4, 76 Freiman, S. W., 124, 127, 147, 163, 197, 206, 214 Fromm, J. E., 86, 116 Fujiwara, S., 98, 119 Fukai, J., 88, 116 Fulton, C. C., 126, 127, 128, 154, 161, 172, 178, 179, 180, 181, 182, 183, 184, 185, 189, 201, 202, 207, 208

309

Furman, E. L., 195, 206 Furuta, A., 124, 201, 207 Fuwa, K., 218, 220, 231, 302

G Galperin, B., 233, 289, 302 Gao, C. F., 127, 162, 163, 164, 174, 176, 180, 183, 187, 190, 208 Gao, H., 124, 126, 127, 128, 153, 154, 154, 161, 172, 178, 179, 180, 181, 182, 184, 185, 189, 201, 202, 207, 208 Gao, W., 295, 302 Garcia, M. H., 226, 241, 304 Gary Leal, L., 224, 227, 228, 304 Giannakopoulos, A. E., 197, 198, 208, 212, 213 Gibson, D. C., 84, 114 Gill, W. N., 106, 108, 118 Glass, K., 29, 76 Glicksman, M. E., 106, 116 Glimm, J., 88, 116 Glowinskiand, R., 98, 118 Goldenfield, N., 107, 117 Gong, X., 178, 190, 208 Govers, G., 279, 302 Govindaraju, R. S., 262, 303 Govorukha, V. B., 190, 191, 194, 195, 208 Greitzer, S. L., 10, 76 Griffin, O. M., 84, 119 Gross, L. E., 124, 127, 209 Grove, J. W., 88, 116 Grubert, J. P., 240, 303 Grzybowski, B. A., 3, 77 Gu, J., 245, 305 Gu, X., 254, 303 Guiu, F., 186, 206 Guo, F. L., 124, 195, 197, 207 Guo, F., 190, 195, 197, 207 Gupta, S. M., 195, 206

H Hack, J. E., 127, 147, 162, 214 Hahn, H. T., 124, 127, 211 Hally, D., 54, 77 Hamilton, D. P., 230, 240, 303 Hammitt, F. H., 86, 118 Han, J. C., 195, 201, 207, 214

310

Author Index

Han, J. J., 126, 128, 154, 160, 180, 185, 187, 189, 206, 208 Han, J., 88, 94, 120 Han, P., 124, 152, 197, 214 Han, X. L., 127, 187, 208, 214 Han, Z. W., 231, 303 Hao, T. H., 178, 190, 208 Hao, T.-H., 128, 160, 162, 174, 176, 208 Hardtle, K. H., 199, 212 Harlow, F. H., 86, 116, 117 Harrison, W. B., 123, 124, 190, 208 Hasebe, N., 156, 206 Haus, M. J., 123, 190, 208 Havelock, T. H., 22, 77 Herbert, J., 190, 208 Herrmann, G., 127, 129, 138, 146, 211 Herrmann, K. P., 190, 191, 194, 195, 204, 208 Heyer, V., 124, 125, 126, 127, 152, 153, 161, 163, 172, 173, 174, 176, 187, 197, 198, 201, 204, 208, 212 Hinch, E. J., 85, 118 Hirt, C. W., 87, 117 Homsy, G. M., 85, 118 Hong, Z. X., 285, 305 Horton, R. E., 259, 303 Hosokawa, S., 98, 119 Hou, P. F., 124, 195, 197, 207 Hou, T. Y., 85, 92, 115, 117 Hsieh, R. K. T., 127, 215 Hu, H. H., 83, 88, 98, 116, 117 Hu, S. L., 190, 212 Huang, C.-h., 278, 303 Huang, S. H., 124, 127, 195, 197, 214 Huang, W., 245, 305 Huang, X. F., 258, 266, 305 Huang, Z. C., 285, 304 Huh, J., 84, 119 Hui, C. Y., 127, 195, 209 Hutchinson, J. W., 171, 178, 184, 187, 189, 209 Hwang, K. C., 127, 190, 195, 209, 210 Hyman, J. M., 83, 88, 117 I Igarashi, H., 124, 125, 127, 197, 214 Ikeda, T., 129, 209 J Jackson, R. G., 240, 303 Jacqmin, D., 95, 117

Jaffe, B., 189, 209 Jaffe, H., 123, 129, 131, 136, 189, 205, 209 Jamet, D., 95, 117 Jeong, J.-H., 107, 117 Jiang, B., 127, 195, 209 Jiang, D. S., 266, 303 Jiang, L. Z., 126, 127, 135, 142, 146, 160, 163, 195, 197, 198, 199, 209 Jiang, Q., 185, 186, 190, 215 Jiaos, J. Y., 259, 305 Jing, K., 265, 302 Johnson, A., 88, 98, 117 Johnson, D. B., 93, 115 Jona, F., 129, 178, 209 Joseph, D. D., 88, 98, 100, 115, 116, 118 Juric, D., 88, 94, 105, 107, 109, 117, 119, 120

K Kachanov, M., 123, 190, 212 Kadtke, H. B., 45, 77 Kahn, M., 150, 206 Kaimal, J. C., 293, 303 Kanai, A., 98, 117 Kang, I. S., 87, 117 Kang, S. Z., 275, 306 Karapetian, E. N., 197, 207 Karihaloo, B. L., 201, 206 Karma, A., 106, 107, 114, 117 Katsura, H., 127, 162, 201, 212 Kawamura, T., 98, 117 Kemmer, G., 146, 205 Ketterle, W., 26, 75 Kevrekidis, I. G., 88, 119 Khanin, K. M., 46, 77 Khazin, L. G., 22, 77 Khutoryansky, N., 127, 162, 174, 176, 201, 209, 213 Kidambi, R., 52, 54, 77 Kimura, Y., 14, 53, 77 Kirchner, H. O. K., 127, 205 Kloosterziel, R. C., 14, 60, 77 Kobayashi, R., 106, 117 Kodama, T., 84, 118 Kodama, Y., 98, 117 Koepke, B. G., 123, 124, 190, 208 Kogan, L., 127, 195, 209 Korepke, B. C., 127, 137, 139, 147, 210 Koshizuka, S., 109, 120 Kostant, B., 53, 76 Kothe, D. B., 93, 114

Author Index Kovalev, S. P., 124, 125, 127, 137, 147, 211 Kuang, Z. B., 190, 212 Kudryvtsev, B. A., 129, 135, 156, 211 Kuijlaars, A. B. J., 2, 78 Kumar, S., 126, 127, 128, 136, 142, 146, 147, 160, 160, 161, 165, 177, 178, 182, 186, 197, 201, 209 Kuo, C. M., 127, 132, 133, 134, 135, 138, 147, 156, 172, 190, 191, 209, 213 Kurup, G. R., 240, 303 Kwok, Y. K., 227, 303 Kwon, J. H., 190, 209

L Ladd, A. J. C., 97, 100, 117 Lafaurie, B., 93, 117 Laflen, J. M., 278, 303 Lahey, R. T., 97, 115 Lai, D., 127, 163, 164, 197, 206 Lamb, H., 226, 303 Landau, L. M., 129, 130, 189, 209 Larson, C. L., 260, 304 Laurent-Polz, F., 60, 77 Lawn, B. R., 124, 127, 206 Leal, L. G., 87, 115, 117, 119 Lebaigue, O., 95, 117 Lebedev, V. G., 16, 76 Lee, J. S., 127, 209 Lee, K. Y., 178, 190, 209, 212 Lee, R. C., 109, 118 Lee, Y.-W., 106, 108, 118 Lei, X. E., 231, 303 Leibovich, S., 224, 227, 228, 304 Lekhnitskii, S. G., 124, 131, 209 Lesieu, M., 233, 291, 303 Levich, V. G., 229, 303 Levin, V., 123, 190, 212 Lewis, D., 53, 77 Li, F. Z., 168, 209 Li, J. C., 218, 220, 224, 225, 226, 227, 233, 235, 242, 263, 265, 266, 283, 287, 289, 290, 293, 295, 302, 303, 304, 305, 306 Li, Q., 190, 193, 212 Li, R. M., 262, 304 Li, S. F., 127, 201, 209 Li, S., 124, 127, 209 Li, X. F., 190, 210 Li, X. L., 88, 116 Li, X. R., 293, 295, 306 Li, X., 97, 118

