Finite volumes for complex applications: problems and perspectives. Vol. 2

Finite Volumes for Complex Applications I1 Problems and Pwspectives Finite Volumes for Complex Applications I1 Spons...

Author: Fayssal Benkhaldoun | Roland Vilsmeier

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Finite Volumes for Complex Applications I1 Problems and Pwspectives

Finite Volumes for Complex Applications I1

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0 HERMES Science Publications, Paris, 1999 HERMES Science Publications 8, quai du MarchC-Neuf 75004 Paris Serveur web : http://www.hermes-science.com ISBN 2-7462-0057-0 Catalogage Electre-Bibliographie Finite Volumes For Complex Applications I1 - Problems and Perspectives Vilsmeier, Roland* Benkhaldoun, Fayssal* Hkel, Dieter Paris : Hermks Science Publications, 1999 ISBN 2-7462-0057-0 RAMEAU : elkments finis, mkthode des analyse numkrique DEWEY : 5 15 : Analyse mathkmatique

Le Code de la proprikte intellectuelle n'autorisant, aux termes de I'article L. 122-5, d'une part, que les O XER

(1)

In (1), the vector of conserved variables is W = (p, pu, pv, E ) T ,the x-component of the flux is = (pu, P ~ 2 p, puv, u ( E p))T. Its y-component has a similar expression. In the problem of interest, the pressure p is related to W via 1 p = (y - 1 ) ( E - -p(u2 v2)) with 7 = 1.4. The solutions of (1) has to fullfill 2 the second law of thermodynamics : we have to have

+

FZ

+

+

dS at

-

+ div (SG) 5 0 in R

where S = -ps (with s = log 2 ) is the mathematical entropy. In [Ta], E. (07) Tadmor has shown that the solutions of (1-2) satisfy the following minimum principle

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2.2 Numerical schemes

2.2.1 Generalities The discretisation of (1) is carried out on a mesh made of triangles. The list of nodes is {Mi)i,l,,,. The generic name of a triangle is T , its vertices are denoted by Mi,, Mi,, Mi,, or 1, 2, 3 when there is no ambiguity. The schemes for (1) are written as

In (4), WF is an approximation for W(Mi, nAt), (Ci 1 is the area of the dual cell associated to node M t l The residual must satisfy div

ghdz := aT

h.li € T

In this conservation relation ghis an approximation of that has to be continuous across the edges of the triangles. Under classical asssumptions, we have a Lax Wendroff-like theorem [AMN] : the scheme, if it converges, converges to weak solutions of (1).In this paper, we follow the approach of Roe-DeconinckStruijs [SDR] via the parameter vector Z = (fi, f i u , f i v , f i H j T : Z is piecewise linearly interpolated on the mesh, and we get 2h.In [AMN] we show that other approximations of 2 enable to put the finite-volume schemes in this residual framework. We introduce Sl ( I = 1 , 2 , 3 ) the inward normal to T opposite to the node - Mi, and A, B the linearised Jacobian matrices computed at the average state Z = Zi, Z i 2 Zi3 . The relation ( 5 ) becomes

+

+

3

where 2Ki = (ni), 2 has real eigenvalues.

+ (ni), B.One can show that K i is diagonalisable and

2.2.2 Design principles The schemes are constructed follwing three design principles -

the scheme is upwind : if Ki only has negative eigenvalues,

'1n particular, we have (Ci( =

5 CT,M,ET (TI.

=0

6

Finite volumes for complex applications

-

the scheme must be linear preserving : if 1,2,3.

aT = 0 then

@T = 0 for i =

the scheme must provide monotone solutions, i.e. without oscillations. The first property is a translation of the physics of the problem. The second one enable to have second order of accuracy a t steady state . This is true for example of the SUPG and streamline diffusion schemes, and not true for standard finite volume schemes. This property can be understood by local truncation analysis on regular meshes [SDR] or on finite element like meshes by variational arguments [Ab]. The last one is really obscure because very intuitive. It is clear for scalar equations, and for system we will translate it formally as a discrete version of (3) : assuming a discrete version of (3), we get a maximum principle on the numerical solution because s is concave. But this has to be understood formally, for now. -

2.2.3 Two example : t h e s y s t e m N and LDA schemes The schemes have been introduced by Deconinck and van der Weide in [DvW] after their scalar versions. We have -

System N scheme

-

. The choices of and N = K;) and N are unique thanks to the conservation constraint (5). System LDA (low diffusion advection) scheme

@?= K+(z, - 2)

(EL,

-1

2

In the definition of these schemes, the matrix N appears. It is the inverse of K,-. This matrix is not always invertible. However, for the Euler equations, one can show [AMN] that it is always invertible except at stagnation points. In any case, one can always give a meaning to K+Z or N a T because the Euler equations are symetrizable, see [AMN]. Hence, there is no problem in the definition of @?or @ f D A . Both schemes are clearly upwind. The LDA scheme is linear preserving contrarily to the N scheme. The N scheme is monotone. This is very obvious for its scalar version because in that case we have

c:=,

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>

with c z j 0. The matrix equivalent formulation involve terms like K:NK~T that are difficult t o handle. In all numerical experiments (with a large variety of geometries and flow conditions) seem to indicate that the system N scheme also satisfies a discrete version of (3). Barth [Ba] has shown that the N scheme, for a linear symetric hyperbolic system is locally dissipative. In [Ab], we show that the LDA scheme is also locally dissipative for a linear symetric hyperbolic system. 3. A positive linear preserving scheme for scalar equation : the PSI scheme revisited

We consider the scalar versions of the N and LDA schemes and the following blended scheme @i = 1@N (1 - l)@fDA.

+

The firt remark is that this scheme satisfies the conservation constraint ( 5 ) whatever I E R. This scheme is upwind by construction since the N and LDA schemes are upwind. We now consider the positivity issue with the same technique as D. Sidilkover, ai = I@:

+ (1- 1)@fDA= 1 + (1 - 1)@ ):;

to (6), the positivity is obtained if 1

( @yA'

+ (1 - 1)- @N

"

. Thanks

,

2 0 for i = 1 , 2 , 3 Ifwe set

'

q D A +

rz = , a solution is given by

@F

1 2 max cp(ri) r,,q'#O

where cp(r) = 0 if r

r 2 0 and cp(r) = else. Simple algebraic manipulations

r -1 shows that if 1 is chosen with the = sign in (7), the scheme is positive and linear preserving. In fact, it is identical to the PSI scheme. 4. A scheme for the Euler equations

Following the same ideas, we consider a scheme written as

where 1 is a matrix. In order t o illustrate the design principles, we consider 1 = 1 = Id where I E R, but a more sophisticated method is developped in IAbI.

8


Conditions TOP Bottom

p 1.4 0.7

p 1

0.25

Mach number 2.4 4

TAB.1: Conditions for the interaction of 2 parallel supersonic flows Let us denote by v = Vws. From

and thanks to the numerical experiments, we assume that the system N scheme has a local minimum principle for the specific entropy. Following the same arguments as for the scalar PSI scheme, the conditions are 1 + (1 - l)ri 2 0, for i = 1,. . . 3 TaLDA

where here ri = vi ' . One can show that this scheme is also locally dissivT@N pative [Ab]. 5. Numerical experiments

We present some results obtained in two different test cases. The first one is the iteraction of two parallel supersonic flows. The conditions are given in Table 1 The mesh is given on Figure 1-a The isovalues of the density are presented on Figure 1-b. A very clear improvement of the results can be observed. The new scheme give monotone results that are more accurate than those of the finite volume scheme (MUSCL extrapolation on conserved variables). The second test case is a GAMM test case : Naca0012, Mach number : 0.85, angle of incidence : 1 degree. We show the isolines of the Mach number (Figure 2-a) and the isoline of the reduced entropy (Figure 2-b). It is clear that the slip line out of the leading edge is improved as well as the entropy profiles. 6. Conclusions We have sketched the construction of upwind residual schemes that are also linear preserving. Some numerical example indicate that these new schemes are more accurate than the now classical finite volume schemes.

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Bibliography

[Abl [AMN]

Pal [Dal

[SDRl

lTa1

IDv wl

R. ABGRALL.Upwind residual schemes on unstructured meshes, in preparation. R. ABGRALL, K. MER, A N D B. NKONGA. A Lax-Wendroff type theorem for residual schemes. In M. Hafez and 5.3. Chattot, editors, Proceeding of a conference for P.L. Roe's 60th birthsday. Wiley, to appear.. T.J. BARTH.Some working note on the n scheme. Private communication, 1996. S.F. DAVIS.A rotationaly based upwind difference scheme for the Euler equations. J. Comp. Phys., 56 :65-92, 1983. R . STRUIJS,H. DECONINCK, A N D P . L. ROE. Fluctuation splitting schemes for the 2d euler equations. VKI L S 1991-01, Computational Fluid Dynamics, 1991. E. TADMOR.The numerical viscosity of entropy stable schemes for systems of conservation laws. Math. Comp., 49 :91-103, 1987. E. VAN DER WEIDE A N D H. DECONINCK. Positive matrix distribution schemes for hyperbolic systems. In Computational Fluid Dynamics '96, pages 747-753. Wiley, 1996.

(a)

FIG. 1: (a)-Mesh for the interaction of 2 parallel supersonic flows ; (b)- Density isolines for the supersonic flows. Top-left : N scheme p E [0.7,1.4], top-right : second order MUSCL finite volume scheme p E [0.689,1.403] , bottom-left : LDA scheme p E [0.615,1.427] ,bottom-right : present scheme p E [0.698,1.402]

10


FIG.2: (a)-Mach number isolines for the Naca0012 test case. Top-left : N scheme M t [0.05,1.394] , top-right : second order MUSCL finite volume scheme M E [0.05,1.422], bottom-left : LDA scheme M E [0.05,1.425], bottom-right : present scheme M E [0.04,1.522]; (b)-Reduced entropy C = V N scheme C E [0,0.038],second order MUSCL finite volume scheme [-0.009,0.039], LDA scheme C E [-0.009,0.091], present scheme X t [-0.0003,0.032]

5-

A Roe-type Linearization for the Euler Equations for Weakly Ionized Gases

Fre'de'ric C O Q UEL L A N C N R S Tour 55-65 5ieme, U.P. M. C. 75252 Paris Cedex 05 Claude M A R M I G N O N O N E R A , BP 72 92322 Chitillon Cedex

This paper is devoted to the numerical approximation of the discontinuous solutions of the Euler equations for weakly ionized mixtures of reacting gases. The main dificulty stems from the non conservative formulation of these equations due to a widely used simplifying assumption. W e show how to derive a well-posed conservative reformulation of the equations from the analysis of the associated convective-diffusive system. We then propose an exact Roe-type linearization for the equivalent system of conservation laws. Our results can be seen as an extension of the classical Roe average, for nonlinearities that cannot be recast under quadratic form.

ABSTRACT

Iiey Words: Convective-diffusive systems. Nonlinear hyperbolic systems. Non conservative products. Shock solutions. Roe-type linearization.

1. Introduction

This work treats the numerical approximation of the solutions of a convectivediffusive system, we write for short as

~ , u + A ( ~ ) ~ , u - ~ , ( D ( ~ ) ~ , u ) ~=>~O( , uZ )E , R .

(1)

This system governs ionized mixtures of reacting gases in thermal nonequilibriurn. Such plasma are studied here in the context of large Mach number flows.

12


The solutions we are interested in, are thus mainly driven by the underlying first order system. The main properties of the extracted first order system

are reported below. This nonlinear system will be seen to be hyperbolic so that its solutions are known to develop, generally speaking, discontinuities in a finite time. But when dealing with discontinuous solutions of (2), a major difficulty arises : there does not exist a flux function, say f , such that A(u) = ~ , f ( u ) . In other words, the hyperbolic system (2) is under non conservation form. It is known that the non conservative products involved in A(u)d,u have no classical sense at the location of a shock since they cannot be given a unique definition within the standard framework of distributions. For this reason, it must be recognized that an additional information is required in order to specify the definition, e.g. the value, of the non conservative product d(u)d,u at shocks. This difficulty has motivated some recent works. We refer in particular to the work by LeFloch [8], DalMaso-LeFloch-Murat [6] where non conservative products are defined on the basis of a fixed family of paths in the phase space : atu [ A ( ~ ) d , u =]0, ~ t > 0, x E R. (3) After LeFloch [8] and Sainsaulieu [ll],the choice of a particular family of paths is dictated by the additional informations brought by the full second order convective-diffusive system (1) (see below for a brief survey). The key feature is that the definition of shock solutions heavily depends on the shape of the diffusive tensor 2) (u) which is modeled in agreement with the physics. These definitions provide us with a relevant setting for defining the discontinuous solutions of the non conservative hyperbolic system (2). Once defined, the first order system is well-posed and its numerical approximation could be tackled. However, two difficulties arise in that way. First, a close formula for shock solutions is in general not available. Furthermore, even when explicitely available, we have illustrated [4] that the error in the discrete capture of shock solutions unacceptably grows with the strenght of the shock. We refer to LeFloch-Liu [9] for an error analysis devoted to the scalar case. At this stage, these two difficulties make the numerical approximation of the (strongly) discontinous solutions of (3) to be virtually untractable. To overcome these two difficulties on the same time, we propose to study the existence of a conservative formulation for system (3) that is compatible with the diffusive tensor 2). That is to say, we ask for the existence of (at least) one change of variables v = v ( u )that brings the non conservative second order

+

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system (1) (with w (u) = 0) to a fully conservative convective-diffusive system

Let us emphasize that not only the first order system must find a conservation form ( P l ) but also that the second order operator must stay under conservation form (P2). Actually, these two requirements (Pl),(P2) ensure the equivalence and V are compatible and the shock of the shock solutions of (3) where solutions of ~ , F ( v )= 0 , t > 0, x E R . (5)

atv +

We refer t o Sainsaulieu-Raviart [12] for a proof. The benefit of such an equivalence is twofold. In a first hand, the shock solutions of (3) are now explicitely given by the Rankine-Hugoniot jump relations associated with (5). In a second hand, Riemann solvers under conservation form can be applied to (5) in order to approximate the equivalent weak solutions of (3). As reported below, a specific family of change of variables turns out to fulfill both (PI) and (P2). For the associated equivalent systenls of conservation laws, we then show how to derive an exact Roe-type linearization.

2. Analysis of the extracted first order system

In this section, we focus ourselves on the definition of the extracted first order system (2), the precise shape of the diffusive tensor V will be discussed later on. We treat mixture of gases made of electrons and n heavy species, ni, 1 5 ni n of them being ionized. All the species we consider are described with the same mean velocity 11. To account for the smallness of the mass ratios M e / M i

0.

(45)

In order to enforce the validity of the two conditions (44) and (45), we now consider the following easy but key lemma for averagings (see also Abgrall [I], Godlewski-Raviart [15]). A v e r a g i n g L e m m a . Let be given ( x L , xR), ( y L , yR) two pairs of real numbers. Let ( T L , T R ) be a n y g i v e n pair of real numbers such that TL TR # 0 . Let U S define the following unsymmetric T-averaging operators :

+

Then, the following identity holds true :

Let us apply for instance the above averaging Lemma to the pairs of interest ( p , X) with the @-averagingwe define by e L = f i l QR = & ,. We easily get from (46) and (47) the well-known Roe identity, we write with classical notations :

Q2 X

X,

fie p =p=

JPLPR

with ApX =

+ -pAX.

(48)

Turning back to the general case, we emphasize that the identity (47) is indeed valid for a n y g i v e n pair (rL,rR). Taking advantage of such a degree of freedom, we introduce below a (wide !) family of unsymmetric averagings that makes always valid the required consistency condition (44). Equipped with these families, we next turn studying how to enforce the hyperbolicity condition (45). To that purpose, the specification of the underlying averaging operator is clearly the central issue. The statements, given below, summarize our main results. They are intended to shed some light on the application of the averaging Lemma we have introduced in order to enforce the validity of both (44) and (45).

24


3.1. Enforcing the Roe consistency condition (44).

In view of the electron gas pressure law (34), the required discrete identity (44) writes :

The above consistency condition involves increments in the conservative variables pYe, p and se while the underlying nonlinearities merely occur in terms of the primitive variables Ye,p and s e / p . To make the needed linearization tractable, we specify in the next statement a convenient family of averages for (42) that shifts (44) from the conservative to the primitive variables. Proposition 12. Let us consider the following averaged forms for the partial derivatives (42) of the electron gas pressure law :

dpe 8P

-

a s s S + E x {yepye-1h(2) - A x p ~ e - l h ' ( ~ ) )(50b) , P P P

-

0

Se dpe = Ye x pye-lhl(-). -

ase

P

Then, for any given pair of states (vL,vR), the consistency condition (44) is p is defined i n (48) : equivalent to the following identity, where -

In order to satisfy (51) and thus (44), it remains to define the unspecified mean values in (50). This is the matter of the next statement. Proposition 13. W i t h the notations of the Averaging Lemma, let us define the averaged forms i n (50), using respectively an arbitrary u-averaging operator :

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and an arbitrary C-averaging operator : C 't

h ( F ) x APrelAp,

if A p

#0, (53~)

C 't

h ( $ ) x yepre-', x

o p~e-lhl(:)

=

otherwise,

'C x A h ( ? ) / ~ h pr.

P

P

i f A$

# 0,

'

(53b) otherwise.

Then, the identity (51) and thus the Roe consistency conditions (37'b)-(37'~) are always valid. In what follows, we shall assume for convenience that both a and ( stay non negative. Equipped with this wide family of relevant averagings, we now turn studying for validity the hyperbolicity condition ( 4 5 ) .

3.2. Enforcing the hyperbolicity condition (45)

The main result of this second step is as follows : Proposition 14. Let us respectively define the a and ( averaging operators i n (521, (53) by

= ( 0 , l ) , if Ye L

< Ye R ,

CL=C(PL), CR=C(PR)

with

((TL, UR)

( 1 , O), otherwise;

(54)

C(p)=pYe-3.

(55)

Then, (54), (55) provide us with the unique pair of averaging operators such that the hyperbolicity condition (45) holds true for any given ( v L ,v R ) E f12. We have the following final statement. Theorem 15. Let us consider the following averaged form of V V 3 ( v ):

{g,

with 5,w,H , s e l f , % e U f } given in (do), with VTPderived from [1] o r [13] and with V v p , constructed from (do), (50), (52)-(53). ) three real Then, for any given pair of states ( v L ,V R ) E R 2 , B ( V L , v ~ admits eigenvalues with a complete set of right eigenvectors. Moreover, this matrix satisfies the Roe consistency conditions (37b) and (37c). Therefore, B ( V L , V R )

-

26


is an exact Roe-type linearization for the system (35)-(36). We emphasize that the averaging operator built on (55) provides one with an extension of the classical Roe-average (48). Indeed, let us compare two of the nonlinearities that arise in the pressure laws : namely pYe x { h ( y ) ) with

(7)').

p x {f Now, it is sufficient to apply to the latter nonlinear expression our formula (55) with the associated exponent a = 1 in place of ye to get &i.

111

R. ABGRALL, La Recherche Agrospatiale, vol 6, 31-43, 1988.

[2]

G . CANDLER 88-0511, 1988.

[41

F . C O Q U E L A N D C . MARMIGNON, Work in preparation.

151

S. C O R D I E R

[61

G . DALMASO, P . G . LEFLOCH A N D F. MURAT, J . Math. Pure Appl., vol 74, 483-548, 1995.

[71

B. LARROUTUROU, Computational methods in applied

AND

ET

R . W . MACCORMACK, AIAA paper

al., Asymptotic Analysis, vol 10, 1995

sciences, ECCOMAS, Eds Ch. Hirsch, Elsevier, 1992. [81

P.G. LEFLOCH, IMA Preprint series No 593, University of Minnesota, 1989.

[9~

P . G . LEFLOCH

[lo]

P . L . R O E , J. Comp. Phys., 357-372, 1981.

11

AND

J. L I U , Math. of Comp., 1994.

L. SAINSEAULIEU, SIAM J . Appl. Anal., 1995.

[I2]

P . A . RAVIART A N D L. SAINSEAULIEU, Mathematical Methods and Models in Applied sciences, 1995.

[I31

J .S. SHUEN

[I41

YA. ZEL'DOVICH A N D Yu. P . RAIZER, Physics of Shock Waves and High-Temperature Hydrodynamic Phenomen, vol 1 and 2, Academic Press, 1966.

[I51

E. GODLEWSKI A N D P. A. RAVIART, Numerical Approximation of Hyperbolic Systems of Conservation Laws, Springer (Eds), Applied Mathematical Sciences, vol. 118, 1996.

ET

al., NASA TM 100856, 1988.

Multidimensional upwind residual distribution schemes and applications

H. Deconinck and G. Degrez von Karman Institute for Fluid Dynamics Sint-Genesius-Rode, Belgium

ABSTRACTA review is made of upwind residual distribution schemes(RDS)for hyperbolic systems for application to compressible and incompressibleflows. First, the design principles of RDS are explained for the simplest case of linear scalar advection. Extension to linear hyperbolic systems is described next. Then, their application to compressible and incompressible pows is discussed, and illustrative examples of applications arepresen fed. Key Words: jnite volume method, finite element method, compressible paws, incompressible po ws

1. Introduction

Multidimensional upwind residual' distribution schemes, which were first introduced by P. L. Roe [ROE 871, have been developed on ideas borrowed from both the finite volume and finite element methods to become nowadays an attractive alternative to either one [PA1 971. The initial motivation for their development was a discontent about some drawbacks of the state-of-the-art finite volume solvers based on 1-D approximate Riemann solvers, namely D by D first order upwinding is very diffusive,

1D Riemann solvers do not capture the real multidimensional flow physics, higher order schemes use wide stencils. The starting point for the development of these schemes was a reinterpretation of 1D finite volume schemes based on the concept of fluctuation [ROE 821. Considering the continuous piecewise linear data representation classically used in finite element methods rather than the discontinuous cell-wise reconstructions used in finite

28


volume methods, the flux difference between two nodes (Fi+, - Fi) is reinterpreted as the flux balance (else fluctuation or residual) over the interval (element) [ i ,i + 1 ] (= J, d F / d x dx), which is to be distributed to the vertices of the element. This interpretation can be carried over to several dimensions, provided the fluctuation is now the flux balance over a simplex (trianglettetrahedron) to which is associated a continuous piecewise linear data representation. Because the elementary discretisation unit is an element rather than an edge or face for the finite volume method, the multidimensionality of the problem can genuinely be taken in to account: in 2D, the fluctuation can be distributed to the three vertices of the triangle rather than to the two neighbouring cells of an edge. Based on this idea, schemes were constructed, which combine a number of attractive features: a much lower cross-diffusion than their finite volume counterparts, due to the genuinely multidimensional upwinding they incorporate, a positivity property which ensures the satisfaction of a discrete maximum principle and consequently the absence of spurious oscillations, (almost) second order accuracy on compact stencils. In addition, the compact discretisation stencil allows for the development of efficient implicit iterative solution strategies [ISS 961 and for an easy parallelisation [ISS 98, vdW 991. The paper starts with the presentation of upwind residual distribution schemes for linear hyperbolic scalar equations and systems, successively. This is followed by a discussion on their application to non-linear problems, specifically to the compressible and incompressible flow equations. Finally, the paper concludes with a few illustrative computational examples. 2. Linear equations 2.1. Scalar advection

2.1.1. Design principles The residual distribution schemes (RDS) have been designed for an optimal discretization of the steady state convection equation

on PI finite element meshes, i.e. triangular (resp. tetrahedral) meshes with a piecewise linear solution representation. Evaluating the residual or fluctuation over an element

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Figure 1: Triangular element with scaled inward normals

T, defined as the integral over this element of the differential operator, i.e.

d

= C , k i u i ,where since both A and V u are constant over the element, one obtains + ki = A . Ziis called the inflow parameter, Ei being the scaled inward normal of the edge opposed to node i (see Fig. 1) and d the dimension of the problem. The method consists in distributingfractions of 4T to the vertices of the element. The resultingdisCrete equations therefore express that the nodal residuals Ri,sum of all contributions from neighbouring elements, vanish, i.e.

3

=Z@T=CpiT@T =O

Ri

T

(3)

T

in which 3'j are the distribution coefficients. On each element T, these distribution coefficients must sum up to one for consistency and conservativity. The different schemes, corresponding to different ways of computing the distribution coefficients, have been designed to satisfy several properties making them optimal: UPWIND CHARACTER (q): No fraction of the element residual is sent to upstream

nodes or pf = 0 when

ki5 0.

POSITIVITY (9): The scheme does not create local extrema or, if we write the con-

tribution to the element residual as

0 'v'j#i.

4:

= p:4T

= C j c i j u j ,we impose cij

Nand hence P ~ ~ P uniquely. commuting systems, the invariance condition is no longer sufficient. The system PSI scheme used in the numerical applications is based on one particular definition of PF)N which satisfies this condition. Further details are given in [PA1 971. The construction of non-linear positive and linearity preserving schemes for systems is still an ongoing research topic. New developments, based on the reinterpretation of the scalar PSI scheme as a blended NILDA scheme are discussed in R. Abgral1's invited paper in this conference [ABG 991.

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3. Compressible flows 3.1.

Conservative linearisation

The residual distribution schemes presented in the previous section have been defined for linear problems. To apply them to non-linear problems such as the Euler equations, an essential ingredient is a conservative linearization, which consists in finding on each triangle T an average state U such that

where

au

-

-

I

=

ax, a,

dn,

Un,dr

so that, locally, the non-linear system of conservation laws dF,/ax, = 0 is approximated by the linear system A e d ~ / d x = e 0. Let's show that, with such a linearization, conservation is indeed satisfied. Summing up the nodal residuals over the domain 0, we have

and the contributions of internal edges cancel out (telescoping property). For the Euler equations, a conservative linearization is easily obtained as a multidimensional extension of Roe's linearization [DEC 931. Indeed, assuming a linear variation of Roe's parameter vector Z [ROE 811 and since both the vector of conserved variables U and the fluxes Fb are homogeneous functions of degree 2 in the components of Z,

where Z =

3.2.

& CverticesZiis the proper average state to ensure conservation.

Transformations and preconditioning

The matrix distribution schemes presented above are invariant under a similarity transformation. It is nevertheless useful to apply a similarity transformation to the lin-

34


earized hyperbolic system in order to achieve maximum decoupling for the following reasons. On one hand, this reduces the computational cost thanks to the considerable simplification of the distribution expressions and on the other hand, this allows to select different distribution schemes on the various decoupled systems/scalar equations. Specifically, with the similarity transformation dU = T a Q , the discrete equations (9) are rewritten T PQiQa T T =o R~= (18)

CT T

where 0;= C i K Q i Q i . Partial decoupling can be achieved if the flux jacobians Al have common eigenvectors. For the Euler equations, the flux jacobians At have one common eigenvector so that it is possible to decouple one scalar equation from the original system, leaving a coupled 3 x 3 system and one decoupled scalar equation du,dpin 2D. The transformation to symmetrizing variables defined as d Q =

(s,

a2dp)' accomplishes this task. The decoupled scalar equation is nothing else than the entropy advection equation, which is well-known to derive from the Euler equations. As shown in [PA1 971, additional decoupling may be achieved by preconditioning, namely the system of equations is rewritten as1

and the residual distribution method is applied to the preconditioned system between brackets. The optimal preconditioning was found to be the van Leer-Lee-Roe preconditioning [vLE 911, which allows to decouple one additional equation, namely the total enthalpy advection equation.

3.3.

Viscous terms

Viscous terms are discretised using the finite element interpretation of the schemes presented in section 2.1.2.. Specifically, the space discretisation of the divergence of the stress tensor V . ? is

Now, since rn; differs from the PI shape function qi only by a piecewise constant, VoT = Vqi, so that, for interior points, the discretisation reduces to a Galerkin finite element discretisation. 'The original conservative variables are used here, but the same procedure can be (and actually is in practice) done for the symmetrizing variables.

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4. Incompressible flows For application to incompressible flows, two avenues have been explored. The first approach, followed by Waterson & Deconinck [WAT97], is based on the fact that the incompressible Navier-Stokes equations are a symmetric advective-diffusive system [DEM 971, which can therefore be discretised using the system distribution schemes presented in section 2.2. using a collocated P1 finite element representation for all variables. The collocated arrangement is made possible by the inherent pressure stabilisation effect of the system discretisation. The second approach, followed by Bogaerts et al. [BOG 981, is based on a stable finite element representation of the velocity and pressure fields (PlisoP21Pl element) which therefore does not require any pressure stabilisation. It results that scalar residual distribution schemes can be used to discretise the momentum equations, i.e. using the finite element interpretation, the discretised equations are

where q.,kis the residual distribution weighting function associated to the velocity node i and to the kth component of the momentum equation, and y j is the shape function associated to the pressure node j. Note that, because of the upwind component of the weighting function upwinding is introduced in the discretisation of the pressure gradient. This turns out to be essential for the accuracy of the method [BOG 981.

5. Applications Capabilities of upwind residual distribution schemes are now illustrated by a couple of computational examples. 5.1.

Znviscid transonicJlow over the M6 wing

The ONERA M6 wing is a well documented testcase for three-dimensional flows from subsonic to transonic speeds [AGA 941. The selected transonic case is M , = 0.84, a = 3.06". The grid consists of 316275 nodes and 1940182 tetrahedra. The far-field boundary is half a sphere with a radius of 12.5 root-chord lengths. The computation of this testcase with the present multidimensional upwind method employed the van Leer-Lee-Roe preconditioning, allowing a hybrid discretization. The decoupled

36


advection equations for entropy and total enthalpy are discretized with the scalar PSIscheme, while the acoustic subsystem is discretized with a blended system LDAINscheme. To give an indication of how the multidimensional upwind method performs compared to standard finite volume methods, the same testcase has also been computed with the Jarneson-Schmidt-Turkel (JST) artificial dissipation scheme, a matrix dissipation (MATD) scheme and the Roe upwind scheme with Venkatakrishnan's reconstruction. Plots for the pressure coefficient Cp and the relative total pressure loss, defined as 1 - p,/p,,, in the 80% span cross-section are shown in Fig. 2. From these figures it is clear that the multidimensional upwind method is much less dissipative in the leading edge region and that it has a better shock capturing. 5.2. Znviscid transonic flow over a complete aircraft

To demonstrate the applicabilty of the present multidimensional upwind method to a complete aircraft configuration, the transonic flow over a generic model of the Falcon 2000 executive jet has been computed. The grid for a half model consists of 45387 nodes and 255944 tetrahedra. The selected testcase corresponding to the cruise condition is M, = 0.84, a = 3.06". The solutions computed with monotone first and second-order multidimensional upwind schemes are compared. For the first order computation the system N-scheme is applied directly to the full Euler equations. The second order computation employs the van Leer-Lee-Roe preconditioning, where the advection equations for entropy and total enthalpy are discretized with the scalar PSIscheme and the acoustic subsystem is discretized with the blended system LDA/Nscheme. Figures 3-4 compare the Mach number and entropy isolines for these two solutions. The better shock capturing and lower spurious entropy generation in the leading edge region of the wing and the engine pylon are clearly observed.

5.3. Incompressible turbulentflow over a backward-facing step We consider now the incompressible turbulent flow over a backward-facing step, experimentally studied by Kim [KIM 781. Calling H the step height, the inlet channel width is 2H, so that the outlet channel width is 3H. The flow Reynolds number is 1.41 10' based on the inlet centreline mean velocity Uo and the outlet channel width. The standard k - E turbulence model [JON 721 is used with wall function boundary conditions. The grid is a triangulated stretched Cartesian grid extending from 3H upstream of the step to 27H downstream. Along all solid walls, the computational domain boundary is set at a distance h = 0.025H from the wall. The grid contains 6247 P1 nodes distributed along 107 vertical grid lines, where each vertical grid line contains 41 P1 nodes in the inlet channel and 61 P1 nodes in the outlet channel. The inlet boundary conditions for u , v and k are taken from the experiment and E

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ym=.s

MDHR ----MD

.-..- - - - .

OW5

-

0

I

0

-1

- , - - _ _-- - -\I ----_ I ,

I

I

0.25

I

I

0.5

I

,

I

I

1

0 75

X

-

0

MDHR

----Roe -0 01 I

0

0.25

,

,

.

l

l

0.5

,

,

,

,

l

0.75

,

,

l

l

J

t

Figure 2: M6 wing: Cp and total pressure loss distributions at 80% span, upwind residual distribution schemes versus finite volume schemes

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(a) system N-scheme

(b) blended system LDAM-scheme Figure 3: Falcon: Mach number contours

FALCON (453?7 nodpp) M1.05 an1 full EuMr N-schme

(a) system N-scheme

FALCON (45386 nodes) M.35 a=l p m n d i m d Euler (WE).

PSI +

L

D

-ma~

(b) blended system LDm-scheme

Figure 4: Falcon: Mach number contours is evaluated using a mixing length model. At the outlet, p is set to 0 and homogeneous Neumann (fully developed flow) boundary conditions are specified for all other variables. Along the wall function boundaries, a homogeneous Neumann condition is used for k and a Dirichlet condition is used for E (= ~ 3 , / ~ k ' / ~ /The ~ h wall ) . shear stress is derived from the law of the wall. Fig. 5 show the flowfield pattern calculated using the LDA scheme. The reattachment length is 6.3H, a value closer to the experimental value of L,/H = 7.0 than

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Streamlines

Figure 5: Turbulent flow over a backward-facing step: mean flow streamlines values obtained by other numerical computations using the same turbulence model, which fall in the range (5.7 L,/H 6.0 [DEM 971). The present method is also seen to capture a small secondary recirculation region at the foot of the step.

r for the duality pairing between H,',/~(I') and H - ' / ~ ( F ) . Further, we denote by n i the unit vector normal to aRi for i = 1 , 2 and pointing outward to the domain. Finally, we split the interface boundary r = r- U r+ where r- = {x E : b(x) . n l < 0) and I?+ = {x E r : b ( x ) . n l 2 0). Note, that this splitting is with respect to the vector n l . An illustration of these notations in 2-D is given on Figure 1. In R1 we use a mixed setting of the problem (1). That is, we introduce the new (vector) variable u = -aVp. To distinguish the solutions in the subdomains we denote by pl = plnl and p, = pl,,. The composite model will impose different smoothness requirements on the components pl and p2. More specifically, we will require that u E H(div, Stl), pl E L2(SZ1), and p2 E Hd (Stz, a0 \ I'). Note that pa is required to vanish on dR2 \ l?. Testing the equation a c l u Vpl = 0 by a function v E H(div, a),using integration by parts, the zero boundary conditions for pl on dR1 \ r , and the fact the trace of pl on I? is the same for the trace of pa on I?, one ends up with the equation,

+

Further, in order to describe the weak form of the equation

V-u

+ V . (bpi) + cop1 = f (x)

in R1

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we need to allow discontinuous functions pl from the space Hfoc: vl E L2(R1) : H,',,(fll)

=

there is a partition K of R1 such that vlIK E H1(K) for all K E K

The functions in H:,,(Rl) have traces from both sides of the interfaces of the subdomains K. Namely, for a given function pl E Htoc(Cll) we denote these traces by p? and pi, where "a" stands for the outward (with respect to K ) trace and respectively, "i" stands for the interior trace. Next, we give the weak form of the above equation. We borrow this formulation from the discontinuous Galerkin methods (see, e.g. [lo], pp. 189-196) by testing the equation by a function wl E H:oc(R1). We note, that this setting is quite similar to the mixed finite element method for convection-dominated convection-diffusion-reaction equations (see, e.g. [12]). Since the functions from Hkc(R1) are piece-wise smooth with respect to the partition K we shall integrate over each K E K and then sum the results. Following [lo] we find first the contributions of the advection-reaction operator Cpl by introducing the bilinear form CK(pl, wl) for any subdomain K E K :

Here n is the outer unit normal vector to d K . Next, we integrate by parts in each subdomain K and sum over all K E K. Thus, for pl , wl E H:oc (01) we get:

Note that this bilinear from is well defined for both continuous and discontinuous functions with respect to the partition K. From this expression we see that if the subdomain K has a sideiface on r- then the trace pi' should be replaced by its counterpart from R2, namely by p2(x). Also on J?- we have w", wl and on dR1- \ r- we take py = 0. Further, for a given function t(x) we denote by t- = min(0, t) and t+ = max(0, t) . Thus, we get the following weak form of the second equation valid for all wl E Htoc(Rl):


56

where

PI

b . V W Idx + ( % P I ,

wl)

for P I :

wl

E H:oc(Q~)

KEK

(6)

and

Finally, testing the equation (1) by a function w2 E H i ( R 2 ; an2\I?), using integration by parts, the zero boundary condition for w2 on d o 2 \ I',and the fact that u . n l = - a V p l . n l = aVp2 . n2 on I',one arrives at,

for all w2 E HA ( R 2 ; an2 \ r). There are various ways one can take into account the influence of the problem in the domain 0 1 on the problem in Q 2 . One of the possibilities, which we shall use further, is to try to make a formulation, which is stable for small diffusion coefficient (or even for vanishing diffusion). In this case it is very important to formulate correctly the boundary conditions. Namely, at the "inflow" part of the interior boundary the solution should be specified from the "outside" data. Taking into account that I?+ is the "inflow1' part of r for the subdomain Q 2 , we add Sr+P I w2 b . n2 d s and subtract its equal Sr+p2w2 b . n2 ds since on I' we have pl = pa. Thus, we get the following form of the last equation:

for all

w2

E H,1 ( Q 2 ; aQ2 \

I?), where

Thus, the coupled system for the three unknowns u E H(div, Q l ) , pl E a02 \ I ') consists of the equations (3), ( 5 ) , and (9)

Hk,(S11) and p2 E H;(n2;

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summarized as:

for all v E H(div, R1), wl E H&,(R1), and w2 E Hi(R2; aR2\I?), respectively. The bilinear forms aij (., .) are defined by (6), (7), (lo), and (1I ) , respectively. 2.2. Well-posed-ness of the composite problem

Here we verify the existence and uniqueness of the solution of problem (12) and its stability in an appropriate norm. For this we shall need some additional notations. Let & = { e ) be the set of edges/faces of the subdomain O1 from K and Lo the set of interior for R1 edges/faces. Recall, that nl and nz are the outward unit normal vectors t o R1 and R2, respectively. For any edge e E Lo denote by n, a fixed unit vector normal t o e and let K,f and KT be the two adjacent to e subdomains from the partition K. For edgeslfaces that are on dRl we shall always assume that n, = n l . Further, denote by [vl] and El the jump and the average of the discontinuity of vl, respectively, along any edge e. More precisely, this is the difference and the arithmetic mean of the traces vl J K a and vl l K ; taken from both sides of e:

Further, we use the following natural norm for vl E H&,(fl~) and v2 E H;(R2; 8 0 2 \I?):

+

(aVv2, Vv2).

(13) All terms in the expression on the right are nonnegative and this defines a norm on the space Hk,(O1) x H;(R2; dR2 \ I?). Note, that under certain conditions on the vector field b this is a norm even if yo = 0.

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The stability of the composed problem (12) is based on the following theorem:

Theorem 1 T h e solution of the problem (1.2) satisfies the a priori estimate:

II~11~2(n~) + II~lllq,nl + Il~2llf,0, 5 cilf IlE, n l

(14)

where the * - n o r m i s defined b y (13). The proof of the above theorem is based on the following lemmas.

Lemma 1 T h e bilinear form (6) defined for vl, wl E H,',,(Rl) can be transformed t o the following form:

Furthermore,

+-2 Ian,,

v; b . nl d s - -

2 Ian,-

v; b . nl ds.

Lemma 2 For all vz € H,'(R2, df12 \ F)

3. Coupling mixed and finite volume approximations of the convectiondiffusion equation Our approximation strategy is based on the finite volume method in the framework studied by Cai [4],Cai, Mandel, and McCormick [6] and also by Bank and Rose [I].

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3.1. An outline of the finite volume element method

We first outline a finite volume discretization method for the case of pure diffusion problem posed on R2,

Here f E L 2 ( n 2 ) is given and q ~ for, the time being is assumed given in the space L 2 ( r ) . The finite volume method under consideration uses two different finite dimensional spaces: a solution space W 2 and the test space W;. The Wz is the standard conforming space of piecewise linear functions over the triangulation E = {T) of f12 into triangles in 2-D and tetrahedra in 3-D (we call them simplices). To introduce the test space W,' we need a dual partition Vz of the domain into the finite (control) volumes V. Let N denote the set of all vertices (nodes) of the trianglesltetrahedra from 72 and let No be a subset of those vertices that are not on the Dirichlet part of the boundary df12 \ I?. In each simplex T E 72 one selects an interior node XT. Next, in 2-D one links XT with the midpoints of the sides of the triangle. In this way the triangle is split into three quadrilaterals. In 2-Dl one can select XT t o be the orthocenter of the finite element T and then the edges of the volume V(x) will be the perpendicular bisectors of the finite element edges (see the right Figure 2). With each vertex x E N of a simplex from 72, we associate a volume V = V(x) that consists of all quadrilateral/polyhedra having x as a vertex (see Figure 2 for finite volumes in 2-D). The splitting of f12 into finite volumes V forms the partition V2 (see, Figure 3). Consider now the test space W2*spanned by the characteristic functions of the volumes V E V2 and that vanish at the nodes N \ No on the boundary 8 0 2 \ r. If one defines the piecewise constant interpolant I; with respect to the volumes V E 112, then the space W; is actually equal to I,*W2 because they have the same degrees of freedom (associated with the vertices x E ni).

The L2(Rz) and H1(R2) norms in W2 are defined in a standard way. We shall need also discrete variants of these norms for functions in W;. First, we define the interpolation operator Ih : Wh+ I-+ Wh by the following natural rule: Ihv2+is the piece-wise linear interpolant of vd over each finite element T E 72. Then we define I l ~ f l l = ~ , (IIhvfl(l,n, ~ This norm is essentially formed by the squared differences of the values of vf a t the vertices of each finite element.

60 Finite volumes for complex applications

Figure 2: A finite volume V associated with a vertex from the primal triangulation. Left: the vertices of V interior t o the triangles T are arbitrary, whereas those on the edges of T are midpoints.

Figure 3: In we use the lowest order Raviart-Thomas spaces over the finite elements T; in R2 we use a solution space Wz of continuous functions that are linear over the finite elements T and a test space W,* of pice-wise constant functions over the volumes V.

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Next, we define the following, in general nonsymmetric, bilinear form on w 2

x

w;:

a2,h(v2,~;) = xENo

=-

/

w;(x)

/

aVp2,h'nds

~ V ( X )

aVp2,h - n w; ds, for all v2 E W2, w; E W;

xEN~av(x) Then the finite volume approximation of (15) is: find v2 E W2 which satisfies the following identity for all w,' t W,';

We note that the integrals over dV for V = V(x) a volume corresponding to a Neumann node x (i.e., if x E I?), contain only the interior (to a2)part of d V . We assume that the triangulation 72 is aligned with the possible jumps of the coefficient matrix a(x), i.e. over each finite element T E 72 the matrix a(x) has smooth elements. Therefore, there is a constant Co > 0 such that for all

where &(x)=

a(s)ds/meas(T), x t T. IT

These inequalities of two d x d matrices with real elements are understood in the sense of inequalities for the corresponding bilinear forms, i.e. a ii(x), iff tTa[ 2 JTii(x)t, V 6 E Rd. Also, the above equality of the matrices a(x) and E(x) is understood in element-by-element sense, i.e. the elements of ii(x) are the mean values over T of the corresponding elements of a(x). Obviously, in case of piece-wise constant coefficients a(x) G(x) and Co = 0. The well posed-ness of the finite volume element approximation follows from the weak coercivity of the bilinear from a2,h(v2,wa)for sufficiently fine partitions 5.We have:

>

-

Lemma 3 Let the partition 72 be so fine that h < l/Co, where the constant Ca is determined in (17). Then the following inequality holds true SUP

zu;EW,'

a 2 , h ( v 2 2 .W')

IlwZflll,h

2 ~ l l v ~ l lfor ~ ,all ~ v2 ~ ,t

with a constant C independent of h.

~2

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In R1 we use the lowest order Raviart-Thomas spaces. Thus, V as a subspace of H(div; Ol), and Wl is the space of piece-wise constant functions with respect t o the partition 71 and therefore a subspace of Hio,(R1) for K = 5 . Thus, the advection term Cpl will be discretized using the pressure space Wl of piecewise constant functions on the triangulation 71. For the discretization in O2 we will use the space W2 c HA(02, an2 \ r) of continuous piecewise linear functions on 72. Further, in the finite volume setting we use as a test space W,+of piecewise constant functions on V2. Thus, applying equation (4) consecutively for (vl, wl) E Wl x W2, and (712,w;) E W2 x W;, respectively, we get the mixed finite element and finite volume approximations, respectively, of the bilinear form corresponding to the first order term. However, like in the standard Galerkin finite element method this approximation of the operator C will lead to central differencing, which in turn will lead to a conditionally stable (only for sufficiently small step-size h) scheme. In order to derive a unconditionally stable scheme we shall use upwind approximation in 0 2 . 3.2. Derivation of the coupled method

Since both vl and wl are discontinuous piece-wise constant functions with respect t o the triangulation 71 the formula (4) is applied in a straightforward manner for K = 71 so we get the following approximations at, and a:, of the forms all and al2, respectively:

+(covl, wl) for vl E WI, wl E Wl and at2(v2,w1) =

J,-

Iiv2

WI

b

e

nl ds for

v2 E W2,

WI

E

Wl.

(19)

Now we find the contributions of the the operator C from 0 2 and we define the approximations of the bilinear forms a21 and a22. We shall simply rewrite (4) for lC = V2:

Since the functions in W2 are continuous then C(v2,w;) is well defined for all v2 E W2 and w; E W2f. Taking into account that the functions in W,* are

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Figure 4: The shaded area is the volume V centered at the vertex x" the doted boundary of V denotes the "inflow" boundary, while the solid one is the "outflow" part; on the various pieces of the boundary dV we have the following approximation of the convection term: on (a, b) , (m, c) , (f,g ) and (9,a): v2(x) = vz (xi); on (b, m) : v2(x) = v2(xO)= v2(C); on (c, d) : vz (x) = 2r2(5O) = vZ(D); on (d, e) : vz(x) = vz(xO) = 'uz(E); on (e, f ) : vz(x) = t12(x0) = v ~ ( F ) . constant over each finite volume V E V2 then the contributions from each finite volume V E V2 are:

Since v2 is continuous then obviously, we have v,O = vi = VZ(X). On the boundary r+ the values v,O are not defined (this is the inflow boundary for 0 2 ) and we shall take them from the corresponding counterpart in 01, i.e. as vl (x). Thus, we split the integrals over dV into two parts and get

Unfortunately, the exact calculation of the first integral in (21) will lead to central differences and therefore to a scheme which is stable only for sufficiently small step-size h. The limitation of the step-size h will depend on the magnitude of the convection coefficient b relative to the diffusion coefficient (matrix) a . For problems with dominating convection this will lead to prohibitively small h. In order to avoid this conditional stability we introduce an up-wind approximation of the integrals. This approximation is done in the following way. We denote by

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V(xi) a finite volume centered at the vertex xi and by V(xo) any of neighboring volumes centered at the vertices xO. Theintegralover dV \I?- is split into subintegrals over the boundaries of V y = V(xi) n V(xO)n T with its neighboring finite (control) volumes and contained in the finite element T. We assume that over each y the function 6(x). n does not change sign (i.e, is either nonnegative or negative). Then on y we use upwind approximation of the following form (for a 2-D illustration, see Figure 4): b ~ n v ~ ( x ) ~ ( b - n ) + t ~ 2 ( x ~ ) + ( b ~ n ) for - v 2 z( ~~~ )y, .

Note, that in the finite volume V(xi) we have I;tvz(x) = v2(xi). Similar equalities are valid for the neighboring volumes V(xo) as well. Thus, roughly speaking the function v2(x) has been replaced by its interpolant in the space of discontinuous functions W,* and then taken the appropriate (up-wind or in the opposit direction of the vector-field b(x)) values at the finite volume interfaces. A particular finite volume in 2-D is shown on Figure 4. Summing for all V E V2 we finally get the following form by taking also into account the diffusion term (16):

for all v2 E W2, wa E W; and the form (ul, W;) =

l+

v1 w;

b - n2 d~ for ui

E Wl, w; E W;.

(23)

The coupled mixed finite elementlfinite volume approximation of the composite problem (12) reads as: find uh E V, pl,h E Wl, and p2,h E W2, such that (a-luh, V)

- ( ~ l , hv , 'v)

-(v

-a:l@l,h,

'

uh, wl)

+ < PZ,h,V

'

n l >I?=

0,

~ 1 )-42(I;tp2,h, wl) = -(f, wl),

(24)

for all v E V, wl E Wl, and wz* E Wz*,respectively. 3.3. Stability of t h e coupled scheme and e r r o r e s t i m a t e

An important feature of the described above discretizations is that the corresponding operator is coercive in an appropriate norm and the method is stable.

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For proving the stability we shall follow the same argument as in the case of original setting. Let as before E = { e ) be the set of edgeslfaces of the elements from 3 and let Eo be the set of interior edgeslfaces. Similarly, = {y) is the set of edgeslfaces, with each y being the boundary of two adjacent volumes Vl E V2 and Vz E V2 contained in a finite element T, i.e., y = V1 n V2 n T. This splitting can be also used in the computational procedure, since it will lead t o element-wise contributions of the convection term to the stiffness matrix. Note, that all edgeslfaces y are in the interior of R2. For the coercivity of the coupled problem we need the following discrete variant of the norm (13):

Here ti is a piece-wise constant matrix with respect t o the partition by (17).

5 defined

Theorem 2 Let h < l/Co, where Co is defined by (17). Then the solution of the problem (12) satisfies the a priori estimate:

where the (*, h ) - n o r m is defined by (25) with respect to the partitions

5 and

72. Proof: As in the continuous case, by testing (24) with v = uh, wl = -pl,h, and w2 = -Ilp2,h we get the equation:

('-luh,

~ h +) a ? ~ ( ~ l ,~h1, , h + ) a:2(pl,hr Iip2,h)

+ a z h l ( ~ ~ ~ 2~, hl , h+) a,h,(pz,h, I i ~ 2 , h = ) (f,

+ (f, Ilp2,h).

(27)

~ 1 , h )

Further, the estimate (14) is a consequence of the simplified form (28) of a?, and (29) of ag2, which are established in the lemmas below. Lemma 4 For any edge/face e denote by n, a fixed unit vector normal t o e and let T: and T; be the two adjacent elements t o e. Similarly, for any

66


edge/face y E denote b y n, a u n i t vector normal t o y pointing t o one of the neighboring volumes V: and V;. Also, [vl] denotes the jump of a function across a n underlined boundary (here we have either e o r y) and Zl denotes the arithmetic m e a n of the jump (introduced in Section ). T h e n ,

=,c 1

J[vl][wlll~.nlds+ c ~ P . ~ , [ ~ I ] s I ~ s

~ c ( ~ I , w I )

e€Eo

+(cOvl,W I )

e€&o

+

S

I?. nl vlwl ds for v l , W I

E Wl.

(28)

Similarly, for all v2 E W 2 , w,* E W,* the following identity is valid (to simplify the expressions we have used the notation v,' = Ilvz):

Proof: The proof of (28) and (29) essentially repeats the arguments of Lemma 1. There is a small difference in the proof of (29) where the integrals over each y E G have been computed by using up-wind approximation.

Lemma 5 T h e following identity i s valid for all vl E W l :

Similarly, for all Iiv2):

7.12 E W2

(here i n order t o simplify we use the notation v,* =

Finally, we have the following error estimate:

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T h e o r e m 3 Assume that the solution p(z) of the problem ( I ) is H2-regular i n 0. Then the solution (uh,pl,h,p2,h) of the coupled mixed discontinuous finite element and finite volume methods (24) converges to the solution (u, PI, p2) of the composite problem (12) and the following error estimate holds true:

The constant C does not depend on h but may depend on the ratios

2 and 2.

4. Bibliography

R.E. BANK A N D D.J. ROSE, Some error estimates for the box method, SIAM J. Numer. Anal. 24(1987), 777-787. F. BENKHALDOUN A N D R. VILSMEIER (EDs), Proc. First Intern. Symposium on Finite Volumes for Complex Applications, July 15 18, 1996, Rouen, France, Hermes, 1996. F. BREZZIA N D M. FORTIN, Mixed and ~ y ' b r i dFinite Element Methods, Springer-Verlag, 1991.

Z. CAI, On the finite volume element method, Numer. Math., 58 (1991) 713-735. Z. CAI, J . J . JONES, S.F. MCCORMICK,AND T.F. RUSSELL, Control-volume mixed finite element methods, Computational Geosciences, 1 (1997) 289-315. Z. C A I , J . MANDEL,AND S . F . MCCORMICK, The finite volume element method for diffusion equations on general triangulations, SIAM J . Numer. Anal. 28 (1991), 392-402.

S.H. CHOUA N D P.S. VASSILEVSKI, A general mixed co-volume framework for constructing conservative schemes for elliptic problems, Math. Comp., 68 (1999). S.H. CHOU, D.Y. KWAK, AND P.S. VASSILEVSKI, Mixed covolume methods for elliptic problems on triangular grids, SIAM J . Numer. Anal. 35 (1998), 1850-1861.

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T. IKEDA, Maximum Principle in Finite Element Models for Convection-Diffusion Phenomena, Lecture Notes in Numer. Appl. Anal., Vol. 4, North-Holland, Amsterdam New York Oxford, 1983. C. JOHNSON, Numerical Solution of Partial Differential Equations by the Finite Element Method, Cambridge University Press, 1987. R.D. LAZAROV,J.E. PASCIAK, A N D P.S. VASSILEVSKI, Iterative solution of a combined mixed and standard Galerkin discretization methods for elliptic problems, (submitted to Numer. Linear Alg . APP~.).

M . LIU, J. WANG,A N D N.N. YAN, New error estimates for approximate solutions of convection-diffusion problems by mixed and discontinuous Galerkin methods, Univ. of Wyoming, Preprint, 1997. I.D. MISHEV,Finite volume methods on Voronoi meshes, Numerical Methods for Partial Differential Equations, 1 4 (1998), 193-212.

P.A. RAVIARTAND J . M . THOMAS, A mixed finite element method for 2nd order elliptic problems, Mathematical Aspects of Finite Element Methods (I. Galligani and E. Magenes, eds), Lecture Notes in Math., Springer-Verlag, 606 (1977), 292-315.

P . A . RAVIARTAND J.M. THOMAS,Primal hybrid finite element methods for second order elliptic equations, Math. Comp., 31 (1977), 391-396. M. TABATA, A finite element approximation corresponding t o the upwind finite differencing, Mem. Numer. Math., 4 (1977) 47-63. C. WIENERSAND B . WOHLMUTH,Coupling of mixed and conforming finite element discretizations, American Mathematical Society, Cont. Math. 218 (1998), 547-554.

Numerical computation of 3D two phase flow by finite volumes methods using flux schemes

J.-M. Ghidaglia Centre de A4athe'matiques et de Leurs Applications E N S de Cachan and C N R S UMR 8635 61 aerenue du Pre'sident Wilson 94235 Cachan Cedex France [email protected] http://www.cmla.ens-cachan.fr/Njmg

We propose here a general class of cell centered finite volume methods specially designed for the discretization of complex models of partial differential equations like those occuring in 3D two phase flow. After a b r k f introduction to these models (the so called averaged models in Eulerian formulatzon), we develop all the tools which are needed in order to arriue to a fully discrete scheme suitable for coding. Hence we discuss conserz~atiuesystems, non conservatzve ones, time discretization, discretization of source terms, of diffusion operators, of boundary conditions, . . . We also briefly dzscuss non conformal meshes. We strongly rely on the concept of flux scheme ,which is, according to us, very well suited for the problems considered here. ABSTRACT

ICey Words : T w o phase pow, Flux schemes, Finite volumes, non hyperbolic convection, Source terms, Footbridges.

1. Introduction Our goal in this contribution is to discuss a class of cell center finite volume methods, on unstructured conformal or non conformal meshes, designed for conlplex models (such a goal was also addressed by T. Gallouet [GAL 961

70


in the previous version of this symposium). We are mainly motivated by the computation of two phase flows occuring in several fields of the nuclear power industry (of course there are many other contexts where multiphase models occur), that is flow of water and its vapor. Models pertaining to this field encompass many difficulties, whose non exhaustive list is follows : (i) the solutions might develop strong gradients, (ii) the equations might involve stiff source terms, (iii) non conservative products might be present in the equations, (iv) the thermodynamics could be highly complex (e.g. phase transition), (v) the total convection operator might be non hyperbolic, (vi) in some part of the flow low Mach numbers can occur, (vii) (viii) and last but not least : there is no "universal model" to work with. Some of these features are present in one phase Computational Fluid Dynamics (lph-CFD), other are typical in combustion but 2ph-CFD often involves toghether all of them. In this paper we shall propose a very general framework for the discretization of such systems. Generality is essential in this context according to (viii). And passing from one model to an other (which might implies even the variation of the number of scalar equations to be considered) should cause only minor changes in the code. Moreover one should ask almost no a priori conditions on the model (besides the fact that it is build upon physical considerations). One of the major by-product of this stategy is that a code derived according to this philosophy can be a tool for the validation of the physical models, and this point is one of the main issues in 2ph-CFD at the present time. Most of the results presented here were inspired by a very fruitful colaboration with the DCpartement Transfert Thermique et Akrodynamique (TTA/D R&D/EDF) and with the Service de Simulation des Systemes Complexes et de Logiciels (SYSCO/DRN/CEA).

Invited speakers

71

2. A s h o r t discussion of t h e m o d e l s We consider the motion of two non miscible fluids, like e.g. air and water or water and steam. Since we are concerned with macroscopic dynamics, we shall consider a continuum model. For single phase flows, except for very hight speed rarefied flows, Navier-Stokes Equations provide a hightly satisfactory continuous model. For two phase flows, a first approach could consists in writting for each phase a Navier Stokes model completed with jump relations across interfaces between the two fluids. This is of course possible for very very simple situations and cannot be envisaged in general. We shall consider here so called "two fluid" models (in Eulerian formulation) obtained t,hrought an ad hoe averaging process. That is at each point in space and time, the modelization assumes that both fluids are present and a parameter (which is also t o be determined) represents the proportion of each fluid. Such a model can be derived by an averaging process and we refer to e.y. Ishii [ISH 751, Ransom [RAN 891, Drew and Wallis [DRE 941 for more details. The models rely on the three usual balance equations (mass, momentum and energy) for each fluid. Denoting with subscripts v and 1 the quantities attached to one of the two fluids (e.g. density, velocity, energy,..) and by a,, a1 the volume fraction of each fluid : a , a[ = 1, these equations reads as follows ( k E {v, 0).

+

Balance of mass.

where pk is the density of the fluid k, uk the velocity of the fluid k and rk the density of the mass transfert to the fluid k resulting from interfacial exchanges with r, + ri = 0. B a l a n c e of m o m e n t u m .

where P is the pressure in the mixture, T k the viscous stress tensor, g the gravity field, Ik the volumic density of the momentum transfert to the fluid k resulting from interfacial exchanges (after substraction of the pressure contri-

72


bution) and I,

+ Ii = 0

Balance of total energy

where e k is the specific internal energy of the fluid k , h k the specific enthalpy of the fluid k , a k Q k the volumic heat transfered to the fluid k , a k q k the heat flux in the bulk of the fluid k , IIk the volumic density of the energy transfered 111= 0 . to the fluid k resulting from interfacial exchanges with II,

+

Once this set of differential equations written, it remains to close the system by adding enought closure relations in order to obtain the same number of equations and independent unknowns. At the present time, as emphasized in the Introduction, it is an open problem. However by using various methods it is indeed possible t o obtain, in definite physical contexts, some closed models from the equations above. These models will in general leave free a few number of constants that are t o be determined by experiments (correlations). Hence the numerical modelisation of two fluid flows must be strongly coupled with experimental programs. Basic models

We give below three typical models (that can be written in 1, 2 or 3 space dimensions). These models are very simple in the sense that they do not involve complicated physical correlations in their right hand side. Actually these models are used in the context of numerical benchmarks designed in order to check that the numerical method used is able to capture some of the important features of the flow considered and also in order to evaluate the numerical dissipation of the method. We refer t o [MIM 991 for a collection of numerical benchmarks.

2.1. A 3 equation model This is the so called homogeneous 3 equation model that we write for the sake of simplicity in 1D : ap ~ ( P u=) 0 , at

-+- ax

Invited speakers

73

Here p represent the density, u the average velocity, u, the relative velocity a heat flux (source between phases, c the quality, L the latent heat and term). See also Example 1.

2.2. A

4

equation model

This two-fluid model reads as :

Here p k represents the density of the phase k , u k its velocity, p is the pressure, and a k denotes the volume fraction of the phase k , ( a , a t = 1 ) . T h e relative velocity is simply u, = u, - ue. Here, the source term is a drag force which is in g equations (9) and ( l o ) , where R b is given by the correlation k = gcrupe the bubble radius and CD the drag coefficient.

+

2.3. A 6 equation model In the previous two fluid model, no energy balance equations were written. This corresponds t o isentropic flows. Here we want t o consider the full set of balance equations (mass, momentum and energy) and this leads to a so called six equations model (which lead in 3D t o a set of 10 scalar evolution partial differential equations). Let us consider the following system of equations :

74


~(@,P,E,) at

a(aepe at

+ diu (a,p,H,u,)

- p-

8%

at

+ d i u (arpeHeui) - p-aae at

-

- agp,y.u,,

=a~p~g.uj.

(16)

We still have to describe the thermodynamics of this "one-pressure" model. To this extent, we assume that two pressure laws are given, one for the "gas" and one for tlie "liquid" phase :

Here e k , denotes the internal phase energy and Ek f ek + $ IUk

12,

Hk = E k + P. Pk

3. The finite volume approach

The goal of this section is to describe the program that lead to the full discretization of the models of Section 2. We shall not present numerical results refering to the bibliography for that purpose : [AGT 991, [BOU 981, [GI 0 such as:

so by Lemma 3.1, with a = &:

the result. For the second part of the lemma, we replace again all "h2" by "h4" in the preceding reasoning. 3.2.3. Proof of theorem 3.1 The error estimate for the pressure is given by Lemma 3.2. For the velocity, we take back the value of a and C2 in the estimate [8], and we obtain:

Therefore the theorem is proved with C = max(C2, C4).

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References [CHO 981 CHOUS.H. AND KWAKD.Y ., A Covolume method Based on Rotated Bilinears for the Generalized Stokes Problem. S I A M J. Numer. Anal., Vol. 35, No 2, p. 494-507, April 1998. [EGH 991 EYMARDR., GALLOUET T. AND HEREINR., Convergence of finite volume schemes for semilinear convection diffusion equations. Numer. Math., Vol. 82, p. 91-116, 1999.

R., GALLOUET T . AND HERBINR., Finite Volume Meth[EGH 971 EYMARD ods. Preprint NO 97-19 L A T P , Aix-Marseille 1, to appear i n Handbook of Numerical Analysis, P.G. Ciarlet, J.L. Lions e d ~ . [FAR 971 FARHLOUL M . AND FORTIN M . , A New Mixed Finite Element for the Stokes and Elasticity Problems. S I A M J. Numer. Anal., Vol. 30, No 4 , p. 971-990, August 1997. [HAN 981 HAN H. AND W U X . , A New Mixed Finite Element Formulation and the MAC Method for the Stokes Equations. S I A M J. Numer. Anal., Vol. 35, No. 2, p. 560-571, April 1998. [HAR 651 HARLOWF.H. AND WELSHJ . E . , Numerical calculation of timedependant viscous incompressible flow of fluid with free surfaces. Phys. fluids, Vol. 8 , p. 2181-2189, 1965. [NIC 921

NICOLAIDES R.A., Analysis and Convergence of the MAC Scheme I. The Linear Problem. S I A M J. Numer. Anal., Vol. 29, p. 15791591, 1992.

[NIC 971

NICOLAIDES R.A. A N D W U X . , Covolume Solutions of ThreeDimensional Div-Curl Equations. S I A M J. Numer. Anal., Vol. 34, No 6, p. 2195-2203, December 1997.

R. A N D TAYLORT . D . , Computational Methods for Fluid [PEY 831 PEYRET Flow. Springer- Verlag, New York, 1983. [POR 781 PORSHING T.A., Error Estimates for MAC-Like Approximations to the Linear Navier-Stokes Equations. Numer. Math., 29, p. 291-306, 1978. [SHI 961

SHIND. AND STRIKWERDA J.C., Convergence Estimates for Finite Difference Approximations of the Stokes Equations. J. Aust. Math. Soc. Ser B, 38, p. 274-290, 1996.

[SHI 971

SHIN D. AND STRIKWERDA J.C., Inf-sup Conditions for Finite Difference Approximations of the Stokes Equations. J. Austral. Math. Soc. Ser. B, Vol. 39, p. 121-134, 1997.

Convergence Rate of the Finite Volume Timeexplicit Upwind Scheme for the Maxwell System on a Bounded domain

Yves Coudihre INSA, cornplexe scientifique de Rangueil, 31077 Toulouse Cedex 4, France P. Villedieu O N E R A Toulouse, 2 Avenue Edouard Belin, 31055 Toulouse Cedex, France

ABSTRACT : W e derive an O ( h l / ' ) error estimate for the upwind, explicit in time, finite volume scheme for Friedrichs systems. Explicit schemes in that case can not be seen as standart time-space finite element ones. Our demonstration is the generalisation on bounded domains of the ideas of Vila and Valledieu for the Cauchy problem i n EXd. It is applied to the case of Maxwell's equations. Key Words: hyperbolic system, Maxwell's equations, finite volumes, error estimates.

1. I n t r o d u c t i o n

We are interested in the approximation by finite volume means of the Friedrichs systems of the form d

dtu+CaAiu=O i=l

(]O,T[xR),

(An - M ) u = 0

4 0 , .) = uo

(10, T [ x d R ) ,

(a),

(1)

d

where Ai are some symmetric matrices,An = x ~ (n ~ being nthe~ unit nori=l ma1 outward to R ) . (1) belongs to the class of hyperbolic systems first introduced by Friedrichs [FRI 581. For M such that ker (An - M ) is maximal positive, it have a unique solution in L 2 ( ] 0 ,T [ x R ) [RAU 851.

126


We shall treat, as an example, the case of the two dimensional Maxwell's equations i n Transverse Magnetic Mode. Our aim is to derive an 0 ( h 1 I 2 ) error estimate for the upwind, explicit in time, finite volume scheme for ( I ) , under very general assumptions on the mesh and minimal regularity hypothesis on the continuous solution u . The discontinuous Galerkin method, applied to the approximation of solutions of (1) is a space-time finite element method, which can be interpreted as a finite volume method, but with an implicit in time discretisation [JOH 871. However, finite volume schemes with time-explicit discretisation can not be interpreted in general as discontinuous space-time finite element schemes associated to a coercive bilinear form. Derivation of error estimates of order h'I2 for such schemes in the case of the Cauchy problem on EXd was first obtained by Vila and Villedieu [VIL 97, VIL 991, using a new technique of demonstration. On bounded domains, the additional difficulty is the discretisation of the flux A n u on the boundary of S1. Here, we propose a general form for the numerical flux on dS1, which guarantee the consistency and the stability of the corresponding scheme. We also prove an error estimate of order h1I2 (as for time implicit schemes). In section 2, we present the Maxwell's equations under the form (1). The scheme is defined in section 3. The actual results are stated in section 4. Sketches of the proofs can be found in section 5. 2. The Maxwell system

As an example, we consider the bidimensional Maxwell system in Transverse Magnetic mode (TM waves). It may be written as a Friedrichs system of the 2 c R2,with form (1) in f

T

The unlcnown is u = [ E H z H , ] (Electric an Magnetic Fields; supposing that c = 1 is the light speed). We shall suppose that R is the bounded domain between an obstacle and an outer boundary. Classical boundary conditions are - on the obstacle (dR1), the metallic condition: E

= 0,

Numerical analysis -

127

on the outer boundary (i3R2),the lznearized absorbing boundary condition of Silver Muller: H A next = E.

These conditions may be stated like in (1) by taking

where the outward normal to 8 0 is next=

[

a

lT

,l?

3. The numerical scheme

Let Th be a mesh of R , composed of polyhedral cells I(. In order to avoid any local degeneracy of the mesh, we assume that there exists some positive constants a and b such that

m(Il'), m(dli'), diam(Ii') denote, respectively, the measure in Eld of the cell K, the measure in I W ~ -of~ the boundary of K, the diameter of K. Let At be a time step, and t n = nAt. We shall approximate the solution of (1) by a piecewise constant function vh such that

The values of vh are calculated according to the following scheme:

B+ and B- denote, respectively, the positive and the negative parts of a symmetric matrix B.

g k e are some numerical fluxes, defined below;

3.1 The interior numerical flux

Let Sj: be the set of the interfaces interior to R. We take the natural upwind scheme on such interfaces:

3.2 The numerical flux on the boundary

128


Let dSh be the set of the edges of the boundary dR. We propose to take the following general form for the numerical flux on the boundary:

VIc E Th, Ve E dSh n dI 0 is a parameter measuring the importance of the stabilization term. It shall have an effect on the CFL condition.

3.3 Case of the Maxwell system

For the maxwell system given by the matrices ( 2 ) , and the boundary conditions (3), a easy calculation shows that, on a boundary edge e:

Metallic boundary condition:

Absorbing boundary condition:

We point out that classical boundary numerical fluxes fall in this class for some particular choices of y [PIP 991. Metallic boundary condition: with the mirror state technique, the flux is given by taking (5) with v k e = [ -E H z H y (if the interior state i s v t = [ E H, H,

1'):

1 2

-(An

lT

+ M ) + y llAn - MI1 Q is replaced by

which is exactly the previous one, for y = llAn - Mll 2 .

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129

Absorbing boundary condition: the flux is given by taking v;"c, = 0 in (5) (no incoming waves) :

1 (An 2

-

+ M) + y []An- Mil Q is replaced by ~

n + ,

which is equivalent to the formulation above, only for the limit case y = 0.

4. M a i n R e s u l t s

Under some regularity assumptions on the uo, the Ai and a, (1) admits a unique solution u E V = CO([OlTI, (H1(Q))m) n C1([O, TI, ( ~ ' ( 0 ) ) " ) [RAU 851. Theorem 1 ( C o m p a r i s o n ) For T such that

> 0,

suppose that v E L2(]0, T[xS1) is

there exists p E V' (the dual space of V) such that VII, E V,

there exists a measure v such that V4 E C1([O, TI x

then

IIu

2

- v l l p ( l o , ~ [ x n5) (v, 4 )

n),

- 2 ( ~ 4, ~1 )

with 4 ( t , x) = T - t.

Theorem 2 ( C o n v e r g e n c e Rates) Under the following CFL conditions: on the interior interfaces, on the boundary edges,

Atm(dK) m(K)

Atm(dIi') m(h')

(1

I1 - &,

(i

1 1 ~ n .-eM ~ I I +27:)

5 27e - & ,

130


the approximate solution vh given by the finite volumes scheme (4) converges to the exact solutiori u of ( I ) , and we have

5. Proofs

5.1 The Theorem of Comparison

An easy calculation yields, for any on 10, T [ x R ,

)

-

(

,

)

4 E C1([O, T] x a)such that $ ( t , z ) 2 0

(sinceM+~~>O,and4>0).

The theorem 1 is obtained by taking 4 ( t , x) = T - t (dtd = -1, and did = 0 ) . 5.2 L2 Stability

The scheme is expressed in a non conservative form, and then as a convex combination of interfaces contributions: At yn+l K -21;-(gke - A n ~ e v k )m ( e ) = n ~ m(e) V + (~e ) , K e€aK

where

1

I vg v>+l(e) =

(XK =

X K ( - A ~ K (~v g) e - v E ) (interior case), - AK ( M e - A n e IIMe - A n , 11 Q e v k (boundary case) 2 -

vk

+

Atm(8Ii')

). Calculating lv>"

m(h') Interior case:

'1

v l ( )

5

IU>

I2

(e)

12,

we find that

- AK ( v z T (-An;.) T

V&

- v k eT (-An;.)

- E X K ( V ~ - v k e ) ( - A ~ K( ~v k) - WE,) , using the CFL condition in a classical way.

vke)

Numerical analysis

13 1

Boundary case:

1 vKn + l ( e l l 2 5

I

V

~ -I ~ ~

K u ~ ~ ~ e v g ,

where

Me = 2Be

-

XKB

~ ~ B(Be~=,

Me - An, 2

+ Y llMe - Anell

Qe)

Summing over the e E dl l of specific heats : P = (7 - 1)Pe. In order to approach numerically the solutions of the gas dynamics equations, we introduce the so-called finite volume method ; space is discretized with a grid j Ax ( j E Z?) and time by multiples n At ( n E IN) of time step At. We search an approximate value WjlZ of the field W(*, 0 ) at particular vertex j Ax and discrete time level n At thanks to the family of numerical n+1/2 ( j E Z, n E IN) (see e.g. Harten, Lax and Van Leer [HLV83]) : fluxes j'j+lj2 1 1 n+1/2 n+1/2 fj-1/2 ) = 0 . In this note, we restrict At (w;" - WjlZ) + Ax ( fj+l12 ourselves to a two-point numerical flux function that is explicit and first order n+1/2 accurate in space and time, e.g. of the form : fj+l12 - @ ( w T IWy+l). -'

134


l We distinguish between two types of numerical flux functions depending of two arguments : on one side, exact or approximate solutions of the Riemann problem ("flux difference splitting") between states WT and W>l (see e.g. Godlewski-Raviart [GR96] for mathematical and numerical context) with the numerical fluxes proposed by Godunov [Go59], Roe [Roe811 and Osher [Os81] and on the other side flux decompositions ("flux vector splitting"). A flux vector spliting, with Sanders-Prendergast [SP74], Van Leer [VL82], Bourdel, Delorme and Mazet [BDM89] or Perthame [Peg11 among others, suppose that f (W) E IR3 explicited in (1) has been the physical flux function IR3 3 W written under the form

-

f ( W ) = f + ( W ) + f -(W) (2) with a set of constraints on functions f + (0) and f - ( 0 ) detailed for example in the book of Godlewski and Raviart [GR96]. For modelling upwinding, the numerical flux admits the following very simple form : @(Wl,Wr) = f+(Wl) + f - ( W r ) . In the context of a stationary aerodynamics problem, Van Leer, Thomas, Roe and Newsome [VTRN87] compare the Van Leer flux vector splitting [VL82] and the Roe scheme [Roe811 that uses a Riemann problem for a linearized equation. They show that in order to give a correct prediction of skin friction and heat flux on the boundary with a relatively course grid, it is possible with the Roe flux difference splitting while it is not with the Van Leer flux vector splitting of the type (3) for convective part of fluid flow. Their conclusion is to reject flux vector splitting methodology if the objective is to predict more than the simple pressure field. l In fact, the problem occurs in the boundary layer. Along the direction x normal t o the boundary, normal velocity u is very small. Then it is natural to study the evolution of a flux vector splitting (3) for the very simple model of a stationary contact discontinuity, i.e. a boundary layer with infinitesimal thickness. I t is a particular problem of decomposition of discontinuity where the given states Wl and Wr define on one hand a velocity field identically null composed by ul for x < 0 and by ur for x > 0 : (4) Ul = ur = 0 , and on the other hand a pressure field denoted respectively by pl for x < 0 and pr for x > 0 without discontinuity : (5) P l = P r = P . The physical solution of such a stationary contact discontinuity does not depend on time : density jump is maintained at the interface x = 0 as long as time variable is increasing and it is the addition of viscous term or of geometrical perturbations like in Kelvin-Helmholtz instability that modify the interface, which is crucial for a correct capture of boundary layers and shear instabilities. In this note, we prove that in a general way if a flux vector splitting satisfies very natural hypotheses of left-right invariance (section 2), then the associated scheme for gas dynamics contains a numerical viscosity essentially proportional t o the jump of density, then of the order zero relatively t o space step (section 3). (3)

l

Numerical analysis

135

2)

Left-right invariance. We consider transformation a of state W obtained by changing the sign of velocity : t (6) f f ( p , p u , p ~=) ~( P , - P ~ , P E ) . Taking into account the particular algebraic form of state W and relation (I), we observe that

(7) f ( f f W ) + Of(W) = 0 ; when we change the sign of velocity, we change the sign of mass flux and of energy flux but we do not the sign of momentum flux. a Because changing the sign of velocity is equivalent to changing the sign of space direction x, it is useful1 t o introduce the normal unitary vector n ( p (u n) , (pu2 p) n , t o this direction (n E { - 1, I ) ) , to set g(n, W)

--

+

+

and also o n = -n. If we change both signs of velocity ( p E p) (u n) ) and of space direction, the mass and energy fluxes remain unchanged but the sign of momentum flux is changed. We have in consequence : f f W ) = ff g(n, W) . Natural extension of this left-right invariance property to the numerical flux can be formalized by setting : if n = +1 @(WL,W,) (9) Q W l , n , W,) = -@(Wl, W,) if n = -1. The left-right invariance for numerical flux consists to remark that if we exchange both left and right states, the sign of their velocity and the normal direction, we only change the sign of momentum flux. (8)

-

Definition 1. Left-right invariance property. @(Wl,W,) satisfies the left-right The numerical flux function (Wl, W,) invariance property if the function 9(., a, a) defined in (9) and operator a defined a t relation (6) and by a n = -n satisfy the condition (10)

9 ( a W , , a n , a w l ) - uQ(Wl, n , W,) = 0 . We remark that consistency condition @(W,W) = f (W) can be translated for the pair (9,g) by the relation 9 ( W , n, W ) = g(n, W ) and in this particular case, relation (8) shows that *(OW, a n , OW) - uQ(W, n , W ) = g(an, a W ) - a g ( n , W ) = 0. This remark establishes a particular case of relation (10) when Wl = W, = W. Left-right invariance of a flux vector splitting.

Proposition 1.

A flux vector splitting (2) associated with a numerical flux function (3) satisfies the left-right invariance property if and only if we have f+(UW) af-(W) = 0 VW. (11) P r o o f of P r o p o s i t i o n 1. We introduce the representation (3) inside relation (10) when n = +1 :

+

9 ( u W r , a n , awl) - o 9(Wl, n , W,.) = = -@(gWr, gW1) - aQ(W1,W,)

136


for each pair (Wl,W,). If we make the choice of two independent states Wl and W, the preceding relation states clearly the relation (11) and when we explicit the action of the operator a on a vector (see relation (6)), we obtain the detail of the algebra for each component, i.e. relation (11). l On the other hand, if condition (11) is satisfied, then the relation (10) is correct for n = 1 ; it remains true for n = -1 because the left member is an odd function of variable n due to relation (9). In consequence the proposition is established. l Proposition 2. Particular case of classical flux vector splittings. Van Leer flux, Sanders Prendergast flux and Boltzmann schemes satisfy the left-right invariance property.

Proof of Proposition 2. l Van Leer flux satisfy the following relation f + ( W )

=

f (W) if M

= -u

2

= 5 (P(PT,P) - P(PL> P) 7 O , '(PT~P) - ' ( ~ 1 ,P) ) (21) '

l

In the case of Van Leer flux vector splitting, relations (12) show

and for a Boltzmann scheme, we have, taking into account (13) and (14), +oo foo pB(p,p)=Jd x(-$!~);.u and r B ( p , p ) = l ( I U ' ) f2d v .

Xx

Proof of Proposition 4. l Relations (19) and (11) show that ff(W) +ufP(W) = 0, aW = W . (22) Joined with relation (2), we have from relation (22) f t ( W ) - a f + ( W ) = f (W) if u W = W and relation (20) is established for function f +(.). The entire relation (20) is a direct consequence of (22). The detail of the computation of numerical viscosity is a consequence of the relations (18) and (20). l Proposition 5. Residual numerical viscosity. If one of the functions p(o, e) and '(0, 0) explicitly depends on density, i.e.

a'

# 0 or -(p, p) # 0 , then the numerical viscosity 80 a0 of a flux vectdr splitting scheme is not infinitesimal for a stationary contact discontinuity, whatever be the size of the mesh. Proof of Proposition 5 is an immediate consequence of Proposition 4 and in particular of the relation (21). l Proposition 6. Approximate Riemann solver. Let a ( * , 0) be one of the three exact or approximate Riemann solvers proposed by Godunov [Go591 (exact solution of the Riemann problem), Osher [Os81] (approximate solver containing only rarefaction waves or contact discontinuity) and Roe [Roe811 (approximate solver containing only contact discontinuities). Then numerical viscosity V(Wl, WT) of such a numerical scheme is null if given states Wl and WT satisfy the particular conditions (4)-(5) of a stationary contact discontinuity. if we have

*(p, p)

Numerical analysis

139

P r o o f of P r o p o s i t i o n 6. In the case of Godunov and Osher schemes, conditions (4)-(5) state that the solution of the Riemann problem is effectively only composed by a stationary contact discontinuity. Due to Rankine-Hugoniot relations for a stationary discontinuity, physical fluxes of the two states Wl and W, are equal and we t have f* == (0, p, 0) = f ( W L )= f (W,). Taking into account relation (7) in Definition 2, the result is established in this case. If we use the Roe flux. we c o m ~ u t ein a first step - /Roe811 intermediate L

velocity u* of a mean state : u*

F

Jpr~+ l 6% = 0 due to relation ,fi " . + ,hz * '

(4) and we evaluate also total enthalpy of this mean state before the calculus of Roe matrix A* that satisfies, taking into account previous expression of f * : A* (Wr Wl) = f (W,) - f (Wl) = 0. The difference (W, - WL) is an eigenvector of matrix A* relatively to the eigenvalue u* = 0 due t o the expression of intermediate velocity u*. In consequence, when we decompose discontinuity Wr - WL on the basis of eigenvectors for matrix A*, we observe that this difference is non null only for the linearly degenerated wave, i.e. on the contact discontinuity itself. Conclusion is then exactly the one done previously for Godunov and Osher fluxes. -

Conclusion. In order t o capture numerically a boundary layer with a finite volume scheme, numerical analysis of the problem of stationary contact discontinuity shows that classical flux vector splitting schemes satisfying the left-right invariance property generates a numerical viscosity of order one relatively t o the jump of densites whereas it is not the case if we use an exact or approximate decomposition of the Riemann problem. This fact founded on very simple algebra shows that Van Leer at a1 conclusion can be extended to all flux vector splittings referenced in this note : flux vector splitting satisfying the left-right invariance is incompatible with viscous computations. The previous remark conducted us during the time of development of the Navier Stokes solver Ns3gr to include the Osher flux whereas the initial choice was the Sanders and Prendergast flux vector splitting. This choice has been performing, even for the resolution of the Euler equations of gas dynamics in the particular case of capturing shear stationary waves (see [DM92]). In an analogous wa,y, the parabolized version Flu3pns (Chaput et a1 [Chgl]) of Flu3c computer software has required the introduction of the Osher flux decomposition in order to simulate flows with a precise evaluation of viscous effects. 4)

5) References. [BDM89] F. Bourdel, P. Delorme, P. Mazet, Proceedings of the 2th International Conference on Nonlinear Hyperbolic Problems, Notes on Numerical Fluid Mechanics, vol. 24, p. 31-42, Vieweg, 1989. [Ch91] E. Chaput, F. Dubois, G. MoulQs,D. Lemaire, J.L. Vaudescal. A Three Dimensional Thin Layer and Parabolized Navier-Stokes Solver Using the MUSCL Upwind Scheme, AIAA Paper no 91-0728, january 1991.

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(DM921 F. Dubois, 0 . Michaux. Solution of the Euler Equations Around a Double Ellipsoydal Shape Using Unstructured Meshes and Including Real Gas Effects, Workshop on Hypersonic Flows for Reentry Problems (DksidBri-Glowinski-Pkriaux Editors), Springer Verlag, vol. 2, p. 358373, 1992. [Go591 S.K. Godunov. A Finite Difference Method for the Numerical Computation of Discontinuous Solutions for the Equations of Fluid Dynamics, Mat. Sbornik, vol. 47, p. 271-290, 1959. [GR96] E. Godlewski, P.A. Raviart. Numerical Approximation of Hyperbolic Systems of Conservation Laws, Applied Mathematical Sciences, vol. 118, Springer, New York, 1996. [HLV83] A. Harten, P.D. Lax, B. Van Leer. On Upstream Differencing and Godunov-type Schemes for Hyperbolic Conservation Laws, SIAM Review, vol.25, no 1, p. 35-61, january 1983. [Os81] S. Osher. Numerical Solution of Singular Perturbation Problems and Hyperbolic Systems of Conservation Laws, in Mathematical Studies n047 (Axelsson-F'ranck-Van der Sluis Eds.), p. 179-205, North Holland, Amsterdam, 1981. [Peg11 B. Perthame. Second Order Boltzmann Schemes for Compressible Euler Equations in One and Two Space Variables, SIAM Journal of Numen'cal Analysis, vol. 29, p. 1-19, 1991. [Roe811 P. Roe. Riemann Solvers, Parameter Vectors, and Difference Schemes, Journal of Computational Physics, vol. 43, p. 357-372, 1981. [SP74] R.H. Sanders, K.H. Prendergast. The Possible Relation of the 3kiloparsec Arm t o Explosions in the Galactic Nucleus, The Astrophysical Journal, vol. 188, p. 489-500, march 1974. [VL82] B. Van Leer. Flux-Vector Splitting for the Euler Equations, Proceedings of the ICNMFD Conference, Aachen 1982, Lectures Notes in Physics, vol. 170, p. 507-512, Springer Verlag, 1982. [VTRN87] B. Van Leer, J.L. Thomas, P. Roe, R.W. Newsome. A Comparison of Numerical Flux Formulas for the Euler and Navier Stokes Equations, AIAA Paper no 87-1104, AIAA 8th CFD Conference, 1987.

Analysis of a Finite Volume Solver for Maxwell's Equations

Fredrik Edelvik Department of Scientific Computing, Uppsala University Box 120, S-751 04 Uppsala, Sweden [email protected]

A finite volume solver for Maxwell's equations is analyzed. The solver shows excellent dispersion characteristics on three difierent uniform triangular grids. Long term stability is achieved for general unstructured grids using a third order staggered Adams-Bashforth scheme for the time discretization. The solver has been hybridized with a finite difference solver and the resulting hybrid solver is shown to be second order accurate for a 2 0 cylinder scattering case. ABSTRACT

Key Words: Finite volumes, Maxwell's equations, Dispersion, Hybrid grid

1. Introduction

The most popular numerical method for solving electromagnetic problems is the Finite-Difference Time-Domain (FDTD) method [TAF 951. It is normally used on a Cartesian grid and the variables are staggered both in time and in space. The main disadvantage with FDTD is its inability t o model complex shaped geometries producing staircasing errors. One way of solving this problem, but still taking advantage of the efficiency of the FDTD method, is to use a hybrid grid solver. A Finite-Volume Time-Domain (FVTD) solver on an unstructured grid is used in the near vicinity of the object and this solver is coupled to an FDTD solver on a structured Cartesian grid that is used in the outer region. This approach was proposed by Riley et al. [RIL 971 and it is also used in the Swedish Computational Electromagnetics project GEMS (General Electromagnetic Solvers). GEMS is a collaborative research and code development project between Swedish industry and academia. This paper con-

142


centrates on the finite volume solver. The solver is described in Section 2 and its dispersion characteristics and stability properties are analyzed in Sections 3 and 4, respectively. In Section 5 some numerical results obtained with the hybrid solver are presented and the last Section summarizes the paper and some conclusions are drawn for future work. 2. The Finite Volume Solver

The finite volume (FV) solver has so far been implemented in 2D for solving the transverse magnetic (TM) Maxwell's equations,

for a linear, isotropic, non-conducting and non-dispersive material. Here Hx and H, are the x- and y-component of the magnetic field, E, is the z-component of the electric field, p is the magnetic permeability and 6 is the electric permittivity. The basis for the FV solver is the following integral form (in 2D) of Maxwell's equations

=-&nx eJA g

d

Edl,

=fin ~ x Hdl,

where A is an arbitrary area and the line integral is taken along the path I' that encloses A and n is a unit normal. In 2D the computational domain is discretized using a staggered unstructured grid, consisting of a primary grid of triangles and a dual grid (cf. Fig. 1). For T M mode, Hz and H, are stored at the nodes, whereas E, is stored at the barycenters of the primary cells. By joining barycenters of neighboring cells the dual grid is constructed during the pre-processing stage. The area integrals are evaluated by the assumption that the electric and the magnetic field components remain constant over a primary and a dual cell, respectively. Simplifying the two integrands in the line integrals implies

where H and E, denote average fields and the sums are taken along the edges with unit vector t of the respective cells. Ad and A, are the areas of the dual

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143

and primary cells. The evaluation of the first line integral is performed by the assumption that the electric field is piecewise constant along each dual edge. Following Riley et al. [RIL 971, the second line integral is evaluated according to

where HL is the FDTD component in the direction nl orthogonal to the dual edge crossing the primary edge t , H i and H I ,are the magnetic field in the nodes of the edge t . Hl is incorporated in the edge projected field value to guarantee that the divergence is preserved on a local cell level. This is very important, since not fulfilling the two divergence relations of Maxwell's equations could cause spurious modes that would destroy the solution. The time integration method that we use is a third order staggered AdamsBashforth (ABSS) scheme proposed by Fornberg et al. [FOR 991 for scalar wave equations. For the TM Maxwell's equations the scheme takes the form

where A and B are operators taking care of the space discretization 3. Dispersion Analysis on Triangular Grids

A frequently used technique t o characterize the errors of a numerical scheme is to use Fourier analysis. Assuming that the fields are periodic in space and neglecting boundary conditions, we make the following ansatz for the three unknown field components

where k, and Icy are the x- and y-components of the numerical wavevector, respectively. The numerical wavenumber k will in general differ from the physical wavenumber k defined by k = w / c , where w is the angular frequency and c is the speed of light. k depends on how well the wave is resolved in the grid, the type of grid and the angle of propagation in the grid. The difference between and k gives rise to numerical phase and group velocities that depart from the exact values, which cause numerical errors that accumulate with distance. In figure 1 the different uniform grids used in the analysis are shown. Note that the normals and edge vectors align in the equilateral and diamond grid. Thus, only the FDTD correction Hl survives expression ( 6 ) , so for those two grids we do not have t o take the magnetic field in the nodes into account. For

144


the one directional grid, however, we have to consider both the magnetic field at the nodes and the FDTD correction. Since the approximation of the dispersion relation is in the midpoint of the respective building blocks shown in figure 1, in the analysis, the E, components are interpolated from the barycenters to the midpoint of the building blocks.

Figure 1: T h e different u n i f o r m triangular grids used in t h e analysis, the equilateral grid, t h e one directional grid and the diamond grid. T h e respective dual grids are indicated by dashed lines. Let us first look at the errors introduced by the space discretization for the equilateral grid. Substituting (9) into (8) and assuming that no error is introduced by the time discretization results in

4 jwEz = -[ s i n ( - k y A / 2 h ) H I €A

+ s i n ( i X A / 4+ , k , A / 4 6 )

HH,

+ s i n (-kXA/4 + ~ A / 4 h H) H ~. ]

(10)

Similarly, from (7) we have for the three edges (see Fig. 1)

jwH1 = 2fi'Ez ---s i n ( - k , ~ / 2 ~ j ), PA

Inserting (11)-(13) into (10) leads to the numerical dispersion relation

Letting A

-+

0 we obtain

i.e. the dispersion relation is approximated to second order accuracy. The same analysis on a square grid with edge length A, where the solver is identical to the

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145

&

FDTD scheme, gives an error constant equal to [TAF 951. The dispersion relation is approximated to second order accuracy also for the grids with right triangles, but since these two expressions are somewhat lengthy they will be omitted here. The dispersion error characteristics can be presented in terms of the error in the phase speed of waves of different grid resolution and propagation angle. Let k, = k cos(cu) and i,= sin(cr), where cr is the angle of propagation relative to the x-axis. Substituting these two expressions into the dispersion relation (14) and solving the nonlinear equation for i,results in a relation between the numerical wave speed, up, and c , since vp/c = k/&. Figure 2 illustrates the numerical phase velocity as a function of propagation angle and grid resolution. The results obtained by the FV solver are in excellent agreement with the true wave speed for all three grids even for a moderate grid resolution of 10 points per wavelength. The dispersion characteristics are clearly better for triangular grids than for square grids.

Equilateral tri grid - U10

Equilateral lri grcd - N10 Diamond tri grld N10 One directionaltrl grid - N10

-

0.901

0 975

0

30

60

W

120

150

180

O'OgO

30

60

W

120

150

180

Figure 2: Variation of numerical phase velocity for diflerent grids as a function of grid resolution and propagation angle. Discretizing the time derivative with ABS3 we obtain the following fully discrete dispersion relation for the grid with equilateral triangles

The time step is chosen according to the stability limit for ABS3 (see next Section) as cAt = &A/7. As in the semi-discrete case we solve (16) for k . Since ABS3 is a third order accurate scheme, the errors from the space discretization will dominate and a plot of vp/c shows almost identical results as in figure 2. If we use the Leap-Frog scheme for the time-stepping we get a small improvement in the dispersion characteristics since the scheme is only second order accurate and thus the errors from the space and time discretization cancel to some extent.

146


4. Stability Analysis of t h e FV Solver

An important issue for explicit solvers across many disciplines is how to achieve long term stability without adding too much artificial dissipation. The stability region for ABS3 in the scalar case is given in figure 3.

,

3

Figure 3: Stability region for third order staggered Adams-Bashforth. The scheme is stable between f12/7 along the imaginary axis compared to the Leap-Frog scheme, which is stable between f2. That implies that we have to use a shorter time step for ABS3. However, the main disadvantage with Leap-Frog is that it is only stable on the imaginary axis and becomes unstable as soon as we have eigenvalues with a nonzero real part, which we are likely to have on unstructured grids and when boundaries come into play. Let us analyze how ABS3 behaves in our case. After some straightforward algebra (7) and (8) can be written in matrix form as

The plane wave ansatz

leads to the following generalized eigenvalue problem

where A, B and 6 are the matrices in (17), respectively. We begin by looking a t the uniform triangular grids used in the analysis of the dispersion error. For the grid consisting of equilateral triangles we get the matrices A and B from (10)-(13). A stability condition for the Leap-Frog scheme on the three triangular grids is easily derived and equals

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147

where the first part in the right-hand side is the stability condition on Cartesian is equal to the shortest edge in the unstructured region. grids and minlength When we solve the generalized eigenvalue problem (18) on these grids we observe that stability is obtained if the time step suggested by (19) is decreased by a factor 6/7, which is exactly the relative difference in stability along the imaginary axis (see Fig. 3) between the two methods. To analyze the eigenvalues for a general unstructured grid including boundaries we can no longer use Fourier analysis. Instead, let

and after some straightforward rearrangements we are able to write (7) and (8) on matrix form as in+'= P ( A , B)Zn. Analyzing the eigenvalues of the iteration matrix P, for the grid shown in figure 4, reveals that if we choose the time step for ABS3 in the same manner as above we get the eigenvalue spectrum shown in figure 4, where all eigenvalues are within the unit circle. If we use the Leap-Frog scheme with the same time step the largest eigenvalue is of the order 1.0003. Hence the Leap-Frog scheme is unstable even for a tinie step well within the stability limit along the imaginary axis.

Figure 4: Eigenvalues of the iteration matrix P using ABS3 for a PMC cylinder scattering case. Primary grid around the cylinder is shown to the right. The long term stability using ABS3 on general unstructured grids around a PMC and a PEC cylinder has also been verified by feeding the FV solver along one of the outer boundaries with a narrow square wave pulse for ten million time steps without any signs of instability. 5 . Numerical Results

The FV solver has been hybridized with a finite difference solver and the resulting hybrid solver has been tested on two scattering cases, a PMC and a PEC cylinder. In both cases a plane Gaussian shaped wave is impinging on the cylinders. In figure 5 the results for different resolutions using the hybrid

148


solver are compared to a well resolved FDTD solution. An analysis of the results reveals that the hybrid solver is second order accurate for both cases. The hybrid code has been verified to be long term stable in the same manner as the stand alone FV solver.

Figure 5: Errors for hybrid solutions of different resolution. PMC cylinder to the left and PEC cylinder to the right. 6. Conclusions

We have demonstrated that our FV solver has excellent dispersion characteristics on three different uniform triangular grids. A third order staggered Adams-Bashforth scheme is used instead of the normally used Leap-Frog scheme. Indications on how to choose the time step for ABS3 on general unstructured grids are given. The solver has been shown to be long term stable for two cylinder test cases. The implementation of the FV solver in 3D is ongoing and the hybrid technique will be used there as well. 7. Acknowledgment

The author would like to thank Ulf Andersson and Gunnar Ledfelt at KTH, Stockholm, who participated in the hybridization of the code and performed the numerical tests in Section 5. 8. Bibliography

[TAF 951

TAFLOVEA., Computational Electrodynamics, The FzniteDifference Time-Domain Method, Artech House, Norwood, 1995.

[RIL 971

RILEY,D. J. et al. , (rV0LMAX:A Solid-Model-Based, Transient Volumetric Maxwell-Solver Using Hybrid Grids n, IEEE Antennas Propagat. Magazine, N" 39, 1997, p. 20-33.

[FOR 991

FORNBERG, B. et al. , ((Staggered Time Integrators for Wave Equations B, submitted to SIAM J. Sci. Comput.

A result of convergence and error estimate of an approximate gradient for elliptic problems

Robert ~ ~ r n a r ,d 'Thierry Gallou#, Raphadle ~ e r b i n ~ Ecole Nationale des Ponts et Chausse'es, Marne-la- Valle'e, France Universite' de Provence, Marseille, France

--

Using a classical finite volume piecewise constant approximation of the solution of a n elliptic problem i n a domain R , we build here a n approximate gradient of the solution. It is then shown that this approximate gradient converges i n Hdi,(R). A n error estimate is given when the solution to the continuous problem belongs to H 2 ( R ) .

ABSTRACT

K e y Words: elliptic equations, finite volumes, gradient, convergence, error estimate.

1. Introduction

As a paradigm of elliptic problems, we consider the Laplace equation -Au(x) = f (x), for a.e. x E R ,

(1)

with Dirichlet boundary condition: U(X)

= 0 , for a.e. x E aS2,

where we make the following assumption. Assumption 1 1. 0 is a n open bounded polygonal subset of R d ,d = 2 or 3,

2. f E L 2 ( R ) .

(2)

150


Here, and in the sequel, "polygonal" is used for both d = 2 and d = 3 (meaning polyhedral in the latter case). Note also that "a.e. on do" is a.e. for the d - 1-dimensional Lebesgue measure on 80. Finite volume methods for Problem (1),(2) have been intoduced by many authors (see [EGH 991 and references therein), with some proofs of convergence and error estimates. Since the approximate solution constructed with a classical cell-centered finite volume scheme is piecewise constant, an approximation of the gradient of the solution may be seen to be more complex than with a finite element method. Indeed, the convergence of a reconstructed gradient has been shown in [CVV 971, for certain quadrangular meshes using a nine point scheme. It has also been shown on certain meshes by rewriting the finite volume scheme as a finite element scheme [ABM 951, [VAS 981, [AWY 971 or a Petrov-Galerkin scheme [DUB 971. Here we show that one may construct an approximate gradient on all "admissible" meshes by using some mesh functions which generalize those used in mixed finite element theory (see e.g. [ROT 911). 2. A p p r o x i m a t e of u a n d Vu

The following definition of admissible meshes for the finite volume scheme includes a large variety of meshes, such as, in two space dimensions, triangular meshes (with a four-points scheme), cell-centered or vertex-centered rectangular meshes (with a five-points schemes) and Voronoi meshes. Definition 1 (Admissible meshes) Let R be an open bounded polygonal subset of R d ,d = 2, or 3. A n admissible finite volume mesh of 0 , denoted by 7 ,is given by a family of "control volumes", which are open polygonal convex subsets of 0 (with positive measure), a family of subsets of contained i n hyperplanes of IR~, denoted by E (these are the edges (ZD)or sides (3D)of the control volumes), with strictly positive (d - 1)-dimensional measure, and a family of points of 0 denoted by P satisfying the following properties (in fact, we shall denote, somewhat incorrectly, by 7 the family of control volumes): (2)

The closure of the union of all the control volumes is

a;

(ii) For any K E 7 ,there exists a subset EK of E such that d K = ?T \ K = .UnEzKu.Let E = U K ~ ~ E K . (iii) For any (K, L) E 72 with K # L, either the (d- 1)-dimensional Lebesgue measure of R n is 0 or R n = i? for some a E E, which will then be denoted by KIL. (iv) The family P = ( X K ) K €is~such that X K E K (for all K E 7 )and, if a = K I L , it is assumed that the straight line DK,L going through X K and X L is orthogonal to KIL.

Numerical analysis

15 1

I n the sequel, the following notations are used. T h e mesh size is defined by: size(7) = sup{diam(K), K E 7). For any K E 7 and a E E, m(K) i s the d-dimensional Lebesgue measure of K (i.e. area if d = 2, volume i f d = 3 ) and m(a) the (d - 1)-dimensional measure of a . T h e set of interior (resp. boundary) edges i s denoted by Eint (resp. Eext), that is Lint = {a E E ; a (f d o ) (resp. EeXt = { a E E; a C dR)). For any K E 7 and a E EK, we denote by dK,, the distance from XK t o a and set TK,, = and we denote by n ~ , , the u n i t vector normal t o a outward t o K.

2,

Under Assumption 1,let 7 be an admissible mesh in the sense of Definition ~ u~ be defined by 1 and let ( U K ) ~( u~, )~, ~, and

A proof of the convergence in L2(R) of the approximate solution u~ to the unique variational solution u E H i ( 0 ) of Problem ( I ) , (2) and error estimates are given for example in [EGH 991. We now build an approximate of Vu, the gradient of the solution of the continuous problem. To this purpose, we introduce, for K E 7 and a E EK the solution $K,,, E H1(K) of the following Neumann problem .d A 4 ~ ,(5) a =4 m ( ~ ,)

for a.e. x E K,

J,

4K,v(x)dx = 0, V ~ K , , ( Y.)nK,,, = 1, for a.e. y E a , V ~ K , , ( Y-) n ~ ,=s 0, for a.e. y E a , @E E K , #~a .

(6)

The functions V@K,, generalize on every finite volume mesh the RaviartThomas low degree finite element. Using the finite volume approximate of the fluxes, these functions allow to build the approximate of the gradient GT, given by uu - U K

V @ K , u ( ~for ) , a.e. x E K.

UEEK

3. Convergence t h e o r e m

The following convergence theorem states the convergence of the approximate solution UT to the continuous one u in L2(R) and the convergence of GT to Vu in Hdi,( 0 ) .

152


Theorem 1 Under Assumption 1, let 7 be an admissible mesh (in the sense of Definition 1) . Let u~ be defined by (3)-(5) and let u E H i ( R ) be the unique variational solution of Problem ( I ) , (2). Then UT + u in L2(R) as size(7) tends to 0. Furthermore, assume that there exist two fixed values C > 0 and M such that the inequalities dK,, 2 Cdiam(K) and M 2 card(EK) hold for any control volume K E 7 and for any a E E K . Let GT be the approximate gradient defined by equations (6)-(7). Then GT converges in Hdiv(R) to the gradient of the unique variational solution u E H i ( Q ) of Problem ( I ) , (2) as size(7) -+ 0 . Sketch of the proof (the complete proof of this theorem is given in [EGH]). First note that for all K E 7 and for a.e. x E K .

and therefore divGT -+ -f in L2(R) as size(7) tends to 0. Assuming only u E H; (R), let E > 0 and cp E C,"O(R) such that 2 I l u - ~ l l ~ l ( n ) E. Using the variational formulation of (6) in each K E 7, one proves the existence of some Fl (M, 5) > 0 and of some F2(R,f , cp, M, C, 7 ) > 0 which tends to zero as size(7) -+ 0 such that

0 which tends to zero as size(7) -+ 0 such that

Using the triangular inequality, one has 116- - V

+

(12)

+ 0.

Using (S), we get

U I I L ~ ( I~ )2 ~llGr - ~ c p l l ~ 2 ( n 2 l ~E.

Choosing size(7) small enough such that F2(a, f , 9 , M , 6 , 7 ) + Fi (M, C)F3(0, f , cp, 7)I E, equations (9)-(12) lead to

which shows that GT -+ V u in L2(R) as size(7) GT -+ Vu in Hdiv(R) as size(7) + 0.

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153

4. Error estimate

Theorem 2 Under Assumption 1 let > 0 and M > 0 be given values and 7 be an admissible mesh (in the sense of Definition 1) such that the inequalities ~ K , C> - Cdiam(K) and M >_ card(EK) hold for all control volume K E 7 and for all u E E K . Let u~ be given by equations (5')-(5). Then there exists c, only depending o n R , u and C, such that

llu - U T I I L ~ ( R )

i c size(7-I

(14)

Furthermore, let GT be defined by equations (6)-(7). Assume that the unique variational solution, u, of Problem ( I ) , (2) belongs to H 2 ( R ) . Then there exists C > 0 which only depends o n R , u, and M such that

0 such that

5. Cornpacity and Convergence We handle now the first part of the conclusion under the convergence of the scheme. For that we need t o use the classical following compactness theorem.

Numerical analysis

177

Theorem 1 (Kolmogorov)

Let 3 a bounded subset of L ~ ( I wsuch ~ ) that

Then for every R C C IRd, F is relatively compact in L ~ ( R ) . We are now able to state the convergence theorem

(T,, 6tm) a sequence of meshes and time steps that satisfies assumptions of proposition 2. Assume moreover that h , tends to zero when m tends to co (which implies that s t , tends to zero ) . Then there exist u E L" ( R x ( 0 ,T ) )such that p ( u ) E L2 ( 0 ,T ,H 1 ( a ) )and up to U T ~ =, u ~ for~ L~ M ( a x ( 0 ,T ) ) weak star topology and a subsequence, lim,,, in Lp(R x (O,T)),V p < a.

Theorem 2 (Convergence theorem, part 1) Let

Elements of proof. Let extend uT,at on iRq+l by zero out of x ( 0 , T ) . From corollary 1 and corollary 2, we directly deduce that for every (J, s ) E RQ+l, llp(ur,at(.+ t, .

+ 2C's + ( 2 ~ l t l m ( a a+) 2 m ( ~ ) M,', s)

+ t ) )- c p ( u ~ , a t ('))lli2(Rm+l) ., i

where M, = max l p ( x )1.

2Cltl(lEI+ 2h)

[O,11

This inequality allow us to apply theorem 1 and we obtain regularity on the limit by looking at accroissements taux which converge to the derivatives in D'. (see for example [EGH 971) 6. Convergence, part 2 Theorem 3 (Convergence theorem, part 2) W e suppose that the assumptions of

theorem 2 are satisfied, and we assume also that there exists B > 0 such that for all mesh T,, the followzng regularity property is satisfied :

Then the function u given i n theorem 2 as solution of [ I ] in the following sense : V$ E Ctejt = {q E C2"(C? x [0,TI) such that V q . n = 0 and q(., T ) = O),

Proof. The convergence of u, to u is strong which implies that f ( u , ) and cp(u,) converge to f ( u ) and p ( u ) . So it suffices to show that

178


Let m fixed and 7 = 7,,St = St,. Let then multiply [4] by St*;, where = $ ( x K ,n S t ) and sum over I( E '7- and n E 10, N ] . We get Tl T2 T3 = 0 , where

+ +

Q :

n=O

KET L E N ( K )

where Bk$ is equal to Q E if given by

and S3 = /a

v k I L5 0 and 42 otherwise.

We compare ?;: to Si

jnvf ( u ) . v*

Classically, (see for instance [YNS 971, [EGH 971) because II, is a regular function and $ ( T )= 0 we get lim (S1- TI) = 0. m--too

Numerical analysis

179

By using estimates [S], [7] and regularity condition [9] we get also

and

ITz-SZI

5 J -

CI($) h .

Then lim IT2 - SS/= lim IT3 - S3J= 0, and the proof is complete m+m

m+w

7. Uniqueness

Let u l et u2 be two solutions of [lo]. We denote by ud = ul - 212. For all w E Cte,t, we have

where F =

f ( ~ 1-) f (

~ 2 )and

u1 - u2 attention with the dual ~ r o b l e m:

= p ( u l ) - p ( u z ) . SO it is natural to pay 211- u2

Fro111 [LSU 681, we can state the following result

Theorem 4 (Existence t o the regularized dual problem) Let F , v and @ be CCO functions under fi x [O,TI, and assume that there exists 6 > 0 such that @ ( x , t )2 6 . Then for every y E C,00 (Q x (0, T)) there exists an unzque solution to [I21 Moreover, we have also the following estimates

Proposition 3 Let $ a solutzon t o the regularized dual problem with second naembey xand M Y , ,%fa, Mv and MF some upper bounds for 1x1, @, IvI et IFI. Then there exzst C ( y , M a , M v , M F , R , T ) > 0 such that

and

Elements of proof. [I31 is a direct consequence of the maximum principle for parabolic equations. For [14] and 1151, we multiply the equation by A$ and integrate over d ! x (0, T). Because of [13], IIV+IIL~(nx(O,T)) is controled by dllA$llLz(nx(o,~)). We complete then the proof by using time and space integrates by part and Young inequalities. -

-

We are now able to give the main result of this secbion

180


Theorem 5 Assume cp-l is an holder continuous function with exponent there exist an unique solution to [lo].

$.

Then

Proof. By theorem 3, there exist a t least one solution to [lo]. We now turn to the study of uniqueness. Let x E C r ( R x (0, T ) )and 6 > 0 (6 5 MG). Oh = max(6, O) is again in Lm(M x (0, T ) )and 6 is a lower bound for it. We don't have regularity hypothesis on Qa and F but we can construct G,, F, and v, some sequences of regular functions on fi x [0, TI that converge to Oa, F and v in LP(M x (0, T)) for p < rn and such that For every n , by theorem 4 there exist a solution $, in C2,'(!,?x [0, TI) to the dual . . problem associated to G,, v,, F, and X . Because the upper bounds of G,, v,, F, and the lower bound 6 of G, are independant from n , estimates on A$, and V$J, are also independant from n , so we get rT r

But Oa - @ 5 6 Ilio,s) because @ and @a are equal on {O 2 6). Then, if we denote by As = { u d # 0) n {O < 61, we get

Because cp-l is an hijlder continuous function with exponent Moreover m(Aa) tends to zero, so that

i, ud 5 6 on As.

Since that is true for every regular function X, the proof is complete. [EGH 971

R. EYMARD, T. GALLOUET, R. HERBINFinite volume methods, Prebublication 97-19 LATP Marseille, to appear in Handbook of numerical analysis, Ph. Ciarlet & J.L. Lions ed., 1997.

[EGH 991

R. EYMARD,T. GALLOUET,R. HERBIN Convergence of finite volume schemes for semilinear convection diffusion equations. Numerische Mathematik 82 : 91-116, Springer Verlag, 1999.

[YNS 971

Y. NAIT SLIMANE Me'thodes de volumes finas pour des probldmes de dzffusion-convection non line'aires. Th6se de l'universite Paris 13, 1997.

[LSU 681

O.A. LADYZENSKAJA, V.A. SOLONNIKOV, N.N. URAL'CEVA. Linear and quase-linear equations of parabolic type. Transl. of math. monographs 23, American Mathematical Society, 1968.

Convergence analysis of a cell-centered FVM

Hans-Peter Scheffler *, Reiner Vanselow " "Institut fur Analysis **Institut fur Numerische Mathematik TU Dresden, Mommsenstr. 1 3 D 01062 Dresden -

A well-known cell-centered F V M with Voronoi boxes for discretizing the Poisson equation is analyzed. To achieve this purpose, a nonconforming F E M is constructed, such that the system of linear equations obtained by using the nodal basis coincides completely with that for the F V M . I n this way, convergence properties of the FEM, which are formulated i n terms of function space norms, can be transformed t o the F V M .