311

Liang, J., 190, 195, 197, 207 Lifshitz, E. M., 129, 130, 189, 209 Ligget, J. A., 262, 305 Lim, C. C., 26, 60, 77 Lima, J. L. M. P., 263, 304 Lin, H., 291, 306 Lin, J. D., 285, 304 Liu, B., 190, 210 Liu, G. N., 124, 127, 195, 215 Liu, H. D., 235, 306 Liu, J. X., 190, 210 Liu, L. C., 285, 304 Liu, Q. D., 191, 210 Liu, Q. Q., 235, 262, 263, 265, 266, 302, 304, 306 Liu, S. H., 285, 304 Liu, Y. B., 276, 304 Liu, Y. J., 124, 127, 195, 215 Liu, Y. M., 85, 120 Lloyd, I. K., 150, 206 Loboda, V. V., 190, 191, 194, 195, 204, 208 Loewenberg, M., 85, 118 Longuet-Higgins, M. S., 84, 118 Lord Rayleigh, , 108, 118 Lothe, J., 127, 142, 185, 205 Louge, M. Y., 86, 118 Lounasmaa, O. V., 17, 77 Lowengrub, J. S., 85, 92, 117 Lowengrub, M., 197, 201, 213 Lu, J. Y., 279, 304 Lu, P., 127, 210 Lu, T. J., 128, 166, 174, 201, 206 Lu, Z. B., 285, 305 Lumley, J. L., 224, 227, 228, 304 Lundgren, T., 100, 116 Luo, B. Z., 232, 235, 304 Lynch, C. S., 124, 125, 178, 184, 197, 199, 210

M Ma, , 156, 206 Ma, L. F., 128, 156, 158, 159, 190, 191, 193, 194, 210 Mahrer, Y., 287, 304 Mai, Y. W., 190, 211, 212 Mao, X., 126, 128, 162, 190, 212 Marden, M., 65, 77 Markenscoff, X., 191, 207 Marsden, J. E., 59, 78 Martens, L. A. A. G., 124, 205 Mason, W. P., 123, 210

312

Author Index

Mataga, P. A., 127, 201, 209 Maue, A. W., 2, 77 Maugin, G. A., 127, 189, 207, 215 Maxey, M. R., 98, 118 Maxey, R., 97, 120 Mayer, A. M., 2, 22, 78 McFadden, G. B., 83, 106, 114, 115, 116, 120 McHenry, K. D., 123, 124, 127, 137, 139, 147, 190, 208, 210 McKean, H. P., 45, 75 McMeeking, R. M., 124, 125, 126, 127, 152, 160, 161, 162, 165, 166, 167, 169, 170, 171, 172, 174, 176, 178, 184, 185, 187, 189, 201, 202, 210 Megaridis, C. M., 88, 116 Meguid, S. A., 133, 190, 193, 207, 210 Mehta, K., 124, 147, 210 Meiburg, E., 85, 118 Mein, R. G., 260, 304 Meiron, D. I., 84, 114 Mellor, G. L., 287, 304 Mertz, G. J., 2, 78 Metais, O., 233, 291, 303 Meyer, L. D., 279, 302 Mikhailov, G. K., 129, 161, 210 Milton, G. W., 197, 210 Minev, P. D., 88, 119 Misawa, M., 98, 119 Mitchell, T. M., 86, 118 Mittal, R., 106, 120 Miyata, H., 98, 117 Miyatake, O., 88, 116 Moeng, C.-H., 225, 304 Moldenhauer, W. C., 279, 304 Molkov, V., 127, 195, 209 Molz, F. J., 289, 304 Monkman, J., 53, 78 Montaldi, J., 26, 65, 77, 78 Moore, I. D., 279, 304 Morikawa, G. K., 34, 78 Mortazavi, S., 98, 118 Morton, W. B., 5, 78 Moser, J., 43, 45, 75 Moulton, F. R., 46, 78 Movchan, A. B., 197, 210 Murray, B. T., 106, 120 N Nabarro, F. R. N., 4, 76 Naot, O., 287, 304

Nardone, C., 93, 117 Narita, F., 126, 127, 160, 162, 190, 197, 201, 210, 211, 212, 213 Narita, K., 160, 190, 210 Nas, S., 88, 94, 120 Nas, Y. J., 88, 94, 120 Needleman, A., 168, 209 Neibling, W. H., 279, 302 Nelsen, R. B., 5, 54, 78 Neumann, H. H., 295, 305 Newham, R. E., 123, 124, 211 Newnham, R. E., 123, 190, 208 Newton, P. K., 52, 72, 77, 78 Nichols, B. D., 87, 117 Nielsen, P., 226, 238, 304 Nino, Y., 226, 241, 304 Nishioka, T., 190, 212 Niu, G. Y., 285, 305 Noda, N., 190, 214 Noh, W. F., 87, 118 Norris, A. N., 201, 211 Nowacki, J. P., 127, 212 Nydahl, J. E., 109, 118 O O’Neil, K. A., 29, 78 O’Neil, T., 59, 78 Oguz, H., 84, 118 Oh, W., 88, 116 Oka, Y., 109, 120 Okamoto, H., 14, 77 Okazaki, K., 124, 125, 127, 141, 147, 197, 198, 211, 214 Oran, E. S., 88, 118 Orszag, S. A., 84, 114, 233, 289, 302 Ortiz, M., 178, 184, 187, 211 Osher, S., 94, 95, 115, 118, 119 Ouyang, B., 283, 293, 303 Ozawa, E., 127, 201, 212 P Pak, Y. E., 124, 125, 126, 127, 129, 130, 131, 135, 137, 138, 139, 140, 141, 142, 145, 146, 147, 152, 155, 156, 161, 166, 170, 172, 174, 182, 187, 197, 198, 201, 201, 202, 203, 204, 211, 213 Palffy-Muhoray, P., 53, 76 Pan, E., 190, 195, 211 Pan, T. W., 98, 118

Author Index Pan, Y., 98, 118 Park, S. B., 124, 126, 127, 131, 135, 136, 138, 141, 142, 143, 144, 145, 146, 149, 152, 154, 160, 161, 163, 165, 166, 170, 171, 172, 173, 178, 180, 182, 183, 187, 188, 189, 198, 199, 201, 202, 203, 211 Park, S. S., 124, 127, 211 Park, S., 124, 127, 178, 211 Parton, V. Z., 126, 127, 129, 135, 156, 161, 163, 210, 211 Patankar, N. A., 83, 88, 117 Patel, B. K., 98, 118 Patterson, J. C., 240, 303 Paw, U. K. T., 295, 302 Pekarsky, S., 59, 78 Perrin, B., 106, 108, 114 Philip, J. R., 259, 304 Phillips, O. M., 226, 304 Piomelli, U., 289, 304 Pisarenko, G. G., 124, 125, 127, 137, 147, 211 Plesset, M. S., 86, 108, 115, 118 Pohanka, R. C., 123, 124, 125, 127, 129, 206, 211 Polvani, L., 2, 78 Ponce, V. M., 262, 304 Poo, J. Y., 93, 114 Popa, M., 178, 211 Porter, A. W., 53, 78 Poulikakos, D., 88, 116 Pozrikidis, C., 85, 97, 118, 120 Pracht, W. E., 86, 115 Prosperetti, A., 84, 97, 108, 118, 120 Puckett, E. G., 94, 119 Q Qi, L. X., 258, 266, 305 Qian, C. F., 127, 172, 215 Qin, Q. H., 190, 201, 211, 212, 214 Qin, S. W., 124, 207 Qu, J., 190, 193, 212 R Raad, P. E., 93, 115 Rajapakse, R. K. N. D., 126, 128, 160, 162, 163, 164, 165, 166, 172, 174, 175, 176, 190, 212, 214 Rallison, J. M., 85, 118 Ramamurty, U., 197, 198, 212, 213 Raman, C., 26, 75 Ramanujapu, N., 109, 119