ABSTRACT

K e y Words: cell-centered finite volume method, Voronoi boxes, convergence analysis, nonconforming finite element method.

1. Introduction Finite Volume Methods (FVMs) are standard methods for finding numerical solutions of partial differential equations. Like Finite Element Methods (FEMs) they can be applied to a wide class of problems over arbitrary domains and allow local refinements of the domain partition. For a given f E L2(fl) we consider the Poisson equation -divgradu= f i n f l c R 2 , u = O onI'=dfl. (1.1) In order to simplify the presentation we restrict ourselves to open, convex and bounded polygonal domains R. The convergence proof of the FVM is based on the following two steps: Description of a nonconforming FEM such that the system of linear equations coincides completely with that for the FVM. Proof of convergence for the corresponding FEM.

182


In distinction to other authors (cf. e.g. [HAC 891) we use a nonconforming FEM basing on dual Voronoi boxes. The dual Voronoi boxes in combination with the choice of special discrete function spaces are well suited for our aim. The linear convergence of the FEM with respect to some energy norm is proved under the assumption that the solution u of (1.1) belongs to H2(R). As usual, some geometrical properties have to be satisfied for the partitions of 0 . A more detailed representation of this subject is given in [VAS 981. Here we give an improved version for the estimation of the consistency error term and discuss convection-diffusion problems, too.

2. The FVM and the corresponding nonconforming FEM

2.1. Box and dual box partitions

In the following, let M = {P) with P E cl (R) be an arbitrary finite set of points. Further, we use the notations Mi = M n R and Ma = M n I?, where m = card (Mi) > 0 and card (Mb) > 1 have to be satisfied. Let IP - Q ( denote the Euclidian distance between two points P and Q.

Definition 1. For P E M the Voronoi box bp is defined by bp = { Z E cl (R) : IZ - PI IZ - QI VQ E M ) . The set B = { b p } of all Voronoi boxes is called box partition.

~ 0 ) and N N i ( P ) = N N ( P ) n R . To define the FEM, which is used for the convergence analysis of the FVM, we need another partition of the domain R which is dual to the box one.

Definition 2. For P E Mi and Q E N N ( P ) the dual Voronoi box dbpQ is defined by

d b p ~= A P E 1 (P, Q)E2 ( P ,Q) U AQEi (P, Q)E2 (P, Q ) . The set d B = {dbpQ) of all dual Voronoi boxes is called dual box partition.

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For the further considerations, the following property is assumed to be satisfied: For all P E Mb and Q E N E ( P ) it holds b p which is obviously equivalent t o f dl?= 1 J dl? V P E Mi

n bQ # 0 ,

(A)

Q€NN(P) b p n b ~

dbp

2.2. The Finite Volume Method

If we integrate both sides of (1.1) over the Voronoi box b, E B we obtain by applying Green's formula the equations - f [n(bp)lT grad u dl? = J"J f dR =: D ( P ) V P E Mi, (2.1) abp

bp

where the vector n(bp) denotes the outer normal direction of b p . Further, on the straight lines bp n bQ the outer normal n(bp) coincides with the vectors

Now, if in (2.1) the arising integrand [epQITgrad u is substituted by the constant finite difference approximation 4 Q ) - ,(PI PPQ and if (A') is used, then we obtain the following well-known cell-centered FVM for the Poisson equation (1.1): Find u V = u V ( M ) E Rm such that L V u V= bV (2.3) , , where the matrix L' and the right-hand side bV are given by ^iPQ -for Q E NNi(P) PPQ and b: = D ( P ) k,Z = 1,..., m. = YPR for = k [ePQITgrad u x

.

LL

I

C,,

R€NN(P)

0 otherwise The point P belongs to the index k and the point Q to the index 1.

2.3. The corresponding Finite Element Method

A weak formulation of the boundary value problem (1.1) reads as follows: Find u E V = HA(R) such that Vv E V. a(u,v) := J'(gradu)Tgradv dR = JJ fwd0 =: d(v) (2.4) R

n


184

For the FEM we define a finite-dimensional space Vh by Vh = {I E L2(n) : 1 i n t (dbPp)€P(P, Q ) Q ~ ~ PE Q dB v is continuous in P E M and v(P) = 0 Q P E Mb) , where P(P, Q) with P = (xp, yp)T and Q = (XQ,YQ)Tdenotes the space F ( p , Q) = span (1, [(XP - XQ)(X- XP) + (YP - YQNY- YP)]), and the function values at the points P E M iare choose as degrees of freedom. For the convergence analysis, we consider the nonconforming FEM: Find uh = u h ( M ) E Vh such that

with

x

ah (uh , vh) := d

JJ ([epQITgraduh) ([epQITgradvh) d o ,

(2.6)

b € d~B dbpq ~

epQ defined by (2.2) and D ( P ) defined by (2.1). The bilinear form ah is also defined on [V @ Vh] x Vh and, because of grad wh = Wh(Q)- wh(P) epQ vwh E P(P, Q) , DpQ which results in [grad vITgrad wh = ([epQITgradv) ([epQITgradwh) VV E H1(dbpQ),W h E P(P, Q) . This implies ah (v, wh) = JJ" [grad .uITgrad wh d o Vv E V, wh E Vh . (2.8)

x

dbpqEdB dbpq

Using the nodal basis functions and denoting the vector of the function values of the solution uh of the FEM (2.5) in the points P E Miwith uE, a linear system of equations arises, which has the form LEuE = bE. (2.9) The stiffness matrix L ~ the , vector uE = uE(M) E Rm and the right-hand side bE are given by = ah(@Q,@p), U ; = uh(P) and bf = dh(@p) k,1 = 1,...,m. Here, the indices are analogously used to Section 2.2. From the special form of the right-hand side dh it follows (cf. [VAS 981)

LE

Theorem 1. The problems (2.3) and (2.9) are equivalent, i.e. the vectors uV and uE coincide.

2.4. Convergence concept for the FVM

The solution of a FVM is a vector in Rm, whose entries can be considered as approximations of u(P), P E Mi, where u solves (1.1).Nevertheless and

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in contrast to some other authors (cf. e.g. [LAM 96]),we prove convergence results for the FVM (2.3) in terms of function space norms. For that reason we need a bijective correspondence between the vector uV E Rm that solves (2.3) and the function uh E Vh which is the solutions of (2.5). Theorem 1 supplies such one additionally satisfying the interpolation property u h ( P ) = u! for all P E Mi. Let now a norm 11.11 on Vh be given that is a seminorm on V $ Vh . Definition 3. For a sequence { M h ) of sets of points satisfying the assumpti) corresponding sequence of approximate ons in Section 2.1 let { U ~ ( M ~be)the solutions defined by (2.3). W e say that the FVM (2.3) is convergent with respect t o ll.llh , i f f llu - uhllh approaches 0 for the solution u of (2.4) and the sequence { u h ) = { u h ( M h ) ) defined by the FEM (2.5).

3. The convergence result

3.1. The first step

To prove convergence of the nonconforming FEM (2.5) the well-known second Strang Lemma is used, which leads to an estimation of the form

with a positive constant C independent of h (cf. [VAS 981). In our application we choose the energy norm 2 ll"llh

= (3.2) with ah given by (2.6),such that the assumptions of the second Strang Lemma are satisfied.

3.2. The second step

To obtain an error estimation it is necessary to deduce bounds for the terms on the right-hand side of (3.1). For the approximation error term inf

vhEVh

1126

- Uh1Ih

we introduce the interpolation operator IIdB : H2(51) n H1(51) C V which is defined by (&B U ) ( P ) = u ( P )

VP E M ,

-+

Vh,


186

and take advantage of inf

IIu

~ h E v h

- vhllh

I 1121 - ~ B u I I ~ .

Standard techniques like that one in [CIA 911 lead to

with a positive constant c independent of h, if it holds u E H2(fl)n Hh (0)and if the partitions of fl satisfy some geometrical properties, which are comparable with the minimal angle condition for a corresponding Delaunay triangulation. The problem, that it holds can be overcome by appropriate spaces and seminorms like and

II[~PQ]

T

Iqliil(dbpQ)= gradqIIO,dbpQ in place of Sobolev spaces like H1(dbpQ) as well as a slight modification of Theorem 15.3 of [CIA 911. Obviously, in that theorem the assumption can be substituted by the weaker one I(I - W(~>l,,,,n = 0 VP E Pk(fl). For the consistency error term

we have to estimate dh (u, wh) = ah(^, wh) - dh (wh) which was done in [VAS 981 in the following way: At first, we obtain

1

+I [wh(P) - wh(Q)] J

b p nbq

[ e p ~ l ~ g r a dd ru.

(3.4)

Thereby, it is used, that the standard bilinear form ah, which is given by the right-hand side in (2.8), is substituted by that one in (2.6). Then IddbpQ(u,wh)l was estimated by applying Theorem 33.1 of [CIA 911, which under the same assumptions as above leads to

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187

again with a positive constant c independent of h But, if we use (2.7), in place of (3.4) we get

hpQ ( u , w h ) = { w h ( Q ) - w h ( P ) ) v d p Q (21) with

Now, lrldpQ( u ) [can be estimated by applying the well-known BrambleHilbert lemma (cf. e.g. Theorem 28.1 of [CIA 911). Together with

this results in the same inequality (3.5), but gives a shorter proof. 3.3. The third step

Altogether (3.1), (3.3) and (3.5) lead to the following convergence result. Theorem 2. If the solution u of (2.4) belongs to H 2 ( R )n H i ( R ) and if the partitions of R satisfy some geometrical properties (for the details cf. [VAS 98]), then there exists a positive constant C independent of h such that it holds

llu - uhlih where

5Cd

max

b p E~d B h

PPQ Iul2,n.

is defined by (3.2) and uh is the solution of the FEM (2.5).

Because of Definition 3 the convergence properties of the FEM (2.5) can be transformed to the solution of the FVM (2.3).

4. Discussion

In [HAC 891 the convergence of a FVM like that, which is given by (2.3), is proved by using triangles for the partition of Cl and the well-known conforming linear FEM. It is an advantage of our approach, that the analysis can extended to convection-diffusion equations of the form div {- E grad u + b u ) = f , where E is a positive parameter and b is a given constant vector. One possibility is the use of a full-upwind technique with the approximation [epQIT{- E grad u bu) E PPQ ~ i: { K ( T [ ~ P Q IU (~P ~) -) K ( [epOITb)U ( Q )

+

PPQ

with the function function K defined by

13P0 E

}

188


K ( t )=

{

ifz20

z

otherwise, which altogeth'er leads to a FVM like that in [MIS 981. Another one is to use exponential fitting with the approximation [epQIT{- E grad u b u) 1 PPQ T % B(- @!2 eTb) U(Q) - B(T e b) u(P) ~

+

-

PPQ

{

E

}

(cf. e.g. [BBF go]), where the Bernoulli function B is defined by

5 . Bibliography

[BBF 901

BANK,R.E., BURGLER, J.F., FICHTNER, W., SMITH,R.K., <Some Upwinding Techniques for Finite Element Approximations of Convection-Diffusion Equations,, Numerische Mathematik, 58, 1990, p. 185-202.

[CIA 911

CIARLET,P.G.,

A s s u m p t i o n 3 g F E ~ l / " ( a R ) , X E R such that v . n / 2 X 0 on dQ. Furthermore, zf v ( x ) . n ( x ) / 2 X = 0 for all x E dR then one assumes the such that zts d-dimensional measurp m ( O ) # 0 and such exzstence of 0 C that div(v)/2 b # 0 on 0 .

+

a

+

R e m a r k 1 Assunzpttons 1 and 3 glue the coercttzvzty of the elliptic operator assoczated t o the varzatzonal equality of ( I ) , (4). It does not need a conzpatzbzlzty wlatzon, so we conszder that, euen i f X = 0 , thls case I S not a Neumann condztzon but a Fourier condztlon. This elliptic problem is then discretized with a finite volume scheme: a "four-point" scheme is used for the diffusion tern1 and an upst,rea~llscheil~efor the convection one. A discrete system is obt,ained for each type of boundary condition. Exist,e~lceaiid uniqueness (for tlie Neumann's boundary c o ~ ~ d i t i o n , the uniqueness is up to a constant like i11 t.he continuous case), of t,lle approxilllat,e solution is proven. If the exact solutioii is assumed t~obe a t least in H 2 ( R ) , one may then esta.blish the convergence of the scheme by proving error estimat,es; a first est,imate in a discrete HA norm is obt.ained. An error estimat,e in the L' nor111 follows with t8hehelp of discrete PoiiicarC inequalit.ies. The convergence of the method for Neumann and Fourier condit,ions requires seine additional work compared t o t,llat of the Dirichlet case. I11 the case of Neuinann boundary condit,ions, a "discrete mean PoincarP" inequality needs to be proven in order t.o obtain an L' error estii11at.e. In t,he case of the Fourier condition, it is interest,ing t o not,e that an art,ificial upwinding has t,o be introduced in the treat,ment of tmheboundary condition in order for the scheine t o be well defined with no additional condition on the mesh. Finite volume schemes for a diffusion convection equation with homogeileous boundary conditions were studied in e.g. [LMSSG], [He931 and [VPLS'L] wit,h different assumptions on the d a t a and the mesh.

2. D i s c r e t i z a t i o n In order to discretize the problem, first we define the mesh D e f i n i t i o n 1 ( A d m i s s i b l e m e s h e s ) A finite volume mesh of R, denoted by

7, zs gzven by a famzly of "control volumes", which are open polygonal (or polyhedral) convex subsets of R (wzth posztive measzlre), a fanzzly of szibsets of 2 contazned in hyperplanes of K t d , denoted by f (these are the edges (zf d = 2) o r szdes (zf d = 3) of the control tloltimes), wzth strzctly posztzue ( d - 1)dzmenszonal measure, and a famzly of points of denoted by P . The finite

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191

zlolume mesh is said t o be admissible if the properties (i) to (iv) below are satisfied and restricted admissible if the properties ( i ) t o (v) below are satisfied. ( i ) The closure of the unzon of all the control volumes is

It;

(12) For any I i E 7 ,there exzsts a subset EK of E such that d l i = U o E z Aa . Let E = U K ~ ~ E K .

r\I 0 a zero contact angle a t the contact line between liquid, solid and gas (cf. [BergGa]. [Ber96b], and [BDPGGSS]). Moreover, this contact line propagates with finite speed, i.e. we are dealing with a free boundary problem. However, if initial d a t a have a nonzero contact angle, the propagation speed may be singular for 1 = 0. no maximum or conlparison principles are known. Those aforementioned issues also mean a great challenge in designing efficient numerical tools. A natural approach t o guarantee nonnegativity of discrete solutions is t o develop a numerical scheme that allows for discrete counterparts of the relevant estinlates i. e. the energy and entropy estinlate - known from the continuous setting. In section 2, we will introduce a n implicit finite volume scherile which gives the perfect framework t o realize this concept. Having presented in section 3 the relevant a priori-estimates in order t o obtain compactness of sequences of discrete solutions, we will show in section 4 that a certain kind of harmonic integral means is the right choice for a n entropy consistent numerical flus that allows for nonnegative discrete solutions. In section 5 we will introduce our method of tinlestep control which is based on a new, esplicit fornlula for the velocity of the free boundary. This allows a tracing of the free boundary reminiscent of laws. On the other hand, the the tracing of shocks in hyperbolic ~onservat~ion very formula for the velocity of the free boundary suggests that for sufficiently smooth initial d a t a a t u a i t t ~ z gtime p h e n o m e n o ~ zoccurs, i.e. there is a slight delay in the onset of spreading. This phenomenon is well know11 for solutions t o second order degenerate parabolic equations, like the porous media equation. We will present numerical silnulations which give strong evidence that it also happens in the case of fourth order degenerate equations.

'

-

2. Deriving the entropy consistent finite volume scheme T h e two major classes of discretizations for evolution problems - finite volume and finite element schemes - have both significant advantages. Finite volume schemes very easily lead tto conservative schemes and illcorporate fluxes on cell faces in a natural way, whereas finite element schemes correspond t o a Galerki~i discretization of the continuous problem and therefore carry strong provisions concerning a convergence analysis. In general, it is unusual to apply finite volume schemes t o fourth order parabolic problems. But. due t o the peculiar diffusive structure of the elliptic term in ( I ) , the thin film equation plays a n

'For a different ansatz to ensure nonnegativity , based on variational inequalities, we refer to [BBG].

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207

esceptionary role - as d o other related higher order equations like the CahnHilliard equation with degenerate mobility, too. A suitable mixed finite volume - finite element discretization is the starting point t o derive a conservative and entropy consistent numerical flux with a n intuitive interpretation of t,he construction. In particular, the entropy consistency guarantees nonnegativity of the resulting numerical solution independent of the spatial resolution. Later on we will derive from the original finite volunle scheme a finite element scheme. which will turn out t o he preferable concerning further investigation in the numerical analysis. For simplicity we assume C2 t o be a n interval in 1D or a polygonally bounded domain in 2D. respectively. We suppose St t o be subdivided into cells. On these cells we suppose the discrete height U to be piecewise constant. We denote the negative Laplacian of the height which physically has the interpretation of pressure by p = - A u . Finite elements allow a straightforward discretization of this Laplacian. Thus we have t o find a suitable finite element mesh and function space for the pressure. We choose linear finite elements on the mesh dual to the finite volume mesh. To be more precise, we start with a simplicia1 grid Th on Q consisting of subintervals, respectively triangles E, on which the discrete pressure P will be defined a s a function in the corresponding linear finite element space vh, where h indicates the chosen grid size. Then a dual mesh is built of open dual cells D,, again intervals, respectively polygonally bounded cells, corresponding t o the vertices x of the primal mesh (cf. Figurel). We define a single dual cell by

D, := {y E St : dist(y, x)

< dist(y, E ) , .i is vertex of the

mesh)

.

In t,he following, discrete functions will be denoted by uppercase letters, in constrast t o lowercase letters for arbitrary functions in the nondiscrete function spaces. T h e discret.e height U will be defined spatially constant on these dual cells. Figure 1 shows a n example of such dual triangulations. To st,art with the

FIGURE 1. A 2D finite element triangulation whose edges are outlined in black and the corresponding dual finite volume mesh indicated by dashed lines. discussion of finite volurne schemes, let us consider a cell D of the dual grid. On this subvolun~ewe can rewrite equation (1) in conservation form

208


where p = -Au and v is the out.er normal on 8D. The right hand side describes the inflow a t the boundary, and M ( u ) G p is the corresponding flux. Thus, a numerical mobility M and a numerical pressure gradient are the main ingredients of a spatial discretization. As already mentioned, with the dual grid a t hand the latter requirement is easy to fulfill. For given U we define P = -Ah[/ on vh,i.e. P is the unique function in vh with

where Zh : C0(!2)-+ vh is the nodal projection operat.or and (-, .)h indicates the well-known lumped mass scalar product corresponding to the integration formula (O, Q ) h :=Jnzh(OQ) . Gradients of P are by construction piecewise constant on elements E and thus almost everywhere on faces F of dual cells D. T o pay account to the fact that the values Uf = U(X&EV)may be different due to the discontinuity of U across cell boundaries, we suppose the discrete mobility M t o be a function M : 1Fi2 -+ 1R; ( U t , U p ) I-+ M ( U + , U-) , where U+, U - are the outer, respectively interior values of U a t the corresponding face. Finally, we can formulate our semi-discrete finite volume scheme

where Q ( U + , U-, V P ) = M ( U + , U - ) V P is the corresponding numerical flux. In our case we suppose that the discrete mobility M ( U t , U - ) is a symmetric which is positive semidefinite and piecewise constant on E E f i . matrix in lfLdxd T h e resulting scheme is known t o be conservative [Kr97] if this flux is symmetric, i.e. Q ( U + , U-, V P ) = Q ( U - , U+. V P ) respectively

Thus, the inflow on F corresponding t o D should coincide with the outflow with respect t o the adjacent element a t the face F. This immediately implies the conservation of mass J"aU dx . Furthermore the flux should be consistent with the continuous flux q = M ( u ) V p , i. e.

where the second term on the right hand side vanishes for decreasing grid size. There is still a great flexibility in selecting a numerical mobility. Let us recall that in case of hyperbolic conservation laws upwind discretization and entropy consistency conditions on the numerical fluxes, i. e. certain monotonicity properties, select the right entropy solution and guarantee moreover that neither artificial oscillations nor nonphysical shocks occur. These ideas carry over to the discrete modelling of thin films. For the trivial choice M ( U + , U-):=M(-)1d nonnegativity of the numerical solution can no longer be guaranteed. Entropy consistency will ensure discrete nonnegativity independent of the selected grid size. In fact, we will be lead t o some type of harmonic integral mean as an appropriate choice. This can

Numerical analysis

209

also be interpreted as a suitable type of upwinding. In the continuous setting an entropy is defined by

Choosing its derivative as a test function in the continuous problem we find that J'G(u) is decreasing with time. This is in analogy to hyperbolic problems, where entropy estimates can be derived by testing the viscous approximated problem accordingly. These entropy estimates carry over to the discrete case provided we define if U+ # Uotherwise. This numerical mobility can be regarded as a function M ( Z h U ) on the primal grid. For the generalization to arbitrary dimensions, we refer t o Section 4. Finally, the semidiscrete scheme can be discretized in time implicitly or explicitly. Therefore suppose [0, T ]t o be subdivided in intervals Ik= (tk, k k + l ] with tk+l = t k r k for time increments r k > 0 and k = 0,. . - , N - 1. We will use backward difference quotients with respect t o time which we shall henceforward denote by d;, respectively. Because of the significant stiffness of our problem we choose an implicit discretization. Otherwise a CFL-type condition r 5 C h4 would entail very small timesteps. In Section 5, we will discuss the selection of appropriate timesteps in detail. These ideas to construct entropy consistent finite volume schemes can be carried over t,o an appropriate pure finite element discretization. Therefore, we consider P and U both as functions in Vh and obtain the following finite element formulation of equation (1) with fully implicit, backward Euler discretization in time:

+

For given UO E vh such that

vh find a sequence (uk,pk)for k = 0, - .. ,N -

1 with

uk,pk E

for all O, Q E V h . Thus, a solution of (3) is obtained solving a nonlinear system of q = d i m v h equations for each time step. Let us define by M h , Lh the standard lumped mass, respectively stiffness matrix and by L h ( W ) the matrix corresponding to the degenerate quadratic form, i. e.

(L,(w)u,v) =

M(W)VUVV~~.

Here we denote the nodal value vector for a function V E Vh by V , and with a slight misuse of notation rewrite L ~ ( w )for L h ( W ) . Then for given Uk E IRq we search U k + l E IRq such that F(Uk+') = 0 for

210


Due t o the absence of Dirichlet boundary conditions, Ah is not injective, i. e. kerAh = { r c-, CIC E IR) . This corresponds t o the observation t h a t AhU =

0. In contrast t o the finite volunle interpretation, the replacement of the exact mobility M ( . ) by a certain quantity M ( U )may now be interpreted as the choice of a specific quadrature t o integrate the elliptic term numerically. For a certain class of grids in 2D (cf. [GRSS]), a tedious but straightforward computation proves the equivalence of the finite volunle and finite element approaches. 3. Existence, stability and compactness of discrete solutions

In this section, we recall the main results concerning existence and regularity. T h e proofs can be found in [GR98]. T h e key estimate for numerical analysis is the following energy estimate:

lT

+

W ( U )1 ~

:s, / v u o ( ~ ) ( ~

~ 1 d' x d f = -

dr.

In 1D a direct consequence of this estimate is a result on Holder continuity w.r.t. time for spatial averages of discrete solutions U which can be cornhilled with the energy estimate t o yield the following pointwise Hijlder regularity:

+

Lemma 3.1: Assume d = 1 and that for integer I , k > 0 with 1 k < N the relation k r 2 h4 holds. Then for u discrete solution (UThr PTh)with 11 M(U,h)(l, M I independently of T,h , there exists a constant C depending such that only on

< ll~~]l

Iu"~(~) forx E

- U"(.)l

< c(kr):

a.

As a consequence, convergence of discrete solutions to a solution in the continuous setting can be proven. For the quite different techniques t o be used in higher space dimensions, we once more refer t o [GR98]. 4. Entropy estimates dimensions

- discrete nonnegativity in arbitrary space

In the case of space dimensions d > 1, the discrete mobility can no longer be given by a scalar valued quantity. This is due t o the observation t h a t on each cell of the dual mesh numerical fluxes coming from different directions have t o be treated differently. It turns out t h a t the right approach is t o choose the mobility a s a field of elementwise constant, symmetric positive semidefinite

Numerical analysis

21 1

d x d-matrices which depends continuously on the discrete function U E v h . To make the mobility matrix consistent with the entropy function, an additional admissibility condition has t o be satisfied: . ~ I ~ ( I : ) o z ~ G ' (=~ V ~U ) , where G'(s) := y ( r ) d r with y(s) = rn(r)-'dr.

Ji

Jl

Here, m is a n appropriate approximation for the continuous mobility JM (for its explicit form depending on the snloothness of M, we refer to [GR98]). Not,e that C: is nonnegative and convex by construction. For nondege~~erat~e reference simplices E(,,, ... ,,,):=convex hull(0, cylel, ,~rded), we verify i~nmediatelythat

(L(~, ['(ate,)

if1 =

(iilij)

i . j = l .... , d

with i f i j =

1 -ds)

4 s )

-1

dij

satisfies the asiollls above. For U ( n k e k ) = I/(()) the definition simplifies to ,TIkk = nz(/[(O)). For elements E which can be mapped onto a reference element E by a rigid transformation x H j. = xo + A - l x , A an orthogonal matrix, the matrix M := AAI.il-' satisfies conditions (ii), (iii). Since A is orthogonal, M is symmetric and positive sernidefinite; hence condition (iv) is satisfied, too. For the general case, we refer to [GR98]. This ~orlst~ruction allows to obtain the following discrete analogue of the entropy est,imate: Lemma 4.1: Let (/I, P) be a solution to the system of equations (3)-(4), and assunre that ( M ,G') is an adnxissible t-ntropy-mobility pair as described above. Then, for arbitrury T = I i r , Ii E IN, the following estimate holds:

As a consequence, the following result on nonnegativity of discrete solutions can be obtained which is in fine accordance with related results in the continuous setting (cf. [BF90] and [Gr95]):

lJs)

Theorem 4.2: (Existence of nonnegative discrete solutions Let Th be an admissible triangulation of S1 and let n > 0 be the growth coefficient of JW in zero. Assume that the mobility .M is monotoneously increasing and z~anzsheson IR- U (0). For arbitrary E > 0, there exists a positive control 0 parameter a 0 which only depends on d , n, E , h and the initial datum uo

>

212


such that: For every 0 < a < a0 discrete entropy-mobility pairs (G,, M,) can be constructed having the property that the corresponding discrete solutions Uyh of equation (3)-(d), W' 0 , satisfy: U$ > - & i f u 0 > O a n d O < n < 2 . > - E if u0 2 uOa n d n = 2. Uyh > a/2 if u0 2 uo and n > 2 . For a proof of this theorem, we refer t o [GR98]. Let us remark that L. Zhornitskaya and A.L. Bertozzi [ZB] who studied finite difference schemes for growth 2 obtained quite similar results on positivity of discrete solucoefficients n tions.

>

+

FIGURE 2. Numerical approximation of the solution t o ut div(u . V A u ) = 0 for initial d a t a given as the characteristic function of a nonconvex set

5. Implementation, timestep control, waiting time phenomenon One of the most important questions with respect t o numerical sinlulat,ions of wetting phenomena is how t o trace the solution's free boundary in an efficient way. In order t o describe the arising difficulties, let us first consider questions of implementation. In each timestep, we have t o solve a nonlinear system (cf. section 2 ) . In fact, we first consider a related semi-implicit system, given by

+

~ = ~ Luk. ~ ) ( ~ d rkM ; ~ L ~ ( W ) A u For the solution of the fully implicit scheme, we apply a n appropriate fixedpoint iteration t o satisfy the original problem with w = (for det,ailscf. [GR98]). Now observing that in the semi-implicit scheme the numerical free boundary

Numerical analysis

213

cannot propagate Inore than a distance h in each timestep, it. is reasonable to choose the time increment r smaller than the quotient -where speed(t) stands for the maximum normal velocity of t,he numerical free boundary a t time t. As a consequence of this special choice of time increment, only a very small number of iterations is necessary to obtain the solution of the fully implicit scheme. Formal considerations - performed in the continuous sett,ing - indicate that the normal velocity V,([(t)) of the free boundary in a point [(t) can be related t o spatial derivatives of u in [(t) according to the following formula: ~ , ( < ( t )= ) lim

~ - r ~ ( t u(t, ) x)

A dv

x E supp(u(t, .)).

,

(7)

This formula has been proved for self-similar source-type solutions in [GR98]. In the framework of the algorithm studied in this paper, we formulate a discrete counterpart of formula (7) in the following way: In a timestep tk, we first determine on each E E Thnumbers

-

M:%lE' (zbllar.

v(tk, E ) :=

A h

1)

if uThIE 2 0 and otherwise.

(%)E

> 0 (8)

with y E (0, 1).

(9)

Then, we define the time increment by the formula rk

:=

0.01

+ m a yhx ~ ~~r (, t kE, ) '

If n 2 1, the results on Holder continuity in space for discrete solutions allow to h-512. give a robust, but coarse upper bound: m a x E ~ T ,v(i, E ) 5 This implies for the time increment:

cIIuIIZZ~~~)

Hence, the assumption r longer.

> h4 in Lemma 3.1 does not mean any restriction any

Fornlula (7) indicates that for sufficiently smooth initial data the velocity of the free boundary vanishes. So let us take R = ( 0 , l ) and as initial d a t a the 5

function uo(x) = ([cos ( $ a x ) ] + ) . We choose M ( u ) = u2 in equation (1) and obtain for t E [O,1] a solution u as shown in the six diagrams on the left of figure 3. From top left linewise to bottom right, they represent six snapshots of u ( t , -) for increasing t,imes t . To have a closer look a t the behaviour a t the free boundary for small t, we depict on the right the function v(t, 2 ) =

log (1020 . u(t, x))

if u(t, x ) > 10WZO otherwise

a t four different times ranging from t = 0.0 in the background t o t = 2.5- lo-" in the foreground. It turns out that for t < 2.6 the free boundary does not move, whereas for larger times the support monotoneously increases. This

2 14


FIGURE 3. Delayed onset of spreading for solutions to the = 0 and sufficiently smooth initial equation ut ( u 2 - u,,,), data(number of gridpoints: 500)

+

gives very strong evidence that also for the thin filrrl equation a waiting time phenomenon occurs. For other simulations illustrating the variety of phenomena encountered in thin film flows, we refer the reader to [GR98] and [Gr99].

J. Barrett, J. Blowey, and H. Garcke. Finite element approximation of a fourth order nonlinear degenerate parabolic equation. to appear in Numer. Mathematik. [BDPGG98] M. Bertsch, R. Dal Passo, H. Garcke, and G . Griin. The thin viscous flow equation in higher space dimensions. Adv. D i n . Equ., 3:417-440, 1998. F. Bernis. Finite speed of propagation and continuity of the interface for thin [Ber96a] viscous flows. Adv. in Diff. Equations, 1, no. 3:337-368, 1996. [Ber96b] F. Bernis. Finite speed of propagation for thin viscous flows when 2 5 n < 3. C . R . Acad. Sci. Paris; Se'r.1 Math., 322, 1996. F. Bernis and A. Friedman. Higher order nonlinear degenerate parabolic equa[BF90] tions. J. Diff. Equ., 83:179-206, 1990. [DPGG98] R. Dal Passo, H. Garcke, and G . Griin. On a fourth order degenerate parabolic equation: global entropy estimates and qualitative behaviour of solutions. SIAM J. Math. Anal., 29, 1998. [Gr95] G. Grun. Degenerate parabolic equations of fourth order and a plasticity model with nonlocal hardening. Z. Anal. Anwendungen, 14:541-573, 1995. [GR98] G . Griin and M. Rumpf. Nonnegativity preserving convergent schemes for the thin film equation. 1998. Preprint No. 569, SFB 256 University of Bonn. [Gr99] G . Grun. On the numerical simulation of wetting phenomena. 1999. t o appear in Proceedings of 15th GAMM-Workshop, Kiel. [Kr97] D. Kroner. N u m e ~ i c a lSchemes f o r Conservation Laws. Wiley and Teubner, Chichester and Stuttgart, 1997. L. Zhornitskaya and A.L. Bertozzi. Positivity preserving numerical schemes for [ZBl lubrication-type equations. SIAM Num. Anal. submitted, 1998. [BBGl

Convergence of finite volume methods on general meshes for non smooth solution of elliptic problems with cracks Philippe Angot Universite' de la Me'diterrane'e, I.R.P.H.E. Chiteau-Gombert, 38 rue F. Joliot Curie, F-13451 Marseille Cedex 20 - E-mail : [email protected]

Thierry Gallouet and Raphakle Herbin Universite' de Provence, C.M.I. - L.A. T . P . , 39 rue F. Joliot Curie, F-13453 Marseille Cedex 20 - E-mail : [gallouet,herbin]@gyptis.univ-mrs.fr

ABSTRACT A model of insulating cracks for elliptic problems is presented and proved to be well-posed. The solution is indeed discontinuous. A finite volume scheme o n general polygonal meshes is introduced to solve such problems. Since n o unknown is required at the fracture interface, the scheme is as cheap as more standard schemes for the same problems without cracks. With weak regularity assu.mptions, uJe establish for discrete norms some error estimates in O(h) where h is the maximum diameter of the control volumes of the mesh.

Key Words: elliptic problems, insulating cracks, discontinuous solutions, finite volumes, fracture resistances, error estimates.

1. Introduction The concept of contact resistance is sometimes introduced empirically for diffusion problems (Fick's law) with imperfect contact, e.g. thermal (Fourier's law) or electrical (Ohm's law) contact resistance, or also hydraulic resistance of fissure for flows in porous media described by the Darcy's law. The objective is t o take account of fault lines or too thin layers cornpared to the largest scale under study. Hence, from this macroscopic scale the solution at the interface is indeed discontinuous. In previous works e.g. [ANG 891, we generalized this concept and formulated for such elliptic problems a mathematical model with discontinuous coefficients which includes a jump transmission condition linking the divergential flux with

216


the jump of the solution a t the interface. It ensures the well-posedness of the associated elliptic or parabolic problems and we proved in [ANG 98, ANG 98b] the global solvability within a variational framework. Besides, we have shown how to use the imperfect transmission problem for fictitious domain modelling with immersed boundary conditions imposed by a penalty method. In that case, we performed the convergence analysis and derived the associated error estimates as functions of the penalty parameter; see also [ANG 971. For the numerical solution on a rectangular mesh, we proposed in [ANG 89, ANG 89b] an original finite volume method, either cell-centered or vertex-centered, and based on the introduction of "fracture resistances" a t the faces of the control volumes. For this scheme, some error estimates in O ( h ) are established in [ANG 98, ANG 98b] and various numerical results have illustrated the capabilities and the efficiency of this methodology. In the present work, we generalize the model in two ways; see also [ANG 991. First, we do the mathematical analysis in the case where the fracture interface C is an open surface strictly included in the bounded domain fl, e.g. without any connection with its boundary = dfl. We prove the existence and uniqueness of the solution for a diffusion-reaction problem. This case is more difficult because the open domain is no longer located locally on one side of its boundary since the fault interface does not divide fi into two disjointed subdomains. Second we extend, in the case of a polygonal interface, the finite volume scheme t o general polygonal meshes, as considered in [EYM 971 or [HE1 87, SHA 961, e.g. triangular [HER 951 or VoronoY meshes; see also [COU 96, HER 96, LAZ 961. The construction of a general admissible mesh is made in such a way that the discontinuity lines of the operator coefficients, and/or the polygonal fracture interface lie on faces of some control volumes. Then, we construct a finite volume scheme including fracture resistances at faces of control volumes, well-suited t o the numerical approximation of the imperfect transmission problem. We show how t o satisfy both conservativity and consistency of the numerical fluxes. Indeed, the numerical scheme is locally conservative by construction. Let us notice also that our numerical scheme inherently involves the locally conservative approximation of the immersed jump condition (3) without using unknowns located on the interface C. Hence, it only uses a four-point stencil for triangular finite volumes in 2-D, or a six-point one in 3-D. This means that the solution cost is as cheap as for a more classical finite volume scheme without any fracture interface, and hence cheaper than with the "double-node" finite difference scheme proposed in [SAM 781.

2. Well-posed elliptic model for insulating cracks Let the domain fi c Rd (d = 2 or 3 in practice) be an open bounded polygonal set, ?= dfl being its boundary, which includes a polygonal interface C C Rd-'.Let us define the open bounded set R such that fi = 0 U C and its boundary F _= d R = f' U C. I t is always possible to prolong C within a

Numerical analysis

21 7

polygonal interface 2 > C which divides the domain fi into two disjointed subdomains R- and R+ such that fi = 0- U 2 U R+. We denote by X - and X + the characteristic functions of R- and a+, respectively. Let n be the normal unit vector on C oriented from R- to R+. For the data f E L2(R), we consider the second-order elliptic problem for the real-valued function u defined in R :

where the symmetric second-order tensor of diffusion a ( ~ i ~ ) l < i , and ~ < dthe reaction coefficient b are measurable and bounded functions verifying classical ellipticity assumptions : a E ( L ~ ( R ) ) ;~ ' l a o

> 0,

V< E lRd, a(x)-

0 such that, for all i, j E x,it

ma.

T E 7 h : m e >O

(A4) There exists a constant C [ 1 , 1 $ ] ~ ,it holds

denoted by h:'.

S " R :

(A3) There exists a constant C

(w.r.t. Qi) normal "u:

rnz1dij

>0

holds

l€[l,le]~ such that, for all i, j E

< Cmeasd (a:')

x,T E %, 1 E

.

(A5) There exists a constant C > 0 such that, for all i , j E [I, it holds ( h 2 ' i 4 5 Cmeasd (a:') .

x,T E x,1 E

A dimensional analysis of the quantities appearing in the Assumptions ( A l ) , (A3), (A5) easily shows that these conditions are not very restrictive.

2. Discretization (Treatment of the trilinear form n ) d

Because of n ( w , u,v ) :=

X((W. v ) u l , u l ) , the description of the discretizaIT1 0

tion can be restricted t o the scalar case. So let w E w ~ ' ( R ) ~be such that

Numerical analysis

227

0

G - w = 0 and define, for u, v E w ~ ( Rthe ) , form

It is not difficult t o see t h a t n, can be represented equivalently a s

Taking into considerat.ion the condit,ion V . w = 0, it is not so far t o omit the last. term, i.e. we get n,(w,

1

U , 21)

1

= -[(V . ( w ~ )v), - ( u , V . ( w , ~ ) = ) ] ?[b(w, u, v) + b(-w, v, u)],

-

2

1 - ( ( V - w ) u , v). This is the starting point 2 for the discretization. Suppose there is some control function r : W -+ [O, 11 satisfying the following properties: where b(w. u ,

11)

:= ( O

- ( W T L )v) ,

-

r ( : ) is isotone for all z ,

343)

is Lipschitz-contilluous on the whole real axis.

l..

Furthermore, set yij := m ~ '

u

-

. w ds and r i j

?. . d . . := r ( 7 ) . We mention

..I

t.hat yij is ant,isyinmetric, i.e. yji = -yij. Moreover, in the definition of yij, the value of J,--u - w ds can be replaced by certain approximation which has to satisfy, among natural error est,imates, the above antisymmetry condition. Then, by standard arguments in the derivation of finite volunle discretizations (cf. [ANG 95]), we can write z,

Thus we get

:C C

n,, ( w , u,V ) M -

iciz

jeA,

[(rij - i ) ( U i -

+

1

- U 8 V3) 1 73. m V. -. uj)(vi - ~ j ) -(ujvi 2

228


sfl,

Now, redefining for wh E Vh := uh, uh E Shl the quantities Y.. .- m- 1 u . wh ds as well as rij, we set 23 'tj TETh :m:>O 11

C

lT

Returning t o the original form n , we set for wh, uh, vh E lTh d

n ( w , u , v ) 63 nh(wh, uh, vh) :=

-

n , s h ( ~ h~,' h~, l h ) 1=1

COROLLARY 1 If the control function r satisfies (P5), then it holds

1

-

Typical examples of the control functions are r ( z ) = ;[sign z

I

scheme), r ( z ) = 1 -

expz - 1

+

+ 11 (full upwind

(exponentially fitted scheme), r ( z ) =

[ i I] (Samarskij's scheme). 2 2 - t 121 Finally, some discrete forms and operators have to be introduced. For u, u E 0

( R ) + S h l , we set ( V u , V U ) ~where ( V u , v u ) := ~

( V u , V V ) ~:= TETh

The extension of these definitions to the case of va valued functio~lsis obvious and will not be denoted separately. By Ih: W;(Q) t Shl, an interpolation operator is denoted, whereas Lh : C ( 1 ) t L,(Q) stands for a so-called lumping procedure. T h a t is, the image of Lh is the subspace consisting of functions being constant on the elements of the secondary partition 7;.Concrete properties of these operators are collected in the subsequent assumptions.

3. Properties of the discrete forms

In order to verify the required properties of Lipschitz-continuity and consistency of n,h, we formulate further assumptions.

Numerical analysis

229

(A6) There exists a constant C > 0 independent of h such that, for all 'uh E Shl,

(A7) For arbitrary p E [I, 61, there exists a constant C vh E Shl,it holds IlvhllO,p,n Cllvhllh

> 0 such

that, for all

0 such

(A8) There exists a constant C all u h E S h l , it holds

-A

Since A

that, for arbitrary p E [I, 61 and for

is non-empty, in general, IILhvhllO,p,nis only a seminorm on Shl.

(A9) There exists a constant C > O such that for all vh E Shl, it holds

> 0 such

(A10) There exists a constant C

that

(i) for all v E W22(R),it holds 11Ihu110,m,n (ii) for all v E w ~ ( R )and all T E 11h"-'11,2,T

0 such

( A l l ) There exists a constant C (i) for arbitrary p E (d, 61,

21

11v112,2,n,

and all T E Th, it holds

that

E Wi(C2) and all T E

Th,it holds:

I(I- I ~ ) U I ~ 5 , ~C, ~T; - ' I I U I I I , ~ , T , 1 = 0 or 1 = 1 , (ii) for all v E W22(R) and all T E Th, it holds

0

is a constant which does riot depend on h.

L E M M A 2 Suppose (A?), (Ad), (.46), (A?'), ('48), (AQ), (.410), ( A l l ) , ( A ) . Then, for any w E b ~ ' z ( R )n ~ V satisfying O . w = 0, any u E 0

w ( R ) n bVi(s2) and any element

holds, where C

>0

tjh

E

Shl

the estimate

is a constant tvhich does not depend on h .

4. Application

The above approach call be used to give an alternative proof of the convergence properties of Schieweck's fanlily of nonconfornling quadrilateral/l~esal~edral elenlents [SCH 971 which find successful applicat,ioll in parallel Navier-Stokes codes. Det,ails for the case of the so-called PI-parametric element are described in [ANG 981. 5. Bibliography

[ANG 951

L. Error estimates for the finite-element soANGERMANN, lution of an elliptic singularly perturbed problem. Iilf-4 J. Numer. Anal., 15, 1995, p.161-196.

[ANG 981

ANGERMANN, L. Error analysis of upwind-discret,izations for the steady-stat,e incolnpressible Navier-Stokes equations. Fakultat fiir Mathematik, Otto-von-Guericke-U~liversitat Magdebnrg, Prepri~itNr. 33, 1998.

[RST 961

ROOS, H.-G., STYNES, M. A N D TOBISKA,L. ivumerical methods for singularly perturbed differential equations. Springer- Verlag, Berlin-Heidelberg-New York, 1996.

[SCH 971

SCHIEWECK,F. Parallele Losung der stationken inkompressiblen Navier-St,okes Gleichungen. Habilitationsschrift,, Fakultat fiir Mathernatik, Otto-von-Guericke-Universit,at Magdeburg, 1997.

A new cement to glue non-conforming grids with Robin interface conditions

Yves Achdou Insa Rennes, 20 Av. des Buttes de Coesmes 35043 Rennes, France

Caroline Japhet, F'rederic Nataf CMAP, Ecole Polytechnique 91128 Palaiseau, France

Yvon Maday Laboratoire d'Analyse Nume'rique, Universite' Pierre et Marie Curie

4, place Jussieu, 75252 Paris Cedex 05, fiance

We propose and analyse a domain decomposition method based on Schwarz type algorithms that allows for an extension to optimized interface conditions on nonconforming grids. We consider the convection-diffusion equation discretized b y a finite volume method. The nonconforming domain decomposition method is proved to be well-posed and the error analysis is performed. Then we present numerical results that illustrate the method.

ABSTRACT

Key Words: Domain decomposition methods, optimized artificial interface conditions, non-conforming grids, convection-diffusionproblems, finite volume methods, parallel computing, High Performance Computing.

1. Introduction

The goal of our project is to design domain decomposition methods based on the use of optimized interface conditions on non-matching grids. The original Schwarz algorithm is based on a decomposition of the domain 0 into overlapping subdomains and the solving of Dirichlet boundary value problems in the subdomains. It has been proposed in [L 891 to use more general boundary conditions for the subproblems in order to use a non-overlapping decomposition of

232


the domain. The use of exact artificial boundary conditions as interface conditions leads to an optimal number of iterations, see [HTJ 881, [NRdeS 951. As these boundary conditions are pseudo-differential, "low frequency" approximations of these conditions have then be proposed, see [D 931, [NR 951, [GGQ 961. In [J 961 approximations which minimize the convergence rate of the algorithm are proposed, and increase dramatically the convergence speed of the method. The mortar method, first introduced in [BMP 891, which enables the use of non-conforming grids, can't be used easily with optimized interface conditions in the framework of Schwarz type methods. The goal of our work is to design and study a non-conforming domain decomposition method which allows for the use of Robin interface conditions (&f a ) for Schwarz type methods. We consider the convection-diffusion equation

discretized by a finite volume method where q and v are positive but arbitrary small and a is the vector field. We first consider the symmetric definite positive case ( a = 0) and then in fj 5 the convective case ( a # 0). 2. Domain Decomposition at the continuous level

Let R be a bounded domain in Rd for d following problem: Find u such that

(-A

+ 7)( u )= f

> 1 and q > 0.

We consider the

in R

u = 0 on a R The domain R is decomposed into N non-overlapping subdomains, 0 = UI j i 5 N f i i . Let a > 0, the above problem is reformulated as a domain decomposition prob) ~-< ~such < N that lem: Find ( u ~ -

An iterative method for solving the above domain decomposition method is:

Numerical analysis

233

The well-posedness and convergence of the above problems and algorithm have been studied in [L 891. We are interested in the discretization of (4) by a finite volume scheme with non matching grids on the subdomains's interface. 3. F i n i t e volume discretization

The scheme is taken from [H 951. On each domain Ri let Z be a set of closed polygonal subsets of Ri such that f i i = U K K ~and ~Eni the set of edges associated with Z , i.e. a set of closed subsets of dimension d such that for any (K, K') E with K # K', one has either K n K' = 0,dim(K n K') 5 d - 1 and K n K ' E Eni. In this case, d K n dK' is denoted by [K, K']. We also assume that no edge intersects both dRi/aR and dRi n d o . We shall use the following notations: Let ~i be an edge of Eni located on the boundary of Ri, K(ei) denotes the control cell K E such that E, E K . & i is~ the set of edges such that dR n d o i = UEE&iDE. Let us recall that a Dirichlet boundary condition is imposed on this part of the boundary. Ei is the set of edges such that dRi/dR = UeEdiE. Let us recall that a Robin interface condition is imposed on this part of the boundary. &(K) denotes the set of the edges of K E EiD(K) = & ( K )n & i is~ the set of the edges of K E Z which are on dR n dRi. Ei(K) = &(K)nEiis the set of the edges of K 6 7;: which are on aRi/dR. N i ( K ) is the set of the control cells adjacent to K : Ni(K) = {K' E Z/ K n K ' € En,). We make the following

x.

A s s u m p t i o n 3.1 W e assume that there exist points (ye)cEEnion the edges (y, E E ) and points (xK)K~-T,inside the control cells such that for any adjacent control cells, K and K', the straight line [xK,xK!] i s perpendicular to the edge [K, K'] and [ X K , XK'] fl [K, K'] = { Y [ ~ , ~ 'and ] ) , for any edge E E &i U &a, the straight line [xK(, Computers and Fluids, 2 2 , p215, 1993.

[DEC 951 DECONINCK, H, PAILL~RRE, H., ROE, P. L., 0 and take E, : S;I -+ ( H ~ ( ( I Rt o~ be ) ) a~ suitable approximation t o the exact evolution operator E ( T ) ,r 2 0. We denote by Rh : S: --+ Si a reconstruction operator, r > p 0 . In the present paper we

>

>

Innovative schemes

291

shall limit our con~iderat~ion to cases of co~lst~ant time step At, i.e. t , = n A t , and of a uniform mesh consisting of d-dimensional cubes with a uniform mesh size h,. Definition 1. Starting from some initial valtre (1' at time t = 0 , the finite volume evolution Galerkin method ( F V E G ) is reczlrsirlely defined by nteans of

where the central di'erence v ( x + h / 2 ) - v(x - h / 2 ) is denoted by 6,v(x) and SXk&(LJn+*) represents an approximation to the edge flu2 difference. The cell boundary zlalue is evolved using the approximate evolution operator E, to t, r and averaged over 0 r 5 A t and along the cell boundary, i.e.

+

un+*

hefollowing section. The

Innovative schemes

301

choice of the q5i+l/2,j is exactly the same like the one described in [YSD99] with the use of the limiter functions g; as

Additional possible choices for this function and a detailt description are given in [YSD99]. The definition of 4' will be give below. The choice of the filter function depends on the characteristic splitting of the flux representation of the gradients. Thus, we consider as elements of

a i + l / 2 , j := (Rftl 2,j)-'(Ui+l J - U i , j ) .

d

The different a epend again on different matrices of the right eigenvectors depending on the gridpoint were they are evaluated. They are given as:

4 f + 1 / 2 , ~ + 1corresponds /~ to ai+l/2,j+l12 and so on The filter function writes as

Hence

with

The a:+l12,1 in (3),(4) are the characteristic speeds of the corresponding Jacobians, which are equal to their eigenvalues. This term can also be weighted with the diffusion coefficient steering the amount of dissipation corresponding to this direction.

/'?i+l12,j

2.3. The structure tensor

We start from the so called structure tensor J t j . The extensive description of its construction and properties can be found in Weickert [Wei98]. We use a

302


smoothed version of the characteristic gradients R-' A U d , where Ua is a presrnoothed version of the d a t a U n . This means convolution with a Gaussian kernel whith convolution scale 6, i.e

Since in the continnous case this is equivalent with solving the heat equation we apply this t o the data UiXjwith stopping time T = i d 2 , where 6 is a parameter which has to be chosen. In order t o remove the small scale oscillations by means of this smoothing technique we define 6 depending on the grid size h := d m . Consequently, the structure tensor reads as

with

where cr1i6 corresponds t o the smoothed data U a . Therbye one can also use a smoothed version of the structure tensor (5)) i.e

which means component-wise convolution with scale u , which denotes the width of the averaging region. In practice we are solving the heat equation for each component seperately.

2.4. The diffusion matrix

After having constructed the structure tensor we are going t o compute the eigenvectors and eigenvalues of (6),

Innovative schemes

303

Therbye the corresponding eigenvalues are given by

Now we construct the dissipation matrix D by introducing the ansatz due to Weickert [Wei98]:

Here V, denotes the matrix of the eigenvectors (7), (8) and A, = diag(ll,l2) represents a diagonal matrix where the diagonal elements have to be calculated in a convenient manner as described below. In order t o recover shocks (or, equivalently, in order to enhance edges) the diffusivity l I perpendicular to edges should be reduced if the contrast XI,, is high. This can be achived by an anisotropic regularization of the Perona-Malik niodel [PMSO] also adapted from Weickert :

The values of m and C, are chosen in such a way that t,he so-called flux @(s) := S ~ ( Sis) increasing in an interval s E [0, A] and decreasing in s E]X, a[. The choices depend on a one-dimensional analysis of the Perona-Malik model. In agreement with Weickert we chose m = 4 and thus C4 = 3.31488. The so-called contrast parameter A, seperating areas with forward (low contrast) from backward (high constrast) diffusion, can be chosen freely. Based on numerical experiments it turns out that for calculations concerning systems it is useful1 t o have an adaptive parameter instead of a fixed one. We calculate for each variable the maximum of the according gradients in every timestep and choose a fixed percent of this maximum to determine the parameter A. Thus, the diffusion matrix (10) can be written as

with coefficients

The superscript 1 reminds that the diffussion where V I , ~= (vllip, matrix (10) and so the structure tensor (5) resp. (6) have to be computed for every characteristic variable seperatly.

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Weickert proved that there is always a finite difference stencil which leads to a stable scheme. Moreover, he was able to show that three directions suffice to guarantee a non-negative discretization, since a negative discretization is equivalent to an ill-posed problem. The discretization of a 3 x 3 stencil reads as follows:

This gives the weighting coefficients for the dissipative fluxes which leads to a steering of the dissipation terms depending on the magnitude of the gradients as described it in the last section.

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3. Numerical results

As an example we use a test case given by LeVeque [Lev931 with initial d a t a

lJl for U,. for

d

m < 0.13, > 0.13.

and Ul = (2,0,0,15)T,U,. = ( 1 , 0 , 0 ,l)T in primitive variables. The solution consist of a shock running outwards followed by a rarefaction wave and a contact discontinuity. A second shock moves inwards towards the center. We use an equidistant discretization with Ax = A y = h = 0.025 and a CFL-Number = 0.4. The smoothing parameters are given with b = 0.25h and u = 0.0 which means no smoothing of the structure tensor takes place. The contrast parameter X is chosen as 0.4 max(X1;,).

Figure 1: Solution of the test case (12) a t time t=0.13

4. Concluison We have extended the characteristic filter approach by Yee, Sandham and Djomehri [YSD99] by an anisotropic directional based diffusion from image processing. Since this integration into the field of numerical conservation laws is a novel approach, the calculations presented here are preliminary results and will need further research. Overall we are quite optimistic concerning the behaviour of the suggested extension of this scheme.

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References [GMS98] Th. Grahs, A. Meister, and Th. Sonar. Image Processing for Numerical Approximations of Conservation Laws: Nonlinear anisotropic artificial dissipation. Hamburger Beitrage zur Angewandten Mathematik, Reihe F: Computational Fluid Dynamics and Data Analysis 8, 1998. (submitted to SIAM J. Sci. Cornp.). [Har78] A. Harten. The artificial compression method for computation of shocks and contact discontinuities. 111. self-adjusting hybrid schemes. Math. Comp., 32:363-389, 1978. [Hart331 A. Harten. High resolution schemes for hyperbolic conservation laws. Journal of Computational Physics, 49:357-393, 1983. [HirSO]

Ch. Hirsch. Numerical computation of internal and external flow, volume 2. J . Wiley & sons, 1990.

[Lev931 R.J. LeVeque. Simplified multi-dimensional flux limiter methods. In M.J. Baines and K.W. Morton, editors, Numerical Methods for Fluid Dynamics 4, pages 175-190. Oxford University Press, 1993. [MS99]

A. Meister and Th. Sonar. Finite Volume Schemes for compressible fluid flow. in press: Surveys of Mathematics in Industry, 1999.

[PM90]

P. Perona and J . Malik. Scale space and edge detection using anisotropic diffusion. IEEE Trans. Pattern Anal. Mach.Intel1, 12:629-639, 1990.

[Roe811 P.L. Roe. Approximate Riemann solvers, parameter vectors, and difference schemes. Journal of Computational Physics, 43:357-372, 1981. [Son971 Th. Sonar. Mehrdimensionale ENO- Verfahren. Advances in Numerical Mathematics. B.G.Teubner Stuttgart, 1997. [Wei98] J . Weickert. Anisotropic Diffusion in Image Processing. B.G. Teubner, Stuttgart, 1998. [Yee85] H. C. Yee. Construction of explicit and implicit symmetric TVD schemes and their application. J. Comput. Phys., 68:151, 1985. [YSD99] H. C. Yee, N. D. Sandham, and M. J . Djomehril. Low-dissipative high-order shock-capturing methods using characteristic-based filters. J. Comput. Phys., 150:199-238, 1999.

Nonlinear projection met hods for multi-entropies Navier-Stokes systems

Christophe BERTHON

Frkdkric COQUEL

ONERA, BP 72, 92322 Chitillon Cedex, FRANCE.

4, place jussieu,

LAN- CNRS, 75252 Paris Cedex 05, FRANCE.

ABSTRACT This paper is devoted to the numerical approximation of the compressible Navier-Stokes equations with several independent pressures. Several models derived in plasma physics or in turbulence typically enter the proposed framework. The striking novelty over the usual Navier-Stokes equations stems from the impossibility to recast equivalently the present system in full conservation form. Classical finite volume methods are shown to fail in the capture of shock layers. We propose a new method, the so-called nonlinear projection operator, for correcting the errors while preserving all the stability properties.

Key Words: Navier-Stokes equations, Non conservative products, Nonlinear projection.

1. Introduction The present work treats the numerical approximation of the solutions of the Navier-Stokes equations for a compressible fluid modelled by two independent pressures, e.g. each of the pressures comes with its own specific entropy. Despites that such models are seen t o exhibit several close relationships with the usual Navier-Stokes system, the fundamental discrepancy stays in the lack of an admissible change of variables that recasts the system in full conservation form. None of the entropy balance equations boils down t o a conservation law involving non conservative products that account for dissipative phenomena : namely the entropy dissipation rates. Such systems occur in several distinct physical settings. They arise for instance in plasma physics and they can be also recognized within the frame

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of the "two transport" equations models for turbulent compressible flows. All these models are addressed below with the assumption of a large Reynolds number. The numerical capture of the viscous shock layers is of primary importance in the present work. Since the Reynolds numbers of interest are large, these layers display the character of a shock wave in that they differ from their end states only in a small interval of rapid transition. Our purpose is to correctly capture the two end states, vr, and V R , of a given shock layer together with its speed of propagation u whitout resolving sharply the viscous layer itself. It is quite well-known that such an issue does not raise special difficulties within the standard frame of the Navier-Stokes equations, e.g. in conservation form. At the very core of this success, is the definition of v R ( q vL) stays free from the entropy dissipation rate and in this sense, its numerical capture stays also free from the rate of numerical dissipation. The situation turns out to be completely different in the setting of the extended Navier-Stokes equations. Its non conservation form makes this time the triple (u; VL,vR) to heavily depend on the precise shape of the diffusive tensor. Such a dependence stays at the basis of recent works devoted to hyperbolic systems involving non conservative products (see Lefloch [6], Dal Maso-LeFlochMurat [2], Raviart-Sainsaulieu [8]). In the setting of numerical methods, this dependence implies that the numerical viscous tensor must fit the exact last one. The negative consequences can be found in the numerical result presented below. In the present work, we propose an approach based on the analysis of the discrete dissipation rates of a given Lax entropy. It turns out that the end states require for their correct capture to prescribe explicitely the Lax entropy dissipation rate. This requirement asks the numerical methods to satisfy in turn an imposed rate of entropy dissipation. This non standard issue is precisely the main motivation of the present work. Let us underline that the results we state below extend in a straightforward way to higher space dimensions, taking advantage of the rotational invariance of the equations. 2. Mathematical model

We consider a gas with density p and velocity u, which is modelled by two independent pressure laws p and p,, associated with two constant adiabatic exponents y > 1 and y, > 1. The system of PDE1s that governs such a fluid model writes :

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where the involved temperatures respectively read T = p/p and T, = p,/p. This convective-diffusive system can be understood as an extension of the standard Navier-Stokes equations when considering an additional PDE for governing an additional pressure. Depending on the closure relations for the viscosities p , p, and the thermal conductivities K and K , , several distinct physical models enter the present framework. In this section, all these transport coefficients are assumed to be fixed positive constants for the sake of simplificity in the discussion. The smooth solutions of system (1) obey additional governing equations as we now state : L e m m a 1 Smooth solutions of (1) satisfy the following conservation law:

+ 5 + 3. Smooth

where the total energy p E is defined by p E = solutions satisfy i n addition the following balance equations

where the specific entropies are respectively given by s := log Consequently, smooth solutions of (1) obey :

:

F,

s , := log

E.

where the right hand side follows under the assumption of two constant viscosities p and p,. The three balance equations (2), (3) and (4) can be proved t o be the only non trivial additional equations for smooth solutions. As a consequence the discrepancy stays in the lack of four non trivial conservation laws. Indeed, none of the equations (3), (4) and (5) boils down t o a conservation law and ( I ) cannot be recast in full conservation form. After the works by LeFloch [6] and Raviart-Sainsaulieu [8], the non conservation form met by (1) makes the end states of shock layers to depend on the closure relations for the coefficients p, p, and K, 6,. In order to assess this issue, let us focuse on the non standard balance equation (5) where by contrast with (3) and (4) which dissipation rates are independently imposed, (5) exhibits a c o m p a r e d rate of both the entropy dissipations. The idenity (5) continues t o play a major role in a general setting since they encode a generalized jump condition which turns out to play a central role for our numerical purpose :

3 10


Theorem 2 Assume that p, p,, rc and n, denote positive constants. Then , associated with the resulting dissipative tensor necessarily the triple (a,v ~vR) obeys the jump relations :

and necessarily satisfies the two entropy inequalities :

Under an appropriate setting (see the companion paper [I]), (7) can be specified as follow :

defining a generalized jump condition where the involved averages find a unique definition (see [I] for details). Remark 3 In view of the jump relations (6) and ( S ) , one of the two entropies, either ps or ps,, must be understood as a nonlinear function of the four remaining independent variables (p,pu, pE, .).

3. Godunov methods with nonlinear projections

For the sake of simplicity , we do not address the discretization and we set K , = 0 We refer the reader to the companion paper [I] for the required discrete formulae.

K

=

3.1. L2 projections (tn 4 tn+'lp)

An equivalent system using the conservation laws for p, pu, pE and the evolution law governing ps, is considered. Notice that the evolution law for ps can be understood as an additional law sat,isfied by the smooth solutions. Choosing consitant formulae, based on a Godunov method for instance, the discrete solution satisfies, formally, the following dissipation rates of entropie :

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As a consequence of the error which occurs in ( l o ) , the expected generalized jump relation (8) is systematicaly violated. The reader is referred to [l]for a rigourous proof and to the numerical results below for an illustration of the negative consequences of such a failure.

3.2. Nonlinear projection methods (tnf

'1-

+ tn+' )

We propose to add an additional step to classical L2 projection methods, the nonlinear projection step, which purpose is t o correct the errors. In order to preserve the required conservation properties, let us define :

Setting v:+' = (p;+"-, (pu):+l'-, ( p ~ ) y + l ' -(ps,):f' , ), to enforce the validity of the generalized jump condition at the discrete level, we propose to seek for ( p ~ , ) ? + ' as a solution of :

The above nonlinear problem in the unknown (ps,):+l can be shown to admit a unique solution as soon as the approximate Riemann solver involved in the first step obeys discrete entropy inequalities for the Lax pair (ps, psu). The nonlinear projection step (12) allows to prove in addition the following stability results : Theorem 4 Under the required CFL restriction, the following discrete entropy inequalities are satisfied :

+

for all strictly decreasing and concave functions and $I. The following maxim u m principles for the specific entropies s and s, are m e t :

Both the pressures p:+l and p,"+' positive.

stay positive as soon as the density pn+' is

3 12


The proof of the above statement is detailled in [I].

3.3 Numerical results The ability of the schemes in the capture of shock layers for (1) is evaluated when testing their sensitivity in the prediction of the end states with respect to the mesh refinement. The initial data is made of two constant states prescribe by

Problems are directly motivated by the three distinct regimes that underly the flow model under consideration and that are dictated by the amplitude of the viscosity ratio p T / p . The viscosities p and /I, are assumed to be positive constants we referre to [I] for the setting of varying viscosities and non zero heat conductivities. conductivities. In all the benchmarks discussed below, the Reynolds number is set at the constant value Rey = lo5.

All the figures assess that the usual numerical strategy (see [7] or [5]concerning the details of this method) grossly fails to restore the correct end states in the three regimes. Turning considering the L2 projection method, the discrete solutions agree with the exact ones only in case A. Such a property no longer holds for problems B and C and consequently large errors occur. These two schemes furthermore suffer from a dramatic sensitivity with respect to mesh refinements for problem C in that discrete solutions do not seem to converge to a given limit function even for the finest proposed grids. By contrast and concerning benchmarks A and B, the discrete solutions stay non sensitive with respect to the mesh refinement but the "limit" function does not coincide with the expected exact solution. Turning considering the nonlinear L2 projection method, it produces approximate solutions that achieve a fairly good agreement with the exact solutions while staying almost non-sensitive with Ax in the three investigated regimes.

Innovative schemes Classical Scheme

L2 Projection Scheme

Figure 1: Problem A : p , / p

Classical Scheme

3 13

Nonlinear Projection Scheme

> 1

Bibliography C. BERTHON A N D F. COQUEL, Nonlinear projection methods for systems in non conservation form. work in preparation. G. DALMASO,P . LEFLOCHA N D F. MURAT,Definition and weak stability of a non conservative product, J. Math. Pures Appl., 74, 483-548 1995.

E. GODLEWSKI A N D P.A. RAVIART, Hyperbolic systems of conservations laws, Applied Mathematical Sciences, Vol 118, Springer 1996. T. Y. HOUA N D P . G . LEFLOCH, Why nonconservative schemes converge to wrong solutions : error analysis, Math. of Comp., Vol 62, No 206, 497-530 1994. A N D C. OLIVIER, On the numerical appproximation of B. LARROUTUROU the K-eps turbulence model for two dimensional compressible flows, I N R I A report, No 1526 1991.

P .G . LEFLOCH,Entropy weak solutions to nonlinear hyperbolic systems under non conservation form. C o m m . Part. Dzff. Equa. 13, No 6, 669-727 (1988). B. MOHAMMADI A N D 0. PIRONNEAU, Analysis of the K-Epsilon Turbulence Model, Research in Applied Mathematics, Masson Eds 1994. P . A. RAVIART A N D L. SAINSAULIEU, A nonconservative hyperbolic system modelling spray dynamics. Part 1. Solution of the Riemann problem, Math. Models Methods in App. Sci.,5, No 3, 297-333 1995.

About a Parallel Multidimensional Upwind Solver for LES

D. Caraeni S. Conway L. Fuchs Division o,f Fluid Mechanics, Lund Institute of Technology SE-221 00 Lund, Sweden

A parallel ,pow solver has been developed at Lund Institute of Technology, ,for Large Eddy Simulations ( L E S ) of turbulent compressible jlows. The numerical algorithm is based o n the Residual Distribution scheme approach. The scheme employs a n extremely compact stencil, while still having second order accuracy, which makes it well suited ,for parallelization. I n this paper we report about the numerical scheme and about the parallel algorithm implemented i n the code. First the performance concerning the parallel scalability is addressed. Finally some results ,for the L E S of the classical channel ,flow are prese,nted.

ABSTRACT

Key Words: Residual Distribution Scheme, Large Eddy Simulations, Parallel Flow-Solver.

1. Introduction The advances made in computer technology over the last years, have led to a great increase in the engineering problems which can be simulated using CFD. The computation of flows over complex geometries a t relatively high Reynolds number is becoming more common using LES, as recent reviews on the subject have shown [PI0 981. LES for engineering applications are typically extremely expensive, requiring huge resources in terms of processor power and memory, and long computational times. When using large parallel computations, LES becomes accessible for many industrial applications. Accurate numerical algorit,hms, well suited for parallelization are needed in order to make LES available for production CFD.

316


2. LES Equations

The expression for the LES equations, can be obtained by " tophat" Favrefiltering of the time dependent compressible Navier Stokes equations, written in conservative form:

The subgrid terms representing the subgrid stress tensor, 7;" = p [ u x j -

Ezj],

&Gj],the subgrid heat flux, H:" = PE [~; and the subgrid viscous work, G;gs = F [ r s i - GUi],a,ll require modelling. In the present simulations, the 7;;' was modeled by using a Sub-Grid Scale (SGS) model, as described below. All the other SGS terms presented above, are not explicitly modeled. 3. SGS Models In the Large Eddy simulation technique, the largest scales of the turbulence are resolved and the effect the smallest scales have on the resolved scales, is modeled. This is done by introducing a SGS model. In the present code, three SGS models have been implemented: the Smagorinsky model, Lilly's dynamic model [LIL 911 and the Dynamic Divergence model (DDM). The DDM model is a novel, anisotropic dynamic SGS model with independently determined coefficients in each coordinate direction. Note that in this case, it is the divergence of the SGS stress tensor, 7:i:,that is modeled [JHF 981 [SCF 981. For the dynamic models, the model parameter is calculated dynamically at each point from the instantaneous flow field. Negative values of this parameter enable back-scatter, which implies that turbulent energy can be transferred intermittently from the small scales to the larger ones. Local filtering, and artificial bounds, have to be used, in order to avoid large oscillations in the model parameter.

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4. Numerical Algorithm

Residual Distribution (RD) or Fluctuation Splitting (FS) schemes date back to the late 80's [ROE 821, when P.L. Roe developed schemes for solving scalar advection-diffusion problems on unstructured meshes. More recently, Van der Weide and H.Deconinck [PDE 971 proposed a matrix generalization of the scalar schemes which can be applied to the solution of non diagonalizable hyperbolic systems. Wood proved [WKK 981 that RD schemes give a better accuracy of the solution, for advection diffusion problems, when compared with the classical finite-volume formulations. Though most applications of RD schemes have been limited so far to steady state computations [DSB 941, we extended their application to accurately simulate time dependent flows [CAO 99][CAB 991. Our code uses a second order in time, implicit scheme of Jameson-type [JAM 911. This is a dual-time step scheme, with sub-iterations to converge the solution at every new real-time step. Multigrid iterations are performed to accelerate the convergence of the solution in these sub-iterations. The code works on an originally developed unstructured mesh, cell-vertex data structure. A hierarchical oct-t,ree organization of the data was employed with a single tetrahedron division rule. This was shown to allow significant reductions in memory usage, [SCL 97].The discretization of the convective term of the Navier-Stokes equations is based on the matrix Residual Distribution schemes approach [PDE 971. A finite element, central-Galerkin scheme has been used to discretize the viscous part, the unstationary term and the source terms, i.e. Coriolis forces, centrifugal forces, etc. The update scheme is u,"' lvk+'- u ': l v k AUi, where the correction AUi is given by:

+

where U = (p, pu3, pE) is the conservative variables vector, j = 1..3, AT is the pseudo-time step, n - is the real-time step index, T is one of the cells (tetrahedral) sharing the node i, i = 1..4 is the node index, V , is the volume of the dual cell Oi, surrounding the i node. Here U,"+lpkis the approximation of u?" at the pseudo-time step k, ,BT is the mod.l;fied matrix distribution coefficient for the convective term, is the advective-residual over the cell T.

where Fj" = ~ j " ( ~ " + lis?the ~ ) advective flux vector, and F," = F,"(U"+ l , k ) is the viscous flux vector, both computed for un+llk,nj,i are the cell-face inward normal vectors, is the contribution of the advective part

,BT@zviscid

3 18


of the Navier-Stokes equations to the nodal residual, i(F;njsi), the contribution of the viscous part of the Navier-Stokes equations to the nodal residual, is the volume average of the unstationary term, on the (T n Oi). We used an implicit second order discretization with finite differences for

[g]jzt:jt

[g]:

A modtfied second order matrix Residual Distribution scheme has been used for the convective part of the Navier Stokes equations. It employs a dynamic correction of the artificial dissipation of the scheme based on the local flow characteristics. When doing LES, the artificial dissipation of the scheme, i.e. the LW scheme [PDE 971 in our case, is not desired and it can be shut off in regions far from shock waves. Close to shock waves, the artificial dissipation of the original scheme, is reintroduced. If strong shock waves are present, a continuous blend between the second order scheme and a positive first order scheme, i.e. the Narrow scheme, is used in the shock wave region. This enabled us to have monotone shock capturing, while still using very small dissipation in the smooth flow regions. The matrix distribution coefficients, P$, are given by:

The matrix distribution coefficients are computed, for the second order scheme, LW:

and for the positive first order, Narrow scheme:

Here v,,ll is a cell-CFL number. Ki are the Jacobians of the convective aFc flux vector relative t o face i, Ki = &nj,i. The coefficients vCe[land w e n d are corrected dynamically. When doing LES, they vanish in regions far from flow field discontinuities. More details about the matrix generalization of the Residual Distribution schemes can be found in [PDE 971. 5. Parallel Algorithm Since the Fluctuation Splitting algorithm only needs to access the first order neighbors to update the solution a t a vertex, the parallelization process is

Innovative schemes

3 19

greatly simplified. If it were necessary to access the second-order neighbors, or even higher-order neighbors, then for any vertex that lies along a boundary between processors, a very elaborate and likely expensive method for communicating the appropriate update information across these inter-partition boundaries would need to be devised. Second order classical finite volume codes require access to the second-order neighbors. However, these algorithms get around the complication of having to store the communication information to reach these second order neighbors across processors by using a two-step update procedure. This requires also two communication cycles per time step [VNK 921, whereas the Fluctuation Splitting algorithm only needs one. This makes the fluctuat,ioil splitting method very attractive for parallel implementation. The Parallel Virtual Machine (PVM) defacto standard has been used to implement the parallel algorithm in our code [CAD 991. The code runs as effectively on large distributed memory machines (Cray T3E, IBM SP2...), on the SGI Origin 2000 servers or on a heterogenous collection of platforms. The code takes advantage of its unstructured grid for realizing an almost perfect load balancing by assigning a number of volumes to compute to each processor, proportional with the processors speed. The parallelization of the code has been done using the SPMD paradigm, i.e. the same code is executed in all processors. The global data structure, i.e. grid, field variables, etc. is decomposed into a number of subdomains, equal with the number of running tasks. While any algorithm for domain decomposition can be employed, so far only one dimensional domain decomposition has been used. This is done in a separate preprocessing step and does not affect the solvers flexibility. Each task runs with its own data structure, which is a disjunct part of the global data structure. The tasks are implicitly synchronized by using blocked data exchange. The algorithm implies at each real time step sub-iterations for converging the solution. In each sub-iteration, the solver: - computes the residual, i.e. the integral of the convective, diffusive, source term and unstationary part of the Navier-Stokes equation, over one tetrahedral cell, looping through all cells. Splits the residual according with the distribution scheme in fluctuations, and sends fluctuations to the nodes, where these are collected in some special fields. - communicates using PVM, with neighboring parallel processes, the fluctuations accumulated in nodes and some other useful information, for nodes laying on the domains interface. - imposes boundary conditions, - loops through all nodes, to update the nodal values using the fluctuations stored in those special fields. As it. can be seen, the algorithm needs no communication while computing and distributing the fluctuations. This step in the solver algorithm requires the longest computational time. It requires only one back and forward communication for each sub-iteration before applying the boundary conditions and updating the nodal values. The volume size of this communication is reduced, since only nodes on the domains interface exchange information.

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6. Numerical Results

LES simulations of complex turbulent flows in engineering applications, using the present code, have been presented in [CAO 991[CAB 991. Here we only address the parallel scalability of code and the accuracy of our LES computations.

6.1. Parallel scalability

The code has been run on an SGI Origin 2000 platform, on up to 8 processors, and also on a cluster of 2 WinNT PC's, using an Ethernet connection. The results, addressing the parallel speed-up are presented graphically in figure 1. It can be seen that the performance, when using the two WinNT PC cluster, has been close to 75%, due to poor communication while using the Ethernet connection. The code proved instead to have excellent scalability on the Origin platform (98%) [CAD 991.

6.2. LES of channel flow

Results for the classical turbulent channel flow are presented. The Reynolds number based on the bulk velocity and the channel height, is 5800. No-slip isothermal boundary conditions have been set at the walls, while periodicity boundary conditions have been used both in the strearnwise (x) and the spanwise (2) directions. Computations have been performed using the dynamic D D M model. The grid size is ( 2 7 4 (6) (2~613)where 6 is the channel height. The grid has (85)(51)(51) nodes and aprox. 1,240,000 tetrahedral cells. The grid is stretched in the y-direction (yi = 6(1 - cos(pi))/2, for pi = ~ (- 1j) / ( N - I), j = 1,2, ...., N. Here N is thenumber of grid points in the y-direction).A uniform grid has been used in the streamwise and the spanwise directions. The flow has been simulated for enough flow-through times (the domain length in the strearnwise direction divided by the bulk velocity) to obtain a statistically stationary turbulent channel flow. Data for statistics are accumulated over the last 15 flow-through times. Planar averages are calculated by averaging for all points on planes parallel to the walls and in time, and the results are presented as a function of wall-distance (y) only, and are compared with experiments (Kreplin, 1979) and with D N S data (Kim, 1987). The averaged non-dimensional friction velocity E, = 0.06156 (i.e. uT/Ub, where Ub is the bulk velocity) compares well with the D N S and the experimental (i.e. u, = 0.0643) result. Figures 2 to 5 show planar averaged time averages of the mean of the axial velocity and Root-Mean-Square of velocity fluctuations (RMS of Favre U",V", W"). The results are normalized using the friction velocity, u,. The results obtained using the DDM model compare well with the DNS and the experimental results. Figure 6 shows an instantaneous dis-

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321

tribution of the skin friction coefficient, Cf.

7. Future Work Future work will include extensive testing of different dynamic SGS models. The goal is to perform parallel LES for complex engineering applications.

[PI0 981

PIOMELLI,U., Large Eddy Si~nulation:present state and ,future directions

[ROE 821

ROE, P .L., Fluctuations and signals. A framework for numerical evolution problems, K.W. Morton and M.J. Baines, editors, Numerical Methods for Fluid Dynamics, Academic Press, 1982.

[PDE 971

PAILLERE,H , et al., Upwind residual distribution methods for compressible flows : An alternative for finite volume and finite element methods, VKI 28th CFD Lecture Series, March 1997.

[DSB 941

DECONINCK,H. et al., High resolution shock capturing cell vertex advection schemes on unstructured grids, VKI Lecture Series, March 21-25 , 1994.

[WKK 981

WOOD, WILLIAMA. et al., Diffusion characteristics of upwind schemes on un,ctructured Triangulations, AIAA 98-2443, Albuquerque, NM 1998.

[JAM 911

JAMESON,A., Time dependent computations using multigrid, with applications to unsteady flows past airfoils and wings, AIAA paper, 91-1596, 1991.

[CAB 991

CARAENI,D. et al., LES of spray in compressible flows on Unstructured Grids, AIAA 99-3762, Norfolk, 1999.

[CAO 991

CARAENI,D. et al., Large Eddy Simulation of the Flow in a Bladed Diffuser, TSFP, Santa Barbara, 1999

[CAD 991

CARAENI,D. et al., Parallel NAS3D: An efficient algorithm for enggineering spray simulations using LES, pres. C1-C, International Parallel CFD'99, 1999, Williamsburg.

[SCL 971

MITRAN,S. et al., Large Eddy Simulation of rotor stator interaction in centrifugal impeller, JPC, Seattle, July 1997.

[JHF 981

HELD,J . et al., Large Eddy Simulation of separated transonic flows around a wing section, AIAA 98-0405, Reno 1998.

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[SCF 981

CONWAY, S. et al., Investigation of the ,flow across a swirl generator using LES, AIAA 98-0921, Reno 1998.

[LIL 911

LILLY,D.K., A proposed modzfication of the Germano subgridscale closure method, Phys. Fluids A, 4:633-635, 1991.

[VNK 921

VENKATAKRISHNAN V. et al., A MIMD implementation o,f a parallel Euler solver ,for unstructured grids, The Journal of Supercomputing, Vol. 6 , 1992.

Fig. 1 Parallel Speedup

. ... .... DDM

Fig 2. Planar Time Averages of U

Fig 3. Planar averages of Unrms

1 2

---

-

3-

B 06

2

D

-

0 6

Exp

0 4

. .. . ... . D

0 2

M

Exp DNS

-DDM

0

0

0.25

05 ylh

0 75

-

1

Fig 4. Planar averages of V1'rms

I

I

0

025

0 5

075

1

yln

I

Fig 5. Planar averages of W"rms

-X

Fig. 6 Skin friction distribution

A higher-order-accurate upwind method for 2D compressible flows on cell-vertex unstructured grids

L. A. Catalano Istituto di Macchine ed Energetica Politecnico di Bari Via Re David 200, 70125 Bari, ITALY E-mail: [email protected]

~~p

-

-

-

~

p

p

~

p

p

- ---

A finite-volume method for the solution of two-dimensional inviscid compressible flows on cell-vertex unstructured grids is presented. The method is based on a novel bi-linear reconstruction of the unknowns and on a standard flux-diflerence-splitting scheme. Moreover, a new approach is proposed to achieve the same higher-order accuracy also near solid walls. The method is validated by computing the inviscid flow i n a two-dimensional cascade i n subsonic and transonic conditions.

ABSTRACT

Key Words: higher-order reconstruction, upwind, unstructured.

1. Introduction

In the last decade, a great effort has been devoted by many CFD researchers to the development of higher-order-accurate upwind solvers on unstructured grids. Different approaches have been proposed, including both finite-element and finite-volume discretizations [BAR 91, DEC 92, CAT 97, HAL 97, SEL 961, the major difficulty being the discretization of the inviscid terms in the conservation equations. Concerning the finite volume discretization, most of the upwind schemes proposed to date are based on a gradient-based reconstruction of the flow variables onto the two sides of the surface which defines the finite volume built around each node. Then, an approximate Riemann solver is applied a t each interface to select the upwind contributions.

324


How to perform a higher-order reconstruction on an unstructured triangular grid is not trivial: if the unknowns are located in the center of each triangle, the finite volume is the cell itself, and the higher-order reconstruction onto its sides becomes a very cumbersome task, because of the difficulty of defining and computing the flow gradients. For such a reason, many researchers have redirect their efforts to cell-vertex discretizations: a dual mesh is first constructed by tracing the medians of each cell, so as to build a finite volume around each node [BAR 93, SEL 96, HAL 971. Clearly, the interfaces are now located at the midpoint of each side. To knowledge of the author, all of these methods base the reconstruction on a nodal definition of the flow variable gradient. Different approaches have been proposed to compute its value from the gradients in the surrounding cells, but all of them appear complex and time-consuming, see e.g. [BAR 93, HAL 971. In this paper, an alternative, much simpler approach is proposed, which also includes a new higher-order near-wall discretization. 2. Numerical method 2.1. Governing equations

The governing equations for two-dimensional, compressible, inviscid flows, are written in integral form as:

as,

U = (p, pu, pv, peO)Tis In eq. 1, n is the inward normal of the contour of S , the vector of the conservative variables and 3.n = [(pv,), (puvn+p72,), (pvv,+ pn,), (phOvn)lTis the flux entering through the unit length of dS. As usual, p is the density, p is the pressure, e0 is the total internal energy and hO is the total enthalpy. Moreover, v will denote the velocity vector, with normal and tangential components v, = v . n and v, = v . s, respectively, and with Cartesian components u and v. The system of governing equations is closed by assuming perfect gas. 2.2. Space discretization

The domain is discretized by means of an unstructured mesh composed of triangles with unknows located at each cell-vertex. In particular, the primitive variable vector Q = ( p , ~ , v , is~ chosen )~ as unknown. A finite volume is constructed around each internal node by connecting the barycenters of two neighbouring triangles, see the node i in fig. 1. An upwind discretization of the RHS of eq. 1 is then obtained as follows: a left state and a right state are reconstructed at the interface (ij) defined on each side connecting the node i

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%.

325

*..----.......'i----------i' ,

Figure 1: Construction of finite volumes for internal and boundary nodes. Determination of the cell C j i . Construction of the auxiliary cells. with each sorrounding node j. Then, a standard Riemann solver is used to select the upwind contributions to the flux. The left and the right states can be accurately reconstructed by employing a bi-linear variation of Q:

In eq. 2, lji and lij are the two opposite vectors pointing from the two nodes to the intersection of the interface with the side, see fig. 1. Concerning the definition of the gradients, first consider the interface (i + 112) of a uniform one-dimensional grid; in such a case, Q k I l 2 is linearly reconstructed as:

It must be remarked that the reconstruction is based on the gradient of Q in the left-neighbouring cell, rather than on the gradient defined in the node i . Similarly, in two dimensions, the gradient ( V Q ) j i (similar arguments hold for ( V Q ) i j )must be defined in one of the cells sharing the node j , rather than in the node itself, as it suggested in [BAR 93, HAL 971. The extension of the previous arguments to two dimensions suggests to define ( V Q ) j i as the gradient in the cell Cji which contains the prolongation of the side (ji), plotted as a dot-dashed line in fig. 1. Clearly, the choice of a cell-vertex triangular grid allows to define a bi-linear variation of Q in each cell, thus defining the cell = (VQ)c,,uniquely: gradient (VQ)ji

326


In eq. 4, Q k is the primitive variable vector in the node k . Moreover, the opposite side k has length s k and inward normal n k . Standard one-dimensional limiters can also be applied to the gradient, but have not been introduced yet. The flux-difference-splitting of Roe [ROE 861 is then used to solve the Riemann problem defined at each interface. The flux is computed as

k=l

In eq. 5, ak, k = 1,...,4, are the intensities of the entropy, of the shear and of the two acoustic waves, and X k , k = 1,...,4, are the corresponding propagation velocities:

Finally, e k , k = 1 , ..., 4 are the eigenvectors which project each wave contribution onto the conservative variable vector:

In eqq. 6 and 7, 6( ) = (

)R -

(

)L,

and - denotes the Roe averages:

2.2. Boundary conditions When a side lies on a boundary, the construction of the finite volumes is completed by tracing one third of the corresponding median and by adding half of the side t o the finite-volume contours of its two nodes: as an example, fig. 1 shows the resulting finite volume associa.ted to the boundary node b. Clearly,

Innovative schemes

327

the flux through each boundary side is computed directly by means of the nodal values; moreover, solid boundaries need only to add the corresponding pressure forces to the momentum equation. Characteristic boundary conditions are then applied to the nodal residuals to ensure that physical boundary conditions are satisfied. The lack of further cells beyond the boundaries make impossible to perform a higher-order reconstruction onto the near-boundary interfaces. The reduced

Figure 2: Computational mesh. accuracy is negligible for far-field boundaries, but a significant amount of numerical entropy would be generated near solid walls. A simple procedure to overcome this problem is here proposed: a row of auxiliary cells is created by adding an isosceles triangle beyond each solid face and by connecting the corresponding auxiliary nodes, see the dashed cells in fig. 1. The states in the auxiliary nodes are updated by imposing the following conditions of Isentropic Simple Radial Equilibrium a t the mid-point of the solid face:

R, being the radius of curvature a t wall. 2.3. Time integration The state in each node is updated by means of a two-stage Runge-Kutta explicit scheme with non-optimal coefficients 0.4 and 1 and CFL number 0.35.

328


3. Results

The rnethod previously described has been applied to the computation of the inviscid flow in a cascade of VKI LS-59 gas turbine rotor blades, for two different values of the outlet pressure. The blade profile has been modified at the trailing edge by adding an artificial wedge (having no load) in order to simulate the presence of the recirculation zone, which cannot be captured by the present inviscid formulation. Fig. 2 shows the grid used in both the cases analyzed, obtained by slightly modifying a structured grid with 129 x 17 nodes (97 nodes are located on the blade surface). Clearly, the resulting quality of the grid is rather poor. The Mach number contours obtained for outlet Mi,

Figure 3: Mach number contours for outlet Mi, = 0.81 ( A M = 0.05). (isentropic Mach number) equal to 0.81 are shown in fig. 3. A good agreement

0

exp. (Kiock)

- computed

Figure 4: Experimental and computed distributions of Mi, on the blade for outlet Mi, = 0.81. between the computed distribution of Mi, on the blade and the experimental one [KIO 861 is visible in fig. 4.

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Figure 5: Mach number contours for outlet Mi,= 1

329

(AM = 0.05).

The transonic flow with outlet Mi,equal to 1 has been then considered: the Mach number contours in fig. 5 show the formation of two shocks on the suction side of the blade, which are both sharply captured. Small oscillations are present and can be eliminated by applying a standard limiter. The surface Mach number distribution shown in fig. 6 is again in good agreement with the experimental data of [KIO 861, except for the discrepancy on the suction side of the blade, due to the presence of a separation bubble caused by the shockboundary layer interaction. Such a phenomenon is obviously missed by the present inviscid analysis.

Figure 6: Experimental and computed distributions of Mi,on the blade for outlet Mi,= 1.

3. Conclusions

A finite-volume method for the solution of two-dimensional inviscid com-

330


pressible flows on cell-vertex unstructured grids has been presented. In particular, a novel approach to the higher-order reconstruction of the unknowns has been proposed, and extended to the near-wall regions by means of a row of auxiliary cells. The method has been validated by computing the inviscid flow in a two-dimensional turbine cascade in subsonic and transonic conditions. The extension to the discretization of the viscous-flow conservation equations can be based on state-of-the-art methodologies. 4. Bibliography [BAR 911

BARTHT. J., Aspects of unstructured grids and finite-volume solvers for the Euler and Navier-Stokes equations, Lecture Series 1991-06, Von Karman Institute, 1990.

[BAR 931

BARTHT. J . , A 3-D least-squares upwind Euler solver for unstructured meshes, Lecture Notes in Physics, 414, Springer Verlag, pp. 90-94, 1993.

[CAT 971

CATALANO L. A. et al., Genuinely multidimensional upwind methods for accurate and efficient solutions of compressible flows, Euler and Navier-Stokes solvers using multidimensional upwind schemes and multigrid acceleration, Notes on Numerical Fluid Mechanics, 57 Vieweg, Braunschweig, Germany,, 1997.

[DEC 921

DECONINCK H. et al., Multidimensional upwind methods for unstructured grids, Agard R-787, May, 1992.

[HAL 971

HALLOL. et al., An implicit mixed finite-volume-finiteelement method for solving 3D turbulent compressible flows, International Journal for Numerical Methods in Fluids, 25, pp. 1241-1261, 1997.

[KIO 861

KIOCKR. et al., The transonic flow through a plane turbine cascade as measured in four European wind tunnels, Transactions of the ASME, Journal of Engineering for gas turbines and power, 108, No. 2, pp. 277-284, 1986.

[ROE 861

ROE P. L., Characteristic based schemes for the Euler equations, Ann. Rev. Fluid Mech., 18, pp. 337-365, 1986.

[SEL 961

SELMINV., FORMAGGIA L., Unified construction of finite element and finite volume discretizations for compressible flows, International Journal for Numerical Methods in Engineering, 39, pp. 1-32, 1996.

A New Upwind Least Squares Finite Difference Scheme (LSFD-U) for ~ u l e r ~ ~ u a t i of o nGas s Dynamics N. Balakrishnan

Praveen. C

Assistant Professor G r a d ~ ~ nSt~rdent te CFD Centre, Department of Aerospace Engineering Indian Institute of Science, Bangalore - 560 012 ABSTRACT A new upwind Least Squares Finite Difference Scheme(LSFD-U) has been developed. The fundamental principle underlying this method is the representation of the derivatives of the fluxes appearing in the conservation laws, using a Generalised Finite Difference strategy based on the method of least squares. This method can operate upon any kind of grid and requires only local connectivity information pertaining to a cloud of grid points around any given node. The results obtained are very encouraging and the use of LSFD-U in the computation of flows past complex configurations is extremely promising. Key Words: Finite difference, Finite volume, Upwind schemes, Hyperbolic equations, Method of Least Squares. 1. Introduction One of the remarkable progresses made in the area of CFD, in recent years is the development of Grid Free Method [I-41 for numerically solving the conservation laws encountered in fluid dynamics. The fundamental principle underlying this method is the representation of the derivatives of the fluxes appearing in the conservation laws using a generalised finite difference strategy based on the method of least squares. This method which can operate upon any kind of grid (structured. unstructured or cartesian) requires only local connectivity information pertaining to a cloud of grid points around any given node. The utility of this method in computing flow past complex configurations is extremely promising. The present work draws its inspiration from the fact that this method has been applied only in the framework of flux vector splitting schemes and not in the framework of flux difference splitting schemes. Here we have attempted to extend the applicability of this Grid Free Method to the framework of flux difference splitting schemes and in the process arrived at an entirely new methodology equally applicable to flux vector splitting schemes. In section 2, we present briefly the details regarding the Least Squares Kinetic Upwind Method (LSKUM) [I-41. In section 3, the new least squares scheme is presented with a brief review of the flux difference splitting schemes. In section 4, we present the difficulties in extending the methodology to 2D and 3D flows, and also present two variations of the scheme, which would circumvent such problems. In section 5, we present the results and discussions. Concluding remarks are made in section 6.

2. Least Squares Kinetic Upwind Method (LSKUM) The kinetic schemes for solving Euler equations of gas dynamics are obtained by exploitin,o the fact that these equations are moments of the Boltzmann equation of kinetic theory for gases. Consider the ID Boltzmann equation : f,+vf,=O (1) where f is the velocity distribution function which is a iLlaxwellian and v is the molecular velocity. The fundamental principle underlying LSKUM is that the discrete approximation to f, appearing in the Boltzmann equation is obtained using a least squares approximation given

332


I, =-

( ~ f AX) ,

(2)

11~~112 where Af,=f,-f,; Ax,=x,-x,, subscript '0' denotes the node under consideration and 'j' its neighbour. The moment of the discretized Boltzmann equation will lead to an upwind scheme for the Euler equation, if the stencil of the grid points to be used in equation (2) is chosen, taking into account the direction of signal propagation. In other words, discrete approximation to f, at any given point is obtained by using the grid points on its left if v>O and vice versa. An interested reader is referred to the papers cited above for a number of interesting developments in LSKUM, such as the use of weighted least squares in equation (2). This idea when extended to the flux vector split framework of Euler equations, given by, U , + F,'+ F,=O (3) where U is the vector of conserved variables and F is the flux vector, the discrete approximation to F,'at any given point will involve grid points to its left, and F; will involve grid points to its right.

3. Upwind Least Squares Finite Difference Method (LSFD-U) Inspired by the fact that the discrete least squares approximation to the derivative F, involves the flux difference term AF, it was thought that it would be appropriate to make use of flux difference splitting in the least squares framework. Before we discuss the details of the present least squares algorithm, we briefly discuss the flux difference splitting scheme as applied to finite volume framework. \

\

\\

I I

\

I I

j-1

m

w

I I 1

Fig 1.

I

//'

I

/

I

0.'

'+,I ../=-='

I

Left State

I

\ \ \ \ \

\

I

i

a

I

I

\\

'\

a

I'

\ \

\

*'

,=

a

j+ 1

Right State

Typical 1D Finite Volume Computational Domain

Fig 1.depicts a typical I D finite volume computational domain. In the flux difference splitting scheme[5], the total flux difference AF = FR- FL is split into a positive part (AF)' corresponding to the right running waves and a negative part (AF)' corresponding to the left running waves, based on a suitable linearization procedure, in such a way that the interfacial flux F, is given by Fj = \

3

+ (AF)

= FR - (AF)+-

1.

(4)

In a finite volume framework, an interfacial flux obtained as an average of the above two expressions is made use of in the state update formula. Now we describe the present methodology. Equation 4 given above clearly suggests that the flux difference between any fictitious interface perpendicularly intersecting the line connecting the two grid points and the points themselves, can be obtained using a suitable linearization procedure. At the heart of the present methodology is the use of such flux differences based on upwinding principle in the discrete least squares approximation to F, appearing in the Euler equations. This leads to an upwind scheme based on the least squares principle. Also, it is not necessary that the present methodology should be used only in conjunction with Fiux Difference Splitting Schemes. The very fact that an upwind estimate of

Innovative schemes

333

the flux at the fictitious interface is all that is required for the determination of flux differences involved in the present method, suggests the use of this method in conjunction with all upwind flux formula. As can be easily seen, the present methodology makes use of a global stencil of grid points in contrast to the methodology described in section 2, which requires an upwind stencil of grid points. Also, we make use of an upwind estimate of the interfacial flux for determining AF's appearing in the least squares formula, in contrast to the explicit use of the nodal values of the fluxes in the earlier framework.

4. LSFD-U in 2D Here we present some of the interesting problems we face when extending the algorithm just described for solving 2D flows. Fig.2 gives a typical 2D stencil of grid points. Let I, represent the fictitious interface drawn across the line connecting the point under consideration

'0'.

and its j'h neighbour, and t?

represent the unit vector along

'oj'.

Representing the flux along oj by

we have

where F and G are fluxes in x and ;directions respectively. Now our job reduces to recovering the information regarding the gradients of the 2D fluxes namely VF, and Vcofrom the many I D flux difference terms given in equation 6. The derivatives F, and G, thus recovered would eventually be used in the state update formula. Equation 6 represents an over determined system of equations and it appears that the straight forward way to obtain the gradients of the fluxes is to minimise E? with respect to the derivatives of the fluxes, where, J

J

=

J

- (

A

n

- G o 64j I n r j

(7)

Unfortunately, simple algebra would demonstrate that such a procedure leads to a singular system. To circumvent this problem we suggest the following two methodologies. 4.1 Metirod I Method 1 draws its inspiration from the work of Ghosh and Deshpande (21. Here we effect locally a co-ordinate transformation (x.y)-t((,q), in such a way that one of the axes (5) coincides with the streamwise direction. It is a well known fact that the fluxes normal to the streamwise co-ordinate direction involve only pressure terms and a global stencil of grid points can be used for approximating the derivatives of such fluxes without loss in stability. If

-F and

represent the fluxes along the new co-ordinate directions

derivatives

5

5

and

I;and q respectively, the

can be obtained using the least squares procedure described in 131,

making use of a global stencil of grid points. Thus the streamwise rotation of the co-ordinates leaves us with a non singular system involving and iq. A simple least squares procedure

6

described in the previous section can be adopted for the estimation of derivatives

and G,,thus

@

and

iq. The

determined are substituted in the state update formula. It should

334


be remarked that in this method, the second and third components of the fluxes represent 6 and q momentum conservation.

and

4.2 Method 2

In this method we locally rotate the co-ordinate system from (x,y) j ( < , q ) , in such a way that

Brl +a5= 0.

where

-

F

and

6

represent the fluxes along the new co-

ordinate directions 6 and q respectively. Note that the second and third components of the flux

-

-

vectors Fand G still represent the x and y momentum conservation. It can easily be demonstrated that a co-ordinate system rotated at an angle P, given by,

2

would satisfy the condition that

+

= 0 . The gradients of the fluxes used in the

estimation of p are obtained using the least squares procedure [3] making use of a global stencil of grid points. This results in an overdetermined system of equations with two unknowns

9 and G- v , which can be solved using least squares procedure. The derivatives

thus determined are used in the state update formula. 5. Results and Discussions The new least squares upwind finite difference method (LSFD-U) is validated using standard ID and 2D test problems. In the computations high resolution is obtained based on a linear reconstmction procedure [6]. Non physical oscillations in the solution are suppressed using Venkatakrishnan limiter [7]. In all the computations presented in this work the fictitious interface is always placed at the mid point of the line segment under consideration. Figure 3 gives the results obtained for the I D shock tube problem of Sod [8]. The results are obtained on a non-uniform grid generated using cosine spaclng for grid points. One hundred grid points have been used in the computation. Roe [ 5 ] flux has been used in these calculations. The grids made use of in the 2D computations are presented in Fisure 4. The grid details are given in Table 1. All the 2D computations have been made with KFVS [9] flux formula.

Configuration Cylinder NACA 00 12

Table 1 No. of nodes 4317 4733

No. of nodes on the wall 160 160

The pressure contours obtained for a low subsonic flow past a cylinder(in the incompressible limit; M,=0.1) are presented in Figures 5 and 6. The wall pressure data are compared with the exact potential flow solution in Figure 7. The Mach contours obtained for subsonic(M,=0.63, angle of attack=lU) and transonic (M,=0.85, angle of attack=lU)flows past NACA0012 airfoil are presented in Figures 8,9,11 and 12. The Cp values are compared with those obtained using the cell vertex finite volume scheme in Figures 10 and 13. The first order results obtained using Method 2 are identical to those obtained using Method 1, and therefore are not presented here. The CL and CD values obtained using LSFD-U are compared with GAMM[IO] results in Table 2.

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Table 2 R.I,=0.63, AOA=Z0

Method I II GAMM

C~

CD

0.3387 0.3343 0.3335

0.00 14 0.0006 0.0000

335

M,=0.85, AOA=IO

cL

0.3772 0.3806 0.3790

CD 0.0550 0.0545 0.0576

From the results it is evident that the new LSFD-U framework is capable of capturing all features expected out of inviscid compressible flows. Also, it should b e remarked that the linear reconstruction procedure developed for the finite volume schemes, when used in conjunction with the present LSFD-U framework, can produce excellent improvement in solution resolution.

6. Conclusions A new upwind least squares finite difference method has been developed. T h e new scheme by the virtue of using a least squares framework can b e considered as a Grid Free Method. It has an added advantage of making use of a global stencil. T h e way the interracial fluxes are calculated in the new scheme resembles that of finite volume method and therefore all t h e developments that have taken place in finite volume method, like the method of reconstruction [6] can b e directly adopted. T h e use of LSFD-U in the computation of flows past complex configurations is extremely promising.

Acknowledgements The authors like to thank Mr J. C. Mathur. NAL. Bangalore. for providing the unstructured mesh generator. The authors like to thank Mr. Krishnakumar, under graduate student, IIT. Kharagpur. for his involvement in the initial phases of development of the code used in the computations, during his stay at IISc, Bangalore. as a Summer trainee. The authors also like to express their gratitude to Mr. Hiwish. R, project assistant. CFD Centre, IISc, for his invaluable help in preparing the manuscript.

References S.M. Deshpande, A.K. Ghosh and J.C Mandal. "Least Squares Weak Upwind Method for Euler Equations", 89 FM 4. Fluid Mechanics Report. Department of Aerospace Engineering. Indian Institute of Science, Bangalore. A.K. Ghosh, "Robust Least Squares Kinetic Upwind Method for Inviscid Compressible Flows", June 1996, PhD thesis. S.M. Deshpande, P.S. Kulkmi, and A.K. Ghosh, "New Developments in Kinetic Schemes", Computers Iblath. Applic., Vol 35, No 112, pp. 75-93, 1998 K. Anandhanarayanan, D.B. Dhokrikar, V. Ramesh and S.M. Deshpande. "A Grid Free Method for 2D Euler Computations using Least Squares Kinetic Upwind Method", Proceedings of the third Asian Computational Fluid Dynamics Conference, pp. 390, vol 2, December 7' - 1 lLh1998, Bangalore, India. Roe, Philip. L. "Approximate Rrimann Solvers Parameter Vectors and Difference Schemes", Journal of Computational Physics, Vol. 43, pp. 357-372, 1981 T.J.Banh " Higher Order Solution of the Euler Equations on Unstructured Grids using Quadratic Reconstruction", AUA-90-0013, 1990 V. Venkatakrishnan. "Convergence of Steady State Solutions of Euler Equations on Unstructured Grids with Limiters", JCP, vol 118. pp. 120, 1995. G.A. Sod, "A Survey of Several Finite Difference Methods for system of Nonlinear Hyperbolic Conservation Laws", JCP, vol 27, pp. 1, 1978. J.C. Mandal and S.M. Deshpande, " Kinetic Flux Vector Splitting for Euler Equations", Computers and Fluids. vol 23. No 2. pp 447-478. GAMM Workshop on Numerical solutions of Compressible Euler Flows, June 1986.

336


Fig 2. Typical 2D grid distribution for LSFD-U

Fig 3. LSFD-U applied to shock tube problem

Fig 4. Unstructured grid used for 2D computations Linear Reconstruction

Fig 5 Pressure contours obtained using Method 1 for M,=O.l

Linear Reconstruction

Fig 6 Pressure contours obtained using Method 2 for kI,=O.l

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First Order

337

Linear R e s a n s t i u s t . a ~

Fig 8 Mach contours using Method 1; M,=0.63 and AOA=2' L~naar Reconstructban

:Reco~ImitC1 - MI M2.I Ore.,

x/c

Fig 9 Mach contours using Method 2 for M,=0.63 and AOA=2"

Fig 10 Pressure distribution for M20.63 and AOA=2"

338

Finite volumes for complex applications F i r i t Ordar

Fig 11 Mach contours using Method I; hI,=0.85 and AOA=1°

Linear Rsconslruc:~cn

Fig 12 Mach contours using Method 2 for M,=0.85 and AOA=1°

Fig 13 Pressure distribution for M,=0.85 and AOA=1°

A finite-volume algorithm for all speed flows F. Moukalled and M. Darwish American University of Beirut, Faculty of Engineering & Architecture, Mechanical Engineering Department, P. 0.Box 11-0236, Beirut, Lebanon. ABSTRACT. A new collocatedfinite volume-based solution procedure for predicting viscous compressible and incompressible flows is presented. The technique is equally applicable in the subsonic, transonic, and supersonic regimes. Pressure is selected as a dependent variable in preference to dens;@ because changes in pressure are significant at all speeds as opposed to variations in density which become vevy small at low Mach numbers. The newlv developed algorithm has two new features; (i) the use of the Normalized Chriable and Space Formulation methodologv to bound the convective flzixes; and (ii) the use of a HighResolz~tionscheme in calculating interface densi@ values to enhance the shock capturing proper@ ofthe algorithm. Keywords; Pressure-based method, All speedflows, High-Resolution algorithm.

1. Introduction

In Computational Fluid Dynamics (CFD) a great research effort has been devoted to the development of accurate and efficient numerical algorithms suitable for solving flows in the various Reynolds and Mach number regimes. The type of convection scheme to be used in a given application depends on the value of Reynolds number. On the other hand, the Mach number value dictates the type of algorithm to be utilized in the solution procedure. These algorithms can be classified into two groups: density-based methods and pressure-based methods, with the former used for high Mach number flows, and the latter for low Mach number flows. The ultimate goal however, is to develop a unified algorithm capable of solving flow problems in the various Reynolds and Mach number regimes. To understand the difficulty associated with the design of such an algorithm, it is important to understand the role of pressure in cotnpressible flow [KAR 861. In the low Mach number limit where density becomes constant, the role of pressure is to act on velocity through continuity so that conservation of mass is satisfied. Obviously, for low speed flows, the pressure gradient needed to drive the velocities through momentum conservation is of such magnitude that the density is not affected significantly and the flow can be considered nearly incompressible. Hence, density and pressure are very weakly related. As a result, the continuity equation is decoupled from the momentum equations and can no longer be considered as the equation for density. Rather, it acts as a constraint on the velocity field. Thus, for a sequential solution of the equations, it is necessary to devise a mechanism to couple the continuity and momentum equations through the pressure field. In the


hypersonic limit where variations in velocity become relatively small as compared to the velocity itself, the changes in pressure do significantly affect density. In this limit, the pressure can be viewed to act on density alone through the equation of state so that mass conservation is satisfied [KAR 861 and the continuity equation can be viewed as the equation for density. The above discussion reveals that for any numerical method to be capable of predicting both incompressible and compressible fluid flows the pressure should always be allowed to play its dual role and to act on both velocity and density to satisfy continuity. Several researchers [KAR 86, RHI 86, MAR 94, DEM 93, LIE 931 have worked on extending the range of pressure-based methods to high Mach numbers. In most of the published work the first order upwind scheme is used to interpolate for density, exception being in the work presented in [DEM 931 where a central difference scheme blended with the upwind scheme is used. The bleeding relies on a factor varying between 0 and 1, which is problem dependent and has to be adjusted to eliminate oscillation or to promote convergence. In the work presented in [LIE 931 the retarded density concept is utilized in calculating the density at the control volume faces. This concept is based on factors that are also problem dependent and requires the addition of some artificial dissipation to stabilize the algorithm (second-order terms were introduced), which complicate its use. To this end, the objective of this paper is to present a newly developed collocated pressure-based solution procedure that is equally valid at all Reynolds and Mach number values. The algorithm will have two new features. The first one is the use of the Normalized Variable Formulation (NVF) [LEO 871 andlor the Normalized Variable and Space Formulation (NVSF) [DAR 941 methodology in the discretization of the convective terms. The second one, is the use of HighResolution (HR) schemes in the interpolation of density in the source of the pressure correction equation and the convective fluxes in order to enhance the shock capturing capability of the method. The increase in accuracy with the use of HR schemes for density is demonstrated by comparing predictions, for the flow over a bump, obtained using the third-order SMART scheme for all variables except density (for which the Upwind scheme is used) against another set of results obtained using the SMART scheme for all variables including density. 2. Finite volume discretization of the transport equations

The conservation equations governing two-dimensional compressible flow problems may be expressed in the following general form:

at

+ V -(p$)= 0 .(r'v$)+Q4

111

where $ is any dependent variable, v is the velocity vector, and p, I?, and Q' are the density (=P/RT), diffusivity, and source terms, respectively. Integrating the above equation over a control volume (Fig. 1) and applying the divergence theorem, the following discretized equation is obtained: datp p $ k ~ + ( ~ e + +J,, ~ w+ J ~ ) = Q @ V

[21

Innovative schemes where J f represents the total flux of 4 across face 'f and is given by J~ = ( P V ~ - T@v() .sf

341

PI

Each of the surface fluxes Jf contains a convective contribution, J: , and a diffusive contribution, J: , hence:

J , = J;

+ J;

where

J;- bv+),.sf J:(-r4v4).s,

[51 The diffusive flux at the control volume face 'f is discretized using a linear symmetric interpolation profile so as to write the gradient as a function of the neighboring grid points. The convective flux across face f can be written as: JF = cnf [61 where Cf is the convective flux coefficient at cell face 'f. As can be seen from [6] the accuracy of the control volume solution for the convective scalar flux depends on the proper estimation of the face value $f as a function of the neighboring $ nodes values. Using some assumed interpolation profile, $f can be explicitly formulated in terms of its node values by a functional relationship of the form: ef = f(e*) [71 where $,b denotes the neighboring node I$ values. After substituting [7] into [6] for each cell face and using the resulting equation along with the discretized form of the diffusive flux, [2] is transformed after some algebraic manipulations into the following discretized equation: =

NWP)

where the coefficients a$ and aRBdepend on the selected scheme and b$ is the source term of the discretized equation. \

Figure 1. Control volume.

3. The NVSF methodology for constructing HR schemes As mentioned earlier, the discretization of the convection flux is not straightforward and requires additional attention. Since the intention is to develop a high-resolution algorithm, the highly diffusive first order UPWIND scheme [PAT 811 is excluded. As such, a high order interpolation profile is sought. The difficulties associated with the use of such profiles stem from the conflicting requirements of accuracy, stability, and boundedness. Solutions predicted with high order profiles tend to provoke oscillations in the solution. To suppress these

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oscillations, the composite flux limiter method [LEO 881 is adopted here. The formulation of high-resolution flux limiter schemes on uniform grid has recently been generalized in [LEO 881 through the Normalized Variable Formulation (NVF) methodology and on non-uniform grid in [DAR 941 through the Normalized Variable and Space Formulation (NVSF) methodology. To the authors' knowledge, the NVSF formulation has never been used to bound the convection flux in compressible flows. It is an objective of this work to extend the applicability of this technique to compressible flows. For more details the reader is referred to [DAR 941.