313

Rao, M., 83, 119 Rappel, W.-J., 106, 117 Rasmussen, H., 83, 116 Ratiu, T., 53, 77 Raynes, A. S., 124, 147, 163, 197, 214 Reece, M. J., 186, 206 Reible, D. D., 218, 220, 305 Rice, J. R., 193, 195, 201, 202, 212 Risso, F., 102, 116 Ritter, A. P., 124, 214 Roberts, M., 26, 77 Robinson, P. B., 84, 118 Rokhar, D. S., 26, 76 Rosenberg, N. J., 231, 232, 305 Ru, C. Q., 126, 128, 147, 154, 160, 162, 177, 186, 190, 212 Rubin, B. S., 197, 207 Ruschau, G. R., 123, 124, 211 Ryskin, G., 87, 119

S Saff, E. B., 2, 78 Sakaguchi, T., 98, 120 Sangani, A. S., 84, 119 Sankaranarayanan, K., 88, 119 Sarin, V., 98, 118 Scardovelli, R., 83, 93, 117, 119 Schaefer, R. J., 106, 120 Schaufele, A. B., 199, 212 Schladov, S. G., 230, 303 Schmidt, A., 106, 119 Schmidt, D. S., 22, 76 Schneider, G. A., 124, 125, 126, 127, 152, 153, 161, 163, 173, 174, 176, 187, 197, 198, 201, 204, 208, 212 Schultz, W. W., 84, 119 Schumann, U., 295, 305 Schwartz, L. W., 85, 115 Scoging, H., 271, 305 Sethian, J. A., 95, 106, 119 Sevostianov, I., 123, 190, 212 Shan, X. W., 88, 119 Shan, X., 88, 119 Shang, J. K., 124, 152, 197, 213, 214 Shankar, N., 124, 214 Shannon, J. P., 86, 117 Sharp, D. H., 88, 116 Shaulov, A. A., 123, 190, 212 Shaw, R. H., 295, 302, 305 Shelley, M. J., 85, 92, 117

314

Author Index

Shen, B., 262, 305 Shen, H. T., 232, 235, 254, 304, 305 Shen, H., 254, 303, 305 Shen, S. P., 190, 212 Shen, S., 190, 212 Shen, W. M., 287, 303, 304 Shen, Y. P., 124, 127, 195, 215 Shen, Z.-Y., 128, 160, 162, 174, 176, 208 Shi, W., 254, 305 Shih, C. F., 168, 209 Shiiba, Y., 88, 116 Shikhmurzaev, Y. D., 92, 119 Shima, A., 84, 118 Shin, J. W., 178, 212 Shin, S. W., 107, 119 Shin, S., 109, 119 Shindo, Y., 126, 127, 160, 162, 190, 197, 201, 210, 211, 212, 213 Shioya, T., 124, 195, 206 Shipley, W. P., 5, 79 Shirane, G., 129, 178, 209 Shopov, P. J., 88, 119 Shrout, T. R., 124, 127, 178, 211 Shyy, W., 83, 106, 119, 120 Simons, D. B., 262, 304 Singh, R. N., 124, 125, 126, 127, 128, 136, 142, 146, 147, 148, 149, 150, 151, 155, 160, 161, 165, 173, 177, 178, 180, 182, 186, 187, 197, 198, 201, 202, 204, 209, 214 Singh, V. P., 219, 222, 305 Smagorinsky, J., 233, 289, 305 Smereka, P., 84, 94, 119 Smerreka, P., 94, 119 Smith, P. L., 123, 125, 127, 129, 211 Smith, R., 83, 119 Smith, W. A., 123, 190, 212 Sneddon, I. N., 197, 201, 213 Snelders, H. A. M., 17, 78 Soh, A. K., 190, 210 Sommerfeld, M., 83, 115 Son, G., 109, 119 Sosa, H., 124, 127, 130, 131, 132, 135, 136, 142, 146, 147, 156, 161, 162, 174, 176, 201, 203, 209, 213 Sou, A., 98, 120 Squires, K. D., 97, 119 Sridhar, S., 197, 198, 212, 213 Steinbach, I., 107, 114 Sternberg, S., 53, 76 Stieltjes, T. J., 66, 78 Stone, H. A., 3, 77, 228, 302

Stow, C. D., 86, 114 Strain, J., 106, 119 Street, R. L., 86, 115 Stremler, M. A., 17, 68, 74, 75, 78 Stroh, A. N., 124, 132, 133, 191, 191, 213 Sun, C. T., 126, 127, 131, 135, 136, 138, 141, 142, 143, 144, 145, 146, 149, 152, 154, 160, 161, 163, 165, 166, 170, 171, 172, 173, 178, 180, 182, 183, 185, 187, 188, 189, 195, 197, 198, 199, 201, 202, 203, 209, 211 Sun, R., 285, 304 Sun, S. F., 285, 304, 305 Sundaresan, S., 88, 119 Suo, Z., 124, 125, 127, 132, 133, 134, 135, 138, 147, 153, 154, 156, 161, 171, 172, 178, 184, 190, 191, 208, 209, 210, 213, 214 Suresh, S., 197, 198, 208, 212, 213 Sussman, M., 94, 119 Sutherland, A. J., 240, 305 Swenson, E. V., 34, 78 Szego¨, G., 20, 20, 78

T Tabeling, P., 106, 108, 114 Takada, N., 98, 119 Tan, X., 124, 152, 197, 213, 214 Tanaka, K., 127, 162, 201, 212, 213 Tang, C. B., 241, 305 Tang, K. L., 276, 304 Tang, L. Q., 258, 266, 302, 305 Tanuma, K., 195, 205 Tauber, W., 88, 94, 120 Taya, M., 123, 124, 207 Taylor, P. A., 225, 225, 302 Tezduyar, T. E., 88, 98, 117 Tiersten, H. F., 129, 213 Thomson, J. J., 2, 2, 78 Thomson, W., 2, 41, 78 Thuneberg, E., 17, 77 Tian, W. Y., 128, 159, 190, 191, 206, 213 Ting, R. Y., 123, 190, 212 Ting, T. C. T., 127, 132, 133, 190, 193, 205, 206, 213 Tkachenko, V. K., 60, 71, 72, 79 Tobin, A. C., 124, 125, 127, 137, 139, 140, 141, 142, 145, 146, 152, 155, 166, 170, 172, 174, 182, 187, 197, 198, 202, 204, 213 Tobin, A., 127, 137, 139, 211

Author Index Tokieda, T., 2, 79 Tomita, Y., 84, 118 Tomiyama, A., 98, 119, 120 Tong, P., 127, 128, 154, 161, 172, 178, 179, 180, 181, 182, 183, 184, 189, 201, 202, 208, 215 Tong, X., 107, 114 Tonhardt, R., 107, 120 Tonon, F., 195, 211 Truesdell, C. G., 97, 120 Tryggvason, G., 84, 85, 88, 94, 98, 99, 102, 104, 105, 107, 114, 115, 116, 117, 118, 120 Tsuji, Y., 83, 115 Turner, J. S., 227, 228, 305

U Uchino, K., 124, 201, 207, 213 Udaykumar, H. S., 83, 106, 119, 120 Ulitko, A. F., 127, 211 Ungar, L. H., 107, 120 Unsworth, J., 123, 190, 205 Unverdi, S. O., 88, 94, 120 Utsumi, T., 95, 120

V Vainchtein, D. L., 29, 75 van Heijst, G. J. F., 14, 60, 77 Vaudin, M. D., 124, 147, 163, 197, 214 Vinje, T., 84, 120 Virkar, A. V., 124, 147, 210 Vogels, J. M., 26, 75 von Ka´rmań, T., 65, 77

W Wan, Z. H., 239, 302 Wang, B. L., 190, 201, 214 Wang, B., 124, 127, 190, 195, 197, 207, 210, 213, 215 Wang, H., 124, 125, 126, 127, 142, 146, 147, 148, 149, 150, 151, 155, 161, 173, 180, 187, 198, 202, 204, 214 Wang, L.-P., 97, 98, 118, 120, Wang, M. F., 280, 306 Wang, M. Z., 176, 190, 208 Wang, T. C., 127, 184, 187, 190, 208, 213, 214

315

Wang, T., 235, 306 Wang, W. Z., 259, 305 Wang, X. D., 127, 190, 214 Wang, Y. J., 245, 305 Wang, Y., 254, 305 Wang, Z. K., 124, 127, 195, 197, 198, 213, 214 Warder, R. B., 5, 79 Warhaft, Z., 228, 305 Warren, J. A., 106, 120 Watanabe, K., 126, 127, 162, 213 Watt, R. G., 232, 305 Welch, J. E., 86, 117 Welch, S. W. J., 109, 120 Wheeler, A. A., 83, 106, 114, 120 White, G. S., 124, 147, 163, 197, 214 Whitesides, G. M., 3, 77 Williams, C. W. M., 14, 77 Willis, J. R., 127, 132, 133, 134, 135, 138, 147, 156, 172, 191, 213 Wilson, J., 109, 120 Wintner, A., 2, 79 Winzer, S. R., 124, 214 Wood, R. W., 3, 79 Woodward, P., 87, 118 Woolhiser, D. A., 262, 305 Wu, C. K., 218, 220, 295, 303 Wu, K. C., 193, 214 Wu, S. Y., 285, 305 Wu, X. F., 190, 210

X Xiao, F., 95, 120 Xie, J. H., 280, 306 Xie, S. N., 258, 302 Xie, Z. T., 226, 233, 287, 290, 295, 303, 304, 305 Xu, H. B., 85, 120 Xu, X.-L., 126, 128, 160, 162, 163, 164, 165, 166, 172, 174, 175, 176, 214 Xu, Z., 124, 152, 197, 213, 214 Xu, Z., 254, 305 Xue, M., 85, 120

Y Yabe, T., 95, 120 Yamada, T., 287, 304 Yamamoto, T., 88, 116, 124, 125, 127, 197, 214

316

Author Index

Yan, W., 127, 162, 201, 212 Yang, F., 127, 214 Yang, W. Z., 266, 305 Yang, W., 124, 125, 127, 153, 154, 161, 178, 184, 210, 214, 215 Yao, D. L., 287, 293, 295, 303, 304, 305, 306 Yarmchuk, E. J., 79 Ye, D. Z., 291, 306 Yoon, H. Y., 109, 120 Youngren, G. K., 85, 120 Youngs, D. L., 93, 120 Yu, D., 84, 120 Yu, S. W., 190, 201, 212, 214 Yuan, F. G., 190, 195, 211 Yuan, Q., 254, 303 Yue, D. K. P., 85, 120 Yushin, N. K., 195, 206

Z Zaleski, S., 83, 93, 117, 119 Zanetti, G., 93, 117 Zapryanov, Z. D., 88, 119 Zemach, C., 93, 114 Zeng, X., 126, 163, 172, 174, 212 Zhang, D. Z., 97, 120 Zhang, J. G., 293, 295, 306 Zhang, K. L., 275, 278, 306

Zhang, M. G., 231, 303 Zhang, R. J., 280, 306 Zhang, T. Y., 127, 128, 147, 154, 161, 162, 172, 178, 179, 180, 181, 182, 183, 184, 189, 201, 202, 208, 214, 215 Zhang, X., 190, 212 Zhang, Y., 185, 186, 190, 215 Zhang, Z. Z., 223, 245, 305, 306 Zhao, M. H., 124, 127, 195, 215 Zhao, Y. T., 190, 208 Zhao, Z., 88, 116 Zheng, B. L., 124, 127, 195, 197, 214 Zheng, F. L., 275, 306 Zhong, Z., 190, 210 Zhou, H., 85, 120 Zhou, J. F., 226, 235, 242, 304, 306 Zhou, P. H., 276, 304 Zhou, S. A., 127, 215 Zhou, X. M., 276, 304 Zhou, Y., 233, 254, 303, 306 Zhou, Z. G., 190, 215 Zhu, M. Y., 83, 88, 117 Zhu, T., 153, 154, 215 Zhuzhoma, E. V., 52, 75 Ziff, R., 26, 76 Zinchenko, A. Z., 85, 97, 120 Zun, I., 98, 120 Zwick, S., 108, 118

Subject Index

B

A

barium sodium niobate, 205 barium titanate, 123, 130 bathymetry, 234 BEC see Bose–Einstein condensates Berenson’s correlation, 112 Bett’s reciprocal theorem, 156 bidispersed bubble size distribution, 103 bifurcation, 60 bilinear element, 271 biological crust skin, 283, 293, 295 biological diversity, 220 biosphere, 289 –290 biotic marine fluxes, 220 boiling, 104 –105, 108 –110, 112 Bose–Einstein condensates (BEC), 26 Bosten lake, 221 bottom boundary layer, 225 bottom sediment flux, 243 boundary conditions electric, 125– 128, 146, 176, 180, 186, 197–198 exact, 128, 163, 174 –177 mechanical, 128, 178, 197 boundary element method, 84 boundary integral method, 84–85 brittle piezoelectrics, 162, 180 Brownian motion, 97, 228 bubbles buoyan, 88, 99 collapse, 84, 85 columns, 103, 112 deformable, 84, 87, 100–104, 109 bubbly flows, 82, 83, 96, 99, 112 buckminsterfullerenes, 53

2D see two-dimensional ABL see atmospheric boundary layer absorption function, 288 acid rain, 220, 231 actuators, 123, 185 Adler–Moser polynomials, 32, 44– 45, 51 advection, 68–70, 72–73, 93, 95, 242–243 adverse human activities, 219 aerosols, 220, 229 air quality, 219 Aksu water Budget Experimental Research Station, 291 albedo, 230 alluvial channels, 240 amplitude evolution, 106 angular... frequency, 11–12, 56–58, 62, 71 impulse, 7–8, 54 velocity, 7, 9–10, 21, 55 anisotropic... piezoelectrics, 126–127, 158 annual precipitation, 259 anti-plane piezoelectrics, 142 anti-plane shear, 135 antipodal vortices, 67 apparent energy release rate, 181–182, 183, 185 arid areas, 283 –286, 291, 293, 300 Artemisia ordosica, 284, 292–293 asymmetric equilibria, 32– 33 atmosphere, 283, 287, 288, 290, 291, 298, 230, 301 atmospheric boundary layer (ABL), 224–287, 289 317

318

Subject Index

Bueckner work conjugate integral (BWCI), 156 buoyancy, 224, 226– 227 bursting events, 241 Businger–Dyer formula, 225 BWCI see Bueckner work conjugate integral

C cadmium sulfide, 205 canopy turbulence, 225 Caragana korshinskii, 292–293 carbon dioxide, 223 carbon-free power stations, 223 cavitation bubbles, 84, 85 CBL see convective boundary layer cellular solidification, 106 center of vorticity, 7, 9, 12, 24, 41 Changxing Island, 234 charge separation, 189 charge-free condition, 135, 152 Chebyshev integration scheme, 187 chemical reactions, 231 chlorine, 231 chlorophyll, 252 Chongming Island, 234 CIP method, 95 clay loess belt, 265 closure integral, 144 closure relations, 96 coal-dominated energy, 222 coalescence, 96 coastal engineering layer, 225 coercive field, 147 collinear vortices, 11 compact tension specimen, 125, 144 –146, 162, 168 –171 complex potential theorem, 131 –132 complex velocity, 65 compressive strain, 150 concentration flow route, 268, 273 condensation, 109 conducting cracks see permeable cracks conjugate integral, 156 constitutive equations, 130, 131 contact crack model, 178 contact mechanical boundary condition, 128 contact zone model, 194–195 contaminated water, 221 continuity equation, 236, 242

convective boundary layer (CBL), 224, 299 Coriolis force, 227 crack-tips dielectric non-linearity, 154 electric non-linearity, 178 energy release rate, 138, 162, 169– 170 mode-mixity, 170, 174 cracks, 121–205 arrest, 147 closure, 144, 166, 186 growth, 147, 149, 172, 174, 178 –179 lengths, 140, 151 propagation, 142 shielding, 184 surfaces, 135 critical fracture stress, 182 critical shear stress, 279, 280 crust skin, 283, 293 –295 crystals, 1–75 cumulative infiltration, 260