4. High resolution algorithm The need for a solution algorithm arises in the simulation of flow problems because a scalar equation does not exist for pressure. Hence, if a segregated approach is to be adopted, coupling between the u, v, p, and P primitive variables in the continuity and momentum equations will be required. The segregated algorithm adopted in this work is the SIMPLE algorithm [PAT 811, which involves a predictor and a corrector step. In the predictor step, the velocity field is calculated based on a guessed or estimated pressure field. In the corrector step, a pressure (or a pressure-correction) equation is derived and solved. Then, the variation in the pressure field is accounted for within the momentum equations by corrections to the velocity and density fields. Thus, the velocity, density, and pressure fields are driven, iteratively, to better satisfying the momentum and continuity equations simultaneously and convergence is achieved by repeatedly applying the above-described procedure. The key step in deriving the pressure-correction equation is to notice that in the predictor stage a guessed or estimated pressure field from the previous iteration, denoted by P("), is substituted into the momentum equations, the resulting velocity field, denoted by v * , which now satisfies the momentum equations, will, in general, not satisfy the continuity equation. Thus, a correction is needed in order to obtain a velocity and pressure fields that satisfy both equations. Denoting the pressure, velocity, and density corrections by P', v'(u', v'), and p', respectively, the corrected fields are given by: P = P ( " ) + P ' , v = v * + v f , and p = p ( n ) + p ' [91 Combining momentum and continuity and substituting P, v, and p using [9], the final form of the pressure-correction equation is: P' P' P' P' P' P' apPi,=aEPi+awPb+aNPh+aSPi+bp where

From [11] it is clear that the starred continuity equation appears as a source term in the pressure correction equation. Moreover, in a pressure-based algorithm, the pressure-correction equation is the most important equation that gives the pressure, upon which all other variables are dependent. Therefore, the solution accuracy depends on the proper estimation of pressure from this equation. Definitely, the

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more accurate the interpolated starred density ( p * ) values at the control volume faces are, the more accurate the predicted pressure values will be. The use of a central difference scheme for the interpolation of p* leads to instability at Mach numbers near or above 1 [KAR 86, DEM 931. On the other hand the use of a first order upwind scheme lead to excess diffusion [KAR 861. The obvious solution would be to interpolate for values of p* at the control volume faces the same way interpolation for other dependent variables is carried. That is to employ the bounded HR family of schemes for which no problem-dependent factors are required. Adopting this strategy, the discretized form of the starred continuity equation becomes:

The same procedure is also adopted for calculating the density when computing the mass flow rate at a control volume face in the general conservation equation.

5. Results and discussion The validity of the above described solution procedure is demonstrated in this section by presenting solutions to the inviscid flow over a bump. The physical situation consists of a channel of width equal to the length of the circular arc bump and of total length equal to three lengths of the bump. Results are presented for three different types of flow (subsonic, transonic, and supersonic). For subsonic and transonic calculations, the thickness-to-chord ratio is 10% and for supersonic flow calculations it is 4%. In all flow regimes, predictions obtained over a relatively coarse grid using the SMART scheme for all variables including density are compared against results obtained over the same grid using the SMART scheme for all variables except density, for which the UPWIND scheme is used. Due to the unavailability of an exact solution to the problem, a solution using a dense grid is generated and treated as the most accurate solution against which coarse grid results are compared.

5.1 Subsonic flow over a circular arc bump With an inlet Mach number of 0.5, the inviscid flow in the channel is fully subsonic and symmetric across the middle of the bump. Isobars displayed in Fig. 2(a) reveal that the coarse grid solution obtained with the SMART scheme for all variables falls on top of the dense grid solution. The use of the upwind scheme for density however, lowers the overall solution accuracy. The same conclusion can be drawn when comparing the Mach number distribution along the lower and upper walls of the channel. As seen in Fig. 2(b), the coarse grid profile obtained using the SMART scheme for density is closer to the dense grid profile than the one predicted employing the upwind scheme for density. The difference in results between the coarse grid solutions is not large for this test case. This is expected since the flow is subsonic and variations in density are relatively small. Larger differences are anticipated in the transonic and supersonic regimes.

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(b) Figure 2. SubsonicJlow over a 10% circular bump; (a) isobars and (b) profiles along the walls. 5.2 Transonic flow over a circular arc bump

Results for an inlet Mach number of 0.675 are displayed in Fig. 3. Figure 3(a) presents a comparison between the coarse grid and dense grid results. As shown, the use of the HR SMART scheme for density greatly improves the predictions. Isobars generated over a coarse grid (63x16 c.v.) using the SMART scheme for all variables are very close to the ones obtained with a dense grid (252x54 c.v.). This is in difference with coarse grid results obtained using the upwind scheme for density and the SMART scheme for all other variables, which noticeably deviate from the dense grid solution. This is further apparent in Fig. 3(b) where Mach number profiles along the lower and upper walls are compared. As shown, the most accurate coarse grid results are those obtained with the SMART scheme for all variables and the worst ones are achieved with the upwind scheme for all variables. The maximum Mach number along the lower wall (z1.41), predicted with a dense grid, is in excellent agreement with published values [DEM 931. By comparing course grid profiles along the lower wall, the all-SMART solution is about 11% more accurate than the solution obtained using SMART for all variables and upwind for density and 2 1% more accurate than the highly diffusive all-upwind solution. 5.3 Supersonicflow over a circular arc bump

Computations are presented for an inlet Mach number value of 1.4. Mach number contours are compared in Fig. 4(a). As before, the course grid all-SMART results (58x18 c.v.), being closer to the dense grid results (158x78 c.v.), are more accurate than those obtained when using the upwind scheme for density. The Mach profiles along the lower and upper walls, depicted in Fig. 4(b), are in excellent

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agreement with published results [NI 821 and reveal good enhancement in accuracy when using the SMART scheme for evaluating interface density values. The use of the upwind scheme to compute density deteriorates the solution accuracy even though a HR scheme is used for other variables. The all-upwind results are highly diffusive.

--

-

--

1-6

'--T

ri

r,

.,

i

I * -

L , .

-

A

-

.

.

._id

- -

X

(b) Figure 3. Transonic flow over a 10% circular bump; (a) isobars using various schemes, and (b) profiles along the walls.

(b) Figure 4. Supersonic flow over a 4% circular bump (Mi,,=1.4); (a) Mach number contours using various schemes, (b) proJiles along the walls.

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6. Concluding Remarks A new high-resolution pressure-based algorithm for the solution o f fluid flow at all speeds was formulated. The new features in the algorithm are the use o f a HR scheme in calculating the density values at the control volume faces and the use of the NVSF methodology for bounding the convection fluxes. Results obtained were very promising and revealed good enhancement in accuracy at high Mach number values when calculating interface density values using a High-Resolution scheme.

7. Acknowlegments The financial support provided by the European Office o f Aerospace Research and Development (EOARD) (SPC-99-4003) is gratefully acknowledged.

8. Bibliography [DAR 941 Darwish, M.s. and Moukalled, F.," Normalized Variable and Space Formulation Methodology For High-Resolution Schemes," Numerical Heat Transfer, Part B, vol. 26, pp. 79-96, 1994. [DEM 931 Demirdzic, I., Lilek, Z., and Peric, M.,"A Collocated Finite Volume Method For Predicting Flows at All Speeds," International Journal for Numerical Methods in Fluids, vol. 16, pp. 1029-1050, 1993. [KAR 861 Karki, K.C.,"A Calculation Procedure for Viscous Flows at All Speeds in Complex Geometries," Ph.D. Thesis, University of Minnesota, June 1986. [LEO 871 Leonard, B.P.,"Locally Modified Quick Scheme for Highly Convective 2-D and 3-D Flows," Taylor, C. and Morgan, K. (eds.), Numerical Methods in Laminar and Turbulent Flows, Pineridge Press, Swansea, U.K., vol. 15, pp. 35-47, 1987. [LEO 881 Leonard, B.P.,"Simple High-Accuracy Resolution Program for Convective Modelling of Discontinuities," International Journal for Numerical Methods in Engineering, vol. 8, pp. 129 1-1318, 1988. [LIE 931 Lien, F.S. and Leschziner, M.A.,"A Pressure-Velocity Solution Strategy for Compressible Flow and Its Application to ShockiBoundary-Layer Interaction Using Second-Moment Turbulence Closure," Journal of Fluids Engineering, vol. 115, pp. 717-725, 1993. [MAR 941 Marchi, C.H. and Maliska, C.R.,"A Non-orthogonal Finite-Volume Methods for the Solution of All Speed Flows Using Co-Located Variables," Numerical Heat Transfer, Part B, vol. 26, pp. 293-31 1, 1994. [NI 821 Ni, R.H.,"A Multiple Grid Scheme for Solving the Euler Equation," AIAA Journal, vol. 20, pp. 1565-1571, 1982. [PAT 811 Patankar, S.V., Numerical Heat Transfer and Fluid Flow, Hemisphere, N.Y., 1981. [RHI 86) Rhie, C.M.,"A Pressure Based Navier-Stokes Solver Using the Multigrid Method," AIAA paper 86-0207, 1986.

Preserving Vorticity in Finite-Volume Schemes

Philip Roe and Bill Morton Dcpar-tment of Aerospace Enganeerzng The linz~~erszty of Mzchzgan Ann Arbor, Mzchzgan 48109-2140 ( '5,4

Department of Mathematical Sciences Universzty of Bath Bath B A 2 YAY United Iizngdorn

ABSTRACTW e discuss the fact that m a n y otherwise accurate finite-volume schemes have a tendency t o yield anomalous solutions i n certain circumstances. These are strongly linked t o the appearance of spurious vorticity. For a model problem, we show that certain finite volume methods i n fact preserve vorticity. Although these are not new schemes, they are not currently fashionable. A possibility exists t o modify t h e m so that they are are of high-resolution and upwind with respect t o acoustic waves. Ii'ey Words: Conservation laws, Euler equations, vorticity

1.

Introduction

Vorticity is a very important aspect of many fluid flows, especially in three dimensions, on account of the great times for which it persists in a highReynolds number flow following its initial creation. The velocities "induced" by vorticity are essential t o the operation of many fluid devices,and the behaviour of those regions where vorticity is concentrated are the key to the generation of sound. For computational purposes, incompressible flows are often formulated in variables t h a t include vorticity, and "vortex methods" t h a t explicitly track

348


concentrated vortex cores are also widely used. For compressible flow, vorticity is hardly less important, but it is rare for it to be included in the computational formulation. Instead, the stress is placed on conservation so as to ensure the correct capturing of shocks. The consequence is that vortical structures are often badly diffused by numerical dissipation and this is a serious impediment to the prediction of complcated viscous flows. We have recently begun [MOR 991 construction of finite-volume methods that exert control over vorticity. We make no attempt to solve the vorticity evolution equations themselves, but try to find schemes such that a correct evolution of the discrete vorticity is an exact consequence of the discrete conservation laws, just as at the continuum level. Another way of expressing this is to say that whatever form of numerical dissipation is employed to stabilise the propagation of waves should have no impact on the advection of vorticity. There is a further motivation, which is that at the level of the Euler equations many otherwise accurate methods, especially those that attempt to minimise numerical dissipation by adopting upwind strategies (so that any disturbance is damped by the smallest damped compatible with its propagation speed) notoriously produce nonphysical solutions (carbuncle phenomenon) under some special (but not uncommon) circumstances. It now appears that these solutions are in fact quite legitimate solutions of the Euler equations, and could only be excluded by some selection principle over and above that of entropy-satisfaction. On the basis of examples encountered to date, all of these undesirable solutions appear to feature much greater vorticity that the anticipated solution. A very simple example is given in Section 2. There is also the related phenomenon discovered by Quirk [QUI 941; that the same schemes that give rise to carbuncles can be "destabilsed" by very small grid perturbations. In simple academic examples it is usually easy to detect that something is wrong with the solution, and to tinker with the code until it goes away, but in situations where the flow is anyway expected to be complex there may be no simple way to detect unwanted phenomena. There is therefore an urgent need to understand these anomalies and to find rneans of avoiding them.

2.

Nonunique Solutions of the Euler Equations

Consider the supersonic flow past a flat-nosed two-dimensional body (although the argument works just as well for a flat-nosed cylinder with axial symmetry). We expect the solution to feature a rather smooth detached bow shock, giving rise to an embedded subsonic region, from which the flow accellerates smoothly to regain supersonic speed. This is shown in the top half of Figure 1. In the bottom half is an alternative solution. Plane oblique shocks originate from a more-or-less arbitrary point on the line of symmetry. Behind them is a triangular region of stagnant fluid extending to the shoulder of the body. Outside of this region the flow behaves exactly as it would if flowing past a triangular wedge (or cone) of solid matter. The pressure, velocity, etc.

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Figure 1. T w o possible Euler solutions for the the flow past a flat-faced body. in that region may be found from the usual compressible tables and formulae. Prandtl-Meyer rarefaction waves return the flow to its original direction and weaken the initial shocks. Such a flow is in fact possible t o create experimentally, and is illustrated in Plate 272 of [VAN 821 which is a schlieren picture taken in a windtunnel. The unusual flow configuration is the result of a thin "splitter plate" placed along the axis AB. Presumably what happens is that the boundary layer on the plate creates vorticity that is then diffused outward. In the limit of very high Reynolds number the vorticity eventually becomes concentrated in the infinitesimal shear layers AC, AD There is no net circulation generated in the flow but each half of the stagnant region (above and below the axis) has circulation around it. This flow is a valid second solution of the Euler equations, providc that the pressure in the stagnant region matches the pressure outside it. The temperature in the stagnant region is arbitrary if we allow that an entropy layer might also exist, so there is a doubly-infinite family of solutions. All the shocks are entropy-satisfying, but under some conditions the shear layers might not be stable. One may now imagine that the corners of the body are steadily rounded off until the nose becomes a semi-circle. The alternative solutions would become the carbuncles. According to Pandolfi and L)'Ambrosio, who have made very detailed observations [PAN 991 the solutions produced in the carbuncle phenomenon are also genuine Euler solutions. One might hope that by solving the Navier-Stokes equations instead the carbuncle would automatically disappear. However, Gressier and Moschetta

350


[GRE 991 report that the Reynolds number had to be reduced t o as little as 100 before this happened. It seems that the mechanisms by which the Euler code introduces the vorticity is stronger than the mechanism by which the Navier-Stokes code removes it. The Quirk phenomenon occurs when a plane one-dimensional shock is propagated along one direction of a rectangular grid. If the grid is indeed perfectly rectangular then one recovers the one-dimensional solution but if one row of grid points, parallel to the direction of motion, is very slightly perturbed, an exponentially-growing pat,tern of high-frequency disturbances is generated that eventually yields a flow pattern not unlike that seen in carbuncles. Robinet et a1 [ROB 991 attempt to relate this to a new observation concerning the linear stabilty of a plane shock. Although this is a classical flow stability problem, they have discovered a 'strange mode' overlooked by previous analysts. They find perturbations of a plane two-dimensional flow by separation of variables. This leads to an eigenvalue problem having linearly independent solutions except for certain combinations of frequency in time and wavenumber in space. Rather unexpectedly, these exceptional solu~ionsinvolve a resonance in which an acoustic mode and a vortical mode become indistinguishable. The linear algebra problem has t o be completed by a Jordan block with a generalised eigenvector. Coupling this with the equations governing the shock perturbation reveals a solution that can grow exponentially with time. Many features of this solution are shown to appear in numerical experiments on the Quirk phenomenon. Experience to date is that all numerical schemes that display the carbuncle phenomenon also display the Quirk phenomenon. There is a proof [GRE 981 t h a t all schemes capable of resolving a parallel shear layer without dissipation (a highly desirable property, and one responsible for the widespread adoption of flux-difference-splitting schemes rather than flux-splitting schemes) will in fact display the Quirk phenomenon. This collection of facts almost suggests a crisis situation. In the next section we present analysis, condensed from [MOR 991 that opens an avenue of escape.

3.

A Model Problem

The simplest model problem to combine wave propagation and vorticity is the system wave equation. We will write this in two space dimensions in the matrix form, using a notation corresponding to acoustic waves in a fluid that is stationary in the mean, with pressure p* and velocity 2 = ( u * ,u * ) , thus

Innovative schemes

Cells Vertices

0

35 1

p, u, v p', u', v'

Figure 2. Grid definition. Here u = ( p * / ( p c 2u) ,* / c ,v * / c ) r ( p , .u, v , ), c is the sound speed in the mean flow, and

Restriction to two dimensions is merely for economy of notation; all of the analysis extends very straightforwardly to three dimensions. We study the wave equation in system rather than scalar (dttu = c 2 V 2 u )form for two reasons. Firstly because this is the form of the wave equation that is hidden inside the Euler equations, whether in their two-dimensional unsteady or in their three-dimensional supersonic steady forms. Secondly, because the scalar form automatically implies vanishing vorticity, whereas the interaction of the waves with vorticity is one of the aspects we want to study. Here the interaction is very simple, as befits a model problem. We easily deduce from (1,2) that

at< = 0 ,

where

< = d X v- a y u .

In other words, there is no interaction and any initial distribution of vorticity is preserved. Maintaining this independence at the discrete level will be our objective.

3..

Discrete Notation

In this paper we concentrate on simple finite-difference formulations on uniform square grids, such that the spacing in the x and y directions is h and the time step is At, with u:,~a discrete approximation located at ( x , y,t) =

352


(ih, jh, n A t ) . The standard discrete differencing and averaging operators are defined by

The points with integer coordinates will be called cells, those with one integer and one half-integer coordinate edges and those with two half-integer coordinates vertices. The variables stored at cells will be (p, u , 7)). The same variables stored at vertices will be distinguished by primes where neccessary. The variables stored at vertical edges will be ( P , U ) and those stored at horizontal edges (Q, V). In a finite-volume interpretation the edge quantities are the fluxes. (See Figure 2) If it is recognised that each application of the above operators moves the mesh values t o a different set of grid points then arbitrarily long products of operators are allowed and all multiplications commute.

3.2.

Conservation Form

In the cell-centred finite-volume method, discrete conservation is ensured by drawing a control volume around the grid point of interest, and writing the update as an integral around this volume. In the generic case of a vector U of conserved variables, with fluxes F, G in the (z, y)-directions respectively, one has

where F*,G* are numerical fluxes evaluated from some formula to be determined. In the present case we can write, with u = cAt/h and following the notation of Figure 2,

It will usefully restrict the schemes to require that they can be written in this form. A second-order scheme of the Lax-Wendroff type follows from taking U , V, P, Q t o be estimates halfway through the time step. However, we will find subsequently that it is a rather special type of conservation form that emerges from the analysis.

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3.3.

353

Preserving Vorticity

The first step in creating a scheme that preserves vorticity is t o define a discrete vorticity, which we do as follows

This will be preserved if

and the condition p,P = p y Q will be met if we take P = p y r l ,Q = p z r l where r1 is some quantity defined at vertices. The only way to define a consistent local pressure while retaining a nine-point stencil is to take

In that case we have

To obtain second-order accuracy, r1 must now be updated to half-way through the time step. The simple formula

is the unique symmetrical formula t o achieve this without enlarging the stencil, leading to

We remark here that a general example of the Lax-Wendroff family will generate vorticity at a rate proportional to h3; this is shown in [MOR 991.

3.4.

Complete Evolution Operator

We construct the matrix operator that will update the solution, so that if

354


certain elements of the matrix Ma a.re already uniquely determined by insisting that the velocities are updated with second-order accuracy while preserving the discrete vorticity (6). We have in fact, from (4) ,(5),(9),(10),

The adjoint property divh = -grad: requires that this matrix be symmetric [MOR 991 and an agument based on maintaining a compact stencil leads to the complete matrix operator

The scheme represented by this matrix has been uniquely determined by the requirements of conservation, vorticity preservation, symmetry of the solution under grid transformations, adjoint symmetry of the discrete operator, and second-order accuracy. However it is not a new scheme. It can be recognised by noting that M A can be factored as

where

and therefore can be written as a two-step scheme. The operation

gives a provisional solution at the vertices. The operation

completes the update by integrating around the vertices. This is in fact the version of Lax-Wendroff known as the Rotated Richtmyer scheme[RIC 621. It is shown schematically on the left of Figure 3. The original motivations for this scheme were compactness, computational economy and stability. In the nonlinear case, as in all two-step Lax-Wendroff schemes, one avoids any multiplication by the Jacobian matrices. The vorticity-preserving property does not seem t o have been previously noticed. It is shown in [MOR 991 that the scheme is stable for the maximum possible CFL range c A t / h 5 1.

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Figure 3. (left) The Rotated Richtmyer scheme. In the first step, symbolised by white arrows, data from the cells is used to create a half time-step solution at the vertices. In the second step (black arrows) integration round the vertices updates the central cell. (right) Ni's cell-vertex scheme. In the first step (white arrows) we zntegrate around the cells to obtain a 'cell-residual', In the second step (bluck arrows) these are distributed to the vertices. 3.5.

Duality

Since bot,h factors of M A depend only on LA they commute, and so the scheme nlay also be written as

In this form it is Ni's cell-vertex scheme [NI 821, in which the variables are usually thought of as located a t vertices of a grid, defining a bilinear interpolant over a square element. T h e first step LA is to integrate d,F dyG over this element and the second step is to distribute this to the nodes of the element. This distribution operation is described by the first factor in (17)

+

4.

Commentary

By choosing a correct definition of discrete vorticity, the results above can be extended [MOR 991 t o the linear wave equation on unstructured two- and

three-dimensional meshes, and to problems with non-constant coefficients for which vorticity should be created. In the latter case there is a discrete Kelvin Theorem giving the growth of circulation around a certain class of contour. All of this is of course only a beginning t o the design of practical schemes, but an important observation is that the vorticity-preserving property does not depend on choosing any particular expression for the quantity r' in (7). It is enough t o compute as a first step any vertex pressure, even a ridiculous one, and then to find the edge presure by averaging along the edges. This gives scope for iritroducing nonlinear methods (limiter functions) t o control the os-

356


cillations around shockwaves without introducing vorticity. I t is beyond t h e scope of this paper t o discuss details, b u t i t is possible t o design schemes t h a t preserve vorticity in two o r three dimensions, a n d which reduce in one dimension t o high-resolution upwind schemes. However, t h e two-dimensional fluxes are computed from six rather t h a n two neighbouring states, which takes the schemes outside the applicability of the results in [GRE 981. Therefore schemes t h a t preserve contact discontinuities b u t avoid carbuncle-type behaviour may still be possible. We are currently investigating such schemes. For t h e fully nonlinear Euler equations, the evolution of discrete vorticity is quite complicated. However, t h e evolution of the curl of the momentum is V x (pv) is rather simple, and t o avoid unwanted solutions i t may be enough t o control this quantity. We hope t o be able t o report shortly on the outcome of numerical experiments.

5.

Bibliography

[MOR 991

MORTON,K. W.. AND ROE, P., "Vorticity-preserving Schemes of LaxWendroff Type", Submitted to SIAM J. Sci. Comp., 1999.

[ROB 991

ROBINET,J-CH., GRESSIER,J., CASALIS,G. AND MOSCHETTA, J-M., "Shock Wave Instability and Carbuncle Phenomenon- Same Intrinsic Origin?", preprint, ONERA Toulouse, submitted to J. Fluid Mech., 1999.

[QUI 941

QUIFX, J. J., "A Contribution to the Great Riemann Solver Debate", Int. J. Num. Meth. in Fluids, 18, pp. 555-574, 1994

[VAN 821

VAN

[PAN 991

PANDOLFI,M. AND D'AMBROSIO,D., "Numerical Instabilities in Upwind Methods: Analysis and Cures for the "Carbuncle" Phenomenon", preprint, Politechnico di Torino, submitted to J. Comput. Phys., 1999

[GRE 991

GRESSIER,J . AND MOSCHETTA, 3-M., "Robustness versus Accuracy in Shock-Wave Calculations", preprint, ONERA Toulouse, submitted to Int. J. Num. Meth. in Fluids, 1999.

[GRE 981

GRESSIER,J. AND MOSCHETTA, J-M., "On the Pathological Behaviour of Upwind Schemes", AIAA Paper 98-0110.

[RIC 621

RICHTMYER, R. D., "A survey of difference methods for non-steady fluid dynamics", NCAR Tech. Note 63-2, Nat'l. Center for Atmos. Research, Boulder, CO, 1962.

[NI 821

NI, R-H,, "A multiple-grid scheme for solving the Euler equations", AIAA Jnl, 20, p 1565, 1982.

DYKE,M., An Album of Fluid Motion, Parabolic Press, Stanford, CA.. 1982

On Uniformly Accurate Upwinding for Hyperbolic Systems with Relaxation

Jeffrey Hittinger and Philip Roe D e p a r t m e n t of Aerospace Engineering T h e U n i v e r s i t y of M i c h i g a n Ann Arbor, Michigan, USA 48109-2140

ABSTRACT T h e design of uniformly accurate, upwind Godunov schemes for hyperbolic systems with relaxation source t e m s is discussed. T h e goal is t o develop upwind methods whose sole time step constraints are due to the advection terms yet which obtain accurate solutions even when the relaxation t e m s are undewesolved. A n archetypal model system is considered, and analysis of the Riemann initial value problem for this model system is discussed. The ideas learned from this are re-enforced by a n asymptotic analysis of general Riemann problems, and these results are described. A strategy for developing a suitable numerical fEux function is then outlined. Key Words: hyperbolic systems, relaxation, numerical upwinding, stiff source terms

1.

Introduction

Many flow problems such as those of dilute gases or fluid mixtures have natural formulations as hyperbolic systems with relaxation source terms. These pose interesting challenges for numerical approximation. The source terms cause the system to be dispersive; generally, both the eigenvalues and eigenvectors of the system are modified by the relaxation processes which drive the system towards equilibrium. Another issue arises when the relaxation source terms operate on much smaller time scales than the advection terms; the problem is then said t o be stiff. Unfortunately, many problems simultaneously exhibit behavior in both the stiff and non-stiff limits, as well as in between. It is therefore desirable to develop high-resolution numerical algorithms which are uniformly accurate a t all scales. For example, if the data are such that

358


the advection terms permit a time step large enough to bring the flow into local equilibrium, one would wish to see an accurate solution of the equilibrium problem emerge without resolving the details of the transition. In recent years, there has been much research into such uniformly accurate methods, although a clearly satisfactory procedure has yet to emerge. Pember's early work in the area [PEM 931 identified many properties desirable for Godunov-type schemes for hyperbolic systems with stiff relaxation but did not resolve the relative significance of the frozen and equilibrium wave speeds. Splitting schemes [CAF 97, JIN 961, have been proposed, but general objections to splitting were made in [ARO 96, ROE 931. These authors proposed characteristic schemes but found no natural way to make them conservative. A conservative method based on an approximate linearization was proposed in [BER 971. This uses a staggered grid t o eliminate the difficulty of resolving the Riemann problem, which no longer has a self-similar solution. However, such schemes may lose definition of linearly degenerate waves, such as contact discontinuities, which are likely to be very important in reactive or relaxing flows. Currently we are attempting to develop an approximate Riemann solver that represents the physics with uniform accuracy and whose computational costs are acceptable. The general behavior of relaxing Riemann problems is now known thanks t o the work of Liu and Zeng [ZEN99], although they have concentrated mostly on qualitative behavior of rather general systems. It is complementary t o our own work which has involved very detailed analysis of a linear model system which we feel is archetypal. This model has an exact (integral) solution for the Riemann initial value problem, which allows one to see how a simplified solution might be developed.

2.

A Model System

To gain a better understanding of the behavior of hyperbolic systems with relaxation source terms, we have studied the linear1 system

where x E R and t E E%+ are the spatial and temporal independent variables, respectively; 6 E R+ is a constant relaxation time; and r E [O,1] corresponds to the equilibrium wave speed. 1 0 n l y the linear problem is considered for the obvious reason that analysis is made tractable. However, this is not the restriction it might seem, as other authors [BER 971 have identified a suitable linearization for nonlinear hyperbolic systems with relaxation. We will demonstrate their requirements in Section 3.2.

Innovative schemes

359

The first equation ( l a ) describes the evolution of the conserved quantity u(x, t). The flux of u, that is v(x, t), has its own evolution equation (lb), and is driven towards equilibrium over a time scale O(E) by the relaxation source term. In equilibrium, this source term vanishes (v E ru), and the system reduces to the single linear advection equation with the equilibrium wave speed r . Near equilibrium, say v = r u + O(c), it can be shown that

which is an advection-diffusion equation. In similar analysis applied to more complex problems, such as moment approximations of the Boltzmann equations, (2) corresponds to the Euler equations, and (3) t o the Navier-Stokes equations.

3. 3.1.

The Riemann Problem

The Model Problem

For upwind schemes, the Riema.nn problem is the heart of the numerical algorithm. We will describe the Riemann problem for the model system and point out behavior that is generic to all relaxation systems. Sett'ing 4, = -u and $t = v allows the system (1) to be reduced to a scalar equation, to which an exact integral solution can be constructed using classical methods. (For details see [ROE 931.) This integral solution is amenable to asymptotic analysis, particularly a t small times, and can also be evaluated numerically to provide a complete picture of the evolution of the solution from the small-time t o the long-time asymptotics. Consider the system (1) in vector form

with the piecewise constant initial conditions

u(x < 0,O) = UL

and

u(x 2 0,O) = UR.

(5)

The discontinuity in value a t the origin has a domain of influence bounded by the frozen characteristics I[) = Ix/tl = 1. Outside of the domain of influence, the initial conditions are constant, and, hence, the system of partial differential equations (4) reduces to a system of ordinary differential equations in time. This system is easily integrated t o find

360


where T = t / ~ The . conserved variable u remains constant while the relaxation variable v decays exponentially from its initial value to the equilibrium value ru. This rapid adjustment of the relaxation variable forms a temporal boundary layer, as is apparent in the contour plots presented in Figure 1. Within the domain of influence, the solution is

where

with q* = t - x, 0, so that Q has a null space Jd(Q), which is the equilibrium manzold in which the equilibrium solution takes place. Let Lo E Rnxmbe the row matrix of the left eigenvectors of Q which span N ( Q ) ; similarly, let Ro E RmXnbe the column matrix of the right eigenvectors of Q which span N ( Q ) . It is easy t o show that the equilibrium (Euler) equation is

Innovative schemes

363

Interface State, ~=0.01,e 0 . 5

Figure 2. T h e interface state for the R i e m a n n problem u~ = (1,5)T and UR = (10,3)T for the equilibrium wave speed of r = 0.5 and relaxation time t = 0.01. T h e solid line is the numerical calculation of the exact solution, the , the dotted line is the small-time asymptotic expansion u p to terms O ( T ~ )and dashed line is the leading-order long-time asymptotic solution. and the late-time solution to the Riemann problem comes from the self-similar solution to this problem with Riemann data LouL,LouR. In [BER 971, it was proposed to select an average matrix Q ( u ~u ,R ) such that L~(Q)A& (Q) is the correct Roe matrix for the equilibrium problem. Because they avoided consideration of the early time Riemann problem, they had no need to select any particular linearization of A , but if this is simply a standard Roe linearization, then isolat,ed discontinuities will be correctly captured in both limits. If n = (m - I ) , the solution is only stable if the equilibrium wave speeds (the eigenvalues of LoARo) precisely interleave the eigenvalues of A [LIU87]. For this case, an equilibrium wave will grow from the smooth flow between each pair of frozen waves, and simultaneously, this equilibrium wave will be diffused. The frozen waves decay unless they carry a jump lying in N ( Q ) ;such waves will be found in both the early and late time solutions. If n < (m - l ) , are no theoretical results, but the general picture of frozen shocks undergoing exponential decay with equilibrium shocks arising from the spaces between them appears t o be correct. At early times one can obtain expansions similar to (12). The k-th frozen wave decays such that

364


where lk and r k are the k-th left and right eigenvectors of the matrix A , and a k ( 0 ) is the strength of the initial jump. The solution between the waves is again piecewise polynomial of degree n at order n. 4.

C o n s t r u c t i n g an Upwind Flux Function The substitution u = e x p { Q ~ ) wtransforms (4) into

Integrating this around a cell of a regular, uniform grid with cell-centered data, and translating back to the original variables, one finds

where t' = t - nAt and f = A u . If the solutions fjf ; ( t ) are known on the interface, this is an explicit formula for the new cell average. The exponential decay in the transient solution is represented explicitly, and within the integral, the exponential factor acts as a filter. To elucidate the weighting in the integrand, let z = Lu, where L is the row matrix of left eigenvectors of Q. This transformation creates a state vector of the n conserved variables (those in N ( Q ) )and of the (m - n) relaxation variables which vanish in the equilibrium limit. Multiplying the update formula by L, one obtains

where f = LAL-'z and A is the diagonal matrix of the eigenvalues of Q. For conserved variables, the corresponding eigenvalues are zero, so the exponential factors in the update (23) are just unity. For relaxation variables, interpreting At as the time step based solely upon the advection terms, the exponential weighting in the integrand is essentially unity over the entire interval if At/&> 1, the main contribution of the integral comes from the vicinity of the end point t' = At. The question now is whether a computationally acceptable approximation can be found for the fluxes on the interfaces, arising from the Riemann problems with data uy, u s l . Clearly this is possible in the two limiting cases At/€ > 1, and if the majority of cells are of one of these types, there will not even be any great expense involved, because a switch simply could be used to select the appropriate formula. These are the limits on which previous work has concentrated, and the present approach deals with them very simply.

Innovative schemes

365

The difficulty is only with those cells for which A t / € = O(1). An empirical blending of the two formulae will probably not be very accurate, because of the non-monotone transition seen in Figure 2. However, Figure 2 also shows that the early-time expansion may have a sufficiently large radius of convergence that it can capture the crucial behavior. We are presently investigating such expansions in the general case. The solution for the jumps is given above, and the solution for the linear variation is also straightforward. The quadratic and higher terms appear more complicated, although there is a straightforward means t o derive them. But even if the formulae prove expensive they should only be needed a t relatively few interfaces.

5.

A Practical Example

Between the continuum (Euler) model of fluid flow and the molecular (Boltzmann) description, there lie many intermediate models. The hierarchy of models devised by Levermore [LEV 961 are of especial interest, because by design, they possess many properties required for trouble-free computation: they are hyperbolic, symmetrizable, entropic, and well-posed. Beyond the Euler equations, the next member of the hierarchy retains ten moments of the Boltzmann equation by assuming that the distribution of the random velocity c is proportional t o exp(-iOijlcicj), where Oij is a symmetric, non-negative 3 x 3 matrix closely related to the temperature. Moments of this distribution give the pressure and temperature as tensor quantities. The relaxation process is simply that the temperature tries t o relax back toward a scalar,

and the eigenstructure of the frozen problem is straightforward (See [BRO 951). We feel that although the procedures outlined in this paper may be expensive in general, they will prove relatively simple for well-motivated physical models. 6.

Conclusions and Future Work

The analysis of a simple linear model system has provided clues towards the development of a uniformly accurate upwind method for hyperbolic systems with relaxation. Specifically, the solution of the Riemann problem can be constructed and analyzed for this model system, and this analysis has identified strategies for designing upwind methods for hyperbolic systems with relaxation source terms. Currently, we are implementing and evaluating these ideas.

366

7.


Acknowledgment

This work is supported in p a r t by a U.S. Department of Energy Computational Science Graduate Fellowship.

8.

Bibliography

[ARO 961

ARORA,M., Explicit Clbaracteristic-Based High-Resolution Algorithms for Hyperbolic Conservation Laws with Stiff Source Terms, Ph.D. thesis, The University of Michigan, 1996.

[ARO 981

ARORA,M. A N D ROE, P., L ' I s ~ ~and e s Strategies for Hyperbolic Problems with Stiff Source Terms", Barriers and Challenges in Computational Fluid Dynamics, V. Venkatakrishnan et al., eds., Kluwer Academic Publishers, Norwell, MA, 1998, pp. 139-154.

[BER 971

BEREUX,F. A N D SAINSAULIEU, L . , "A Roe-type Riemann Solver for Hyperbolic Systems with Relaxation Based on Time-Dependent Wave Decomposition", Numer. Math., 77, 2, 1997, pp. 143-185.

[BRO 951

BROWN,S . , ROE, P . , A N D GROTH,C., LLN~merical Solution of 10Moment Model for Nonequilibrium Gasdynamicsl', AIAA Paper 95-1677, June, 1995.

[CAF 971

CAFLISCH, R . , J I N , S . , A N D RUSSO,G., "Uniformly Accurate Schemes for Hyperbolic Systems with Relaxation", SIAM J. Numer. Anal., 34, 1, 1997, pp. 246-281.

[JIN 951

J I N , S., "Runge-Kutta Methods for Hyperbolic Conservation Laws with Stiff Relaxation Terms", J. Comput. Phys., 122, 1, 1995, pp. 51-67.

[JIN 961

J I N , S. A N D LEVERMORE, C., "Numerical Schemes for Hyperbolic Conservation Laws with Stiff Relaxation Terms", J. Comput. Phys., 126, 2, 1996, pp. 449-467.

[LEV 961

LEVERMORE, C., "Moment Closure Hierarchies for Kinetic Theories", J. Stat. Phys., 83, 5/6, 1996, 1021-1065.

[LILT871

T.-P. LIU, " Hyperbolic conservation laws with relaxation",

Comm.

Math. Phys, 108, 1, 1987, pp. 153-175. [PEM 931

PEMBER,R., "Numerical Methods for Hyperbolic Conservation Laws with Stiff Relaxation: 11. Higher-Order Godunov Methods", SIAM J. Sci. Comput., 14, 4, 1993, pp. 824-859.

[ROE 931

ROE,P. A N D ARORA,M., "Characteristic-Based Schemes for Dispersive Waves: I. The Method of Characteristics for Smooth Solutions", Numer. Methods Partial Differential Equations, 9, 5, 1993, pp. 459-505.

[ZEN991

ZENG,Y., "Thermal Nonequilibrium and General Hyperbolic Systems with Relaxation", preprint, University of Alabama at Birmingham, 1999.

Implicit Finite Volume approximation of incompressible multi-phase flows using an original One Cell Local Multigrid method

Stkphane V I N C E N T and Jean-Paul C A L T A G I R O N E Avenue Pey-Berland BP 108 33402 Talence Cedex France

The numerical simu~ationof multi-phase flows involving stretching and tearing of interfaces requires accurate tools, able to describe near the free surface the different scales of the flow which results from the development of instabilities. O n fixed Cartesian mesh, a n original local multigrid method, which refines the grid at the cell scale and adapts i n time and space, is proposed. A n implicit Finite Volume solver, coupled with a T V D - V O F like interface capturing method, is carried out on each grid level. The method is validated and discussed o n analytical velocity fields and Rayleigh-Taylor instabilities. ABSTRACT

Key Words: free surface flows, multigrid method, implicit Finite Volumes


The numerical simulation of multi-pha.se flows with strong stresses acting on the interface is classically achieved by the implementation of fixed Cartesian rrieshes with an interface tracking method (Marker [DAL 671, VOF [YOU 821 or Level Set [SUS 971). However, due t o the memory limit of supercomputers and the computational time, the numerical simulation of non-symmetric threedimensional free surface flows is restricted t o problems where the length-scales of the phenomena occurring near the interface arc close. To limit the computation node far away from the free surface and concentrate the calculation points on the interface, an original One Cell Local Multigrid method (OCLM) is proposed. Starting on a coarse grid Go which corresponds t o the physical domain 0, a refinement criterion R, is defined t o detect the points t o be refined on

368


0.

It should be noted that a full tensor for the absolute permeability K is considered and the approximation obtained in (4) is such that 0 < D- 5 Dml15 D+ < m which is an important property for the analysis of this FVS. In fact, we can write the coefficient DM,[ in the following forme:

where L and L' E 8T such that L n M = 0 and L'

__+

n Mr = 0,hence

---+

Furthermore, we have = &xlxM . adj [I 0.

+

The sound speed c of the fluid is the square root of this quantity. Generally, a drift flux is added to the set of equations (1)-(3) to incorporate the effects of the slip velocity between the phases. This will however not be the case in the present study.

2.1 Equation of state The equation of state consists of three zones: a liquid, a two-phase mixture, and a vapor zone. The liquid and the vapor zones are described by usual single fluid equations of state. In the two-phase zone, each fluid is supposed to be at saturation, so that the density pe, p, and the enthalpy he, h, of each phase are functions of the pressure only. The density p and the enthalpy h of the mixture satisfy : h, - hl - hv - h h - hl -----. P Pe PU

+

The transitions between the mixture zone and the two others occur along the saturation curves. The equation of state is continuous across this curve but not continuously differentiable. This fact is clearly visible in Fig. 1.

Fields of application

397

Figure 1: Specific volume l l p versus enthalpy h for the water equilibrium EOS, at pressure p = 7MPa. The liquid zone corresponds to lower values of h , and the vapor zone to the higher values of h. In the two-phase zone in-between, l l p varies linearly with h . Note that the derzuatiues of the equation of state are descontinuous across the transitions. 3. The preconditioned finite volume scheme 3.1. Some notations

We rewrite the Euler equations (1)-(3)with the compact form:

dtU

+ div F ( U ) = 0.

(4)

We consider a triangulation of the computational domain by polygonal cells. For a cell I1

ac'

) 7( 2 ,R (t) , 1) + - D"

(T ( 2 , t ) )

ac" Y&-

(T'

= (C; ( T )- C:) .TdR (t)

R (4 t) 1

(7)

2.3 Energy Balance

We evaluate the overall energy balance at the sample macroscopic level, for which we take into account the heat released at the mesoscopic scale - i.e. particle scale- by the exothermic mass diffusion processes described below.

2.3.1 Carbidization Domain , Zone I We denote by the rate of heat released during carbon formation - by diffusion process in zone I-, defined by

2.3.2 Dissolution Domain , Zone II the rate of heat released during carbon dissolution We denote by by diffusion process in zone 11-, defined by

:[lR"'

dQ" (t) = 4aNTi dt

A H ~ C " ( r , r, t ) r2 dr

I

440


2.3.3 Global Heat Balance

ac

We consider the cylinder domain = [0,Rc] and write for V z E [0,Rc] The heat balance takes into account the heat released locally at the particle level for the two diffusion processes by:

aT P - C P( z ,~t ) = div(X grad T ( z , t ) )

+

dt

dt

2.3.4 Boundary Conditions Symmetry Condition at the center of the cylinder

Heat supplied by the furnace maintained a t temperature T ,

3. Numerical Scheme

We integrate from t , to t,+l, the three different balances, assuming that a non uniform temporal mesh is given. We use a variable stepsize backward difference formula of order 1, which is well suited for the numerical treatment of stiff equations. We focuss on the numerical methodology used to discretise the mass balances, written in conservation form, on a variable domain. In order to track efficiently the variation of the computational domains Ql ( t ) and Q2 ( t ), we have considered the integration of the mass balance on a cell ICi ( t ) = [ri ( t ), ri+l ( t ) ] which evolves with time and denotes its boundary 8Ii.i ( t ) = {ri ( t ), ri+l ( t ) ). We adopt a cell-centered formulation. 3.1 Integration on a moving mesh

dC

- ( z ,r , t )

dCl =

L"" J,.,,,

div ( D (T ( z ,t ) ) grad C ( 2 , r, t ) ) dQ (13)

The transport theorem of Reynolds [N'KO 941 affirms that: d

C ( r ,r , t ) r2dr

ac

- ( z ,r , t )

r2dr+

J

aK.(t)

C ( z ,r , t ) r 2dr -dt dt


441

+c( 2 , ri+1 ( t ), t ) ri+l (t)' Vi+l ( t ) dt - C ( z , ~i( t ), t ) ~i (t)' vi ( t ) dt where vi+l ( t ) = and vi ( t ) = are the velocities of the extremities of the interval1 on which the mass balance is integrated. We outline the procedure, and omit the details of the algebraic computations. We integrate the ( z , r, t ) from its definition previous relation from t , t o t,+l and substitute into the mass balance. We apply Green's Formula for the discretization of the divergence of the mass fluxes. We apply the first order accurate backward Euler scheme for the discretization of the right hand side integrals. We denote A t , = - t,. and my = mes (I{:) = ~ ' " + l ( ' " ) r2dr = ri+l (t,) 3 - ri (tn13],the r,(tn) volume of the cell and end up with,

&

We point out that the following term, which takes into account the transport of the concentration field, due to the variation of volume of the integration cell appears :

(

(

-

1

i

(

n

1

) 1

I

)

)

ri+i ( t , + ~ Vi+l ) ~ (tn+1) A t ,

ri (tn+ll2~i (tn+i) Atn

(16)

Since C is constant in cell Ii'; (t,) , we may use "local information" i.e. related to this cell, and therefore write the following continuity relation ( j

i

t n +t n + )

C

(

We now have the following relationship

i

(

+

)I

)

=c

1

(17)

442


After some algebra and leaving the convected mesh terms in the right hand side, we are led to a three diagonally dominant matrix system which can be efficiently inverted by the classical -double sweep- Thomas Algorithm. At this stage we have made no assumption on the nodes velocities. 3.2 Nodes Velocity

We have to prescribe the speed of the mesh; this is realized through a linear speed formula, according to [MUR 591, taking into account the fact that the velocity at the interface between domain I and I1 is known from the stefan problem equation. 3.3 Boundary Conditions

An important issue is the discretization of the boundary conditions on a moving mesh, because they will lead to a different set of equations that will influence significantly the behaviour of the numerical solution, if a wrong formula is supplied, for example consider:

C ( 2 , R ( t ) , t ) = C i =Constant We integrate the equation on cell [rI (t,) , r ~ + (t,)] l assuming that CIKn= C i , therefore (z, r , t )(

K?

( t n )= = 0 and

the equation is,

Using the previous arguments on the continuity of the concentration field we get


443

In fact the variation of the computational volume can be understood as a source term.

3.4 Time Step Adaptation

The stiffness of the problem means that once the heat released by the chemical kinetics is enough, a sharp rise of the temperature occurs inside the sample from the exterior surface to the interior of the cylinder. A time-stepping adaption procedure t o dynamically reduce the time-step with respect t o the temperature field behaviour has been considered.

5. Numerical simulations Ignition temperature is reached, when a sharp increase in temperature field occurs. This ignition process depends on particle size, initial thickness of T i c interfacial layer, heating rate, and value of T,. We present below a sample of such temperature profile.

Temperature evolution of different positions in the cylinder Tfurnace = 1200 K 2000.0

Figure 2: Temperature profile T ( R c ,t)

444


V . , Mathematical modelling and numer[AOU 981 . AOUFIA . , ROSENBAND ical simulation of Titanium/Carbon Ignition , p d International High-Energy Materials Conference and Exhibit, Dec 8-10 (1998). [LAP 961

. LAPSHINO.V. A N D OVCHARENKO A.E. , A mathematicalmodel of high-temperature synthesis of the intermetallic compound Ni3Al during ignition , Combustion, Explosion and Shick Waves , Vol. 32, No 2,(1996),158-164.

[MUR 591 MURRAYW.D. A N D LANDISF . , Numerical and Machine Solutions of Transient Heat-Conduction Problems Involving Melting or Freezing , Transactions of the ASME, (May 1959), 106-112.

[N'KO 941 N'KONGAB . AND GUILLARD H . , Godounov type methods on nonstructured meshes for three-dimensional moving boundary problems , Comput. Meths. Appl. Mech. Engrg. 113 (1994). [SEP 951

SEPLYARSKII B.S. AND GORDOPOLOVA I.S., Ignition of condensed systems interacting through a layer of high-melting products , Combustion, Explosion and Shock- Waves, Vol. 31, N04, (1995),405-410.

Two Perturbation Methods to Upwind the Jacobian Matrix of Two-Fluid Flow Models.

K u m b a r o A., T o u m i I. CEA Saclay, DRN/DMT/SYSCO F-91191 Gif-sur-Yvette Cedex, France [email protected] C o r t e s J. CEA Cadarache, DRN/DTP/SMET F-13108 Saint-Paul lez Durance, France

W e examine the eigenstructure of a two-fluid model. Because of complex interphase interactions, perturbation methods are used to get approximations of the eigenelements. W e compare two significant perturbation methods. Mathematical and numerical results are provided.

ABSTRACT

Key Words: two-phase flows, eigenstructure, upwinding, perturbation methods.


In this paper, we examine the eigenstructure of a two-fluid model, derived from [TOU 961, when regularising terms are added t o it. We think that such a study is of relevance to construct upwind schemes. Unfortunatly, it turns out t o be a difficult task in two-fluid systems. Hence, several approaches are proposed t o get practical approximations of the eigenelements : A numerical algorithm. However, it may be expensive in CPU time for 3D finite volume calculations, although improvments are in progress ([ALO 981). Also intensive use of symbolic calculation packages may help obtain closed form related expressions.

446


A perturbation method using a small parameter. We distinguish -

-

:

the density perturbation method to analyse the eigenstructure when the density ratio is small ( E = 5 . Evaluating Cb from equation [8] and substituting it in equation [7], the net flux J finally becomes:

Equation [ l o ]is the general form for non uniform condition. The expression for uniform condition is obtained setting J = O in [lo], this is equivalent to fix the sediment concentration at the fist grid node. Equation [lo]can be used once the reference level b, and the concentration Cb,m,,, are specified. To evaluate the bed concentration we follow [RIJ 841 which gave the following expression:


479

[Ill in which the mobility parameter T is:

where o'is the standard deviation of effective bed-shear stress, T ~ ,and , ~Tb,C;cr,~ are the instantaneous critical bed shear stress, respectively, along and counter local flow direction, J, and Jz are integrals representing the pick up action of the bed shear stress. The non dimensional grain is:

in which D5,is the particle size, s is specific density (pJp), g the acceleration of gravity and v the kinematic viscosity coefficient (Np). 4. Results

In order to test the proposed boundary condition we compare numerical results with experimental data obtained by [ASH 821 (as reported by [CEL 84]), by [RIJ 811 and [RIJ 851. The first two data set ([ASH 821 and [RIJ 811) refer to experiments in which initially clear water flow, passing over a sand layer, entrains sediment into suspension until the full transport capacity is reached. The second data set ([RIJ 851) refers to measurements of sediment concentration profiles in steep sided trenches. Table 1 contains all the relevant information for the simulated cases.

Table 1. Relevant dutu for simulated cases

480


For sake of simplicity following [RIJ 841 the reference level b has been set equal to 0.01d where d is the flow depth. Inlet boundary (x=O) are of fully developed channel flow. Comparison of calculated and measured sediment concentration profiles allows the assessment of the boundary condition capability to simulate localised effects, furthermore from a practical point of view it is also interesting to verify the capability of the model as a whole to reproduce the total suspended sediment discharge. Figure 1 depicts comparison of calculated sediment concentration profiles with experimental data [RIJ 811, [ASH 821, the general agreement is fairly good. Figure 2a. for the previous data, shows comparison of bottom sediment concentration, it gives an idea of the boundary condition capability to reproduce the pick up action by the flow, particularly noteworthy is the agreement between numerical results and experimental data in the fist three sections where stronger is the flow non uniformity. In order to assess the model capability figure 2b reports comparison of calculated and measured total suspended load, also in this case the agreement is certainly encouraging on the model effectiveness to reproduce real life cases. The capability of the model to reproduce sediment concentration adjustment in case of non constant depth is another interesting test on the way to simulate morphological evolution. Figure 3 and 4 depict sediment concenuation profiles and total suspended load for a steep sided trenches. This case is particularly demanding because the model has to simulate. in a short stretch, both net deposition, in the enlargement zone, where the flow decelerates and net entrainment in the zone where the flow is under a strong acceleration.

5. Conclusions This paper reports preliminary results of a study on boundary condition for sediment concentration. Numerical results shown fairly good agreement with experimental data. Further research is needed in order to assess the capability of the model to reproduce more complex situation as such as three dimensional case and morphological changes. 6. Acknowledgements

The financial support for the research presented in this paper has been provided by a grant of the Italian Ministry for University and Scientific and Technological Research (M.U.R.S.T.) in the framework of the 1997 National Research Project (P.R.I.N.) "Swirling, turbulent and chaotic processes - Water works and Environmental applications".


48 1

7. Bibliography [CEL 881 Celik I. and Rodi W., Modelling suspended sediment transport in non equilibrium situations, Journal of Hyclraulic Engineering ASCE Vol. 114, No. 10 (1988) [JAY 691 Jayatilleke C. L. V., 'Ilne influence of Prand~lnumber and surface roughness on rlze resi.rtance of the Irr~ninursublayer to momentum and heath transfer. Prog. Heat Mass Transfer, Vol. 1, p.193 (1969) [LAU 741 Launder B.E. and Spalding D.B., 7ize numerical compula~ionof turbulent flow, Comp. Meth. in Appl. Mech. & Eng., Vo1.3, p269, (1974). [NAO 971 Naot D. and Nezu I., Wall funclions for the calculation of turbulent 3 0 sediment transport in open channels X X W IAHR Congress Vol. 2 San Francisco USA pp. 1268-1273 (1997) [PAT 721 Patankar S. V. and Spalding D. B., A calculation procedure for heat, mass and momentum transfer in three dimensional parabolic flow.^, Int. J. Heat Mass Transfer Vol. 15 pp. 1787 (1972) [PAT 801 Patankar S. V., Numerical heal transfer and fluid flow, Hemisphere publising corporation, Taylor & Francis Group, New York (1980) [RAU 901 Raudkivi A.J., Loose Boundaiy hydraulic.^, Pergamon Press, 3th edition (1990) [ROD 801 Rodi W., Turbulence models and their application in kydrcutlics, state-of-the-artpaper. International Assoc. for Hydr. Res., Delft, The Netherlands (1980) [SHI 951 Shiono K., Falconer R. A,, Berlamont J., Elzier M. and Karelse M., A note on statified flow in compound channel, Hydra 2000 XXVIth IAHR Congress, 3B8 pp. 134-139 (1995) [RU 841 Van Rijn L.C., Sediment transport. part II: Suspended load transport Journal of Hydraulic Engineering ASCE, 110 (1 1). pp. 1613-1641 (1984) [RU 861 Van Rijn L.C., Mathernutical modelling of suspended sediment in non-unifol-m flows Journal of Hydraulic Engineering ASCE, 112, pp. 433-455 (1986) [VER 951 Versteeg H. K. and Malalsekera W., An introduction to Computa~ionalFluid Dynamics,thefinite volume method, Longman Scientific &Technical (1995)

Figure I . Sediment concentrution profiles

a ) Ashidu [ASH 821 6) van Rijn [RIJ 811

482


........................................................................... Near-Bed Concentration 0

operator is defined as follows;

<x>=

0 x

when when

x x

0.

and


509

The total infinitesimal strain is {A€) = [L]{Au), where {Au) is the incremental displacement.

4. Vertex-based Discretisation Erllploying the constitutive relationship of the previous section in equations (1) and ( 2 ) , and assuming the boundary conditions as described by equation (3) are directly satisfied by the vector {Au), . - the method of weighted residuals can be applied to the equations to obtain the following weak form of the equilibrium equation [ZT89];

where [W] is a diagonal matrix of arbitrary weighting functions. At this point the unknown displacement can be approximated as [ZT89]

{Au)

2

{Ah) =

E[N]~{AB)~ =E[I]N~{AB)~,

(6)

j=1

j=1

where { A u ) ~is the unknown displacement at the vertex j , N j is the shape function associated with the unknown displacement and [I]is the identity matrix. The displacenlent approximation can be introduced into equation (5) if the arbitrary weighting functions [W] are replaced by a finite set of prescribed for each vertex i [ZT89, OCZ941, functions [kV] = Cy=l[W];,

i=l,n.

for

(7)

Equation (7) can be expressed as an incremental linear system of equations of the form [Ii]{Aii)-{ f ) = {O}, where [Ii']is the global stiffness matrix, {Aii) is the global displacement approximation and { f ) is the global equivalent force vector and can be formed from the summation of the following contributions;

[Ii]ij =

l, L,,

Lut lu,

[L w ] ~ [ D ] [ L N dQ ]~ -

+

[W]'tP dl. -

[RW]T[D][LN]~ dl.