D deformable boundaries, 97 deformable bubbles, 100 degradation, 185 dendrites, 106– 108 density bulk, 241 buoyancy induced convection, 226 –227 piezoelectrics, 129–130 salt water, 242 sediment, 279 deperiodizing limit, 67, 68 deposition, 229 depth-averaged mean velocity, 236 desert invasion, 222, 284 dielectrics, 162, 174 diffusion, 228, 243 dilute flows, 97 dimethyl sulphide (DMS), 220 dipoles, 185 direct radiation, 230 directional solidification furnaces, 106 Dirichlet-Riemann boundary value problem, 194 discharge rate, 264 discharge/charge separation, 189 dislocations, 4 dispersate particles, 229 disperse flows, 96, 112

Subject Index displacements, 132, 164, see also electric displacement dissipative eddy viscosity, 289 DMS see dimethyl sulphide dodecahedron, 62 domain integral technique, 168–169 domain switching, 184, 198–199 double-rings, 25 Dou’s formula, 279 drafting, 100 driving forces, 124 drop-former type rainfall simulator, 272–273 drops, 82, 84–87, 92, 96 drought, 219, 221–222, 283, 284 dry density, 279 dry sand layers, 293–295 dryland farming, 276, 280 Dugdale model, 153 –154, 178 –186 dust storms, 219 dynamic fractures, 201, 208, 128, 179, 180 dynamic overland flow hydraulics, 268 –274

E earthquakes, 219 ebbing, 251 ecological succession, 232 eddies, 225, 289– 92 EDIF see electric displacement intensity factors eigenfunction expansion forms (EEF), 156–158 eigenstates, 51 ejection events, 241 Ekman effect, 227 elastic constants, 180, 181, 204, 205 elastic dielectrics, 162 elasto-electric fields, 198 electric boundary conditions, 122, 126, 127, 146, 176, 180, 202, 214 electric dipoles, 184, 185 electric displacement, 173, 174– 175, 189 electric displacement intensity factors (EDIF) bimaterials, 135, 193 electric fields, 175 –176 impermeable cracks, 122, 135, 155, 160 permeable cracks, 161, 162, 166, 175, 202 weight functions, 159 see also stress intensity factors electric enthalpy density, 130 electric fields

319

anisotropic materials, 127 bimaterials, 193 crack growth, 172, 174 crack lengths, 148 –151 electric displacement intensity factors, 175–176 energy release rate, 139–140, 143, 144, 146 fatigue crack growth, 147 ferroelectric twins, 185 impermeable cracks, 122, 135, 155, 187 induced crack closure, 128, 186, 212 induced stresses, 162 mechanical loading, 151 mechanical strain energy release rate, 144, 145–146, 188, 201, 205 microcracking, 152, 154, 178, 180 near-tip microcracking, 185, 188, 189 piezoelectrics, 126 polarity, 148 shielding, 150 strength, 145–146, 187, 199, 200 subcritical crack growth, 147 three dimensional cracks, 195, 198– 200 vector, 130 electric loadings compact tension specimens, 168 –170 crack arrest, 147 impermeable cracks, 160, 135, 161 mechanical strain energy release rate, 145 near-tip microcracking, 180 permeable cracks, 172 piezoelectrics, 125 strip electric saturation, 178, 180–181 see also mechanical loadings electric non-linearity, 178 electric potential, 163 electric saturation, 178 –186 electrical energy release rate, 166 –167 electrical polarization, 180 electrical yielding, 178–185 electroelasticity formulation, 181 electromechanical enthalpy density, 129 electromechanical properties, 204 electron plasma evolution, 29–30 elliptical crack, 172 energy absorbance, 152, 160 cascade drain, 227, 289– 290 consumption, 223 cycling, 283 density, 129

320

Subject Index

equation, 104– 105, 108 J-integral, 152 production indices, 223 transport, 224, 228–229 energy release rates (ERR) bimaterials, 194 –195, 196 crack-tips, 138, 162, 169 –170 impermeable cracks, 138 –156, 160 mechanical strain, 143, 144, 156 near-tip microcracking, 187–190 permeable cracks, 165–172, 173 piezoelectrics, 126 strip electric saturation, 178–185 enthalpy density, 129, 130 entrainment flux, 241, 243 entropy cascade, 227 equations of motion, 53, 63– 64 equilibrium, vortices, 3–4, 20– 24, 32–47, 55–68 erosion, 256, 257, 258, 259, 265, 268– 270, 272, 274 –278, 280 –283 ERR see energy release rates estuaries, 235, 239, 240 –242, 245, 252 eutrophication, 232 evaporation, 110, 286 evapotranspiration, 286, 292 –293 exact boundary conditions, 163 exact electric boundary condition, 128, 174–177 external forcing, 232

F failure, piezoelectrics, 124, 182–183 far-field loadings, 164 fatigue, 147, 184 ferroelastic switching zones, 153, 161 ferroelectric materials, 128 actuators, 185 crystals, 178 switching zones, 154, 155 twins, 185 field experiment, 290–291 field intensity factors see electric displacement intensity factors; stress intensity factors FIFE see First ISLSCP Field Experiment film boiling, 109–110 film rupturing, 92, 95 finite angular frequency, 56 finite Reynolds number, 86– 88, 97, 98, 100

First ISLSCP Field Experiment (FIFE), 290–291 fixed grid, 85, 88, 92 flash flocculation, 229 flaw, 165–166 floating magnets, 2 flocculation, 229, 239 floods, 219, 221, 222 flowline, 270, 275 flows bubbly, 82, 83, 96, 99, 112 concentration, 272–273 fluids, 84–89, 91– 96, 98, 99, 101, 104, 106–109, 111 inertial, 82, 84– 85, 97, 98, 100 inviscid, 84 –85 multiphase, 81–89, 91–93, 95 –97, 99, 101, 103 –105, 107, 109, 111– 114 overland, 258, 259, 261, 262, 264, 268, 270, 272, 274 potential, 84 –85, 97, 100 rill, 276 –277, 278– 279 rotating systems, 227 route concentration, 273 Stokes, 82, 84–85, 97, 98, 100 vector field, 245 fluid flow, 81, 98, 104, 106, 107, 113, 114 fluid-carried mass, 224 force-restore method, 285 forests, 289, 295– 297, 298, 299 fossil fuels, 219, 220, 222–223 fractures, 121– 205 criterion, 183–184 load, 145 strength, 187 toughness, 125, 141, 146, 170, 174 free-surface flows, 93, 114 friction velocity, 245 frictionless contact zone model, 194 –195, 196 front tracking, 88, 94– 95, 107, 109 fronts, 250–251 Froude number, 262

G GDP see Gross Domestic Product generalized stress concepts, 132– 134, 157 generating functions, 19 generating polynomials, 32, 36, 45–46, 49 geometry, vortices, 11, 27, 30, 52, 71 geophysical fluids, 227, 233, 289

Subject Index geosphere, 290, 298, 301 geostrophic flows, 227 ghost fluid method, 94 Gibbs–Thompson Equations, 105 global carbon cycling, 220 global energy release rate, 122, 154, 201 gradient flocculation, 229 Grashof number, 227 greenhouse gases, 219–220 Green–Ampt model, 259–261 grid lines, 87, 88 Griffith crack model, 135 Gross Domestic Product (GDP), 223 ground heat flux, 288 growth bubbles, 81, 82, 84, 85, 87, 92, 96–104, 108–112 cracks, 147, 149, 172, 174, 178–179 microstructures, 106

321

atmospheric flows, 287 boiling, 109 bubbly flow, 83, 99 hoop stresses, 122, 136–137, 142, 161, 165, 173 humidity, 287 hurricanes, 219 HWR see high water rapid HWS see high water slack hybrid methods, 88 hydraulics, 268–274 hydro-power stations, 222, 223 Hydrological Atmosphere Experiment (HAPEX), 290, 301 hydrology, 221, 249, 250, 252, 265, 290, 291, 301 hyperbolic planes, 73–74

I H Hamiltonians hyperbolic plane vortices, 73–74 periodic parallelogram vortices, 72 periodic strip vortices, 63 point vortices, 4– 5, 10, 14, 47, 52– 54, 73 stable vortex equilibria, 27 vortices on a sphere, 52–56, 58–60, 65 Hamilton’s principle, 130 HAPEX see Hydrological Atmosphere Experiment heat capacity, 285, 287 –288 heat equations, soil, 288 heat transfer, 82, 96, 104, 106, 109, 111, 113, 288, 300 Heaviside functions, 89 –90, 92 HEIFE see Heihe River Basin Field Experiment Heihe River Basin Field Experiment (HEIFE), 290– 291 Hele-Shaw cells, 85 helium superfluids, 26–27 Hengsha Island, 234 Hermite polynomials, 19, 32 Heron’s formula, 10 hexahedrons, 60 high water rapid (HWR), 235 high water slack (HWS), 235 hillslope flows, 258 –265, 268 –274 homogeneous...

ice sheets, 232 icosahedrons, 60, 61 identical vortices, 17 –20, 32–39 IGBP see International Biosphere –Geosphere Program impermeable (insulating) cracks, 128, 134–161 bimaterials, 191 eigenfunction expansion forms, 156 –159 energy release rate, 138–156 exact electric boundary condition, 174–177 J-integral, 138 –43 mechanical strain energy release rate, 143–156 near-tip microcracking, 187–190 penny-shaped cracks, 198 permeable cracks, 153 weight functions, 156–159 indentation cracks, 139 –150 indoor artificial rain experiment, 280 industrial emissions, 220 inelastic deformation, 198–199 inertial flows, 82, 84–85, 97, 98, 100 infiltration, 259 –261 initial water content, 263 insulating cracks see impermeable cracks integrated models, 233 intensity, 251 see also electric displacement...; stress... interface tracking, 88, 94–96, 107

322

Subject Index

internal energy density, 129 International Biosphere– Geosphere Program (IGBP), 290 interrill erosion, 278 invariant vortex patterns, 12 inviscid flows, 84 –85 ions, 27

J J-integral crack-tip enclosing contours, 168 impermeable cracks, 138 –143, 159 mechanical strain energy release rate, 151–152 near-tip microcracking, 185, 188 permeable cracks, 166, 172 piezoelectrics, 125 Jacobi theta function 72 jump condition, 91

K Ka´rmań vortex streets, 25, 60, 63 Kelvin’s variational principle 7, 22, 26, 32 kinematic wave theory, 261–265 kinetic energy, 295, 298 kinking, 161 Korteweg– deVries equation, 45

level-set methods, 94–95, 106, 109 Liapunov stable configurations 22 lift forces, 101 light transmittance, 252 linear elastic dielectrics, 129 Linear Elastic Fracture Mechanics (LEFM), 124 linear impulse, 7, 12, 54 Linear Piezoelectric Fracture Mechanics (LPFM), 124, 125, 135 linear stability, 22 lines, identical vortices, 17–20 liquid-vapor phase changes, 105, 108–112 load ratio, 195, 196 loadings remote, 135, 173 uniform far-field, 164 see also electric...; mechanical... local energy release rates, 179, 183, 185, 189 local polarization switching, 184 Loess Plateau, 222, 256–283 long-range periodicity, 70 longitudinal mean salinity distribution, 249 Los Alamos Catalog, 26, 29, 31 low water rapid (LWR), 235 low water slack (LWS), 235 LPFM see Linear Piezoelectric Fracture Mechanics LWR see low water rapid LWS see low water slack

L

M

laboratory runoff generation experiments, 272–273 Lagrange integral time scale, 228 Lagrangian density, 129 –130 Lagrangian grids, 87 Laguerre polynomials, 19 lakes, 232 landslides, 219 large eddy simulation (LES), 289–290 latent heat, 288 Lattice Boltzman method (LBM), 88, 97, 98 lead zirconate titanate, 203 –204 lead zirconate titanate (PZT), 123, 130 LEFM see Linear Elastic Fracture Mechanics Lekhnitskii’s complex potential theorem, 131–132 LES see large eddy simulation

M-integral, 159 MAC see Marker-And-Cell macrocracks, 123, 178 magnetized electron plasma evolution, 29 magnets, 2 –4 Malmberg– Penning trap, 29 manifolds, 52–74 mantle convection, 227 mapping vortices, 72 marker functions, 92– 96 marker points, 88, 94–96 Marker-And-Cell (MAC) method, 86 –88, 93 mass conservation, 90 –91, 94 mass transfer, 104, 285 mass transport, 228–229, 235–241 mean flow, 245 mean salinity, 249– 250

Subject Index mechanical boundary conditions, 128, 178, 197 mechanical driving forces, 124 mechanical energy, 152 mechanical energy release rate, 166–167 mechanical loadings, 124 compact tension specimens, 168, 170 crack arrest, 147 electric field, 151 impermeable cracks, 134, 135 permeable cracks, 175 strip electric saturation, 178 –186 see also electric loadings mechanical strain energy release rates (MSERR), 143, 156, 160, 161, 187, 189 mechanical yielding, 179–184 mechanical-electrical coupling, 123 mesh-free methods, 109 microcracks, 123 –124, 155, 178 microstructures multiphase fluid flow, 82, 83, 97, 106–108, 113 piezoelectrics, 189, 197 Mie scattering, 230 minerals, 231, 232 mixing index, 240, 249, 251, 254 mixing length, 288 mode-mixity, 171, 174 models contact crack, 178, 186 contact zone, 194–195, 196 Dugdale, 154, 178, 180 Green– Ampt, 259 –261 Griffith crack, 135 Philip’s, 259 momentum equations, 242 momentum transport, 228 Monin–Obukhov length, 225 morphotropic lead zirconate titanate, 123 Morton’s equation, 5 motion, equations, of, 53, 63–64 mouth bars, 234, 254 moving phase boundaries, 91 MSERR see mechanical strain energy release rates multi-peak rainstorms, 266–267, 270 multi-vortex equilibria, 59–63, 70 multifluid flow computations, 83–114 multiphase flows, 81–114 disperse flows, 96, 112 one-field methods, 88

323

phase changes, 104 Reynolds number flows, 86

N natural disasters, 219 Navier–Stokes equations, 86 –91, 98, 108, 287–288 near-tip... dipoles, 185 Dugdale model, 178 –186 electric fields, 126, 135 electric saturation yielding, 185 mechanical fields, 126 microcracking, 178, 180, 187, 189 stress fields, 135, 136, 187 nested polygons, vortices, 23 –26 net radiation, 230 neutral boundary layer, 224– 225 neutral vortex triple, 14 nitrogen, 231 noise, 27 –28 non-dilute flows, 98 non-linear electric boundary condition, 177–190 noncondensable gases, 109– 110 normal electric displacement, 163– 164 normalized Reynolds stress, 295, 297 Northwest Institute of Water and Soil Conservation, 276, 280 nuclear power stations, 223 nucleation, 109, 124 Nusselt number, 111– 112 O octahedrons, 61 one-fluid methods, 81, 83, 88–96, 104 –105 organic matter, 231, 232 orthogonal polynomials, 17 oscillatory boundary layer theory, 226, 236 oscillatory indices, 192 overland flows, 258, 262, 300 oxides, 231 ozone layer, 231 P parallel cracks, 140 Park-Sun fracture criterion, 182 –183 partially stratified estuaries, 239, 240