[RW]:[D]{~~UP) d r j

and

(8)

5 10


Figure I: 2D control volumes, (a) overlapping F E and (b) non-overlapping FV. where Ri is the control volume associated with the vertex i and is the boundary of the control volume.

ri = I?,

U

rt,

4.1. Standard Galerkin FE Method In the standard Galerkin FE method the weighting function associated with a vertex is equal to the shape function of the unknown associated with that vertex [ZT89, Hir88, OCZ941, [WIi = [NIi. The shape functions describe the variation of an unknown over an element and there can be a number of elements associated with each vertex. Hence, it is apparent that control volumes described by weighting functions of this form will always overlap. This is illustrat,ed in Figure l ( a ) for a simple two dimensional case of two adjacent nodes i and j , where the control volumes Ri and R j have contributions from all the elements associated with their respective vertices i and j . Hence, for the standard Galerkin FE method the contributions as described by equations (8) and (9) are

where [BIi = [ L N ] ; . It is important to note that if the boundary of the control volume, such as that described by ri in Figure l ( a ) , coincides with the external boundary of the domain, the shape functions are not necessarily zero along that part of the boundary. Thus, if a flux is prescribed such as a traction this will not necessarily disappear and may contribute to the equivalent force vector as described in equation (11). Additionally, the symmetrical nature of the stiffness matrix as indicated by equation (10) should be noted. The Galerkin approach is


511

accepted as the optimum technique for treating physical situations described by self-adjoint differential equations, particularly those in solid mechanics, as the inherent symmetrical nature is preserved by the choice of weighting functions [ZT89, OCZ941. 4.2. Vertex-based FV Method In the vertex-based FV method the weighting functions associated with a vertex are equal to unity within the control volume, [WIi = [I],and zero elsewhere. This definition is equivalent to that for the subdomain collocation method as defined in the standard texts [Hir88, ZT891. Though it is important to note that weighting functions defined in this manner permit a variety of possibilities with regard to the control volume definition [OCZ94]. This is because the weighting functions are not restricted to to a direct association with the cell or element as in the Galerkin case. This is an important consideration and requires the recognition of the vertex-based FV method as a discretisation technique in its own right [Hir88]. For the vertex-based FV method the contributions as described by equations (8) and (9) are

It is important to note that the traction boundary conditions can be applied directly as another surface integral, but in the previous Galerkin approach an additional surface element is generally included on the domain boundary. A non-overlapping control volume definition suitable for a vertex-based FV method is illustrated in two and three dimensions in Figures l ( b ) and 2(a), respectively. The Figures illustrate the assembling of vertex-based control volumes from their required sub-control volumes [Tay96]. Additionally, the asymmetric nature of the contributions to the overall stiffness matrix as described by equation (12) does not ensure that symmetry will always be preserved. For this reason FV methods were initially argued as being inferior, but in the light of recent research where different control volume definitions have been proposed, the extent of this inferiority has come into question [OCZ94, Zie95, BC951. 5. Results and Conclusions In this section the vertex-based FV method is applied to a three dimensional validation problem and compared with the standard Galerkin FE method. The non-linear solution procedure adopted in for both these methods is based upon


512

Conuol volurnc

Yf

Sub-conuol

L I

1

--_

_

,'

, /

Vertex

X

InlegraUm pa1nI.i 3

4

Figure 2: (a) 3D assembly of FV sub-control volumes and (b) spherical vessel. that of Zienkiewicz and Cormeau [ZC74, Tay961. Both methods utilised an explicit technique with regard to time stepping of the Perzyna equation (4). It is important to note that the FV solutioil procedure only differs from that of the FE in contributions to the global equivalent force vector and the global stiffness matrix. Hence, allowing an accurate comparison of the two methods [Tay96]. The methods are compared with regard t o accuracy and computational cost. They are also analysed for a variety of meshes with different element assemblies.

5.1. Test case: Internally pressurised spherical vessel For this validation problem a thick walled spherical vessel, consisting of an elastic-perfectly plastic material, undergoes an instantaneously applied internal pressure load. The pressure load is 30 dNmm-', the Youngs modulus and Poisson ratio required to define the elasticity matrix are 21,000 dNmm-' and 0.3, respectively, and the yield stress is 24 dNmm-'. This problem is rate independent and the final solution is equivalent to that of an elasto-plastic analysis [ZC74]. A closed form radial solution is available [Hi150]. Numerically the problem can be modelled in three dimensional Cartesian coordinates, with the displacement componeilts fixed to zero in the relative symmetry planes. The spherical vessel is then reduced to an octant as illustrated in Figure 2(b)'. Examples of meshes consisting of linear tetrahedral (LT),bilinear pentahedral (BLP) and trilinear hexahedral (TLH) elements are illustrated in Figures 2(b)', 2(b)3 and 2(b)4, respectively. Firstly, the problem was analysed with a series of meshes consisting of TLH elements. The hoop stress profiles, along the radii, as obtained from one of the numerical analyses are plotted and compared against the reference solution in Figure 3(a). The profiles illustrate the stress in the plastic and elastic regions, and the radial extent of the plastic region according to the analytical solution. The close agreement of the two methods is illustrated. However, it


IS Oee 10

10 Hoop 8-1

e

Hoop s(rcs.

5 (Wmm-=)

e

5 13

rfi

0

I

150 Radial dialrow (m)

Radial dirlrocc lmm)

Figure 3: (a) 950 TLH and (b) 4,800 LT elements. is important to note the closer agreement between the reference solution and the FV method when a coarse mesh is employed. These observations may be associated with the higher order, trilinear nature of the elements employed in the three dimensional analysis at this stage. With regard to the FV method, the implementation of pressure loads (tractions) will involve bilinear face elements for T L H elements. Hence, when considering the application of pressure loads for the two methods as described in equations (11) and (13), the contributions are different due to the individual weighting technique associated with each method. Furthermore, the weighting technique employed for the FV method may be more complementary, when applied generally, as all the terms are integrated conservatively at a local level. Conversely, for the FE method the weighting is not locally conservative which may introduce errors when pressure loads are employed. These conclusions are tentative and rely on the interpretation of the present observations, but they agree with the findings of other researchers [Whe96] and strongly suggest that further research of the FV method is worthwhile. Secondly, the problem was analysed with a series of meshes consisting of B L P elements and there was much closer agreement between the methods [Tay96]. This is attributable to the lower order, bilinear nature of the element concerned and the linear nature of the triangular faces over which the pressure loads were applied. As illustrated in Figure 2(b)3 the BLP elements are orientated so the pressure load was prescribed over a triangular face. This was an outcome of the automatic mesh generator employed [Fem] and it is possible to further study the element when pressures are applied to the bilinear, quadrilateral faces, though it was not studied in that research. Thirdly, the problem was analysed with a series of meshes consisting of LT elements. The hoop stress profiles from one of the analyses are plotted in Figure 3(b). There is complete agreement between the methods with regard to LT elements as the global stiffness matrices and global force vectors constructed by the two methods are identical. This is a consequence of the linear nature of

5 14


Figure 4: (a) CPU times on a SPARC 4, 11OMHz. both the element concerned and the triangular faces over which the pressure is applied. It is possible to demonstrate this equivalence analytically [Tay96] by extending to three dimensions, a two dimensional analysis which has been applied to elastic problems involving linear triangular elements [OCZ94]. Finally, the methods were compared with regard to computational cost. Considering LT elements, as the matrices are identical and symmetric a conjugate gradient method (CGM) is applicable in both cases. As illustrated in Figure 4(b), the FV method (FV-CGM) requires more CPU time than the FE method ( F E C G M ) even when the same linear solver is employed. This is expected as the FV method visits six integration points, while the FE method visits a single Gauss point when adding contributions to the linear system of equations [Tay96]. Considering TLH elements, the geometrical nature of this validation problem prohibits an orthogonally assembled mesh. Hence, for the FV method a bi-conjugate gradient method (Bi-CGM) is required due to the asymmetric nature of the coefficient matrix obtained [Tay96]. Conversely, for the FE method a CGM is sufficient as the matrix obtained is symmetric. These requirements agree with the discussions in the previous section. As illustrated in Figure 4(a), the FV method (FV-BiCGM) requires approximately twice the CPU time as the FE method (FE-CGM). This is also expected due to the computational requirements of the two different linear solvers employed. Also for TLH elements, the FV method visits twelve integration points per element, while the FE method visits eight Gauss points per element. Hence, it can finally be concluded that any improvement in accuracy obtained by employing the vertex-based FV method must be offset against the extra computational cost required. Bibliography [BC95]

C. Bailey and M. Cross. A finite volume procedure to solve elastic solid


5 15

mechanics problems in three dimensions on an unstructured mesh. Int. Journal for N u m . Methods i n Engg., 38:1757-1776, 1995. [DM921

I. Demirdzic and D. Martinovic. Finite volume method for thermo-elastoplastic stress analysis. Computer Methods i n Applied Mechanics and Engzneering, 109:331-349, 1992.

[FBCLgl] Y.D. Fryer, C. Bailey, M. Cross, and C.-H. Lai. A control volume procedure for solving the elastic stress-strain equations on an unstructured mesh. Appl. Math. Modelling, 15:639-645, 1991. [Fem] [HH95]

Femview Ltd., Leicester, UK. F E M G E N / F E M V I E W .

J.H. Hattel and P.N. Hansen. A control volume-based finite difference method for solving the equilibrium equations in terms of displacements. Appl. Math. Modelling, 19:210-243, 1995.

[Hi1501

R. Hill. The Mathematical Theory of Plasticity. Clarendon Press, Oxford, UK, 1950.

[Hir88]

C. Hirsch. Numerical Computation of Internal and External Flows: Fundamentals of Numerical Discretisation, volume 1. John Wiley and Sons, 1988.

[OCZ94]

E. Onate, M. Cervera, and O.C. Zienkiewicz. A finite volume format for structural mechanics. Int. Journal for Num. Methods i n Engg., 37:181-201, 1994.

[OH801

D.R.J. Owen and E. Hinton. Finite Elements i n Plasticity: Theory and Practice. Pineridge Press Ltd., Swansea, UK, 1980.

[Pat801

S.V. Patanker. Numerical Heat Transfer and Fluid Flow. Hemisphere,

[Per661

P. Perzyna. Fundamental problems in visco-plasticity. Aduan. Appl. Mech.,

Washington DC, 1980. 9:243-377, 1966. [Tay96]

G.A. Taylor. A Vertex Based Discretisation Scheme Applied to Material Non-linearity within a Multi-physics Finite Volume Framework. PhD thesis, The University of Greenwich, 1996.

[Whe96]

M.A. Wheel. A geometrically versatile finite volume formulation for plane

[Whe99]

M.A. Wheel. A mixed finite volume formulation for determining the small strain deformation of incompressible materials. Int. Journal for N u m . Methods i n Engg., 44:1843-1861, 1999.

[ZC74]

O.C. Zienkiewicz and I.C. Cormeau. Visco-plasticity-plasticity and creep in elastic solids-a unified numerical solution approach. Int. Journal for Num. Methods in Engg., 8:821-845, 1974.

[Zie95]

O.C. Zienkiewicz. Origins, milestones and directions of the finite element method - a personal view. Archives of computational methods in Engg., 2:l-48, 1995.

[ZT89]

O.C. Zienkiewicz and R.L. Taylor. The Finite Element Method: Volume I : Basic Formulation and Linear Problems. Magraw-Hill, Maidenhead, Berkshire, UK, 1989.

elastostatic stress analysis. Journal of strain analysis, 31(2):111-116, 1996.

Control volumes technique applied to gas dynamical problems in underground mines

Elena Vlasseva Associated Professor Department of Mine Ventilation and Labour Safety University of Mining&Geology "St. Ivan Rilski" Sofia 1100, Bulgaria

ABSTRACT. Paper presents an application of Control Volumes' Method in the field of underground mining. Modeling deals with noxious gas distribution (concentration) it1 time and space along mine roadway or network. Models take into account variable velocity, density and outside gas inflows in mass balance of airflow. Different source functions, which due to technological or accidental reasons, provoke occurrence of transient process, are presented in the paper. Numerical and computer models are verifed with real data, obtained unfortunately from great disaster, which took place in one Bulgarian underground coal mine in 1997. Models serve as a tool used to reveal the circumstances lead to that disaster. Modeling results might be also applied to help introduction into practice of engineering solution, which can be analyzed due to their practical implementation. Key words: control volumes; gas distribution; underground mining

1.

Introduction

Air flows in underground mines are of great importance for human lives underground. Normally flows transfer not only fresh air but different impurities, liberated from rocks or as a result of mining operations. Keeping of these impurities into safety limits is one of the main task for underground staff. Once liberated into underground air, gas is distributed along ventilation paths in difficult to predict ways and time duration. Gas might be observed on unlikely for common understanding places. By that reason preliminary knowledge about gas dynamics is extremely valuable. One possible approach might be real mine experiments. This approach however faced great difficulties due to problems with simulation of realistic gas liberation characteristics. Problems with control of such simulation cannot be neglected as well. All the above points show that mathematical and computer

5 18


modeling can be of favor in prediction of gas distribution along ventilation paths. They can reveal specific relationship between gas liberation and its influence on underground air flows.difficu1ties with control and safety of the experiment. In that connection mathematical and computer modeling in close with reality decsription is very important and liable technique. It can reveal specific relationships between gas source and its influence on ventilation system and its important parameters. The process of gas liberation and distribution in a single path and its mixing with fresh air is presented by convection-diffusion equation with source function, varaible velocity and density of air gas mixture. It is solved simultaneously with continuity equation. Control volumes method with exponential scheme is applied for numerical modeling.. Errors due to discterization and limits of method for this class of problems are outlined. Solutions for one ventilation path are addapted to a network modeling where interrelation between air flows with different characteristics are of a great importance to the whole process of safety working conditions in mine. Results of modeling are validated with practice in the following ways: > it was used as investigation tool for one methane explosion which took place two years ago in Bulgaria. Revealing of most likely reasons for its occurrence became possible; P it was also applied to in one case study for inertization with nitrogen of a typical mine configuration. This technical activity is performed to supress mine fire development.

1.

Governing equations and source functions

Let us assume mine roadway with characteristics shown on figure 1. Air flow is well developed in direction of coordinate s. Length of roadway (L) is times greater compared to its width or cross sectional area (f) and this give a ground to present mathematical model into one dimensional way. Independent variables are: s - along the length of mine roadway, m; 7 - time for tracing the process, s.

rr;i0 ulrl -

c . ~ )

I

-

.-

woo

c,ir)

C/O.

N

.. .

2

x,

-P

&'

-L

1

U

Figure 1. Physical model.

I

I

I _r 1

L

(-

5

r:

Figure 3. Discretization of calculation area

Known functions are: u ( s , ~ -) velocity of air gas mixture, rnls;


5 19

q(s,z) - gas liberation, m3/s; p(s,2) - density of air gas mixture, kg/m3. Unknown function is C(S,T) - concentration of gas impurity into air gas mixture. Gas distribution in thus describe physical model obeys mass conservation for gas impurity and air and equation of state for air gas mixture:

a~ +-~ ( P u )-- P, q -

as

a7

(1)

f~

a ( ~ C ) a ( ~ u C ) - D, 37 as P +

a2

(puc)

as2

-

P, (4 f~

(2)

(3) =In the above written equations index g referred to gas impurity, index m - to air gas mixture. Initial and boundary conditions are given with the terms: C(s,O) = Co(s) (4) Pm

ac I.;r =O

C(0, z) = CH( T ) ;

(5)

ax

qk qL=2 e

a

Source functions q(s,z) which present gas liberation or provocation of transient process are presented on figure 2. Their characteristics may change either in time or in space or combinations of both. They are chosen on typical behavior of gas sources: a - gas liberation from walls or during coal transportation; b, c - gas liberation from already mined areas; d - point gas source; e - gas liberation due to repeatable operations (diesel power, blast work); f - gas liberation from mined areas with increase of its surface; g - gas desorption from newly opened surfaces.

qk* qk qLrqL q l ~qL > b

f

L

C

d

L

g

h

Figure 2. functions

Typical

source

520


Combinations of above presented source functions are also possible. For istance: a + d - point gas source with constant emission rate in time; c + f - distributed along the length source with variable in time emission rate; b + g - linearly distributed source along the length with pick in time and then gradually decreasing emission rate; d + e - point source with periodically changeable emission rate. Presented on figure 2 source functions can be easily approximated with polinom or with sum of functions. Numerical treatment of above presented mathematical model - expressions (1-3) plus (4-5) makes possible taking into account all written into model variations.

2. Numerical and computer modeling 2.1. Schemes, mesh-type, discretization

Mathematical model (1-5) is solved by application of Control Volumes Method [PAT84]. Presented on fig. 1 physical model is transformed into regular mesh of control volumes (fig. 2) with size As. They are defined by mesh points and control volumes boundaries. Number of points are Concentration

C(S~,T~)is

defined

I:,;

N, = - + 1 . in

mesh

points,

air-gas

flows

?

Exponential profile [PAT84) for concentration variations is assumed. Following the above points numerical analogue, binding three adjacent points (W,P,E) can be written in the way:

where:

e D

e D

Upper index 0 referred to a previous time step. Boundary conditions (4-5) are transformed into numerical schemes in the same way by assuming first order boundary condition at left boundary (s=O) and second order - at the right boundary.


521

Full procedure on transformation of differential problem into numerical schemes is given in [VLA93]. Having expressions (7) for each three adjacent points linear system with three diagonal matrix for all points i ~ ( t N,) 1 is obtained. Its solution under TDMA gives concentration in each mesh point and in any time moment C(si,rj).

2.2. Approximation, stability, convergence, errors, application limits Numerical schemes (7) approximate differential problem (1-5). This statement was proved by application of Taylor's series. Approximation error is from first order in regard of T and from second order in regard of s. Numerical scheme (7) is absolutely stable. It was proved by applying Matrix criterion [SHISS] by presenting numerical scheme in way from one time layer to the next one and examining eigen vectors and their eigen values [SHI88]. Convergence is a concequence from approximation and stability, following Lax theorem [SHI88]. I I Applicability limits to gas dynamical problems are investigated. For most common parameters - velocities, lengths etc. error analysis was performed. On figure 4 relative errors (numerical/exact solution) in regard of

u AT As

Cu number ( CU = -).

lc-A

10.'

1

!i

Figure 4. Relative errors due to discretization

3.

umber of

mesh points, even limited to the minimal range (3) lead to 8% errors, which for purpose of gas dynamical problems is agreeable. Physically unlikely results has not been obtained.

Modeling validation with practice

Presented herewith mathematical model and its numerical interpretation reach their computer realization in both aspects: for a single roadway or set of sequential ones [VLA93]; for a complex network, where solutions for a single roadway were harmonized at places where they cross each other [STE87]. Modeling was applied in numerous engineering problems from practice. Two of them are presented in this paper: investigation of one methane explosion; technological solutions for inertization with nitrogen of fire zone. 3.1. Methane explosion


522

Severe methane explosion took place on the 1-st of September 1997 in "Ivan Russev" coal mine. Operations in one section were canceled due to annual holidays from IS' August to 1'' September. As a result great amounts of methane were accumulated. Degazation (transfer of this gas out of mine) was to be performed. Mine does not have remote control system and no data (besides miners1 evidences) was available. As a result of incorrect activities of miners, performing degazation, explosion took place causing death for more than 10 workers.

/if

to Ill section

Q3

+

Figure 5. Path in mine for distribution of methane during degasing Investigation about circumstances [MIC98] was performed by application of

U

428

o

t

o

2

J

~

@

Y

11mrm

3

m

m

m

w

1

l I

10

'I

10

75

93

l l m e mln

Figure 7. Concetration at the entrance of Figure 6. Air volumes at the entrance of observed path observed path Changes in cross sections can be seen there. Transient process was initiated by variable air flow (figure 6) and methane concentration (figure 7) at starting point 0. They reflect workers actions, such as: variable cross-sections along the route (fig. 11); assembling of ventilation curtain to direct higher air flow towards the gassed section; switching on/off of booster's fan operation, causing changes in ventilation conditions (air volumes

Q, ,Q2 and Q,


523

variable methane concentration (C2) of in-flowing air into 503 crosscut (Q2)

70 6~

so h

40

:, ,'I I

t

20

10

Or,

ZO

1)

U1

-

a

TR.

KO

lcx,

120

Explosion took place between points 5 and 6 and modeling results (figure 8) show the same - high methane concentration with more than 7 minutes duration stay at the place of explosion. Unfortunately ignition source was also available.

Figure 8. Concetration in time of observation in some points of path

3.2. Inerting with nitrogen of fire zone In some cases during mine operations evidence for [MIC97] development of mine fire can be observed. Then one of anti-fire measures is to inertize atmosphere at a danger zone so that oxygen does not be available to support burning. The model presented herewith can deal with one impurity into air-gas mixture. Inerting of air however presumes more than one ipurity, namely: methane inflow from mined zone (points 5-6 on figure 9) and from mine workings (points 1-2-3-4 on figure 9); oxygen from ventilation flow and from injected technical nitrogen; nitrogen from air and from injected technical nitrogen. In order to evaluate concentration of the above mentioned three gas components via a model constructed for a single component, the author has applied consecutive diffusion mixing. It pressumes appropriate definition of transporting medium and impurity as well as suitable presentation of gas sources ~ ( s , z.) Calculational passes three stages (methane release and distribution, nitrogen outflow from gob area, nitrogen injection at a given place in the panel and its further distribution in the area which must be inert). For any of these stages computer modulus were developed - METHANE, INERT-GOB and NITRO. Common input data for the three computer programs are geometric characteristics of mine workings (fig.9). The three programs interact and their incorporating in the total inverting strategy makes it possible the composition of general program procedure INERTIZATION r & 12 4 3 . 2 Numerous solutions were performed, 4 corresponding to specific fire situations. rtant parameters were needed to inert the observed 4

4".

34

3

Figure 9. Object for inertization

concentration of flammable gas in all points of observed area (this


524

information is very important in order to keep the atmosphere out of explosion); effectiveness of operations performed on fire supression. On figure 10 is shown methane concentration along the path with length 1500 m. Figure 11 shows time delay in inertization from point 2 and 3.

--

0

U E. u

I2

h OU

-

--

Elapsed T i m e , m l n

i)

Figure 10. Time delay in inertization

h)

L e n g t h (I-12-2-3-34-4), m

Figure 10. Methane concentration

4. Bibliography

[VLA93]

[MIC98]

[MIC97]

[PAT841 [SHI88] [STE87}

Vlasseva E.D., Mathematical Modeling of Convection-diffusion Processes in Underground Mines, Ph.D. Thesis, UMG, Sofia, 1993, pp. 200 Michaylov M.A., E.D.Vlasseva, Simulation Analysis of Methane Explosion, Second International Symposium on Mine Environmental Engineering, July 29-31 1998, Brunel University, UK,p. 1-16. Michaylov M.A., E.D.Vlasseva, Modeling of Preventive Nitrogen Inertization in Underground Mines, 1.5'~Mining Congress of Turkey, 6-9 May 1997, Ankara, pp. 203-210. Patankar S., Numerical Methods in Heat Transfer, Sovremennoe Mashinostroene, Moskow, 1984, p. 150 (Russian translation) Shi D., Numerical methods in Het Transfer Problems, Moskow, Mir, 1988, p. 544 (Russian translation) Stefanov,T.P., E.D.Vlasseva, E.E.Arsenyan, Unsteady Gas Flows in Mine Ventilation Networks, 22 International Conference of Safety in Mine Research Institutes, Beijing, China, oct.1987,pp. 115-124.

Simulation of salt-fresh water interface in coastal aquifers using a finite volume scheme on unstructured meshes B. Bouzouf, D. Ouazar LASH, EMI 14 Av. Ibnsina Rabat, Maroc I. Elmahi IVG, University of Duisburg, Germany

ABSTRACT This paper is devoted to the numerical study of seawater intrusion into coastal aquifers. The cell-centered finite volume method is adopted here to solve the set of simultaneous partial differential equations describing the motion of saltwater and freshwater separated by a sharp interface. These equations are based o n the Dupuit approximation and are obtained from integration over the vertical dimension. I n order to have fEexibility upon complex configurations domain, non structured grid meshing is utilized. To approximate the diffusion fluxes, GreenGauss type reconstruction, based on Diamond cell and least square interpolation, is performed. The model is first validated using academic test case studies with known close from solutions. A real case study concerning the Gharb aquifer in North West of Morocco is carried out to show the overall trend of saltwater fronts.

Key Words: Coastal aquifers, seawater intrusion, finite volumes, unstructured meshes, Green-Gauss reconstruction. 1. Introduction The modelling of groundwater in coastal aquifers is an important and difficult issue in water resources. The primary difficulty resides in efficient and accurate simulation of the movement of the saltwater front. Freshwater and saltwater are miscible fluids and therefore the zone separating them takes the form of a transition zone caused by hydrodynamic dispersion. For certain problems where the transition zone is relatively small compared to the aquifer extent and thickness, the simulation can be simplified by assuming that two fluids are immiscible and separated by a sharp interface (sharp interface model). This last assumption, together with the Dupuit approximation, permits the integration of the equations in the vertical direction [BEA 791. The objective of this paper is to present a cell-centered finite volume based approximation to calculate the position of the sharp interface. This class of methods

526


is becoming one of the commonly used techniques for partial differential equations in engineering calculations and computational physics. Their popularity is due mainly to their ability to faithful to the physics conservation and the possibility of solving the problems on complex geometries. The diffusion contributions here are approximated by using Green-Gauss type interpolation. This technique is found to be very robust, it can be used on general mesh, not satisfying necessarily Delaunay condition on the triangulation. Time integration is performed by an explicit Euler scheme in order to keep the memory requirement reasonable. 2. Mathematical Model We assume here that the saltwater and freshwater are separated by a sharp interface, thus two domains are considered. For each flow domain the equation of continuity may be integrated over the vertical dimension reducing the determination of the position of the interface to a 2D problem. The system of equations can be written as follow ([ESS 901):

o f and R,

represent here the fresh and salt water flow domain respectively, K f z and K S z (respectively K f y and K S y )are the hydraulic permeabilities in the fresh and salt water in x-direction (respectively in y-direction), h f and h, are the heads, B f and B, are the thickness of fresh and salt water zone and n is the porosity. Pf , where p f and p, are the specific weights in fresh We note also by b = Ps - P f and salt water, and by 0 for confined aquifer 1 for unconfined aquifer Invoking continuity of the pressure at the interfaces, the interface elevation can be calculated from the freshwater and saltwater heads by

The system (1) represents two coupled parabolic partial differential equations which should be solved simultaneously for the freshwater head ( h f ) saltwater head (h,). Once these values are known, the interface elevation ( C ) can be obtained from (2). The set of the equations (1) can be written in the vectorial form:


with W = ((Sf Bf

+ n(a. + 6))hf - n ( l + 6)h,,

(S,B,

+ n ( l + 6))h,

-

527

nbhf)t

3. Finite volume discretization To solve the system of equations (3) we have considered a triangular cellcentered finite volume formulation ([EBVGPH 9911, where the state variables Wp are the average values for the cells a t time level n:

Integrating eq. (3) on a control volume Ci yields in explicit formulation:

w;+'

=

w:+

At

[R,(Wn) n,

+ R, ( W n )n,] d a + Area(Ci) S(W:)

(4) where A t is the time step. The discretization of equation (4) requires the approximation of terms such as

l,,

( B ~ K L , ,,do ~ )

where 1 = f , s and

and

l,,(& 2) K'

n~ do

rij is the interface separating two cells Ci and C j .

Y. Coudiere et al. [CVV 961, have studied an elliptic problem -div(AVu) = f ulr = 9

in St C lR2 over I' = ail

(5)

Where A is a symmetric definite positive matrix with coefficients a,j in C1 (01, f E CO(R) and g E C2(I'). They have used a Green-Gauss type interpolation to construct the gradients at the interfaces of the mesh. The gradient on each edge is approached by the Green theorem and then a first order Gauss quadrature formula, for which requisite values at the vertices P are interpolated from the states on the neighbourhood of P. The weak consistency of this scheme was

528


proved under some assumption on the weights of the interpolation. We took inspiration from this scheme for devising our numerical procedure and discretize the diffusive contributions. We begin firstly by writing

One constructs the co-volume Cdec centered at the interface F i j and connecting the barycenters Gi and G j of the triangles that share this edge and the two endpoints N and S of rij (see figure 1).

Figure 1: Diamond shaped co-volume

ah1 To calculate - - I r i j , the divergence theorem is applied to the co-volume ax surrounding rij,which gives

Cdec

represents an edge of the co-volume Cdec and n,, is the axial component of the outward unit normal to E .

E

If we note by e = [ N 1 N2], , One can write also

Where h l N , and hlN, are respectively the values of hl on the nodes Nl and N2 of the edge E . The data a t the centers Giand G j are known exactly while the data at the vertices N and S must be determined by some interpolation procedure. For one node P


529

of the mesh, one utilizes a linear approximation v of hl on the set of cells which share the vertex P.

Where V ( P )is the set of triangles K surrounding P, h l K the head a t the center of triangle K and Q K ( P )are the weights of the interpolation corresponding to the node P. In order to ensure the consistency of the scheme, the weights of the interpolation are calculated using a least square procedure (see [CVV 961 for details). 4. Model validation To verify and validate the numerical solution obtained from the finite volume model, numerical simulations have been compared to existing analytical solutions. 4.1 Steady state

Two cases have been checked: confined and unconfined aquifer. For both of them the initial values of hf and ( are arbitrarily fixed. The analytical solutions are as follows: Unconfined aquifer ([VER 681, [VN 751):

with qo = 1 0 m 2 / d a y ,

P

= 0.741 and K = 6 0 m l d a y .

Confined aquifer ([GLO 591, [RH 621):

with go

=

5 m 2 / d a y and

zt = - l o r n .

4.2 Unsteady state Keulegan [KEU 541 presented an analytical solution for the interface in a confined aquifer of uniform thickness:

with D = 1 0 m

,

n = 0.3 and K = 39.024mlday.


Figure 2: Comparison of analytical and numerical solutions for unconfined aquifer.

Figure 3: Comparison of analytical and numerical solutions for confined aquifer.

Figure 4: Comparison between analytical solution and finite volume method.


53 1

The numerical solutions are in good agreement with the analytical solutions as depicted in figures (2), (3) and (4). 5. Application to the Gharb basin, Morocco

We have applied the finite volume based model to the Gharb aquifer which is located in North West of Morocco. The surface area of the coastal Gharb is estimated to about 4000 km2. It belongs to the structural domain of the Morocco Atlantic plain.

Location of the Gharb basin In figure ( 5 ) , the areas of pumping in the Gharb aquifer are depicted. Figure (6) shows the actual front saltwater corresponding to pumping schemes of Figure (5). 6. Conclusion

Characterization of certain coastal aquifer systems may be accomplished by assuming that saltwater and freshwater are separated by a sharp interface. Invoking the Dupuit assumption and performing a vertical integration results in quasi-three-dimensional, the equations may be solved to give freshwater head, saltwater head and interface elevation. Cell-centered finite volume scheme on a unstructured mesh is used to approximate the partial differential equations. Comparisons of the finite volume approach adopted in this paper, with known analytical solutions have shown close agreement. The model was also applied to a real case concerning the Gharb aquifer in North West of Morocco.

BIBLIOGRAPHY [BEA 791

BEARJ., Hydraulics of groundwater, McGraws-Hill, New York, 569 pages, 1979.

[CVV 961

COUDIERE Y ., VILAJ . P. A N D VILLEDIEU P ., Convergence of a finite volume scheme for a diffusion problem, F . Benkhaldoun

532


3

4

.

8 4.2 4.4 4 . 6

1

3.8

4

4.2 4.4 4.6

I o5 I o5 x(m> Fig.5 Areas of pumping in GHARB Fig.6 Simulated freshwater and saltwater xcm)

aquifer

interface

and R. Vilsmeier eds, Finite volume for complex applications (Hermes, Paris), pp. 161-168, 1996. [EBVGPH 991

ELMAHII., BENKHALDOUN I?., VILSMEIERR., GLOTHO., PATSCHULL A. A N D HANELD., Finite volume simulation of a droplet pame ignition on unstructured meshes, J . of Comput. and Appl. Math., Vol 103, 1, pp. 187-205, 1999.

[ESS 901

ESSAIDH. I., A quasi-three-dimensional finite difference model to simulate freshwater and saltwater flow i n layered coastal aquifer systems, U.S. Geological survey Water-Resources Investigations, Report 90-4130. Menlo Park, California, 1990.

[GLO 591

GLOVERR. E . , The pattern of freshwater pow i n a coastal aquifer, J . of Ground Water Resour., 64, pp. 439-475, 1954.

[KEU 541

KEULEGAN H. G., A n example report on model laws for density current, U.S. Natl. Bur. of Stand., Gaitherburg, Md, 1954.

[RH 621

RUMMERR. R. AND HARLEMAN D. R., Intruded saltwater wedge i n porous media, U.S. Geol. Surf. Prof., Paper 450-B, 1962.

[VER 681

VERUIJTA., A note o n the Ghiben-Herzberg formula, IASH bull. 13, pp. 43-45, 1968.

[VN 751

VAPPICHA V. N. AND NAGARAJA S. H., Steady state interface i n coastal aquifer with a vertical outfEow face, National Symposium on Hydrology, Rurkee, India, 1975.

Progress in the flow simulation of high voltage circuit breakers X. Ye, L. Miiller, K. Kaltenegger and J. Stechbarth ABB High Voltage Technologies Ltd., 5401 Baden, Swilzerland ABSTRACT In this paper progresses in thephysical and numerical modelling, which lead to inzprovement in the accuracy and capability of simulation for capacitive switching design oj circuit breakers, are introduced Numerical results and measurement results are conzpared and discussed One inzportant progress lies in the successful treatment of the artificial viscosity. To maintain the numerical stability but at the same time to keep the artrJicial viscosity so small that the physical viscosity is not distorted, the upwind biasing essentially local extremum diminishing (ELEDJ scheme has been adopted and improved. Another progress is associated the moving grids technique, where additional terms have been added to the governing equations for the moving grid without deformation and new grid lines are added or renzoved in the deformed moving grid by the solver in the progress of the calculation. The Chimera boundary interpolation method is used to enable the communication between two blocks with relative motion. Key Words: circuit breaker, moving grids, upwind biasing scheme, Chimera boundary

1. Introduction

In a high voltage circuit breaker a gas with good dielectric and thermodynamic properties such as SF6 is used to extinguish the electric arcing which occurs as electric contacts move apart. The current interruption can only then be realised. The capacitive switching, i.e., the current interruption at high voltage but with low current, represents one important case among various cases of tests and design. The ability to perform capacitive switching without electric breakdown is one of the defining parameters for the speed of the circuit breaker and therefore strongly cost relevant. Furthermore, only the density of gas @) and its electric field strength (E) play an important role in gas breakdown between the contacts of the circuit breaker (s. Fig. 4). As the criterion of the ratio of E/p for gas breakdown is well known, CFD can be employed to simulate the flow field and to subsequently produce the distribution of gas density in the circuit breaker during the design. It is obvious that for such simulations, a high degree of accuracy of flow calculation is required.


Since later 1980's researchers and engineers have begun to apply CFD tools in the development of circuit breakers. Most of them, e.g. [CLA 971, concentrated their efforts on using the flow simulation with implemented arcing models to investigate the ground flow effects in circuit breakers, e.g., pressure build-up in pressure chamber, plasma jets, and to identify the limit and capability of CFD tools. Their efforts to verify CFD tools were, however, constrained mainly in the comparison of the simulated pressure build-up in the pressure chamber with experiments. There are only few works (s. [TRE 911) done for enhancing the accuracy of CFD tools for predicting dielectric strength in capacitive switching and for verification of CFD tools in the regions of nozzles and electrode contacts. As a result, the ability and accuracy to predict the dielectric strength of a circuit breaker with CFD tools have been not satisfactory. The following two points remain unclear. 1) How can the moving parts be treated correctly and which numerical methods are to be introduce to obtain a sufficient high accuracy? 2) How can a CFD tool be verified for predicting dielectric strength in capacitive switching? In this paper, these questions are addressed, with concentration on the simulation of cold SF6 gas with moving electric contacts and further on the prediction of the dielectric strength of a circuit breaker during the capacitive switching. The cold gas simulation is not as trivial as to be anticipated, because the low energy level of cold gas forces a CFD code to consider, with sufficient accuracy, all macroscopic and microscopic flow effects, examples of which are: flow separation, influence of wall and viscose layer and their transient evolution, turbulence transport and suck effect caused by moving contacts. Further, to guarantee the numerical stability, artificial viscosity must be introduced, and hence a sophisticated scheme must be used to keep the artificial viscosity so small that it can maintain the numerical stability effectively but does not confuse the physical viscosity on a viscose layer. A sophisticated moving grids technique has to be introduced both for obtaining a high accuracy and for performing an efficient computation. In section 2, the mathematical models and numerical methods, such as ELED scheme, moving grids technique and Chimera boundary, are introduced. In section 3, examples are presented and discussed for flow simulation where the code verified through the comparison with experiment. Subsequently, the method and example are introduced for coupling of the flow field and the electric-static field in circuit breakers with the consideration of the influence of roughness. Finally, in section 4, our results are concluded with further improvement suggested. 2. Mathematical models and numerical methods

2.1 Mailtematical models

For the fluid flow in a circuit breaker it is necessary to use the complete NavierStokes equations in their time averaged form, i.e. the so called Reynolds equations


535

with an adequate turbulence model. The fluid flow will be then governed by the Navier-Stokes conservation equations for mass, momentum and energy. These equations have the differential form as shown in eq. [I].

with the variables

with the shear stress T and heat flux J a s follows:

where p,, p, are molecular and turbulent viscosity, A heat conduction coefficient, O temperature, Pr Prandtl number, V Nabla operator, Re Reynolds number, W velocity vector. These equations can be transformed to curvilinear coordinate system with the transformation {={(x,y) and v=q(x,y). To close the equation system, the standard k-E turbulence model (s. [LAU 741) was used to obtain the turbulent viscosity p,.

2.2 Numerical methods

The basic equations are solved with a multi-block finite volume Runge-Kutta multisteps time-marching method. The code (HT206) was previously applied for fluid flow in turbo systems (s. [SCH 911 and [SCH 981) and has been extended for the development of circuit breaker. The numerical methods of the code, which are relevant to the flow simulation of circuit breaker, are described as follows: 2.2.1 ELED scheme

The spatial discretisation of the present code is based on a high order non-oscillatory scheme, which consisting of central discretisation and artificial viscosity in the following form (s. [JAM 8 11):

536

Finite volumes for complex applications E(4)

where q denotes the variables in the equation [I] and E and I+: are the coefficients of the scheme. If the formulation of these coefficients is taken from [JAM 811, it is then the classic Jameson-Schmidt-Turkel (JST) scheme. A more advanced formulation of these coefficients is introduced in the present code, that is the essentially local extremum diminishing (ELED) scheme of [JAM 941: ,+;

with

Q I + - , =R(Aq I + :,,Aq 1- , ) '

C

R(% v)

a

2

1

The valuation of the wave speed I + , in 4 and q direction here will be treated generally for variable q overall in the flow field as

where f and g are convection terms in 5 and q direction, U and V are defined in eq. [3] and [4]. Based on our experience the numerical constants r, E, C, and C2 can be , C2=1.5 - 8.0. This formulation is defined to be selected as ~ 1 . 5~, ; 1 0 " ~CI=2.0, scheme which satisfies the condition that in the limit as the mesh width Ax + 0, local maxima are non-increasing, and local minima are non-decreasing.

2.2.2 Moving grids technique To simulate the flow field containing electrical contacts moving during the current separation, two types of moving grids are generally used. The type I moving grid moves only its form and the number of grid will be not changed as shown in Fig. la. In contrast with the type 1, the type I1 moving grid will be expanded or compressed by adding new grid lines to or removing existing ones from left orland right side as shown in Fig. lb. Some methods have to introduced to treat both types of moving grids correctly. additional grid ITS inserted Chimera

a. Type I: whole grid moves

b. Type II: only the grid line on grid side moves

Figure 1. Two types of moving grids


537

2.2.2.1 Treatment of moving grid type I

In the type I moving grid, the variables will be kept in the same cells during the movement at a time step. This will produce error if no additional measures will be taken, because the positions of these values in the flow field have been changed through the moving of the grid cells. To avoid such errors, additional terms in eq. [ l ] must be introduced. In the case that a block of grid moves with a speed of x, and y, as its x and y components, the velocities U and V in the curvilinear coordinate system with 5 = t(x,y,t) and q = q(x,y,t) will then be (s. [STE 781):

cr

tr

where =-xkx - yJy and =-xJX - y J y , then the convection term in eq. [l] after the transformation to the curvilinear coordinate system is

2.2.2.2 Treatment of moving grid type II

For the type I1 moving grid, the movement only involves the two side grid lines, i.e., almost all the cells of this grid are not in motion, therefore, there is no need to apply the additional terms. However, because of the change in the cell size at every time step, there would be implicit unphysical energy and mass loss as grid compresses and unphysical energy and mass increase as the grid expands, if the values of energy and mass after the change of cell size were taken simply from the old before the change of cell size. Therefore, sophisticated treatment of the value of energy and mass is very important to get the accuracy of calculation. As grid expands or compresses, the value of the density and the total energy in the changing cell after the change of cell size will be then

where x, ,-xi is the cell size, new and old denote the values after and before the change of cell size.

538


2.2.3 Chimera boundary

Because of the movement relative to neighbouring grids, the moving grids will have discontinuity of grid line at interfaces with their neighbouring grids. Hence, the Chimera interpolation method (s. [STE 871) will be applied to enable the information exchange between two grids with discontinuity of grid lines. The basic concept is that two layers of ghost cell of a block will overlap with the neighbouring block. To obtain values in the ghost cell, the values of the nearest cell centres of the neighbouring block are interpolated to the cell centre of the ghost cell (s. Fig. 2). The values in the ghost cells can then be used for flux building. Same as the central discretisation, the Chimera interpolation is non-conservative, however, the upwind character is obtained through the ELED scheme.

Figure 2. Overlapping blocks for Chimera interpolation

3. Results and Discussion

3.1 Improvement of numerical scheme

To investigate the improvement of the numerical scheme especially in its capability to resolve the flow discontinuity and to keep a small disturb in shear layer, two test cases are considered.

a) Shock tube problem

b) Fully developed turbulent pipe flow

Figure 3. Two test cases for numerical schemes


539

The first test case is the shock tube problem as described in [SCH 981. In Fig. 3a, the pressure distributions calculated with the JST and ELED schemes are compared with the exact solution of [HIR 901 at the time t=6.1 ms. As can be seen, the scheme of JST oscillates strongly just before the shock front whereas the ELED scheme resolves the discontinuity without oscillation. The use of ELED scheme resulted in only 7 percent increase in the computing time. The other test case is the fully developed turbulent pipe flow as shown in Fig. 3b, where the calculated fully developed radial velocity distributions, which is normalised by the velocity in the middle of pipe Urn,are compared with the experiment results of [NIK 321. It can be seen that the near-wall shear layer is strongly disturbed by the JST scheme, while the result of ELED scheme agrees very well with the experiment.

3.2 Verification with experiments of a circuit breaker

Fig. 4 shows a schematic diagram of the core part of a circuit breaker. To interrupt the current, the electrical contact "finger" will be moved together with isolating and auxiliary nozzles toward left, while the electrical contact "plug" will stay still. The gas in the pressure chamber (not illustrated) will be then compressed through this motion and flows from left into the isolating nozzle, there will be a highly transient and transonic flow. from pressure chamber

-------

\ moving elec. contact (finger)

\ fixed elec. contact (plug)

Figure 4. Schematic diagram of the simulated circuit breaker To verify the code for the capacitive switching design, measurement of static pressure on the 6 points showed in the Fig. 4 was carried out for cold. These 6 measured points are located in different flow regions. O and O are in the diffuser region with flow separations; O is in the geometric throat; 0 and O are in the channel flow region with boundary layer character; 8 is behind the shock front and presents the pressure lost over the shock. Therefore, the values at these 6 positions reflect all flow details. For the calculation, an inlet was defmed as shown in Fig. 4 and the pressure measured in the pressure chamber was used as the inlet boundary condition, so that the leakage in the pressure chamber which is difficultly to be estimated can be ignored. In Fig. 5 the simulated pressure distributions at O to 8 are

540


time in ms

0.0

12.0

24.0

36.0

time in ms

48.0

time in ms

0.0

12.0

24.0

36.0

time in ms

time in ms

48.0

0.0

12.0

24.0

Figure 5. Comparison ofpressure in circuit breaker: simulation, 0 measurement

-

36.0

time in ms

49.0


541

compared with the measurement. The simulation results agree very well with the measurements. The discrepancy of P5 is caused by the selection of the inlet where the boundary layer begins downwards from the start point in the reality.

3.3 Coupling of the electrical andflow fields Based on the test results on a circuit breaker where breakdown voltages were measured in dependence of travel positions, the present code can be verified through comparison of the ratio E/(pkJ with the critical value of 1480 kvm2/kg for 6 bar absolute filling pressure of SF6 from the streamer theory (s. [BEY 861). In this case E/(pk) must be higher than the critical value, where k, is a roughness factor which accounts for the microscopic effects which intensify the local field strength and is obtained from an semi-empirical function of local density and roughness. The electrical field strength E was calculated with the ABB electrical field program ACE. The gas density p was obtained from the flow simulation with the present code. Fig. 6 shows the distribution of E/(pkr) in the circuit breaker. The maximum of this ratio lies on the surface of the plug contact and it overruns the critical value slightly. The simulation results correspond very well with measured breakdownlhold values of the applied voltage observed in measurements.

Figure 6. E/(pkJ distribution and its maximum at trave1=115 mm

4. Conclusion

The progresses in the numerical methods, including numerical scheme and moving grid technique, have been made, leading to the development of the code presented in this paper which is able to fulfil the requirements of capacitive design of high voltage circuit breaker. The effectiveness of the code is confirmed through the following facts: (1) The adoption of the more advanced ELED scheme resolves flow discontinuity efficiently; (2) the physical viscosity is not confused; (3) the calculation results agree very well with the measured results; (4) the calculated

542


density w a s coupled with electrical field strength, and the resulted ration o f W(p kJ predicted the gas breakdown correctly. T h e code will b e further improved and developed, in particular, arcing model will b e implemented and verified, so that it can b e used t o predict the dielectric strength under high temperature a n d pressure.

5. References [BEY 861 Beyer. M., Boeck, W., Moller, K., Zaengl, W.: Hochspannungstechnik, SpringerVerlag, Berlin Heidelberg New York, 1986 [CLA 971 Claessens, M., Moller, K.. Thiel, H.G.: A computational fluid dynamics simulation of high- and low-current arcs in self-blast circuit breakers, J. Phys. D: Appl. Phys. 30, p. 2899-2907, 1997 [HIR 901 Hirsch, C.: Numerical computation of internal and external flows, Vol. 2, John Wiley & Sons, 1990 [JAM 811 Jameson, A,, Schmidt, W., Turkel, E.: Numerical solutions of the Euler equations by finite volume methods with Runge-Kutta time stepping schemes, AIAA paper 8 1-1 259. January, 198 1 [JAM 941 Jameson. A,: Analysis and design of numerical schemes for gas dynamics 1: Artificial diffusion. upwind biasing, limiters and their effect on accuracy and multigrid convergence, Int. J. of Computational Fluid Dynamics, August, 1994 [LAU 741 Launder, B.E., Spalding, D.B.: The numerical computation of turbulent flows, Computer Methods in Applied Mechanics and Engineering, Vol. 3, p. 269-289, 1974 V I K 321 Nikuradse. J.: GesetzmilPigkeit der turbulenten Stromung in glatten Rohren. Forsch. Arb. 1ng.-Wes. Heft 356, 1932 [SCH 911 Schafer, 0 . : Application of a Navier Stokes Analysis to turbomachinery bladecascade flows, 1 9 ' ~International Congress on Combustion Engines, CIMAC, Florence, 1991 [SCH 981 Schafer, 0. et al: Last advances in numerical simulation of aerodynamic forces on turbine blades of turbochargers for pulse charged engines, 22nd CIMAC , International Congress on Combustion Engines, 19-21 May , Kopenhagen , 1998 [S'FE 781 Steger, J.: Implicit finite-Difference simulation of flow about arbitrary twodimensional geometries, AIAA Journal, Vol. 16, No. 7; July. 1978 [STE 871

Steger, J., Benek, J.: On the use of composite grid scheme in computational aerodynamics, Computational Methods in Applied Mechanics and Engineering, Vol. 64, NO.1-3, 1987

[TRE 911 TrCpanier, J.Y. et al: Analysis of the dielectric strength of an SF6 circuit breaker, IEEE Transaction on Power Delivery, Vol. 6, No. 2, April, 1991

River valley flooding simulation

Francisco Alcrudo Area de Meccinica de Fluidos Maria de Luna, 3 C P S - Universidad de Zaragoza 50015 Zaragoza, S P A IN

--

ABSTRACT

D a m break flood wave propagation along a reach of a river valley located i n the Italian Alps is mathematically modeled with package S W Z D that solves the nonlinear Shallow Water equations. Simulation results are compared with data obtained from a physical model of the river valley operated by ENEL (Italy). The dificulties encountered during the modelisation process and the solutions adopted are explained in this paper. K e y Words: D a m break, Flood, Shallow Water, Physical Model

1. Introduction

Considerable efforts are being presently devoted to the validation of numerical models describing dam break flows, mainly due to the need for modern risk assesment and mitigation tools. Real life experimental data concerning actual dam break or severe flooding are very difficult t o obtain because of the unpredictable nature of the phenomenon. However, measurements obtained from reduced scale physical models can provide excellent validation information because the experimental conditions can be more precisely defined. The work reported in this paper colicerns the comparison of the simulation results obtained with SW2D program [ALC92] with measurements of the flooding experiments carried out by ENEL (Italian Utility Company) in a reduced scale physical model of the Toce river. The physical model is some 50m long by l l m wide and is built mainly in concrete (see figure 1). It represents a 5km long reach of Toce river which is located in the italian Alps.

544


*-

4b

TOCE RIVER VALLEY "

,-

--,

(Physical Model 1:100)

9nr'

Figure 1: Digital Terrain image of Toce river valley physical model (ENEL) The model reproduces many details of the actual valley geometry including the river bed and some villages, hydraulic structures and a reservoir located in the middle of the reach that depending on the intensity of the flood is overtopped and eventually filled with water across its embankments. The upstream end of the river reach model is connected to a small water tank fed by a hydraulic pump. Flooding is initiated by starting the pump that rapidly fills up the tank, overtops the entrance to the reach and rushes downstream. The pump capacity is such that the process takes place very rapidly thus simulating an abrupt irruption of water into the valley model. ENEL personnel located water stage probes at 32 different positions in the model valley. Among them one in the river bed at the entrance section and another one in the middle of the feeding tank that can be used to impose the boundary conditions together with the pump discharge versus time that was also recorded. Experiments were carried out for two flooding intensities: The first one such that no overtopping of the reservoir takes place (peak discharge of 0.21 m3/s)and the second one with reservoir overtopping (peak discharge of 0.36 m3/s). Measured water stage readings at several probe locations were compared to water levels obtained with SW2D model for different friction coefficients and flood intensities and overall satisfactory agreement was found. The physical model geometry was distributed by ENEL as a Digital Terrain Model (DTM) covering the model area at regular intervals of 5cm, therefore specifying the bottom elevation function z ~ ( x9), in some two hundred thousand points. The simulations reported here were run on a platform of comparable computing power to that of a Pentium processor. In order to have reasonable run times (a few hours) the size of the DTM grid had to be coarsened by a factor of three.


545

2. Mathematical model It is commonly accepted that bulk flood flow can reasonably be well described by the non linear Shallow Water equations that simply express conservation of mass and momentum in the plane of movement of water. Since they are obtained after and integral mass and momentum balance in the horizontal directions (or by averaging the Navier-Stokes equations across the vertical) no information regarding vertical velocities is obtained. Usually no flow shear forces are taken into account when the problem is convection dominated as it is the case in severe flooding. Fi-iction forces with the bottom are accounted for by empirical formulae such as Manning's or Chezy's. The Shallow Water equations can be written in integral conservation vector form as follows:

1

~ at / U ~ V + / [ F . ~ , + G . ~ ~ HdV ] ~ S = Here t represents time, dV an elementary volume and n, and n, the cartesian components of the normal vector to the elementary surface area dS enclosing the considered volume. Think that in 2-D a volume means in fact an area and an area is actually a line. U is the vector of conserved variables and F and G are the cartesian vector fluxes of mass and momentum.

Here h, u and v represent water depth and the two cartesian velocity components respectively and g is the acceleration of gravity. Source term H accounts for bed friction and bottom slope:

where n is Manning's friction coefficient. The mathematical model SW2D solves the Shallow Water equations in two dimensions by means of a finite volume spatial discretization in multiblock structured meshes coupled to an explicit two step time integration scheme. This is done by applying equation (1) to every cell of the computational domain in the usual Finite Volume approach. Cells can be quadrangles of arbitrary shape but sound judgement has to be exercised so as to avoid very distorted or stretched control volumes that may degrade the overall accuracy. Numerical fluxes are evaluated at cell faces through MUSCL variable extrapolation with limiting to enforce monotonicity. After variable extrapolation, Roe Riemann solver is applied a t each cell interface. Bottom slope and bed friction represented by Manning's formula are spatially integrated pointwise.

546


Figure 2: Cartesian g r i d used i n the computations totalling 22000 points Once the spatial discretisation has been done the solution is advanced in time by a predictor-corrector sequence. Source terms are implicitly time integrated with no extra cost because the operator remains diagonal if they are pointwise spatially discretised. Details of the algorithm can be better found in [ALC98]. 3. Boundary conditions

Proper computation of the flow variables a t the upstream end in order to reproduce the correct flood characteristics at the inflow of the model valley is crucial if good agreement with downstream located probes is sought. Downstream boundary conditions do not exert such a strong influence on the global flowfield mainly because water leaves the reach in critical or supercritical conditions. Available initial data from ENEL were the inflow rate, Q ,the reading of the water level probe located in the inflow tank, named S1, and the reading of the water level probe located at the inlet section, named S2. After considerable efforts it was determined that good agreement with experimental data at the entrance could be found only by imposing a subcritical inflow condition based on the available total head at the feeding tank. Despite the advise given by ENEL that flow conditions are critical at the entrance section, the model could not be run under this assumption. Numerical experiments showed that failure to accept critical flow at inlet was due to a slight adverse slope in this area that led to flow reversals, because critical flow can only be reached at the top of an upslope. Since subcritical inlet conditions require that two flow variables be imposed, they can be implemented by either imposing flowrate, Q, water level at the inflow section (S2 probe reading) or far upstream on the reservoir (S1 probe reading) together with the inlet angle, a.


547

Imposing flowrate is not adequate in a two dimensional computation involving an irregular inlet section because it is difficult to obtain an appropriate criterion t o distribute the available discharge among the inlet section cells. Using the water level a t the inflow section is an interesting option but it is better to use the water level a t the feeding tank (S1 probe reading) as the Total Head available in order to have water level a t the inlet section (S2 probe) as an accuracy check. It must be borne in mind that the inflow rate, Q, can also be used as a check. Due to the size of the tank with respect to the entrance to the reach, the velocity in the former can be considered negligible, and the reading given by probe S1 is considered as the total head h~ that is available at the inflow section. This can be written as follows: (h~)inflow= (h +

T)+ u2

v2

= (h),, inflow

being (h),, the reading of probe S1 a t the considered time and the other variables with subindex inflow are evaluated a t every cell of the inlet section. Also from the outgoing bicharacteristic (see [ALC92] one has:

where u is the velocity vector and n the locally outward pointing unit vector. Subindex bich corresponds to the expression transported by the outgoing bicharacteristic from the inside of the computational domain. Once water depth and modulus of the water velocity at every inlet section cell are determined from the above equations the two cartesian components can be computed if an inflow angle, a, is imposed. In the tests run ct was varied from zero to a few degrees with no significant changes in the computed results.

4. Testcases

Besides the two inflow hydrographs (of different intensities) tested by ENEL, several simulations were performed varying Manning's friction coefficient above and below the value of n=0.016 suggested by ENEL. Also and more importantly, runs were made both with and without the buildings composing the valley villages. The DTM geometry did not contain buildings, but these could be included by modifying the bottom surface function z B ( x ,y) appropriately. However, due t o the low resolution of the grid used, buildings are represented very roughly as figure 1 shows: Villages can be seen as groups of mushroom like sprouts. Their influence on the solution is nevertheless very substantial. For every run made, great attention was paid to matching the inflow rate supplied by the pump (Q) and the water level measured by the probe located in the river bed a t the inlet section (S2 probe). This guaranteed that at least the inflow flood wave was close to the actual one. Figures 3 and 4 show the

548


INFLOW RATE S W D EXPERIMENTS x BUILDINGS n4.015 + BUlWlNGS nd1.02 0 NO BUILDINGSn=O.02

'

.