324

Subject Index

particle-turbulence interactions, 241 particles, multiphase flows, 85–86, 97–99 passive scalar model, 228 penny-shaped cracks, 195, 198 periodic domains, 84, 99, 101–102 periodic parallelograms, 71 –73 periodic strips, 63–71 permeable (conducting) cracks, 128, 161 –174 bimaterials, 191, 194–195, 196 electric fields, 170–172 energy release rate, 165 –172 exact electric boundary condition, 175–176 field intensity factors, 164–165 impermeable cracks, 152, 153 perpendicular cracks, 140 phase... angles, 249 boundaries, 84, 105, 108, 109, 111 change flows, 104, 91, 112, 113 dependence, 257 field methods, 93– 95, 106–107 Philip’s model, 232, 259 photosynthesis, 252 phytoplankton, 230, 252 piezoelectrics, 121–205 bimaterial systems, 190, 195, 196 constants, 180 dynamic fractures, 201 electric boundary condition, 174, 190 exact electric boundary condition, 174–177 impermeable crack, 128, 134–161 non-linear electric boundary condition, 177–190 permeable cracks, 161–174 three-dimensional cracks, 195–201 plane piezoelectrics, 131, 134–136, 154, 156 plants, 283, 284, 285–295 plasma excitations, 29–30 plastic behavior, 4 plastic yielding, 180 Platonic solids, 60–62 plume, 101 point particle approximation, 97, 98 point singularities see point vortices point vortices, 4 –6, 10, 12, 73–74 polar vortices, 60 polarization, 180, 184, 189 pole decomposition equations, 45 poled ferroelectric materials, 139 –143, 162, 180

poled piezoelectrics, 174 poling, 130, 135, 139 poling processes, 126–127 polygons, 20– 33 polynomials Adler–Moser, 33, 43, 44 –45, 51 generating, 45–46, 49 Hermite, 18 –19, 26 identical vortices, 17–20 Laguerre, 19 stationary vortex patluns, 42–47 translating vortices, 49 ponding infiltration, 261 pool boiling, 109 population indices, 223 pore sizes, 141 potential flows, 84– 85, 97–98, 100–101 potential temperature, 287 precipitation, 259 Preissmann implicit finite difference scheme, 243 pressure, 109– 110 process-based approaches, 223–233 propagation speed, 10, 124, 137 pseudo-orthogonal properties, 158–159, 194 pseudo-traction-electric displacement (PTED), 187 PZT see lead zirconate titanate

Q quantum of circulation, 27 quartz, 130 quasi-crystals, 70 quiescent flow, 99, 110

R radiation, 230 rainfall Chinese environmental problems, 220 hillslope runoff erosion, 258–265 intensity, 281–283 types, 265, 266 –268, 269, 270 water erosion, 258 rainstorms, 266–268, 269, 270 ramp patterns, 295, 299 rapid water stage, 235 Rayleigh scattering, 230 Rayleigh– Taylor instability, 84, 86, 92

Subject Index relative equilibria, 4, 12, 58, 60 remanent polarization vector, 189 remanent strain, 189 remote electric fields, 187–189 residual stress, 127 resuspension, 240–241 Reynolds Averaged Navier– Stokes equations, 287–288 Reynolds numbers, 84, 86, 88, 93, 97– 99, 100, 226 Reynolds stresses, 100, 295, 297 Riemann surfaces, 73 rigid lody rotation, 17–20 rigid particles, 98 rill erosion, 275–283 rill flows, 276–277, 278–279 rings double, 25 numbers, 32 vortex equilibria, 30 –31, 60, 61 Rossby number, 227 rotation flows, 227–228 uniform, 71 vortices, 9, 12, 17–39 see also angular frequency; angular velocity runoff discharge, 249 estuarine flow, 235 formation, 258– 265 generation, 256–283 mass transport, 235–238 salt fronts, 251 tide decomposition, 236 turbidity maximum zone, 254– 256 vertical salinity, 252

S Saint Venant equations, 242, 262 salinity distribution, 252, 254 hydrological factors, 249 sediment flocculation, 239– 240 time series, 247 variation, 245, 249 vertical estuarine flow, 242–243, 252, 253 Yangtze river estuary, 243–244 salt front variations, 250 –251 salt water density, 242 salt-wedge estuaries, 239

San Bei windbreak, 283 saturated water content, 263 saturation limit, 180 saturation temperature, 105 SBL see stable boundary layer Schro¨dinger’s equation, 51 scouring, 258 sea water intrusion, 233, 235, 249, 254 sediment concentration, 235, 239, 242, 243 dry density, 279 entrainment flux, 241, 243 flocculation, 239 –240 laden boils, 240 resuspension, 240–241 transport, 233–256, 279 SGS see subgrid scale shallow water, 242, 262 Shapotou Desert Research Station, 291 sharp cracks, 173 sharply stratified estuaries, 239, 240 shear anti-plane shear, 135 flows, 84, 97, 101, 229 impermeable crack, 135 stress, 263 –264, 279– 80 velocity, 295 sheet flow, 258, 259, 268 shielding, 150, 184– 185, 189 SIFs see stress intensity factors silt loess belt, 265 simple flows, 84–86 simple harmonic oscillation, 237 simulated flow vector field, 245 single phase flows, 96 single-peak rainstorms, 266– 267, 269 singular stress-electric field, 193 singularity indices, 192 slack water period, 235 slit cracks, 172, 173 slopes gradients, 262, 264, 266, 271, 273, 276–277, 281–283 lengths 263–264, 281–283 surface flow lines, 273–274, 275 Smagoringsky constant, 290 smart materials, 123 snow avalanches, 219 soil characteristic parameters, 266 erosion, 222, 256–283 heat equations, 288

325

326

Subject Index

infiltration, 259 –261 moisture, 294 water deficiency, 263 water equations, 288 soil –plant–atmosphere as a Continuum (SPAC) system, 285 –290 solar... constant, 230 orbit, 232 radiation, 230 solid particles, 85, 96 solidification, 82, 83, 95, 104 –107 solute distribution, 106 SPAC see soil–plant–atmosphere as a Continuum specific humidity, 287 spherical bubbles, 97, 99, 102–104, 108 spiral configurations, 62–63 stability, vortices, 27 –29, 30– 31, 60, 71– 73 stable boundary layer (SBL), 224–225 stable dry density, 279 staggered configurations, 23 –24 staggered grids, 87, 93, 95 statics, vortices, 2 –12 stationary configurations, 4 stationary equilibrium, 11, 55, 57 –58, 63 –68 stationary patterns, 39–47, 70–71 step (Heaviside) functions, 89–92 stereographic projections, 54 Stokes flow, 82, 84–85, 97, 99, 100 storm surge, 219 strain, 127, 143– 155, 160–161, 187–189 stratified estuaries, 239, 240 stratosphere, 231 streamline patterns, 68–69, 73 stress, 157 critical fracture, 182–183 critical shear, 279, 280 electric field, 193 ferroelectric twins, 185 fields, 135, 136, 187 function, 243 hoop, 136–137, 142, 165, 173 impermeable cracks, 156 incipient function, 243 mechanical, 162 near-tip microcracking, 189 piezoelectrics, 127, 132 –134 residual, 127 Reynolds, 100, 295, 297 shear, 263 –264, 279– 280 subgrid scale, 289

tensile, 199– 200 tensor, 289 stress intensity factors (SIFs) bimaterials, 193, 194–195, 196 impermeable cracks, 135 –137, 161 mechanical strain energy release rate, 152 permeable cracks, 162, 172, 173 piezoelectrics, 125, 126 plane piezoelectrics, 154, 155, 157 weight functions, 156 see also electric displacement intensity factor strip electric saturation, 154, 178 –186 Stroh’s complex potential theorem, 132, 133 subcritical crack growth, 147 subgrid scale (SGS), 289 sulfur, 223, 231 superfluids, 3, 26–27 surface ponding, 258 surface tension, 88, 90–93, 104–105, 113 suspension, drops, 85 sustainable development, 218 switching zones, 153–154, 155 symmetry, 23 –25, 31– 32, 34–36