, 0.WO

1,

,

25

, ,

,,

,

, ,

,

, , ,

,

:,, 0

50

75

-

Bul:T

SOLID EXPERIMENTS BUILDINGS n=O.O15

,

1W

,

NO BUILDINGS n.0.02

125

150

175

Figure 3: Inflow rate and S2 probe reading for the moderate flood event

0 350

0 300 i

r 0

0 250

BUILDINGS n.0015 BUILDINGS n=0 02 NO BUILDINGS n.0 02

0 ZW

0 1%

0 tm

SOU0

0 050

r + 0

25

50

75

1W

125

150

175

25

50

75

1W

EXPERIMENTS

BUILDINGS n;0 015 BUILOINGS rr002 NO BUILDlffiSD002

125

1%

175

Figure 4: Inflow rate and $2 probe reading for the severe flood event comparison between computed and measured inflow discharge (left) and water depths in centimeters (right) at probe S2 (located just at the inflow section) versus time for the two tested flood events. Computational results (crosses, plusses and circles) correspond to different model options as shown. Although agreement in both flow and water depth is quite good, runs with buildings follow better the experimental values. Figures 5 and 6 show the comparison between calculated and measured water levels at probes S4 and P8 for both flooding events. Both probes are located around the central part of the valley, S4 being about 5m and P8 some 16m downstream of the inlet section. Although runs without buildings show larger errors than those including buildings (especially at probe S4) the situation is reversed at other probes not shown here for lack of space. Finally figures 7 and 8 show the same comparison at probes named P13 and P21. Both lie close to the river bed. P13 is located in front of the central reservoir about 21m downstream of the inlet section while P21 is located some 7m further downstream. Overall agreement at the considered locations can be judged acceptable.

Fields of application 0075

.. ,

.

.,

,

.,.

.

. , .., ., . .. ,. 54

,

,

,

,

o

,

'

549

m

OlM

0 I00

0 050

SOLID EXPERIMENTS x BUILDINGS n=O.O15 +

0 25

50

75

100

BUILDINGS n=0.02 NO BUILDINGSn.O.02 125

150

SOLID EXPERIMENTS BUILDINGS 0=0.015

x +

-

0 25

175

30

75

100

BUILDINGS nd02 NO BUILDINGSn.0 02 125

150

175

Figure 5: Comparison at S4 and P 8 probes for moderate flood

SOLID

EXPERIMENTS

SOLID EXPERIMENTS x BUILDINGS n-0015 r BUILDINGS n-002 0 NO BUILDINGSn-0 02

0 WO

I 25

50

75

1W

125

150

115

25

50

75

lm

I r25

r50

175

Figure 6: Comparison at S4 and P 8 probes for severe flood

0 I50 0 150

o1m 0

tw

0 050

0OM

S X l O EXPERIMENTS x BUILDINGS m 0 1 5 + BUILDINGS h 4 02 0 NOBUILDINGS h = M 0 MO

SOLID EXPERIMENTS x BUILDINGS n-0015 BUILDINGS n=OM 0 NOBUILDINGS n-0 02 OWO

25

Y)

75

101

I25

lso

175

I

Y)

75

rm

125

150

Figure 7: Comparison at P I 3 and P21 probes for moderate flood

175

550


0 150

0 IM

0 050

SOLID EXPERIMENTS BUILDINGS n=0015

SOLID EXPERIMENTS BUILDINGS ~ 0 0 1 5 r BUILDINGS -0 02 0 NOBUILDINGS n;O 832 i

0000

50

25

75

100

125

1%

175

25

50

15

>W

125

t50

175

Figure 8: Comparison at PI3 and P21 probes for severe flood 5. Concluding remarks

Although discrepancies between measured and computed water levels can be important a t certain probe locations, the mathematical model used provides a reasonably accurate description of the two flooding events considered. Due to the very valuable assistance that this kind of tool can provide in tasks such as land use and emergency planning or risk assesment studies, it seems worthwhile to carry out further validation and improvement work. 6. Acknowledgements

The author would like t o thank ENEL and especially Dr. G. Testa for providing the experimental data and clarifying many technical questions. Finantial support provided by the European Union under CADAM concerted action is also gratefully acknowledged.

7. References [ALC 921

[NUJ 951

[ALC 981

ALCRUDO F., Esquemas de alta resoluci6n para el estudio de flujos discontinuos de superficie libre, Ph.D. Thesis, Universidad de Zaragoza, 1992 NU JIC M . , Efficient Implementation of non-oscillatory schemes for the computation of free surface flows, Journal of Hydraulic Research, 33, No. I , 1995, p. 101-111 ALCRUDOF . , Dambreak flood simulation with structured grid algorithms , Proceedings of the 1st C A D A M (Concerted Action on D a m Break Modelling) Meeting, (1998), Published b y the EU, in press.

Modelling vehicular traffic flow on networks using macroscopic models. J.P. Lebacquel, M.M. Khoshyaran2

' CERMICS-ENPC. FRANCE. email: [email protected]. TASC. USA.

ABSTRACT: In this paper, we describe a macroscopic model for vehicular traffic flow, with several extensions, resulting in a flow model on a network. These extensions require the introduction of link boundary conditions, partial flow dynamics and intersection models. Some numerical schemes based on the Godunov scheme are proposed for the discretization of the model. Key Words: Godunov scheme, Traffic flow, LWR model, partial flows

1

Introduction

Macroscopic modelling of vehicular traffic flow goes back to the pioneering work of Lighthill and Whitham [LW 551 and Richards [Ri 561, which introduced the celebrated LWR (Lighthill Whitham Richards) model of traffic on an infinite track. This model relies on the continuum hypothesis, i.e. the asumption that vehicular traffic can be described by macroscopic variables, the density K ( x ,t ) , the flow Q ( x ,t ) , and the speed V ( x ,t ) , as functions of the position x and the time t. These variables are related by the following equations: aQ - 0 -+= dt= KV aK

-

V = V, ( K , x)

conservation equation definition of v behavioral equation.

(1)

or simpler:

Qe and V, represent the equilibrium flow-density resp. speed-density relationships ( Q ,( K ,x) dgfKV, ( K ,2 ) ) . Their aspect is the following:

552

_


Qmax (x)

...---

k i t (x)

-

Km(x)

vm v,,,, (XI - - - - -

-

,

K krit(x)

Km&)

Of course, considering the continuum hypothesis, the above LWR model (1) should be considered as a phenomenological model, but it is usually accepted that it provides a reasonably good description of the dynamics of traffic flow a t a space scale of a hundred meters and a time scale of 10 seconds. Actually, the LWR model (I),also refered t o as the first order macroscopic traffic flow model, constitutes but one among several competing approaches to macroscopic traffic flow modelling (see [LL 991 for a general discussion). Other notable modelling approaches include the second order macroscopic traffic flow models (see [Le 951 for an overview and [Sc 881 for relations between second and first order models) and the kinetic traffic flow models [PH 711, [Ph 791, [He 971. Neither experimentation nor theory has provided arguments strong enough to support one model unambiguously. There is also no real consensus concerning " the exact functional form of the equilibrium relationships, but the shapes suggested in the above illustration are generally accepted, up t o a few variations. In the sequel, we shall concentrate on the LWR model, which is simple, enjoys obvious physical meaning, and provides results generally in good agreement with measurements.

2

The Godunov scheme for the classical LWR model on the line

The entropy solution of (1) is the only solution considered usually in the literature on traffic flow modelling. In entropy solutions, the decelaration of trafic generates shockwaves, whereas the acceleration of trafic induces rarefaction waves. Entropy solutions are also characterized by the fundamental fact that they maximize locally the flow [Le 961. The Godunov scheme [GR 911, [Kr 971 provides a numerical solution of the classical LWR model, as shown in [Le 961, [Da 951. This solution is satisfactory for applications: it approximates the entropy solution. Let us introduce the equilibrium supply and demand functions:

(the symbols + and - represent right- and left-hand limits). The following illustration describes these functions, that represent respectively the greatest possible inflow (supply) and the greatest possible outflow (demand) at point x.


553

With these notations, the expression of the Godunov scheme is straightforward:

[

3 3.1

=

~f

+

y

- Q:)

(conservation equation) (flow equation), QI = M i n [A, (K:, i ) , C , (Kf,, ,i 1) (4) ef with as usual (i) d= [xiPl, xi] the cell i of length li, t the index of the time-step, K t the average density in cell i a t time t At, Qi the average flow a t point xi during time-step t. The flow equation expresses that the flow is the minimum between the downstream supply and the upstream demand. The flow equation in (4) is the expression of the analytical solution of the Riemann problem, which can be obtained even if the upstream and downstream equilibrium relationships differ [Le 961. The flow maximizing property of the entropy solution is crucial for this result. The practical necessity for considering space-dependent and even space-discontinuous equilibrium relationships is obvious: lane drops, intersections provide contexts for such discontinuities. ~ ; + l

+ 1

Extensions of the basic LWR model Link boundary conditions

For applications, extensions of this basic LWR model are indispensable. A first and obvious generalization concerns the extension of the model to networks, which implies two steps: the definition of proper boundary conditions for links, and the description of intersections. The equilibrium supply and demand concepts provide the proper framework for the definition of link boundary conditions. Considering now a link such as the following:

the boundary conditions are the upstream demand A,(t) and the downstream supply C d ( t ) [Le 961, [LK 981. The link inflow Q(a, t ) at any time is the minimum between the link supply C, ( K ( a + , t ) , a ) and the upstream demand Au(t). Similarly, the link outflow Q(b, t) a t any time is the minimum between the link demand A, (K(b-, t ) , b) and the downstream supply Cd(t). Thus: Q(b,t)

= M i n [C, ( K ( a + , t ) , a ) , A u ( ~1 ) =Min[A,(K(b-,t),b)

(5)

554

3.2


Partial flows on links

Before considering intersections, it is necessary to consider partial flows on links. Indeed, the fashion in which the traffic flow separates (according to preselection lanes) or does not separate (so-called FIFO flow) in the incoming links of an intersection determines the way in which the intersection works. Further, in many advanced applications, various categories of users must be considered: users would be distinguished according to destination, information availability, path, etc. The macroscopic variables are disaggregated according to some assignment attribute d:

K = C Kand~

Q

=

~

Q

~

The partial flows and densities are related by the trivial relationships: -0 x+x aKd

aQd

Qd = K d V d

conservation equation for Qd, K d

definition of speed Vd of the partial flow .

(6)

These equations must be completed by a phenomenological model. The simplest possible model is the so-called FIFO model:

(vehicle speed independent of attribute d). This model results in a straightforward advection equation relative to the composition coefficients x d def - Kd/K of the flow:

in which the global velocity of the flow V results from the resolution of (2)). A more realistic model is the lane assignment model [LK 98). In this model, vehicles may have restricted access to lanes according to the assignment attribute d. Let I be the set of lanes, Idthe set of lanes accessible to vehicles d, yiKmaxthe maximum density of lanes i, K4 the density of vehicles d in lanes i. Then the K4 are the unknowns of the lane-assignment problem and are subjet to the following constraints:

The K~ constraints express the split of K d into the K!, and the yiKmax constraints express that the total density in lane i cannot exceed the maximum density yiKmax of this lane. The Kd constitute the dynamic data and the


555

constitute the geometric data of the lane assignment problem. The unknowns K,d can be determined by solving either

yi and

(maximizing locally the total flow), or

(Wardrop optimum), subject to constraints (9) in both instances. The meaning of (11) is to assign users t o lanes in such a way that all users having the same attribute d have the same speed on all the lanes they use effectively (otherwise, users would switch lanes in order t o drive faster: this is an individual optimum). These lane assignment models result in systems of conservation laws for which approximate Riemann solvers are under study. A simple case (2 user types, 2 lanes) was analyzed in [Da 971. Supply-Demand models for partial flows can also be defined. The principle is to calculate partial supplies and demands for all superscripts d and to determine the corresponding partial flows by comparing partial demands to partial flows. Partial demands are defined as:

which is a FIFO-like model. The partial supply model really defines the user behavioral model. Let us first define coefficients ,Bd which determine the maximum density (i.e. pdKma,) of the lanes available to vehicles d. If we refer to the notations of the preceding subsection,

We propose the following two models for the partial supplies: Model 1: Cd(x,t) = PdC(x,t) (linear model), Model 2: Ed((z, t) = p d E e , x) (homogeneous section model) Model 1 is extremely simple but allows K d to exceed Pd~,,,. Model 2 does not have this drawback but still does not take partial flow overlapping into account as precisely as the lane-assignment models, since the data formed by the Idsets and the coefficients yi has been simplified and only coefficients pd are left. It would also be natural to define the partial flows Qd as:

(w

Nevertheless, since

EDdis usually >

1 (because of the partial flow overlap-

d

ping), it is possible that partial flows calculated according to (13) satisfy:

556


thus implying a model inconsistency. This inconsistency can be resolved by using the following expressions: Qd = Mas

M i n ( C d , a d )C M i n (z6,n6)]

[c,C s

Q

= M i n [ C , C M i n ( ~ " A ~ 1. 6

The partial flows Qd can be viewed as solutions of the following program: d

0

5 Qd 5 Min [Ed,Ad]

def with y~ some concave increasing functions such as: qd(q) = q - M i n $ d , A d j . The above functions p d are not intrinsic (they depend on the local partial supplies and demands instead of the local geometric attributes such as the pds or the maximum flow). Other functions (entropic functions for instance) would yield the same expressions while being intrinsic.

3.3

Modelling intersections

Modelling intersections is more difficult, since intersection models are phenomenological by nature. They describe for instance, in the case of a merge, bhe local priority rules, or the gap acceptance process. Two modelling schemes can be considered. 1. Modelling intersection as objects of finite extension, by trying t o remodel produce the movement dynamics. This was the idea of the STRADA [BLLM 961, in which exchange zones generalize cells inasmuch as they behave similarly but are endowed with several entry- and exit- points and provide upstream demands respectively downstream supplies for downstream respectively upstream cells or exchange zones. Exchange zone models are discrete by essence. 2. Pointwise intersection models. These were considered in [Le 961 [LK 981 and derived from zone models by letting the zone extension become vanishingly small. The study of these models is the subject of ongoing research. Let us still give one example and consider the node depicted hereafter. Let us denote Cj(t) the supply of exit link (j) of the intersection a t the node point, and Ai(t) the demand of the entry link (i) of the intersection at the node point. A proportion yij of users about to exit link (i) chooses link (j) (the coefficients yij are called assignment coefficients and must be considered exogeneous to the flow model). Thus the partial demand of traffic from link (i) to link (j) def is given by: Aij(t) = yijAi(t). Split supply coefficients pij (depending on the link geometry) can be introduced in order to disaggregate the link supplies Ci(t). We can deduce partial supplies Cij(t) by applying for instance model def 1 (the simplest): Xij (t) = PijCj(t). Since it is possible and even likely that


Pij > 1, a formula similar to

557

(14) should apply, with the same rationale, to

i

yield the partial flows Qzj(t) : (i) + ( j ) :

4

Discretization of partial flows in links and intersections

Let us consider the Supply-Demand model for partial flows (notably equations (12) and (14). If we consider two consecutive cells (i), (i l), the following relationships result (discretized model):

+

Cf+, = C, [K:+,, i + 11 (expressing the traffic supply of cell (i function of the cell mean density K;+, ,

+ 1) as a

+

' Ct+l,d = Pi+l,d Ci+l

or Cf+,,, = Pi+l,d Ce [Kl+l/Pi+l,d,i I] (expressing the partial supplies, according to partial supply models 1 or 2, A: = A, [Kt,i] (expressing the traffic demand of cell (i) as a function of the cell mean density K:, = ( K & / K f ) At = X;,~A;,expressing the partial demands according to the FIFO-like model, -

, expressing the partial flows between ,I ( c : + ~,A:,&)] ,&

Min (Ef+,,,&

Qt'd - M a r

[xi+,

,)

c + 1

in

.c

cells (i) and (i + l" according ) t o (14),

Kt+' z,d = Kt, equation, Kf = 6

+

I,

[Qt

K:,&and Q: =

-

Q;,,], expressing the discretized conservation yielding the total cell density and flow as

6

the sums of partial densities and flows. The similitude between the intersection model (16) and the Supply-Demand partial flow model (12), (14) is evident, thus the discretization of the intersection model follows the same lines as the discretized Supply-Demand partial flow model and need not be described in detail here.

558


5

Conclusion

The development of extensions of the basic LWR is still an ongoing process. The only intersection model for which there exists any kind of experimental support is the Supply-Demand model [LK 981, and link partial flow models are still tentative. Nevertheless, suitable discretized models should be developed, in order to be able to choose between alternative modelling schemes. The finite volume method, combined with the search for analytical solutions, seems to be the best approach to the investigation of numerical solutions of the LWR model and its extensions.

References [BLLM 961 C. Buisson, J.P. Lebacque, J.B. Lesort, H. Mongeot. The STRADA model for dynamic assignment. Proc. of the 1996 I T S Conference. Orlando, USA. [Da 951 C.F. Daganzo. A finite difference aproximation of the kinematic wave model. Transportation Research 29B. 261-276. 1995. [Da 971 C.F. Daganzo. A continuum theory of traffic dynamics for freeways with special lanes. Transportation Research 31 B. 83-102. 1997. [GR 911 E. Godlewski, P.A. Raviart. Hyperbolic systems of conservation laws. SMAI. Ellipses (Paris). 1991. [Kr 971 D. Kroner. Numerical schemes for conservation laws. Wiley Teubner. 1997. [He 971 D. Helbing. Verkehrsdynamik. Springer Verlag. 1997. [Le 951 J.P. Lebacque. L'Cchelle des modeles de trafic: du microscopique au macroscopique. Annales des Ponts. 1st trim., 74: 48-68. 1995. [Le 961 J.P. Lebacque. The Godunov scheme and what it means for first order traffic flow models. Proc. of the 1996 ISTTT (J.B. Lesort ed.). 647-677. 1996. [LK 98) J.P. Lebacque, M.M. Khoshyaran. First order macroscopic traffic flow models for networks in the context of dynamic assignment. E U R O Work Group on Transportation 1998, Goteborg (Sweden). CERMICS Report. To be Published. [LL 991 J.P. Lebacque, J.B. Lesort. Macroscopic traffic flow models order. 14th ISTTT. Accepted for publication. 1999.

:

a question of

[LW 551 M.H. Lighthill, G.B. Whitham. On kinematic waves 11: A theory of traffic flow on long crowded roads. Proc. Royal Soc. (Lond.) A 229: 317-345. 1955. [Ph 791 W. F. Phillips. A kinetic model for traffic flow with continuum implications. Transportation Planning and Technology, Vol. 5, 3, pp 131-138. 1979. [PH 711 I. Prigogine and R. Herman. Kinetic theory of vehicular traffic. American Elsevier, New York. 1971. [Ri 561 P.I. Richards. Shock-waves on the highway. Op. Res. 4: 42-51. 1956 [Sc 881 S. Schochet. The instant response limit in Witham's non linear traffic model: uniform well-posedness and global existence. Asymptotzc Analyszs. 1, pp 263-282. 1988.

Finite volume method applied to a solid/liquid phase change problem

El Ganaoui M., Bontoux P. IRPHE- Universite' d'Aix-Marseille I I I M T , 38 Joliot Curie 13451, Marseille Mazhorova 0. Iceldysh institute Moscow

A second order accuracy method of time and space based on finite volume approxzmatzon i n a fixed m,esh zs developped for Navier-Stokes and energy equations extended to solid/liquid phase change problems. This fixed grid method validated with respect to an interface tracking method i s able to describe the interaction of steady and oscillatory melts with the interface during Bridgman crystal growth.

ABSTRACT

Iiey Words: Finite volume, phase change, interface.

1. Introduction Free and moving boundary problem requires the simultaneous solution of unknown field variable and the boundaries of the domains on which these variables are defined. Phase change during directional solidification of semiconductor crystals by the Bridgman technique is a typical example of such a complex process. Each method of solution must solve the appropriate heat, mass and momenturn transfer equations and deternliiles the melt solid inter-

560


face. It was still necessary to satisfy the Stefan or similar derivative condition on that boundary. Furthermore it was sometimes be difficult or even impossible t o track the moving boundary directly [CRA 841. The possibility, therefore, of reformulating the problem in such a way that the transmission conditions at the interface are implicitly bound up in a new form of the equation, which applies over the whole of a fixed domain. The moving boundary appears, a posteriori as one feature of the solution. One possibility of reform~lat~ing the problem is to introduce an enthalpy function in the energy equation and a porous model in the momentum equation. The enthalpy function is the sum of the specific heat and the latent heat required for the phase change. In the momentum equation, we assume that the liquid turn to solid in an intermediate region t o be a porous medium. In this way on prescribing a Darcy source term the velocity value arising from the solution of the momentum equation are inhibited, reaching values close to zero on complete solid formation [VOL 801. The coupled enthalpy porosity model gives a single set of homogenous NavierStokes and energy equations adapted to the problem of phase change during directional solidification [MOR 991. The finite volume method is validated with respect to an interface tracking method [ELG 961. It uses a fixed grid and the interface position is given from the thermal field (solidification isotherm). The resulting interface shape is also studied and some insight on cristal constitution are given.

2. Formulation For directional solidification, A cylindrical ampoule with radius R and length L contains melt and crystal. The ampoule must be moved relative to a prescribed external temperature gradient. This motion of the ampoule is acounted for by supplying a melt to the top of the computational space at a uniform velocity Ut and with drawing cristal from the bottom with the same velocity. The heat transport between the furnace and the ampoule is modelled with a re scribed furnace temperature profile with three zones, cold (T = T,), adiabatic (aT/dn = 0 ) and hot one (T = T h ) . The length L , the velocity a / L and the thermal difference Th - T, are used as reference scales to give dimensionless form of the variables ( x ,u , 0 ) . x ( r , z ) represents the courant point with radial r and the axial z components, u = (u,, u,) represents velocity with radial and axial coordinates u, and u,, respectively. Liquid, solid and intermediate medium are distinguished by the suffixes 1, s and sl. For the energy equation a continuous enthalpy function is introduced :


561

where E is a prescribed small regularisation of the temperature, fl a mesure of liquid fraction and Ste = Lf/c(Th- T,) is the Stefan number. T h e corresponding enthalpy is continuous and piecewise linear. In this way the energy equation takes the following form in all the dimensionless domain 0 < r < 1 et 0 1 < -1

[GI

3.2.2. AUSM Flux-Vector Splitting (LRouiSteffen) The Advection Upstream &litting Method was originally proposed by Liou and Steffen in the early 90's [LIO 931. First, the inviscid flux vector is split into an advective and a pressure term. These two terms are then split separately, leading to the following expression for the flux a t the cell interface

where Ai+; (0) = {e}i+l - {*Ii and Mi+; = M: + Mt;, . The split Mach numbers M & are definded according t o van Leer's splitting as described above. The pressure flux terms are assumed to be governed by acoustic wave speeds. An expression using second-order polynomials of the Mach number is proposed for the pressure splitting:

3.2.3. A USMDV Flux-Vector Splitting (Wada/Liou) Yet another approach for the splitting of the inviscid flux vector as a mixture of a more FDS based scheme and a more FVS based one was proposed by Wada and Liou [WAD 941 and is called AUSMDV.

Complexity, performance and informatics

675

where s is a switch as a function of the local pressure gradient.

)

where

K = 10

[lo]

It is clear t o see that the AUSMD and AUSMV differ only in the treatment of the term pu2 in the x-momentum flux. The corresponding expressions are:

The velocity splitting within the AUSMDV is similar to the original proposal of van Leer extended by terms designed t o capture stationary and moving contact discontinuities. UL;IUL~]

+

51

if Cm

PLL+/~LI

otherwise

-7

[I31

where CYL

=

~ ( P I PL ) + (P/P)R

r

=

C ~ R

2 ( ~ / ~ ) ~

(PIP)L + ( P / P )R

[I5] and c,, = m a x ( c L , c R ) The pressure splitting is the same as in the original formulation and finally, the mass flux for the AUSMDV is (P/P)L

3.2.4. HLLE Flux-Difference Splitting The most simple flux difference splitting scheme was proposed by Harten, Lax, van Leer and modified by Einfeldt [EIN 911. The solution of the whole Riemann problem is replaced by a model consisting of three constant states separated by two shocks yielding for the flux a t the cell interface

676


where the Af denote the speeds of the fastest and slowest wave in this model. 3.2.5. Roe Flux-Difference Splitting

A more sophisticated flux difference splitting scheme was developed by Roe [ROE 811. Roe replaces the approximation of the solution of the nonlinear Riemann problem by the exact solution of the linearized problem that has to be extended by the approximation of discontinuous solutions.

A,: is the propagation speed of the k-th wave of the linearized Riemann problem and lk, r,: are the corresponding left- and right-eigenvectors. The so constructed formulation of the flux function consists of a central part supplemented by an upwind term, which has to be computed using average values of the conservative variables. One possibility for the construction of these average values is Roe-averaging, see Grotowsky [GRO 941. A problem appears if one of the eigenvalues changes its sign. For centered expansion fans with sonic point the scheme then leads t o an expansion shock and generates non-physical solutions such as the "carbuncle phenomenon'' when calculating hypersonic blunt body flows. To circumvent this difficulty Harten proposed a modification of the modulus function in Eq.(18)

I&(

=

+G) lXk

1

for / A k / < 6 else

where 6 is a small number often referred as "entropy fix". For this method there exists an alternative way of increasing the formally order of accuracy in space, that was originally proposed by Harten and Yee. This Ansatz comprises an approximation of the truncation error of the first order scheme and is known as "modified flux approach" [GRO 941.


677

4. Results First, two-dimensional Riemann problems as they have been investigated by P. D. Lax and X.-D. Liu [LAX 981 are considered. In these problems a quadratic solution domain is divided up into four quadrants and the initial data are constant in each of them. The initial conditions are restricted such that only one elementary wave, a shock, an expansion, or a contact discontinuity appears a t each interface. Depending on the initial conditions complex interactions between the different waves are evolving in time. Fig.2 shows the results obtained with the different upwind methods for the problem number 13. The initial data for this problem consist of two stationary contact surfaces and two moving shocks. The computational domain has been discretized with 300 grid points in each direction in an equidistant manner. The density distribution is displayed. Obviously that the overall best results were obtained with the Harten / Yee FDS and the AUSMDV in connection with van Leer's flux limiter. The moving shocks are sharp, the stationary contact discontinuities correctly captured. Also the secondary contact surfaces and the small vortex structure in the center are fairly resolved. Use of the original AUSM scheme leads to spurious oscillations behind shocks and contacts due to the non-monotonous behaviour of this scheme across discontinuities. Finally, the van Leer / Hanel FVS and the HLLE are characterized by the unappropriate capturing of stationary contact surfaces and, in addition, as a consequence of the higher amount of artificial viscosity inherent to both schemes the resolution of the secondary contacts and the small vortex structure is worse, too. Fig.3 shows an investigation of the influence of three different flux limiters (Roe's "minmod" and "superbee" limiter and the one of van Leer) on the results for two different problems, what is often neglected, and a surprisingly strong influence is obvious that was not expected. In dependence of the limiter location within the TVD region a dramatic improvement of the resolution of the moving contacts in the Lax 5 problem and a better resolution of the triple points in the Lax 12 was detected. In addition, in both cases the use of the "superbee" limiter leads to small oscillations as a consequence of the vicinity t o instability of this limiter, so that perhaps the van Leer limiter should be the best choice. As some kind of a grid sensitivity analysis Fig.4 shows solutions for the Lax problem number 5 on a refined grid (700 grid points in each direction) and in nice agreement with the work of Lax and Liu small vortex structures appear that could not be resolved with any method on the coarser grid. The fact, that the Harten / Yee FDS resolves only one vortex structure may be due to some kind of a direction dependency in the way of increasing the formal accuracy in space. In conclusion again the influence of the flux limiter is clear to see when looking at the pressure distributions obtained with the van Leer / Hanel FVS. Use of van Leer's limiter yields a vortex resolution almost comparable to the AUSMDV solution whereas the "minmod" limiter nearly prevents any vortex to appear.

678


Finally, Fig.1 shows first results of a threedimensional intake flow simulation using the Harten / Yee method. On the upper left hand side of this figure the result for the density distribution for the two-dimensional case is plotted. For the three-dimensional case the interior part of the intake is assumed t o have sidewalls beginning a t the intake lip. On the lower left hand side of the figure the density distribution in the symmetry plane of the inlet is showed. The phenomena in front of the intake lip are obviously exactly the same as for the two-dimensional case, but also the shock positions and angles as well as the separation region on the upper side of the interior part are a t least similar, because the three-dimensionality does only less influence the effects in the symmetry plane. Finally, on the right hand side of the figure two cross sections in the interior part are plotted, which show the three-dimensional effects in the lower corners produced by the two shocks originating from the intake lip and the sidewalls. 5. Conclusions

Numerical solutions of 19 two-dimensional Riemann problems using different upwind methods have been realized. Three of them are presented. The overall best results were obtained using the Roe FDS in the Harten / Yee formulation and the AUSMDV. The original AUSM scheme showed a nonmonotonous behaviour across discontinuities resulting in pressure and density oscillations behind them. Furthermore, a surprisingly high influence of the flux limiter was detected. So deficiencies of the schemes exhibiting more numerical viscosity could partly be cured by an appropriate limiter. Finally, first promising results of a three-dimensional intake flow simulation using the Harten / Yee method were presented. Unfortunately a considerable high value for the entropy-fix had to be chosen for reasons of stability, so that subject of future work in this project will be simulations using the AUSMDV, since it proved t o be comparable accurate in the resolution of even complex interaction phenomena a t a lower computational cost and higher stability. 6. Bibliography

[LIO 931

LIOUM.-S., STEFFENC. J., , J. Comp. Phys. 9 2 , pp. 273-295, 1991.

[GRO 941

GROTOWSKY I. M . G . , ~ E i nnumerischer Algorithmus zur Lijsung der Navier-Stokes-Gleichungen bei ~ b e r -und Hyperschallmachzahlen, PhD Thesis, RWTH Aachen, 1994.


679

[HAE 891

SCHWANE R . , HANEL,D., , AIAA Paper 89-0274, 1989.

[LAX 981

LAXP. D., L I U X.-D., >, SIAM J. Sci. Comput. 19, No. 2, pp. 319-340, 1998.

[ROE 811

ROE P. L., <Approximate Riemann Solvers, Parameter Vectors, and Difference Schemes >>, J. Comp. Phys. 43, pp. 357372, 1981.

[VKE 981

VAN

[VLE 791

VAN LEER B.,

1. For

the upper limit of the timestep, the solution will also' + co. Explicit treatment is the best choice for point A, with the time step calculated from e q [71. < 0 ) is studied. The As second characteristic example, point B ( S > 0 and situation does not differ much from point A, except that within a linear approximation, a 'natural' additional upper time limit can be proposed. With Taylor's expansion, the (qYt1 - 4"). As we source term S is approximated as ~ ( 4 " ~3' )S(4")

+ $$Imn

712 want


S(4"11)=

0, this results in

mntl

=

1

mn - w 47%, which corresponds for

&=

80

. The corresponding dn+' I + . coincides with point B' in figure 1. If by a non-linear theory a better estimation 4 can an explicit treatment [6] to a time step

-

a a+

be found, the needed time step should be given by eq. [7]. In most cases, however, some hypotheses have to be made to obtain this non-linear approximation. To ensure a stable iteration process, the maximum time step is taken as the smallest time step from the linear and the non-linear analysis:

An implicit treatment [4] always fulfills the condition IGI > 1, and moreover guarantees IGI < LYI for all AtI. Therefore implicit treatment is preferable for point B. An infinite time step, however, corresponds to a solution 4n+1= 4B',which can be far away from 4,. The time step to be taken for a non-linear approximation is given by eq. [7]. For the same reasons as in the explicit discretization, the most conservative time step is taken:

4

Point C (S < 0 and < 0) is very similar to point B. The only difference is that the source term is negative, so that the desired amplification 0 < G < 1. The conclusions and time step restrictions are the same as in the previous case. For the same reason, the situation of point D corresponds to point A. Again, the difference is that, as S(4)< 0 the amplification should be 0 < G < 1. Thus it is seen that neither of the two possibilities (explicit or implicit treatment) can fulfill all the three mentioned requirements on itself in any of the four situations. The introduction of a time step restriction seems therefore necessary. For an explicit treatment this is obvious, since an infinite time step leads to an infinite (positive or negative) value of 4.For an implicit treatment, however, it can be necessary, too. In < 0, implicit treatment is practice, however, it is difficult to approximate G. When more robust than explicit treatment. In order not to counteract the implicit treatment, equations [8] and [9] are slightly altered. In [8], the sum is taken instead of the maximum. A closer look at eq.[7] reveals that this corresponds to dropping the term 22 Id,"

in [9]. Based on this discussion, the time step to be taken in any case is given by:

$$

> 0, and an implicit combined with an explicit treatment of the source term S(4)if one if a < 0. It is clear that the practical use of equation [lo] requires a reasonable + approximation of G.

a


713

2.2. Coupled system The source terms in two-equation turbulence models are in general strongly nonlinear and coupled. Two major methods exist to handle the source terms. In the first method [VAN 86, WIL 931, the positive parts of the source terms are not linearized and treated explicitly. The negative parts are linearized in such a way that negative real eigenvalues are obtained. They are treated implicitly. The second method [MER 931 starts with the construction of the exact Jacobian of the source terms: a Depending on its eigenvalues, this matrix can be split into a pos84'

$ %.

itive and a negative part: = + According to the previous analysis, the distinction between positive and negative here is made on the basis of the real part of the eigenvalues. The negative part is then treated implicitly, the positive part explicitly. An additional time step restriction is in principle necessary for robustness. For k-E or k-w based turbulence models, this time step can be determined from a simplification of the turbulence equations ([MER 991). Numerical results, however, show that for realistic flow situations this time step restriction is less stringent than time step restrictions coming from the convective or the diffusive terms. These restrictions were not encountered in the previous analysis because there the equation was simplified towards one containing a source term, but no convective or diffusive terms.

3. Numerical results Both source term discretization methods are numerically investigated. The lowReynolds k - E model by Yang-Shih and the low-Reynolds k - w model by Wilcox are studied for incompressible flows. A multistage time stepping scheme is used to reach the steady state solution [VIE 981. From now on, the method with the approximated Jacobian will be called 'approximate method', whereas the other method will be called 'rigourous method'. Fig. 2 shows the convergence evolution for a fully developed channel flow. The convective terms and the diffusive terms are set to zero, so that the source system's behaviour can be studied. Within the approximate method (curves 1 and 4), there is always a time step restriction, resulting from the implicit treatment of the negative parts of the source terms. For the rigourous method, a calculated time step restriction has to be introduced (curves 3 and 6), or strong underrelaxation is necessary, resulting in a worse convergence (curves 2 and 5). This shows the (theoretical) necessity of a time step restriction in order to retain a robust method. Fig. 3 shows the convergence history for a flat plate flow on a stretched grid (145x89 points). An alternating line solver is used. The approximate method performs equally well as the rigourous method (curves 1 and 2, and curves 4 and 5). The reason is that the convective and diffusive time steps are more restrictive than the source term time step so that they cover the influence of the source term discretization. For the same reason, the introduction of a calculated time step restriction is not necessary here (curves 3 and 6). Fig. 4 shows convergence results for the k - E model for a flat plate flow (193x97 gridpoints) using multigrid. The cpu times for the Navier-Stokes equations and for the turbulence equations have to be added to obtain the global cpu time. A substantial im-

714


provement is seen with the rigourous method when both the Navier-Stokes equations and the turbulence equations are solved on the coarse grids (curves 5 and 6). Curves 3 and 4 correspond to the Navier-Stokes equations solved with MG and the turbulence equations only on the finest grid. Curves 1 and 2 show the result for all the equations only solved on the finest grid. However, no convergence is obtained when the first method is used in combination with full multigrid (curves 7 and 8). This indicates the superiority of the second method. For the k - w model, the difference between the two methods is much smaller. Both methods converge with multigrid, again leading to a substantial improvement over single grid calculations (results not shown). Fig. 5 shows convergence results for the k - E model for a backward-facing step flow. Similar conclusions as for the flat plate can be drawn. However, now the first method also converges with multigrid. Similarly as mentioned in [GER 971, the flat plate flow seems more demanding for the multigrid technique. Similar results are obtained for the k - w model (fig. 6).

Figure 2: Convergence history for channel flow.(l: k - E, approximated Jacobian; 2: k - E , exact Jac.; 3: k - E , exact Jac., with time step restriction; 4: k - w, appr: Jac.; 5: k - w, exact Jac.; 6: k - w, exact Jac., with time step restriction)

Figure 3: Convergence history forflat plateflow with an alternating line solver:(l-6: see above)


715

Figure 4: Convergence acceleration with the multigrid technique for jlat plate flow.(l,2: residual of NS, resp. turbulence, eqs., all solved single grid; 3,4: res. of NS, solved with MG (4 grids), and turb. eqs., solved single grid; 5,6: res. of NS with MG ( 4 grids), and turb. eqs. with MG (4 grids), rigourous method; 7,8: res. of NS with MG ( 4 grids), and turb. eqs. with MG (4 grids), approx. method)

Figure 5: Convergence acceleration with the multigrid technique for BFS jlow.(l-8: see above)

4. Conclusion A method of source term discretisation is presented that is robust and, most importantly, allows the use of the multigrid technique for convergence acceleration. The method is independent of the turbulence model.

5. Acknowledgements The first author is aspirant at the Flemish Science Foundation (F.W.O.).

6. References

7 16


Figure 6: Convergence acceleration with the multigrid technique for BFSflow. (1-8: see above) [GER 971

Gerlinger P. and Briiggemann D.", Multigrid Convergence Acceleration for Turbulent Supersonic Flows, Int. J. for Numerical Methods in Fluids, volume 24,p. 1019-1035, 1997.

[MER 931

Merkle C.L., Weiss J. and Venkateswaran S., Efficient Implementation of Turbulence Modeling in Computational Schemes, Proc. Second U.S. National Congress on Computational Mechanics Washington, D.C., August 1993, 1993.

[MER 991

Merci B., Steelant J., Vierendeels J., Riemslagh K. and Dick E., Treatment of Source Terms and High Aspect Ratio Meshes in Turbulence Modelling, Proc. 14th AIAA CFD Conference, Norfolk City, June 1999, in press, 1999.

[STE 941

Steelant J. and Dick E., A Multigrid Method for the Compressible Navier-Stokes Equations Coupled to the k - E Turbulence Equations, Int. Journal of Numerical Methods in Heat and Fluid Flow, volume 4 (2), p.99-113, 1994.

[STE 971

"Steelant J., Dick E. and Pattijn S., Analysis of Robust Multigrid Methods for Steady Viscous Low Mach Number Flows, Journal of Computational Physics, volume 136, p. 603-628, 1997.

[VAN 861

Vandromme D. and Ha Minh H., About the Coupling of Turbulence Closure Models with Averaged Navier-Stokes Equations, Journal of Computational Physics,volume 65, p. 386-409,1986.

[VIE 981

Vierendeels J., Riemslagh K, and Dick E., A MultigridSemi-Implicit Line-Method for Viscous Incompressible and Low Mach Number Compressible Flows, Proc. of the 4th ECCOMAS Computational Fluid Dynamics Conference, Athens, John Wiley, p. 1220-1225, 1998.

[WIL 931

Wilcox D.C., Turbulence Modeling for CFD, Griffin Printing, Glendale, California, 1993.

Comparison of numerical solvers for a multicomponent , turbulent flow

E m m a n u e l l e X e u x e t C.E. M. I. F. Evry A l a i n Forestier C.E.A. Saclay J e a n - M a r c H Q r a r d E. D.F. Chatou

This contribution's topic is the resolution by different numerical solvers of a multicomponent, compressive, turbulent pow. The unique associated Riemann Problem's solution is identified thanks to an entropy characterization. An exact Riemann solver is implemented and called by Godunov scheme. Some numerical simulations are introduced to exhibit a comparison between Godunov scheme, Vfroe-nc and Rusanou scheme.

ABSTRACT

Key Words: Turbulent flow - Godunov scheme - Linearized solvers


In this contribution is exhibited the resolution of the hyperbolic system which describes a compressive multicomponent turbulent flow. The model is written for a polytropic isentropic gas. With compressive flow, Favre's average is used to select a mean flow and a turbulent one. In this work, we are interested by the one order closure model and particularly the coupling between turbulence and pressure. Reynold's tensor is described through the turbulent kinetic energy It' of the mixture. The system is closed thanks to the li' evolution equation. 2. A t u r b u l e n c e m o d e l t o describe m u l t i c o m p o n e n t flows

The average variables describing the flow are : (p, p a , pu,li')

71 8


The density of the mixture is noted by p. pa stands for the density of one of the fluid, with a the mass fraction of one component of the mixture. U stands for the velocity and K for the turbulence of the mixture. Setting W = (C, IO 2a(Pa) PaPa(Pa)

+ VXl.rE < O and

vX5.r; > 0

(10)

+

The treble characteristic field is linearly degenerated : t t vXl.rtl = VX2.r2 = VX3.r3t = VXq.r4 =O

(11)

The solution consists in at most six constant states separated by shock waves, rarefaction waves or contact discontinuities. The rarefaction curves are :

IGpS

(a, p,u,,u,, I 0, u, = u , ~ ,li = 7 ,

P;

720


( a , p , u nu,, , K ) ,a = ad, u, = U r d , p

> 0, I< = 7 , ~ r 3

Shock curves are :

4 ~- P i (a,p,un,UT,IC)= , a a g , u , = u r g , p >O,Iil= 4 ~ 1- P un = ung -

(a, p , ~ n U ,T , IC),a = ad, U , = u7d,p

Un

= Und

4 ~- Pr I(r, > 0, I< = 4 ~r P

-

+

I{,)

+(P

-

P,)]

PPr

The contact discontinuities verify [un]= 0

and

2 11'

+PI = 0

(16)

2.3. Entropy characterization and uniqueness of the solution

The generic form of the mathematical entropy is :

pU2 p = C1(T

+

1

=dp p2

+ IC) + C2p + C3pa + C4pU + C sPC,I + C6

(17)

Our system ( S ) admits two supplementary conservative variables :

In keeping with the second thermodynamic principle, a p convex entropy is growing on a physical shock.

The equivalence between Lax inequalities and compressive shock is shown. But the equivalence between entropy shock and compressive shock is demonstrated for only the physical entropy E . 3 has no physical sense, because its growing on


721

1-shock curve implies incompressive shock. The computation of the Riemann solution on the entropy shock curves ensure the uniqueness of the result. 3. Other numerical methods and preferences

The advantages of the exact solver are well known : positivity respect, entropy solution. But we have to balance these advantages by the fact that the method uses more CPU than linearized solver which doesn't require analytical computations. Then we introduce different schemes to analyze where a method is more or less efficient than an other. 3.1 A linearized solver : Vfroe-nc Vfroe scheme was introduced by Faille, Gallouet, Masella in 1996. It is based on a local resolution of a linearized Riemann problem. The numeric flux is defined, like Godunov scheme, by the physical flux computed at the interface solution of the linearized problem. An extension of this scheme was introduced by Buffard, Gallouet and Hkrard [BUF 981. Vfroe-nc uses the nonconservative variables to preserve Riemann invariants through contact discontinuities. Thus we prefer the variables (P(pcr),U ) to (pa, pU). 1 - Yl Y,. With Y = ( a ,u,, u t , I cl),

where cj, q denote piecewise constant approximations of the exact solution c on the elements T j , G . Let hki, := minjern diam (Tj) and let x j be the circumcenter of triangle T j . Define the distance between the circumcenters of two neighbouring cells T j , Z E 7" as djl := Ixj - xll and let hjl denote the maximum of the diameters of those cells. We assume that there exists an a > 0 such that we have for all hj := diam (Tj) aha

5 ITjl,

a ldTjl

5 hy-'

and

ahj

5 djl

(6)

for all j , l E I . For any j, 1 E I n and t , E JRf let gjn, : R2 3 R be a C1 numerical flux, satisfying the following conditions for all w, v,w', v' E [A,B].

where L, is a given constant and njl denotes the outer unit normal to Sj1 with respect to T j . Now the upwind finite volume scheme for computing the approximate solutions to (4) is defined by

for all n E (0, ...,N ) and j, 1 E In. Here N ( j ) denotes the indices of the neighbouring cells of Tj. Given the discrete values c? let us denote the approximate solution ch : JRd x [O,T]+ W. by ch(z,t) := c7,if z E Tj,tn t < t n f l . For the time step Atn we assume the following CFL-condition

,:z :I .::-.,-.* ,

,

..-- .........

, . --.------. . . . . . .. .-: ~.~ .. -,in preparation.

[PAK 961

N. G . PANTELELIS, A. E. KANARACHOS, ,Int. J. Numer. Meth. Fluids, 22 (1996), p. 411-428.

[WRO 971

J . W U , H. RITZDORF,K. OOSTERLEE,B. S T E C I ~ E LA., SCHULLER, ->,Int. J. Num. Meth. Fluids,24 (1997), p. 875-892.

Dynamic mesh generation with grid quality preserving met hods

Andreas Wick and Frank Thiele Hermann-Fottinger-Institut fur Stromungsmechanilc

Technische Uniuersitat Berlin 10623 Berlin, Germany

A B S T R A C T T h e various methods employed so far t o accomodate computational grids

t o a moving boundary often fail t o preserve grid quality. In this paper, a detailed analysis o f thzs issue is conducted. Based o n the findings a new method i s presented that performs substantially better. T h e theoretical results are confirmed by numerical studies. I i e y Words: m o v i n g boundaries, deforming grids, d y n a m i c meshes, spring analogy

1. Introduction

The incorporation of moving boundaries and deforming domains has become an important task in numerical flow simulation. There are many applications in which this issue must be faced. To give a few examples, we can mention aerodynamic design optimization, fluid-structure interaction, free-surface flows and multi-element airfoils with moving flaps. In order to take into account the effects of time varying domains in an adequate and straightforward manner numerical algorithms are commonly based on the Arbitrary Lagrangian-Eulerian (ALE) formulation of the conservation laws (see e.g. [ANJ97]). There is no need for special search and interpolation algorithms as is the case with some other methods (e.g. [KA095]). Compared to a standard finite-volume method that operates on a fixed grid (pure Eulerian approach) the numerical solution of the ALE equations additionally requires at. each time step new grid point coordinates and the corresponding velocities with which the cell surfaces move.

778


While the consistent and efficient computation of the grid velocities has been investigated thoroughly [ZHA93], the adjustment of the mesh to moving boundaries is still an open problem, at least for Navier-Stokes computations which require grids with very high aspect ratios. Specifically, for nontrivial geometries with boundaries exhibiting strong curvature or sharp bends, overlapping of mesh cells is likely to appear. In this case, a local or global remeshing becomes necessary and may induce unwanted interpolation errors. Therefore, of the methods proposed in literature, in section 2 we revisit shortly those having the potential to preserve the quality of the initial grid for the whole mesh evolution. Besides the well-known spring analogy model [BATSO], these are methods based on differential equations [ILI94, CRU96, LOE961. The numerical implementation of the equations is addressed in section 3. In section 4 the methods are subjected to a deformation analysis which reveals serious weaknesses. The observed deficits also become apparent in numerical test cases. In section 5 the conclusions that can be drawn are summarized. 2. Grid generation methods

2.I Problem formulation

We assume that we start with a high quality grid at the first time level. Our aim is to find grids for the proceeding time levels that fit the prescribed boundary and that possess a quality comparable to the initial grid. Specifically, cross-sections of coordinate lines must not occur. To accomplish this, we have to inspect a very basic geometrical configuration, typical both for unstructured and structured grids. It is a line segment, connecting two neighbor vertices, which is subjected to a deformation. The difference between the coordinates at the new time level and the preceding time level defines the displacement vector.

We hope to find a displacement vector w that leaves the length of all line segments unaltered while transforming the grid of a former time level to a new one. Of course this is only possible if the prescribed boundary displacements correspond to a rigid body motion. If the boundary moves in another way, causing the interior domain to deform, the resulting stretches have to be equidistributed in a certain way. Otherwise a high local concentration of stretches will lead to a quick degradation of the grid quality.

Adaptivity, tracking and fitting

779

2.2 Methods A very well-known method is the spring analogy model. The lines connecting neighbor vertices are seen as tension and/or torsion springs. The imposed boundary displacement generates forces in the springs, leading to a new equilibrium state. By minimizing the energy of the spring system, the grid is optimally redistributed. A particular simple model that gave good results in many practical applications ([BATSO, FAR941) has been proposed by Batina [BATSO].

V is the set of vertices and N ( i ) the set of neighbors of i. In a structured 2D grid the set of neighbors N(i) consists of the east, west, south and north nodes. Despite some desirable properties of a numerical scheme derived from this equation, it has some flaws that gave rise to the development of alternative methods. Ilinca [ILI94] proposed t o solve a Laplace equation for the grid velocity i. By an integration over the time step the displacement field is obtained. This numerical integration necessitates an assumption regarding the time dependence of the grid velocity (e.g. j. = const, for tn < t < tn At). The error introduced thereby leads to a time step restriction. It is therefore a better choice to use a Laplace equation for the displacement itself.

+

The introduction of a diffusion coefficient in this formula allows for more control over the grid evolution.

Crumpton and Giles [CRU96] choose the diffusion coefficient as a function of the cell volume. They argue that small cells are most endangered with regard to crossing of grid lines. Lohner and Yang [LOE96] prefer the diffusion coefficient to be connected to the wall distance, since large stretches are supposed to be close to the wall. While the cell volume is an intrinsic quantity of the discretisation, the wall distance is not, and as a consequence it might not be available. To keep the formulas valid for cases as general as possible we set

for the diffusion coefficient. The function Vol(xi) returns the volume of the dual grid cell which is associated with vertex xi. The Laplace equation [3] can also be interpreted as a special case (E = const., divergence free displacement field) of the following equation:

780


This formula is also closely related to equation [2]. Similar to the derivation of the spring analogy model, the fact that the minimization of an appropriate energy function leads to optimally redistributed grids is exploited. But while the spring analogy is based an a discrete model now we apply a continous model. That is to say we consider the domain 0 as a Hookean medium with zero Poisson's ratio. Then equation [6] represents the well-known Navier's equation of the classical linear theory of elasticity (see e.g. [ERI62]). Following the reasoning of Crumpton and Giles the Hook medium should be stiff for small cells. The most simple way to do this is to take the reciprocal value of the cell volume as elastic modulus E.

The coefficients of equations [4] and [6] are now strongly dependent on the chosen discretisation which might be considered as a disadvantage. A better choice should be to link the elastic modulus to the stretches whereby equation [6] becomes nonlinear. However, due to limited space we restrict ourselves to linear methods here.

3. Solution Procedure The discretisation of the differential grid generation equations are based on the Finite Volume Method (FVM). To allow complex geometries to be accounted for, a block-structured approach with general non-orthogonal coordinates is adopted here. The strong conservation form of the equations is retained by expressing the vectors and tensors in Cartesian components. The displacement vector is assigned to the vertices ("Cell vertex scheme"). First derivatives, appearing at cell faces, are approximated by central differences leading to a second order scheme. The numerical stability is enhanced by using the deferred correction approach which makes the scheme partly implicit. The system of linear equations obtained from the discretisation is solved by an iterative ILU method [ST068]. 4. Theoretical and numerical analysis

With regard to the grid quality the methods presented in section 2 have to meet two requirements. The main task is the equidistribution of stretches in a manner that avoids high local concentrations of them, since they are responsible for the overlapping of grid cells. Even more fundamental is the preservation of rigid-body displacements. It is evident that a method should not create stretches for boundary displacements that allow for a rigid displacement of the whole domain. It is important to note that these two basic requirements are sufficient to maintain a high grid quality, provided that the deflection of the


781

boundary is small. This is normally the case as the movement of the boundary is subdivided in successively applied steps in order to resolve the time scales of the fluid flow. For small deformations there is a fundamental theorem which states that a general deformation can be considered as resulting from a linear combination of a rigid body translation, a rigid body rotation and stretches along the principal axes of strain ([ERI62]). If a method behaves well for these three types of deformations it is also able to deal with a general deformation since the latter is nothing but a superposition of the elementary ones. In the following section the equidistribution of stretches is further investigated. 4 . 1 Disc with a hole subjected t o a rotated boundary

The undeformed domain is covered by orthogonal grid lines. For the ratio of inner radius ri, and outer radius rout a value of 0.4 is chosen. The outer boundary is rotated in clockwise direction for an angle of a = 40'. Discretisations of the deformed disc with 20 x 10 grids are displayed in figure 1. For this

Figure 1. Deformation of a grid with constant spacing in radial direction by rotation of the outer boundary in clockwise direction; (a) Laplace equation, (b) Navier's equation with E = V o l ( x i ) - l , (c) spring model problem equations [3], [4] and [6] (provided that the diffusion coefficient and the elastic modulus are constant) have an analytical solution. It can be obtained by first transforming the equations from the cartesian coordinate system ( x , y) to a polar coordinate system (9,r ) and then exploiting the rotation symmetry of the probiem. Inserting the analytical solution in equation [l]yields the grid coordinates for the deformed domain: (;)n=(;)n-l+(cosp

where

F=-

(

sinp

-sinp cosy

--

--

r

4 rout 25 r

)(

F(cosa-1) Fsina

782


As a measure of grid quality we use the angle between the grid lines after the deformation. It is obtained from the updated grid point coordinates. The angle is depicted in figure 2a and 2b as solid lines. The exact solution of equation [3] is very well represented by the numerical solution (circels in figure 2). We can therefore assume the discretisation to be fine enough. For the first test series a discretisation with constant cell volume V ( a i ) is selected. As can be seen in figure 2a, if the coefficient in the equations [4] and [6] are determined from the cell volume, the solutions coincidence with the analytical reference solution and hence lead to bad grid quality. While the discretisation with constant cell volume is rather academic, it is more natural to use constant spacing in radial direction. The same equations then give almost satisfactory results. While the Laplace equation leads to a worst angle of 16O, it is greater than 30' for Navier's as well as the diffusion equations, whereby the latter performs slightly better. It is important to note here that the distribution of stretches due to the coefficients [5] and [7] is further improved when the mesh is more refined towards the walls. For the two discretisations investigated, the spring model also preserves the grid quality in a satisfying manner, as can be seen in the figures l c and 2.

Figure 2. Angle over normalized radius; (a) constant cell volume, (b) constant spacing in radial direction

4.2 Solid Body Rotation of a disc

As already mentioned, it is important to examine the performance of the methods introduced in section 2 for the case of rigid-body deformations. To do so, the displacement vector that corresponds to a rigid-body deformation of the entire domain is inserted in the equations. In the case of rigid-body translation, the displacement vector is constant, which is indeed a solution of the equations [2], [3], [4] and [6]. Let us now take a closer look at the rigid-body rotation. The domain S2 shall rotate by an angle of a . The position vector X


783

of the deformed domain is obtained by letting a rotation tensor a act on the position vector x of the undeformed domain.