T Takramagan desert, 222 Tammes problem, 53 tangential electric field, 163 Tang– Cunben’s formula, 241 Tarimu river, 221 Taylor theory, 228 Taylor’s column, 227 Teitema lake, 221 temperature, 105, 287, 295, 299 Tenggeli desert, 284 tensile strain, 150 tensile stresses, 199–200 tension, 183 compact piezoelectrics, 125, 144–146, 162, 168 –171 surface, 88, 90–93, 104 –105, 113 terrestrial processes, 283–297, 298–299 tetrahedrons, 60, 62 thermal contours, 295, 299 thermodynamic potential, 129 thermohaline circulation, 226–227 threads, 92 three-dimensional bubbles, 99 –100, 101, 102

Subject Index three-dimensional cracks, 195 –201 three-point bending, 125, 183 three-vortex systems, 10– 17, 35–36, 58–59, 72 Tibet plateau, 220 –221 tides bottom boundary layer, 225– 226 estuarine flow, 235– 237 longitudinal mean salinity, 249 mass transport, 235–237 salt fronts, 251 turbidity maximum zone, 254– 256 vertical salinity, 252, 253 Tkachenko’s equation, 41, 42, 43, 45 topology changes, 92 tornado, 219 torrential rains, 221 total potential energy release rate (TPERR), 138 tracking methods, 88, 94 –96, 107 traction-free conditions, 135 transition zones, 94 translation, vortices, 11, 47 –52 transport energy, 224, 228–229 mass, 228–229, 235 –241 momentum, 228 sediment, 233–256, 276 –279 triangle, 11–12, 35, 36 tripole, 14–15 tsunami, 219 tumbling, 100 turbidity flocculation, 239 maximum zone, 250 –251, 252, 254–256 radiation, 230 turbulence arid areas, 286 –288 atmospheric boundary layer, 224 atmospheric motion, 286 canopy, 225, 286, 287, 289, 290, 295, 299 diffusion, 228 edd, 289, 290 geophysical fluid, 227 resuspension, 240 single phase flows, 96 vortices, 2–74 two-dimensional vortex motion, 4– 75 two-fluid flows, 85, 93 two-vortex systems, 10, 12, 56–57, 72 typhoons, 234

327 U

ultraviolet radiation, 231 UNEP see United Nations Environment Program uniform far-field loadings, 164 uniform rainstorms, 267 uniform rotation, 71 uniform translation, 51 unit discharge, 263 –264, 272 United Nations Conference on Environment and Development (UNCED), 218 United Nations Conference on Human Environment (UNCHE), 218 United Nations Environment Program (UNEP), 218, 283 unpoled piezoelectrics, 125– 126 unstable vortex equilibria, 28–29 unsteady phase boundaries, 108

V vacuum interfaces, 163 vapor, 104, 105, 108–111 vegetative canopy turbulence, 289, 295–297, 298, 299 velocity angular, 7, 9– 10, 21, 55 canopy turbulence, 295 –297, 298, 299 depth-averaged mean, 236–238 diffusion, 228 friction, 245 impermeable cracks, 137 multiphase flows, 92, 96, 104, 107 oscillatory boundary layer, 226 rainfall-infiltration-runoff generation, 263–264 shear, 295, 299 time series, 245, 246 vertical estuarine flow, 236–238, 242, 243 vortices, 11, 47, 49–50, 63 Yangtze river estuary, 243 vertical estuarine flow, 235, 242 –245 vertical salinity distribution, 252, 253 Vickers indentation, 125, 139 –141, 142, 197–198, 199 viscosity, 91, 226, 289–290 visibility range, 220 VOF see volume-of-fluid

328

Subject Index

void fraction, 100– 101 volcanoes, 219 volume-of-fluid (VOF) method, 87, 93–96, 98, 109 volumetric heat capacity, 285, 288 volumetric soil water deficiency, 263 –264 volumetric water content, 288 vortex crystals, 1–79 vortex streets, 19, 25, 60, 63 vortex stretching, 227 vortices, 1 –79 cylinders, 63–71 environmental mechanics, 227, 228 equilibrium, 3–4, 6, 21– 24, 30–31, 32–47, 55–68 geometry, 30–31 hyperbolic planes, 73–74 identical vortices, 17, 19–22, 32–39 periodic parallelograms, 71– 73 periodic strips, 63–65, 67, 68 polygons, 22–24, 26, 32 quasi-crystals, 70– 71 relative equilibria, 4, 12, 45, 58, 60 rings, 25, 30, 60, 61 spheres, 52–62 statics, 2– 12 stationary equilibrium, 11, 57–58, 63– 68 stationary patterns, 39, 44 strengths, 7–8 torus, 72, 74 translating patterns, 47–52

W wall streaks, 241 water absorption function, 288 Aksu water Budget Experimental Research Station, 291 biological processes, 230–231 cycling, 283, 285–286 deepening projects, 234 depth, 263–264, 273 –274, 275

erosion, 258 light transmittance, 252 loss, 256 plant requirements, 292 –293 quality, 230 resources, 220 –221 surface waves, 225–6, 237 –238 Waterway project, 233–256 wave theory, 226, 261 –262 wavenumber, 226 WCRP see World Climate Research Program wedge effect, 198 –199 Weierstrass zeta function, 72 weight function method (WFM), 156 –159 Weinan plain, 266 well-mixed estuaries, 239, 240 WFM see weight function method Williams eigenfunction expansion form, 156 Williams type energy release rate, 191 wind, 222, 225, 227 windbreaks, 283, 284 wobbling mode, 102 –103 work conjugate integrals, 156 World Climate Research Program (WCRP), 290–291 World Wars, 122–123 Wronskians, 45–46

Y Yalin’s formula, 279 Yangtze river estuary, 222, 233 –256, 257, 298 Yellow River, 221, 257, 265 –266 yielding, 122, 128, 178 –185, 201 Yucheng Experiment Station, 291, 301

Z zero-circulation, 38 zeta function, 72 zooplankton, 232, 252


Erratum ‘Fracture of Piezoelectric Ceramics’

by T. Y. Zhang, M. Zhao and P. Tong (Advances in Applied Mechanics, 38, pp. 147 –289, 2001) In Eqs. (2.37), (2.39), (2.68) and (2.70), the term 2D1 Ei ni should be changed to þfni Di;1 and in Eqs. (2.56) and (2.58), the term 2Dp1 Ei ni should be changed to þfni Dpi;1 . Thus, Eqs. (2.37), (2.39), (2.56), (2.58), (2.68) and (2.70) should read as ð J¼ ð fn1 2 sij nj ui;1 þ fni Di;1 ÞdG; ð2:37Þ G

J¼

ð G

J¼

ð G

J¼

ð

ð f p n1 2 spij nj ui;1 þ fni Dpi;1 ÞdG þ p

G

ðwn1 þ sij;1 nj ui þ fni Di;1 ÞdG;

ðw n1 þ

spij;1 nj ui J¼

þ

ð

2

P

lEij u; j ui;1 þ fpEj u; j1 dP; ð2:56Þ

ð P

ui lEij u; j1 2 fpEj u; j1 dP; ð2:58Þ

p

G

J¼

fni Dpi;1 ÞdG

ð

ð G

ð2:39Þ

ð f n1 2 sij nj ui;1 þ fni Di;1 ÞdG;

ð2:68Þ

ðwp n1 þ sij;1 nj ui þ fni Di;1 ÞdG:

ð2:70Þ

Acknowledgement: Thanks to Prof. C. M. Landis for identifying the errors in eqs. (2.37) and (2.39).


329


Advances in Applied Mechanics. v. 39

Advances in Applied Mechanics. v. 37

Advances in Applied Mechanics. v. 38