With zn s X and

sn-'

r x we find for the displacement vector:

Here 6 denotes the identity tensor. Taking into account that gz = 6, it is clear that the displacement vector as given by [lo] is the solution of the Laplace equation [3]. This is not the case for the diffusion equation. If equation [lo] is inserted it remains:

This equation can be considered as a linear, homogeneous system of equations which determines the components of the gradient However, these components are already determined by equation [5]. In general this components will neither be zero nor will they have other values that fullfil equation [ll]. To render equation [ll]valid, the expression in brackets therefore has to vanish. For a Cartesian coordinate system this condition reads expanded:

vk.

cosa sin o

-sina cosa

) ( i !) -

=0

This equation is not fulfilled unless the rotation angle is zero. Hence we conclude that equation [4] is not able to represent rigid-body rotation. Note that it makes no difference here if the coefficient k is determined from the deformed grid, as proposed by Crumpton and Giles. A similar analysis for the Navier's equation leads to: cosa = 1, which is approximately fulfilled for small rotation angles. We conclude that the Navier's equation is able to represent rigid-body rotation, provided that the rotation angle is small enough. It remains to examine the spring model. Inserting equation [lo] into equation [2] we obtain:

The sum vanishes only if the whole domain is covered by straight coordinate lines, because in this case the contributions from opposite neighbor nodes cancel each other out. This is, of course, an unacceptable constraint on the spatial discretisation of complex geometries. Consequently, the first bracket is required to vanish which leads to the same condition as for the diffusion equation. Hence also the spring model cannot correctly represent a rigid-body rotation. The foregoing theoretical investigations are supported by numerical computations that are performed on a grid with constant spacing in radial direction,

784


Figure 3. Grids generated for revolved boundaries; (a) diffusion equation with k = V o l ( x i ) - ' , (b) Navier's equation with E = V o l ( x i ) - ' , (c) spring model

which was also used in the previous section. Both the inner and the outer radius are rotated in clockwise direction by an angle of 40'. The Laplace equation and also diffusion and Navier's equations reproduce the desired rigid-body rotation, provided the coefficients k and E are constant respectively. A slight deviation from orthogonality can be observed in figure 3b. This grid was generated with the Navier's equation with a variable coefficient k. For smaller angles this deviation disappeared. The use of the diffusion equation with variable coefficient led to a significant deviation from orthogonality (fig. 3a). Also the spring model fails to reproduce the rigid-body rotation, as can be seen in figure 3c. This weakness is probably the origin of the problems that were encountered by many authors (e.g. [RAU93]) when using the spring model. The degradation of the grid quality is expected to be larger the more the mesh lines deviate from straight lines in other words for strongly curved coordinate lines. 4 . 3 A complex test case

After we investigated how the different grid generation methods perform for elementary deformations, we turn now to a general deformation. The geometry of this test case consists of a channel that contains a triangular shaped obstacle which fills one third of the channel high. In figure 4, magnifications of the mesh around the upper corner of the triangle are shown. They are generated by different methods in the following manner. First the triangle is rotated around its center in clockwise direction by an angle of 15'. Subsequently, the interior grid points of the initial grid are adjusted to the new boundary, which yields a new grid. Then the triangle is moved back to its original position again. From the grid produced in the latter stage and the known deflection of the triangle, a new grid is generated that fits the boundary of the initial state. It is clear that after this succession, the initial grid should be retrieved, or else the method employed to generate the grids is not quality preserving.


785

The spring model fails completely, as it leads to overlapping mesh cells (figure 4c). In contrast, both diffusion equation and Navier's equation generate grids that can be used for fluid flow calculations. However, only the Navier's equation leads to a full recovery of the initial grid. When the diffusion equation is used the near wall orthogonality gets lost, as can be seen in figure 4a. This confirms the observations of the previous section.

Figure 4. Detail of the grid around the upper corner of the triangle after a succession of boundary displacements; (a) diffusion equation with k = V o l ( z i ) - l , (b) Nauier's equation with E = V o l ( x i ) - ' , (c) spring model

5. Conclusion

Many authors observed that the methods employed so far for generating grids for complex time varying domains fail to preserve grid qualtity. By means of a deformation analysis, we were able to give an explanation for this. The spring model and the diffusion equation lack the ability to reproduce rigid body rotations. Problems will occur in applications where the boundary or a part of it rotates, e.g. with an oscillating airfoil. The Laplace equation, on the other hand, doesn't give satisfactory results in the presence of moving boundaries that exhibit strong curvature or sharp bends, as high concentrations of stretches are not equidistributed in a sufficient manner. On the basis of our findings! we proposed a new method that resolved these problems. This new method can be seen as a link between the other methods. It is important to note that the analysis conducted in this paper is valid in a very general setting. It does not rely on a particular coordiante system nor on the structured spacial discretisation. Hence, the drawn conclusions hold also for the threedimensional case and for unstructured grids. There are no significant differences with regard to the computational expense required by the different methods. However, in comparison with the cornputational expense of the fluid flow calculations, the overhead incurred by the grid generation is still too high. The implementation of more efficient solution procedures will be subject of further research.

786


Acknowledgment T h e first author wishes t o express his appreciation for the financial support t o this work provided partly by the Dr. Fritz Walter Fischer-Stiftung a n d t h e DFG (grant No. A F 3112-2).

Bibliography [ANJ97]

ANJUA. et al., 2-d fluid-structure interaction problems by an Arbitrary Lagrangian-Eulerian finite element method, Int. J . Comput. Fluid Dyn., 8(1), p. 1-9, 1997.

[KA095]

KAO K.-H. and LIOU M.-S., Advance in overset grid schemes: From Chimera to DRAGON grids, A I A A Journal, 33(10), p. 1809-1815, 1995.

[ZHA93]

ZHANGH. et al., Discrete form of the GCL for moving meshes and its implementation in CFD schemes, Computers Fluids, 22(1), p. 9-23, 1993.

[BATSO]

BATINAJ.T., Unsteady euler airfoil solutions using unstructured dynamic meshes, A I A A Journal, 28(8), p. 1381-1388, 1990.

[IL194]

ILINCAA., Calcul des Ccoulements compressibles tridimensionnels sur des maillages en mouvement et adaptifs, thhse de doct,orat, Ecole Polytechnique de Montreal, MontrCal Quebec, Canada, 1994.

[CRU96]

CRUMPTON P.I. and GILESM.G., Implicit time accurate solutions on unstructured dynamic grids, A IAA-95-1671- CP, p. 285-293, 1995.

[LOE96]

LOHNERR. and YANGC., Improved ALE mesh velocities for moving bodies, Communications i n Numerical Methods in Engineering, (12), p. 599-608, 1996.

[FAR941

FARHATC. and LANTERIS., Simulation of compressible viscous flows on a variety of MPPs: computational algorithms for unstructured dynamic meshes and performance results, Comput. Methods Appl. Mech. Engrg., 119, p. 35-60, 1994.

[ERI62]

ERINGENA.C., Nonlinear Theory of Continuous Media, McGraw-Hill Book Company, 1962.

[ST0681

STONEH.L., Iterative solution of implicit approximations of multidimensional partial differential equations, S I A M J. Numer. Analysis, (5), p. 530-558, 1968.

[RAU93]

LEE-RAUSCH E.M. and BATINAJ.T., Calculation of AGARD wing 445.6 flutter using Navier-Stokes aerodynamics, A IAA-93-34 76, 1993.

A Finite Volume Method for Steady Hyperbolic Equations M. J. Baines, S. J. Leary, M. E. Hubbard Department of Mathematics The University of Reading Reading, RG6 AX, UK

1 Abstract A finite volume method is presented for steady conservation laws on unstructured meshes which incorporates mesh movement. The method can substantially improve the resolution of sharp features (contacts, shocks) by solving the problem on an optimal mesh. Results are presented for a number of steady state test problems, including scalar advection and the shallow water equations in 2-D. For scalar equations there is a close relationship with the method of characteristics [5]. For shocked flows we describe a discontinuous least squares method which uses a nonlinear shock jump residual to adjust the mesh.

2

Introduction

Finite volume methods for these equations include the cell-based multi-dimensional upwinding schemes. These schemes have been very successful in producing good approximate solutions to the above problems [I] but there is still the possibility of even greater resolution of sharp features by a careful deployment of the mesh [2]. A great deal of effort has been put into mesh refinement near shocks using mesh subdivision, but similar improvements in resolution can also be obtained much more cheaply by minor adjustments to the mesh (see e.g.[3]). In this paper we consider a descent approach using a least squares measure of the residual.

3

Fluctuations

We are interested in systems of steady conservation laws such as the Euler Equations or the homogeneous Shallow Water Equations, of the form

where

788


In integral form this becomes

where dS is measured into the arbitrary surface S. In determining approximate solutions U of these equations we may associate with the differential form (1) the residual R = divF(U) while with the integral form (2) we associate the fluctuation

where ST is a triangle. In fluctuation/distribution schemes such as the Multidimensional Upwind schemes a weighted amount of qh in each triangle is added to the values of the solution a t its vertices. The weights may be chosen so that the schemes are conservative, positive and linearity preserving. In particular, conservation is assured if the weights in each triangle sum to unity. In the Least Squares method, which minimises

the use of descent techniques t o achieve the optimisation also adds weighted amounts of qh in each triangle to the values of the solution at its vertices. However, this time the sum of the weights is zero and the method is not conservative in the usual sense. The procedure has the effect of redistributing solution values.

4

Counting Issues

On a computational mesh the number of equations given by (2) is equal to the number of cells in the mesh while the number of unknowns depends on the number of nodes in the mesh. In general these are different. When the number of equations exceeds the number of .unknowns, as for example on an unstructured triangular mesh, it is impossible t o choose the unknowns to satisfy all the equations. Moreover in any iteration of distribution type in which fluctuations are added to the vertices of the mesh with weights, convergence does not imply that the fluctuations vanish because there is a null space. Likewise, in the least squares approach the norm (4) cannot be driven down t o zero because of the existence of the null space. We are interested in allowing the coordinates of the vertices t o be additional unknowns of the problem. Typically this will make the number of unknowns exceed the number of equations. In that case, setting all the equations to zero does not determine the unknowns and there are then many solutions which make the fluctuations zero. A fluctuation distribution type of iteration will yield one of the many solutions at convergence.


789

A unique solution may be obtained if the number of unknowns is equal to t.he number of equations. For a scalar equation and a smooth solution this may be achieved on a triangular (or tetrahedral) mesh by including a sufficient number of coordinates per node in the list of unknowns. The fluctuation may then be driven to zero by a fluctuation distribution scheme and the accuracy of the approximate solution depends on the validity of relying on the fluctuation as a measure of the error and the coarseness and/or connectivity of the mesh. We shall discuss the use of Least Squares as a fluctuation distribution scheme in this context. Recall that such a scheme has the advantage of a norm to minimise but is not conservative in the usual sense.

5

A Scalar Problem

We first consider the scalar two-dimensional equation

where g is divergence-free. In the least squares method we minimise (4) over both the nodal values Uj and the nodal coordinate N j in the direction perpendicular to a. We use a steepest descent method of the form

to achieve the optimisation, where Y may be either Uj or N j and T is a relaxation factor. Fastest convegence occurs when the sweeping takes into account the hyperbolic nature of the original equation. We take scalar examples in which is a function of x or a function of U . (i) g = (y, -2) in a rectangle -1 5 z 5 1,O 5 y 5 1 with initial data

u = { 01

-0.6 5 x 5 -0.5 otherwise

on one inflow side and zero on all others. Results are shown in Figs.1 and 2. As expected the norm (4) has been driven down to machine accuracy and the solution shows the characteristics. The redistribution effected by the Least Squares optimisation is attempting to equidistribute q5 amongst the triangles [I], driven by overwriting the inflow values. (ii) Consider now the system in which A is a function. of u, corresponding to the Shallow Water Equations. We discuss two regimes corresponding to subcritical and transcritical flow. We consider flow in a channel with a bump. In the subcritical case we obtain good results by the Least Squares method which are comparable with Multidimensional Upwinding. The mesh does not alter much when nodes are allowed to move. In the transcritical case however the results are poor and we turn our attention to a variation of the method which uses degenerate triangles.

790


6

Use of Degenerate Triangles

In the presence of shocks or contact discontinuities least squares methods tend t o give oscillatory solutions. A possible elaboration is to introduce degenerate triangles at the shock and to use the Least Squares method with moving nodes to adjust the position of the shock. In effect it is a form of shock fitting. Note that the fluctuation is square-integrable because 4 is always bounded, even at shocks where U is discontinuous. On the other hand, the residual is not square-integrable as a result of divF(U) being unbounded at shocks. T h a t is, on a non-degenerate mesh the l2 norm of the residual is well-defined but on a degenerate mesh it is not. However, the norm (4) is always well-defined. We show results from three examples. x 5 0 , 0 5 y 1 with initial data (iii) g = (y, -2) in a square -1

u -ai+i 5 ai 5 2.ai+l 2

-

U U, E [min (Ui, U i + l ) , max(Ui, Ui+l)]

(16) 5.3 Unrefinment procedure

If Qiis smaller than @,

, then we fuse !2i,j,k U Qi+l ,jIkintoone single volume,


83 1

A

along the x-axis, and compute the new value Y )by:

U of the field U ( for example T,

A

We have a linear convex combinaison and therefore

U;E [min (U,, Ui+l), max (Uill J ; + l ) ]

6.0 N u m e r i c a l s i m u l a t i o n s a n d discussion

All the numerical experiments presented in this paper were conducted on a Power Challenge and on the Origin 2K of Crihan. The various programs were coded in C programming language and were run in 64-bit precision. We present a table of the computational gain obtained by the use of adaptive methods. Id

N=100 nodes

N=200 nodes

N=400 nodes

N=100 nodes,At adapt



N=100 nodes

N=200 nodes

1053 s

9033 s

K=100 nodes,At adapt


N=400 nodes 13779 s N=400 nodes,At adapt

269 s At adapt,,Ax adapt 240 s N=100 nodes 10530 s N=100 nodes,At adapt 2955 s At adapt,,Ax adapt 2500 s

1060 s

6510 s

N=200 nodes

N=400 nodes

32000 s N=400 nodes,At adapt 9140 s

K=400 nodes,At adapt

4" At adapt,Ax adapt 2d

3d

150 000 s 42 000 s

F i g u r e 1. Temperature profile along X axis F i g u r e 2 . Te,mperature profile along X , Y- axis

7.0 C o n c l u s i o n a n d p e r s p e c t i v e s

A variety of simula tion codes in (ID, 2D and 3D) has been written to investigate numerically the relationship between heat supply - surface of application and combustion front propagation[AOU 99E]. Work is in progress to determine the most efficient matrix solver in two and three climensions. Acknowledgment Extensive numerical computations were performed on Origin 2K1 through grant 199009 from CRIHAN.

832


[AOU 99A]

AOUFI A., VREL. D., PETITET J . P . , "Modeling of SelfPropagating Synthesis of T i c " , Intern. Journal of SelfPropagating Hagh-Temperature Synthesis, 6, 1, 41-53, (1997).

[AOU 99B]

A.AOUFI," Self-Adaptive Finite-Volume Scheme for reactiondiffusion equations with Phase Change. Application to the numerical simulation of one-dimensional self-propagating fronts", submitted to Journal of Computational and Applied Mathematics. A.AOUFI, "Fully Adaptive Finite-Volume Scheme for reaction-diffusion equations with Phase Change. Applicatioil to the numerical simulation of bidimensional self-propagating fronts", submitted to Int. J . Num. Meth. Eng.

[AOU 99D]

A. AOUFI,"Three Dimensional Fully Adaptive Finite-Volume Scheme for Reaction-Diffusion Equations with Phase Change. Application to the numerical simulation of self-propagating fronts", submitted to Applied Numerical Mathematics. A. AOUFI,D. VRELA N D J - P PETITET," Bidimensional Numerical Analysis of Critical Parameters for Stable Combustion Synthesis Study of T i c " , Gordon Research Conference on Oscillations and Dynamic instabilities in Chemical Systems, June 6-11, 1999, I1 Ciocco, Barga, Italy. Accepted for Presentation (Poster).

[BON 731

BONACINA C., COMINI G . , FASANO A . A N D M . PRIMICIERO ,"Numerical Solution of Phase-Change Problems", Int. J . Heat Mass Transfer., 16, 1825-1832, (1973).

[GAL 921

T . GALLOUET, "M6thodes des Volumes Finis", Collection Problkmes No11 Linkaires Appliquks, CEA-EDF-INRIA, 2830 Octobre 1992.

[SHY 941

W. SHYYAND M . M . RAO," Enthalpy based formulations for phase change problems with application to g-gitter", AIAA paper No. 93-2831.

[VER 941

VERWER J . G . "Gauss-Seidel iteration for stiff odes from chemical kinetics", Siam J . Sci. Computing. Vol 15. No. 5, pp. 1243-1250, September 1994.

[VRE 951

VREL D., LIHRMANN J . M . , PETITETJ . P . , "Synthesis of titanium carbide by self-propagating powder reactions: part 1, enthalpy of formation of TIC", J. of Chem. and Eng. Data, 40, 280-282, 1995.

Mathematical and numerical modeling of a two-phase flow by a Level Set method. Sandra ROUY1 and Philippe HELLUY2 DCN INGENIERIE C E N T R E SUD S / D LSM B.P. 30 83800 Toulon Naval F R A N C E I S I T V - Laboratoire Mode'lisation Nume'rique et Couplages B.P. 56 83162 La Valette du Var CEDEX F R A N C E

ABSTRACT This study is devoted to the numerical simulation of a two-phase pow. Because of its formal simplicity, a Level Set method has been chosen. The interface is thus located by the zero level set of a smooth function which is convected at the speed of the fluid. A conservative system can also be written which is solved by the Finite Volume method. Special care is needed for the pressure law at the interface between the two fluids. Some numerical results are presented. Key Words: two-phase Bow, finite volume, level set, conservation laws, Roe's method

1 Introduction The purpose of this study is t o describe the separated and time-dependent gasliquid flow which appears in the cooling chamber of a gas generator where the temperature can reach 2500 K and the pressure 180 bar. The time scale is about one microsecond. We have two objectives. The first one is t o follow the evolution of density, velocity, pressure and temperature in both fluids. The second one is to accurately capture the shape of the interface separating the two fluids. The aim of this research is also to implement a simple method in order to achieve these goals. The most common model for describing two-phase flows is the two-fluid flow model obtained via a space and/or a time average of the local instantaneous equations of conservation laws for each phase (ref [SAIN 931). These equations have then to be coupled with some transfer terms between the phases. This model is especially concerned with dispersed two-fluid flows because the geometry of this type of flows allows a formulation of these transfer terms. In this way a system is obtained which can be easily solved.

834


On the contrary, when dealing with separated two-phase flows, it is more judicious to use an interface tracking method. Two different approaches are possible: the lagrangian approach and the eulerian approach. In the lagrangian methods the interface is located by marker particles (ref [UNV 921). These methods are accurate and close to physical phenomena. However they require a lot of particles when the interface is complex. Moreover regridding algorithms must be employed in order t o properly track the interface. The eulerian methods are divided into two categories: the VOF methods and other related methods (ref [HIR 811) and the LEVEL SET methods (ref [MUL 921). The Volume-Of-Fluid methods introduce a new variable: the volume fraction of one of the fluids defined in each cell of the grid. This function is equal to 1 if the cell is filled with fluid and equal to 0 if it is empty. For a mixed cell which contains both fluids: the VOF function lies between 0 and 1. A specific numerical process is used to compute the transport of this quantity and to ensure the conservation of both liquid and gas phase mass. This process requires to reconstruct the interface a t each time step. Different techniques of reconstruction, more or less complicated, have been proposed. These methods are especially used for incompressible fluids with free surfaces. In the Level Set methods the interface is represented as the zero level set of a smooth function 4 defined on the entire physical domain and satisfying a transport equation. This function is initialized as the signed distance from the free surface. In this formulation the interface can merge or break up with no special treatment. It presents several advantages: there is no need to reconstruct the interface or to regrid the neighbourhood of the discontinuity. Moreover it can be easily generalized to three dimensions. For these reasons, we have chosen a Level Set method. This type of method seems to be the simplest way t o treat the interface. In all the methods described above, it is very difficult and sometimes impossible to deal with two-phase flow with a large density ratio. Our main concern is to overcome this difficulty.

2 2.1

Governing equations Equations of motion

We consider all phases as a single fluid which is supposed t o be compressible and inviscid ; we neglect gravity, phase changes and surface tension. The fluid is then governed by Euler equations:

where p is the density of the fluid, d the velocity, E the specific total energy, P the pressure and e the internal energy with the relation E = e $u2.

+


2.2

835

The level set formulation

The interface r is then located by the zero level set of 4 (ref [MUL 921). I? is defined by r = {x/4 (x, t ) = 0). The level set function is positive in the gas and negative in the liquid. Hence we have:

> 0, ifx in gas < 0, ifx in liquid Since the interface moves with the fluid, q5 satisfies the transport equation 4t + + d . V 4 = 0. This equation is not conservative. In order to have a conservative equation, we consider quantity p4 which satisfies the conservative equation:

2.3

Closure of the system

The system is thus the following:

In order to close this conservative system, an equation of state is needed which must be relevant to the different states of the fluid. This is the difficult point of the modeling. We use the ideal gas equation of state for the gas and the Stiffened-gas equation for the liquid (ref [COC 961). Therefore the pressure depends on the density, the internal energy and 4. The equation of state is then:

P = P (P, e l 4) = a (4)Pe + 0(4) P = (y - 1) pe with y = 1.4 if 4 > 0 P = ( N - 1) pe - NP, with N = 5.5 and P, = 4.9211538 Pa if

4 0) =

uR

T .

with f (U) = (fl (U) , f 2 ( u ) l T , U = (p, p u ~pu2, , PE, p41T , u = ( u l , u2) 1s the velocity of the fluid, f l (U) = ( p u l , ~+ ~ P, ? P U I U Z , ( P E P ) u l , ~ 4 ~ and 1 ) ~

+ + P, ( P E + P ) u2, ~

(U) = (PW, pu1u2, pu; 4 ~ 2 ) ~ Because of its simplicity we decided to use Roe's method. This method consists in a linearization of the above system that leads to solve (v is the outward normal to the interface): f2

Let us recall the main features of this method. A system of type (2) admits a Roe-type linearization, if for any (UL,UR) in E x E , there exists two matrices A1 (UL,UR) , A2 (UL,UR) , such that the three following points are verified: (i)

fi (UR) - f~(UL) = AI (UL,UR) (UR - UL)

(UR) - f 2 (UL) = A2 (UL,UR) (UR - UL) (property of conservativity) the Jacobian matrix of f l (ii) A1 (U, U ) = ,U (property of consistency) the Jacobian matrix of f 2 A2 (U, U) = (iii) for any direction u = (ul, u2)T E R2 such that Iul = 1, the Roe matrix A, (UL, UR) "in the direction of v" defined by vl A1 (UL, U~)+v2A2(UL, UR) has real eigenvalues with a complete family of eigenvectors (property of hyperbolicity). f2

In practice, the Roe matrix is determined by computing an averaged state between UL and UR. We apply a change of variables W + U(W) such that U ( W ) and f ( U ( W ) )are homogeneous quadratic functions of W and so we can write the following relation:

A, (UL,UR) = A,

(U) with

= U (W*) and W* =

(WL

+ WR).

In our case, we can prove the existence of the Roe matrix defined by (iii) where JZ, fi$)T with f i = the average state is given by: U = (fi,

m,

fiz,

$ (fi + 6) and T = (flzi+6in) f i + & for r = U I , u2, H, 4.


837

Then Al and A2 write:

-

0 1 1--2 -uI2tZaIuI K(2-5) --u1~2 '112 ~(-~l++alfil~)

0

-

Al

(U) =

-

-(YUZ

-

z-m2

u1 -au1u2

0 (a+l)E

4

0

0

--

-

-4G

-

0 a

-

0 0 0

o

-

u1 -

The pressure law coefficient E is equal t o y - 1 in gas and to N - 1 in liquid. The main difficulty here is to properly calculate a in such a way that it produces a correct pressure at the interface between the two fluids. Several choices will be discussed in section 4. The eigenvalues and the corresponding eigenvectors of Roe's matrix are given by: X1 = u.u - c, A2 = A3 = X4 = 7i.u , X.5 = 7i.v C , 7-1

= 1

-

u

-

H

-

-

E

l

)

-

+

T

,

7-2

= (l,Ei,%,;

(24-2),o)

T

,

T

rg = (0, -u2, vl, u1% - VZG, 0) , rq = (0,0,0,0, l)Tand r5 = ( ~ , K + F U ~ , G + ~ ~ ~ , B + E U . U ~ ~ ) ~ . that is to say in

E is the sound speed and is equal to

+ P,) +f .

and in liquid i? = J N ( P gas c = enthalpy which satisfies the relation H = E

/p

, H is the

total specific

Roe's scheme associated to the linearization is then:

x*

represents the summation over all edges e of the control volume E,, n, where is the normal direction t o the edge and U," is the approximation of the average value of U (., t ) on the control volume. The numerical flux is given by: 4 (Up, Ujn, n e ) = A, (Up, Ujn) Ujn + A;= (UF, Ujn) U p A+ and A- are defined by the relation Af = TA*T-~ where A* = diag Xf = max(0, A) and A- = min(0, A).

(At),

For stability, a CFL condition has to be respected: E m a x IXkI

la&] 5 1

838


4

Results

In order to validate Roe's solver, we compute the well-known test case of the shock tube which reduces to a one dimensional Riemann problem:

There are three characteristic families corresponding to the distinct eigenvalues. We can explicitly compute the one-parameter families of shocks, simple waves and contact discontinuities. The family of shocks is computed thanks to the Rankine-Hugoniot jump conditions which are the following: ff

[PI = [PUI

a [pu] =

+

[P pu2]

a [PEI = [ ( P+ PE) ul 0

[ P ~ =I [ P 4 4

The families of simple waves and contact discontinuities are computed thanks to the Riemann invariants which are u and P for the contact discontinuities and s (the entropy ), u f a c and C$ for the simple waves (ref [SMO 941). We obtain via this resolution an interesting information: the level set function is equal to zero at the contact discontinuity. An example of the solution is shown on the following figure.

' PI

contact discontinuity

x As a first validation, we consider a two gases flow where the pressure laws are different. Density, velocity and 4 are represented on the following figures.


density

velocity

pressure

level set function

839

We can see that the capture of the contact discontinuity is not satisfying. Moreover, this leads to oscillations (ref [COC 961). Another point is that we have to introduce an intermediate equation of state when # L and # R have different signs. We have considered two different equations: an interpolation between the two laws and an "upwind" law. This intermediate law must provide relevant pressures. This point becomes especially critical when we consider the gas-liquid flow. Indeed computations in this case are not successful. Moreover as we said previously, the water pressure may become negative when the internal energy is too small. This adds another difficulty.

840


5

Conclusions

The level set method is a simple method to track the interface. Indeed we obtain a conservative system which can be easily solved by the Finite Volume method. Moreover, no special treatment is needed for tracking the interface. However in the case of a gas-liquid flow, we have t o overcome the fact that we have two different types of equation of state. An intermediate law has to be introduced, which keeps physical sense. Moreover, the Stiffened gas equation has to be modified t o avoid negative pressure. Finally, in order t o properly capture the contact discontinuity, a second order scheme a t least must be employed. Some results with a gas-liquid flow will be presented during the symposium.

6

References

[SAI 931 SAINSAULIEU, L.-An Euler System Modelling Vaporising Spray-Dynamics of Heterogeneous Combustion and Reacting Systems,1993,Vol.l52,pp.280-305. [UNV 921 UNVERDI, S.O. and TRYGGVASON, G.-A Front-Tracking Method for Viscous, Incompressible, Multi-fluid Flows-Journal of Computational Physics- 1992,Vo1.100,pp.25 37. [HIR 811 HIRT, C. W. and NICHOLLS, B.D.-VOF Method for the Dynamics of Free Boundaries-Journal of Computational physics,l981,Vol.39, Ni.l,pp.201-225. [MUL 921 MULDER, W., OSHER, S, and SETHIAN, J.A. -Computing Interface Motion in Compressible Gas Dynamics-Journal of Computational Physics,l992, Vo1.100,pp. 209-228. [COC 961 COCCHI, J.P., SAUREL, R. and LOR AUD, J.C.-Treatment of Interface Problems with Godunov-Type Schemes-Shock waves,l996,Vol.5,pp. 347-357. [GOD 961 GODLEWSKI, E. and RAVIART, P.A. -Numerical Approximation of Hyperbolic Systems of Conservation laws-Applied Mathematical Sciences,l996- 509p. [SMO 94) SMOLLER, 3. -Shock Waves and Reaction-Diffusion equations- SpringerVerlag,1994-632p.

Multiresolution analysis on triangles: tion t o conservation laws

applica-

A. Cohen, S.M. Kaber, M. Poster Laboratoire d'analyse nume'rique Universite' Pierre et Marie Curie (Paris 6), France http://www. ann.jussieu.fr

A multiresolution algorithm is coupled with a finite volume scheme t o solve scalar bidimensional conservation laws. The originality resides i n the adaptivity of the multiresolution decomposition, which takes into account the possible appearance of discontinuities and their displacement. Numerical simulations on triangular meshes point out the advantages of the method i n terms of CPU and memory costs. ABSTRACT

Key Words: Multiresolution - finite volumes - adaptive scheme.

1. Introduction

We are interested in this work in solving conservation laws on polygonal domains. It is well known that such equations can develop localized discontinuities in finite time. In such areas where the solution is not smooth a fine resolution is necessary and furthermore, higher order schemes for flux computations are necessarily nonlinear - including for instance E N 0 reconstruction. These are very costly techniques and it seems reasonable to use all available information on the local smoothness of the solution to decide whether they should be used or not. This can be done within multiresolution framework. The original idea of combining the advantages of this method - data compression, smoothness indicators - into conservation laws solvers in order to reduce the number of flux computations is due to Harten [Har94]. Initially implemented in one dimension, it was then extended to bidimensional cartesian grids, and smooth deformations of rectangular grids, (see references in [CDKP99]). Preliminary approach for unstructured grid is explained in [Abg97]. In the case of triangular meshes, which are more flexible in modeling complex geometries, a complet,e implementation with a detailed analysis of the encoding/decoding algorithm can be found in [KP99, CDKP991. In this former approach, the solution is represented everywhere on a uniform fine grid and the multiscale analysis is used to speed up the flux computations in the smooth areas. What we propose here is a fully adaptive scheme which

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make use of the compressed solution - encoded at each time step by its most significant coefficients. This approach gives way to new difficulties, for instance in the analysis of the stability and precision. These points have been studied in detail in the one dimension case in [CKMP99] and are currently under investigation in the triangular meshes case. The outline of the paper is as follows: we recall in section 2 the multiresolution and the finite volume algorithms for triangular meshes which are then combined to produce the adaptive scheme presented in section 3. Numerical simulations and analysis of performances on tests cases follows in section 4.

2. Multiresolution analysis and finite volumes

We briefly describe a multiscale transformation of a function described by its mean - and not point wise as usual - values on a triangular mesh. The equation is defined on a polygonal domain fl which is discretized by a coarse grid R0 made of No triangles. A hierarchy of nested grids Re (0 5 C 5 L) is built. The grid Re+' is obtained by dividing each triangle of Re into four smaller triangles by joining the three edges midpoints. Triangles from the level C are denoted by Tf for 1 lc 5 Ne = 4 e ~ o We . denote by iif the average of a function u on the triangle Tf and iie = (iif):~~. The mean values tie+l of a function u being given on the fie+' grid, the mean values on the coarser grid fle can be computed by

with G i j = T j - Ti and a distance weighting exponent q 0. The formulation is one-sided due to the restricted consideration of neighbours in Epa,(i). Differentiation and minimization yields the following system (exemplary for 2-D):

C

1 lrsiij/r

j€EpOr(~)

(

( ~ x i j ) ' ~ x iAjy i j ) (Gi,z) A ~ i j A ~ i j( ~ ~ i j Gi,y ) ~

( G j - Gi) Aai,

C

=i€Epor(i)

16ij/9

(flyij)

allowing the computation of a one-sided gradient VG? at iteration level n.


853

S t e p 2: Again with least squares, but now projected in the direction of the gradient V G r , the following ansatz is obtained:

where

EhYp(i) = S P P ( i )

n

u

SP-Ii(k)

k € S K L ( L P ( i ) - 1j)

Considering a @ h y p / d ~ ~ n=t 0 l )for minimization and setting C = 1 yields:

(8)

For the nodes i E S P - L ( l ) this approach must be modified, since the position G = 0 on the edges k E (SIq-P(i)nSI

IGnlanlq l i t 6 ,+ 0 Also 1 , since the addend mmln is contained. A d d i t i o n a l e r r o r correction: The above described normalization can still alter the position of a discontinuity. Since depending on the curvature, a shift of the front line appears, an additional correction method is required. For this, the exact position of the front before normalizing the nodes i E S P L ( 1 ) is stored and compared t o the position after the normalizing step. Considering a linear representation of G over the edge, an error correction is done, computing an increment AGk for the edges, that satisfies:

where Al,,,,, is the shift after the iteration in the first level is com leted. ) Since several edges k E SIC-L(0) may be appended to a node i E S P - ~ 6 the corresponding correction is taken as average. Although this additional correction is simple, its effect is very remarkable as shown in figure 3. 3.3 P r o p a g a t i o n s p e e d , back influence a n d o v e r r u n n i n g n o d e s In the previous chapter a propagation speed Zwas used, without stating how this is obtained. Considering a passive transport, the carrier speed is easy to


Figure 3: Passive convection of a circular line on a field with constant carrier speed. Cornparison with and without additional error correction. Computed solutions.

access. For active discontinuities, the propagation speed depends on the corresponding values at both sides of the discontinuity. Consider as example a shock, whose propagation speed can be computed upon the pressures at both sides. For the edges k SIC-L(0) the left and right states can be computed upon projected variables from the ending nodes. Since the position and motion of discontinuities can be treated as described, it is now interesting, how the discontinuity affects the surrounding fields. In the present work, a flux separation scheme was used, not requirering a sub-cell resolution. In this method the original control volumes of the mesh are preserved. Referring back to figure 1, each edge of the mesh carries a segment of two adjacent control volumes. In smooth regions, a projection from the nodes storing the variables to the cell interface is performed and a flux comof the fluxputed in a central or upwind manner. The flux is Figure 4: then used for both the adjacent control volumes. For ail edges k E SIC-L(0) this is no longer the case. The flux for the "left" side is computed using the "left" side projection, and the the flux for the "right" side with the right side projection accordingly, see figure 4. This method is very simple and can be employed on any grid. As a node of the mesh gets overrun by a front, the variables stored at the corresponding nodes switch to the other side. At present these new values are obtained by extrapolation from the the surrounding neighbours at the new side. Note, that this method does not ensure conservation. However, a fully conservative variant of the method is currently in development and expected to be available soon. -

4. C o m p u t a t i o n a l e x a m p l e s The above presented methods have been developped without any preference for a specific application. However, the examples below are related to fluid dynamics. The figures 5 and 6 show computed results for a shock tube problem where both, the front shock and the shear layer are tracked. The computation


855

was performed on a 2-D unstructured mesh with triangular elements. As a second example, the inviscid, transient computation to steady state for a cylinder at Ma, = 2 is shown. As initial condition a straight shock is located in front of the cylinder. The computation is performed on the grid shown in figure 7. Figures 8, 9 and 10 show the tracked position of the bow shock, the corresponding subset grids and isolines of density for the initial condition, an intermediate and the converged solution.

expansion ..

...--.. i....

~ s o l ~ nof es

contact

shock

..

Figure 5: Shock-tube example: Isolines of density, subset meshes and location of shock and contact discontinuit~es1n1t ~ a condition l ( t o p ) ,after 300 time steps (below) and zoom of the contact discontinuity after 300 time steps (left). These figures are computed results.

- tntt~alcond!l~on

-lnldal condlbon loo timesleps

100 limesteps 200 timesteps 300 tlmssleps

- rI - y :-

2.0.

Figure 6:

-:>

4.0

.

2.0

.

Shock-tube example: cut through a 2-D solution at different time levels.

References

[KER 881 A. KERSTEIN,W. ASHURST,F . WILLIAMS:Field Equation for Interface Propagating in an Unsteady Homogeneous Flow Field. Phys. Rev. A , uol. 37, pp2728-2731, (1988).

[MUL 921 MULDER,OSHER,SETHIAN:Computing Interface Motion in Compressible Gas Dynamics. J. Cornp. Phys. , vol. 100, pp 209-228, (1992). P. SMEREKA, S. OSHER:A Level Set Approach for Comput[SUS 941 M. SUSSMAN, ing Solutions to Incompressible Two-Phase Flow. J. of Comp. Physics, vol. 114, pp 146-159, (1994).

[GLO 971 0 . GLOTH,R.VILSMEIER,D. HANEL:Object oriented programming for com~ u t a t i o n a lfluid dynamics, HiPer' 97, Krakow, Poland, (1997).

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Figure 7:

Top: computa-

tional mesh

Figure 8: Series at top, right: Tracked position of the bow shock, initial (left), intermediate (middle), and final state (right). Figure 9: Series at right: Subset meshes, initial (left), intermediate (middle), and final state (right).

Figure 10: Series at right: Density isolines, initial (left), intermediate(middle), and final state (right).

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A Stabilized Version of the Wang's Partitioning Algorithm for Banded Linear Systems * Velisar Pavlov Center of Applied Mathematics and Informatics, University of Rousse, 7017 Rousse, Bulgaria

Abstract The parallel partitioning algorithm of Wang for arbitrary nonsingular banded systems is stabilized. Some numerical experiments (including random matrices) are presented.

Key words and phrases. Iterative refinement, parallel partitioning method, banded systems, perturbation. AMS(M0S) subject classification. 65G05,65F05,65Y05

1

Introduction

Banded systems of linear equations appear in many problems and are the computing time consuming kernels of various applications. The systems arise either directly, as in the difference approximations of ordinary differential equations, or after suitable rearrangement of equations and unknowns, as in finite element methods for elliptic problems. Such systems we can solve in parallel by the so called partition methods. A typical member of these methods in the case of tridiagonal systems is the method of Wang [8]. This method gives an efficient parallel algorithm for solving such systems. Full roundoff error analysis for the whole algorithm in the case of nonsingular tridiagonal matrices is presented in [lo]. Generalized versions of the partitioning algorithm of Wang for banded linear systems are presented in [2, 61. Full roundoff error analysis in this case can be found in [ll]. In this work it is shown that the algorithm is numericaly stable for some special classes of matrices, i.e. diagonally dominant (row or column), 'This work was supported by Grants 1-702197 and MM-707197 from the National Scientific Research f i n d of the Bulgarian Ministry of Education and Science.

860


symmetric positive definite, and M-matrices. Unfortunately when the matrix (even though well conditioned) of the system does not belong to the above mention classes, the algorithm can breack down or behave poorly. In our paper we present a stabilized version of the generalized Wang's algorithm for arbitrary nonsingular banded linear systems. Let the linear system under consideration be denoted by

+

where A E Rnxn,which bandwith is 2 j 1. For simplicity we assume that the number of superdiagonals j is equal to the number of subdiagonals. The partitioning algorithm for solving (1) can break down when it is necessary to divide by numbers which are less than a certain limit 6. In such cases we improve the algorithm perturbing the inputs or intermediate data. But the result which we get is perturbed. In order to make the solution more accurate we use iterative refinement (see [3]). Hence, it is necessary to solve (1) several times with different right hand sides. A similar perturbation approach is used in [l]for a Strassen-type matrix inversion algorithm, and in [4] for a fast Toeplitz solver. The convergence of the iterative refinement is analysed in [7,91.

The outline of the paper is as follows. Section 2 presents the partitioning algorithm. In the next section we consider perturbations and iterative improvement of the solution. Finally, in Section 4 we present some numerical experiments (including random matrices) in MATLAB.

2

The partitioning algorithm

For simplicity we assume also that n = ks - j for some integer k, if s is the number of the parallel processors we want to use. Let us note that our assumptions are not essential for the consideration. We partition matrix A and the right hand side d of the system (1) as follows:

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861

where Bi E ~ ( " - j ) ~ ( ~ - j i) ,= 1 , 2 , . . . ,s , are band matrices with the same bandwith as matrix A , &,Ti are matrices of the following kind

whoseelementsa(i-l)k+l,~~~l ~ R j and finally

~

aj i k, , b i k , ~ ke R j X j , i =1 , 2,..., 3 - 1 ,

Now we define the following permutation

[ l : k - j , . . ., ( i - 1 ) k i - 1 : i k - j , . . . , ( s - l ) k + l : s k - j , k - j + l : k , . . . ,i k - j + l : i k , . . ., ( s - 1 ) k - j + l : ( s - l ) k ] , of the numbers [ I , . . . ,sk- j ] , and denote the corresponding permutation matrix by P . By applying this permutation to the rows and columns of matrix A we obtain the system

A=

A P x = Pd,

PAP^

=

All

A12

where All = diag{B1, B 2 , . . . ,B,) E ~ " ( " - j ) ~ " ( " - j ) ,

ak

Ck

0

."

0

. . . a2k

C2k

"'

0

A21 = 0

... a s - 1

C(s-l)k

.

"

here Aal E ~ j ( ~ - l ) ~ ~ ( " jA22 ) , = diag(bk, bak,. . . , b(,-l)k), E The algorithm can be presented as follows. Stage 1. Obtain the block LU-factorization

A=

( All

-421

A12 A22

by the following steps:

)

=

LU =

( All A21

0 Ij(s--1)

)(

Is(;-j)

R S

)

0

862


1. Obtain the LU-factorization of All = PILIUl (with partial pivoting,

if necessary). Here P1 is a permutation matrix, L1 is unit lower triangular, and Ul is an upper triangular matrix with diagonal elements u,(1), u2(1), . . . ,u (1) , ( ~ - ~ using ), Gaussian elimination (with pivoting, if necessary). 2. Solve AiiR = A12 using the LU-factorization from the previous item, and compute S = A22 - A21R, which is the Schur complement of All in A. Now let us notice that when we solve (2) it is necessary to divide by u,!l) for i = 1,. . . , s ( k - j ) . In this case if the blocks Bi (one or more) are singular, then at least one of the quantities u!') becomes very small or zero and the algorithm can break down, or behave poorly. In order to avoid this dangerous situation we propose to perturb them with 6, where b is sufficiently small. The implementation of this idea is presented in the next section. Stage 2. Solve Ly = d by using the LU-factorization of All (Stage 1). Stage 3. Solve Ux = y by applying Gaussian elimination (with pivoting, if necessary) to the block S.

3

The Stabilized Algorithm

As was noticed in the previous section the algorithm can break down, or behave poorly, when ujl), for i = 1,. . . , s(k - j ) and are zero or small. So, we can perturb them in such a way that it would be away from zero. The stabilization step can be summarized as follows:

if (1u!')1 < 6) if (\u!')l = 0) "jl) = 6 ; else ujl) = ujl) end end

+ sign(uj1))6;

In this way we shift ujl) away from zero. Hence, the algorithm ensures that we do not divide by a small number.

From the other side the obtained solution is perturbed. Then we apply the usual iterative refinement from [3], with some modification:

= 2; f o r m = 1,2, . . . ?-(m-1) = b - A X ( m - 1 ) ; (A + A) y(m) = r ( m - l ) . x(m) = x(m-') y(m); end x(0)

+

The difference here is that instead of A we solve perturbed systems with the matrix A+A, where A is a diagonal matrix with all such perturbations, and li: is the result of the perturbed algorithm before the iterative refinement is applied. We note that, when 6 = 6LZ (in double precision), in practice the perturbed solution is very close to the exact one and we need usually only one or two steps of iterative refinement, depending on what accuracy we require. Here by po we denote the machine roundoff unit. Taking into account [7] the condition of convergence of iterative refinement is Ccond(A)6 < 1, where cond(A) is a condition number of matrix A and C is a constant of the following kind

4

Numerical Experiments

Numerical experiments in this section are done in MATLAB,where the roundoff unit is po zz 2.22 x 10-16. The exact solution in our examples is x = (1,1,.. . , I ) ~by , Nit we denote number of iterations, and we measure two types of errors: 1. The relative forward error

where P is the computed solution; 2. The componentwise backward error (see [ 5 ] )

BE = max l 10;

Let us consider the following examples. Example 1. Let A be a matrix of the following kind:

This matrix is very well conditioned. In our tests we set E equal to 0 and 10-12. Hence, the blocks B, become singular. We report the results with different values of 6 in Table 1 (when E = 0) and in Table 2 (when E = lo-''). When S = 0 we obtain the original algorithm without stabilization. We see that the original algorithm can break down (when E = 0) on account of dividing by zero, or behave poorly (when E = 10-12). At the same time the stabilized algorithm gives much better results. Let us note that at most two steps of iterative refinement are enough and FE, BE are sufficiently small when S = and that only a few perturbations are necessary. Example 2. In this example we generate 1000 random matrices for each version of the algorithm (original and stabilized), where k = 7, s = 15,j = 2.

Table 2: The forward and backward error and the number of iterations of Example 1, when E = 10-l"

Orig. Stab.

Average F E Worst FE Average F E Worst F E Average N;t Worst Nit

6=0 0.12 2.77

6=

6 = lop8

6 = lo-'

x lo-13 2.42 x 10W1".22 x 10-1".23 3.11 x lo-' 2.52 x 10-l2 6.72 x lo-'' 1.24 1.12 1.41 2 2 2

Table 3: The forward error and the number of iterations for the random matrices of Example 2.

Then the blocks Ba are matrices of the following kind:

where i is a random integer = Now let us fix ,!$> = pr;J' = 3/!2 = ,8g) which belongs t o the range [I,151) to make the block Bi singular and hence the original algorithm numericaly unstable. We report the results in Table 3 for different values of 6, again. The second and the fourth rows of Table 3 present the average forward error for the two versions of the algorithm, and the third and the fifth rows contain the worst forward error which is obtained during the series of experiments. The average and the worst number of iterations we present in the sixth and the seventh rows of Table 3. The tests with random matrices show that the optimal value of 6 is again. So, if we compare both algorithms, we see that the stabilized algorithm is significantly better than the original one. In conclusion, the algorithm behaves well in pracrice, and we need to do more research for its theoretical justification.

866


References [I] Balle, S., Hansen, P., Higham. N.: A Strassen-type matrix inversion algorithm for the connection machine. APPARC PaA2 Deliverable, Esprit BRA Contract # 6634; Report UNIC-93-11, UNIaC, October 1993. [2] Conroy, J.: Parallel Algorithms for the solution of narrow banded systems. Appl. Numer. Math. 5 (1989) 409-421. [3] Golub, G., Loan, C. van: Matrix Computations. The John Hopkins University Press, 1996. [4] Hansen, P., Yalamov, P.: Stabilization by perturbation of a 4n2 Toeplitz solver, SIMAX, 1999. (to appear) [5] Higham, N.: Accuracy and Stability of Numerical Algorithms. SIAM, Philadelphia, 1996. [6] Meier, U.: A parallel partition method for solving banded linear systems. Parallel Comput. 2 (1985) 33-43. [7] Skeel, R.: Iterative refinement implies numerical stability for Gaussian elimination. Math. Comp. 35 (1980) 817-832.

[8] Wang, H.: A parallel method for tridiagonal linear systems. ACM Transactions on Mathematical Software 7 (1981) 170-183. [9] Yalamov, P.: Convergence of the iterative refinement procedure applied to stabilization of a fast Toeplitz solver. Proc. Second IMACS Symposium on Iterative Methods in Linear Algebra, Eds. P. Vassilevski and S. Margenov, IMACS (1996) 354-363. [lo] Yalamov, P., Pavlov, V.: On the Stabilty of a Partitioning Algorithm for Tridiagonal Systems. SIAM J . Matrix Anal. Appl. 20 (1999) 159-181.

[ l l ] Yalamov, P., Pavlov, V.: Stability of a partitioning algorithm for special classes of banded linear systems. Preprint N 38, University of Rousse, March 1998. (submitted to LAA)

On Jeffreys Model of heat conduction

by Maksymilian Dryja and Krzysztof Moszyliski University of Warsaw Banacha 2, 02-097 Warszawa Poland

ABSTRACT We present differential equations describing the Jeffreys Model of heat conduction, and certain corresponding numerical models. We define some suitable weak form of Jeflreys equations with boundary conditions of low regularity, (defined by L2 functions), and formulate existence and uniqueness theorem for this generalized differential problem. We give also theorem on unconditional stability of the numerical model and propose its finite-volume interpretation. Some results of numerical computations, compared with experiments with heat waves are presented.

1.

Introduction

Classical Fourier law relates a heat flow Q and temperature T in a very simple way: Q = -kAT, where % is a positive constant. However, it is well known that the classical heat conduction equation based on this law does not describe well enough all phenomenons observed in experiments. Such phenomenon is, for example, so called heat wave incitated in thin metallic layer by a short heat impulse. Jeffreys model is based on a somewhat more complicated relation than that given by the Fourier law: it is certain integro-differential relation which involves so called relaxation time ([I],[2], [3],[4]). Due t o the modified Fourier law, Jeffreys model fits better to experiments with heat waves: numerical computations based on this model show quite good consistency with experimental results. The heat impulses imitating the heat waves, which we observe in numerical experiments, are represented in the numerical model by the boundary conditions for Jeffreys differential equations. We apply very sharp, and even discontinous heat impulses. Differential equations of the Jeffreys model are in fact very


868

simple, and the only difficulty in present problem was caused by the assumed low regularity of the boundary conditions.

2. Jeffreys Model and its weak form

General equations of Jeffreys Model can be presented in the following form:

where the scalar function T ( t , x) is the temperature, the vector-function Q(t, a) is the heat flow, t E R+, x E R , R C R3, D and ti are positive coefficients (in general constant). In fact we are interested in one-dimensional model, because it corresponds to known results of experiments with the heat waves. The one-dimensional equations are of the following form:

with t E [0, t,,,], x E [L, PI, T(0, x) = To(x), Q(0, x) = Qo(x), T(t , L) = 4 ( t ) , Q ( t ,L) = $(t), T(t, P) = Q(t, P ) = 0, where To, Qo, 4 and $ are given functions. Functions To and Qo are initial conditions, while I$ and $, involved in the left boundary condition, define the heat impulse. Since we assume that 4 and $J are in L'(L, P), it is clear that we have to formulate the boundary value problem for equation (1) in a weak form, which makes possible to prove the existence (and uniqueness) of solution of assumed regularity. Our boundary value problem is defined on the bounded interval [L, PI, therefore it is easy to avoid non-homogenity of the boundary conditions. Let us

and

and let ?(t, x) = T(t, x) - d ( t , x) and &(t, x) = Q(t, x) - $(t, x). Functions T and Q satisfy nonhomogeneous Jeffreys equations and homogenous boundary conditions. Further, we introduce new functions

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and

1

869

t

s ( t , x) =

Q(S, x)ds.

Using these two functions we have the following boundary value problem:

St

+ DR, + S - KS,, = g

(21,

with zero initial and boundary conditions, where f and g are in terms of 4 , $, To and Qo. In order to state the weak formulation for equations (2), we are proceeding in quite standard way. We introduce two bilinear forms: a : (L2(L,P) x H ~ ( LP, ) ) x (L2(L,P ) x H ~ ( LP, ) )

b : (L'(L, P ) x L2(L,P ) ) x (L2(L, P ) x L2(L, P ) )

-+

R,

R,

where:

b(R, S; V, W ) =

LP

[DRV + SWIdx.

Let:

-

1 = -[D P-L

L-P -1L , + ( s ) ~ s Jt

-

(P - x)rn(t)~+TO(X),

t

J +(s)ds 0

- ( P - x)($(t)

+

1

d(s)ds)l+ QO(z).

We define now the space H of pairs of functions (R, S), in which we want to find solutions of our problem. Let H be the space of all pairs of functions ( I Z , S),

such that:

870


Here

11 . 110

is the L2(L, P)-norm.

Weak formulation of Jeffreys Model We are looking for a pair ( R , S ) E H such that for almost all t E [0, t,,,], , all V E L2(L, P ) and W E H ~ ( LP):

for

holds. The following theorem establishes an unique solution of the problem (3)

Theorem 1. Assume that:

TO E L2(L, P ) ,

then in H there exists a solution ( R , S ) of the variational equation (3). Let U c H be the subspace of H of such pairs ( R , S) that Rl S x E C(0, tmax, L ~ ( LP, ) ) ,

s E C(O1 tmaz, H,'(L,

P)).

If a solution ( R , S ) of (3) is in U , then it is unique in U Proof of this theorem will appear in [5]. Now, let us give some comments on the above result. First of all it implies that the solution ( R , S ) of equation (3) is such that: Rt = T - 4 E ~ ' ( 0 t,,,, , L 2 ( ~P)) , and

= Q - II E L~(o,~,,,, H , ~ ( Lp, ) ) , exactly what we need for our solution (TI&).Moreover, the function T with respect to 2, is in L2(L, P ) , hence it should only weakly depend on the function 4 defining the boundary condition for it. This fact was observed during experimental computations. St

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871

3. Numerical model In fact the used numerical model is an implicit finite difference scheme on the rectangular grid of points (tn,xk), where t, = rn for n = 0, I , . . . , N with t,,,=Nr,x~=L+khfor k=O,l,...,M+l,andP=L+(M+l)h . On this grid is defined the following finite difference scheme:

with the following initial and boundary conditions:

where T p , Q; are values of the grid functions in the point (t,, xk), Afk = f k t l - f k , Vfk = f k - f k - 1 , X = and p = &. The next step is to prove the stability of the scheme (4). In order to avoid strong regularity hypotheses concerning functions q5 and $ such as, for instance, boundedness of divided differences and so on, we proceede in similar way as in the section 2. We begin with functions ? ; = T; - 4(tn, xk), and Q; = Qp - $(in, xk), where d ( t , z) and $(t, x) were defined in the section 2. Further, we define:

Using that, the scheme is defined as:

where:

+X

= T o 4 ( ~ % +-1 Q!-')+

872


and

The grid functions R; and Sr are subject to homogeneous Dirichlet boundary conditions, and the following initial conditions:

and

si

= T ( Q O ( x k ) - $(o,

zk)).

Let

Rn = [Ry,R;, . . . , R

M ] ~ ,

and

For the scheme (5), the following stability theorem can be proved:

Theorem 2

+-

If p = is arbitrary, but constant, independent of T and h , then for any m, 0 5 rn 5 N - 1, solution ( R n ,Sn) of the problem ( 5 ) satasjies the inequality:

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873

where K depend only on 4, $, T o , and Q O , and does not depend on divided differences of these functions. Proof of this theorem will appear in [5] together with proof of theorem 1.

Since

Theorem 2 implies in particular that if ,u is arbitrary, but constant, then Tc and Q; are bounded by Ii' in the sense of discrete L~ norm. This means unconditional stability and, by Lax Theory, convergence follows.

4. Numerical experiments There have been done a lot of numerical experiments involving equations (4). Further we compared graphs obtained by computation with those obtained by pl~ysicalexperiments. We succeeded t o choose good enough values for coefficients D and K . We made computations with Fourier model as well. Conclusions are as follows:

1. Jeffreys model fits quite well t o experimental results. T h e heat waves appear in both cases: when the impulse is introduced by initial or by boundary conditions. 2. If we see some kind of waves in Fourier model (it is possible only when the impulse is introduced by boundary condition), they travel with visibly too great speed, and their amplitude vanishes very quickly.

3 . Coefficient D plays the most important role from the point of view of the speed of the heat waves. For more informations see [4]

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References [l] Brorson S.D. Fujimoto J.G. Ippen E.P. "Femtosecond electronic heattransport dynamics in thin gold films" Phys. Rev. Lett. 59 (1987),1962-1965.

[2] Joseph D.D. Preziosi L. "Heat waves" Rev. Mod. Phys. 61 (1989),41-73 [3] Joseph D.D. Preziosi L. Addendum to the paper "Heat waves7' Rev. Mod. Phys. 62 (1990),375-391 [4] Moszyriski K . Palczewski A. "Assymptotic analysis of heat propagation models" submitted to Archives of mechanics

[5] Dryja M. Moszynski K. "On Jeffreys model of heat conduction" Prepared for publication

Investigation some method of cavitating jets

S.A. Ocheretyany and V.V. Prokof'ev Department of unsteady hydrodynamics, Institute of mechanics Moscow State University, 11 9899, Michurinski pr. 1, Moscow, Russia

The problem of multiphase liquid flow after ventilating cavity with internal pressure exceeding the ambient pressure and with low concentrations of gas (or vapor) bubbles i n liquid is considered. The two-dimensional Bow considered corresponds i n one case to Chaplygin-Kolscher scheme, and i n another one-to the problem of two symmetric jets collision with a cavity formation. The velocity profile at large distances behind the cavity is determined. The most intensive disturbances in the gas-liquid mixture Pow are introduced by the gas injection into liquid i n the neighbourhood of the tail point of the cavity. And the injection of small gas bubbles is most eficient there. The vapor injection into a cavity is not eficient for increasing the liquid flow momentum, but i t can be used for generation of a cavitating jet. The account of multi-velocity effects i n the two-phase medium equations is essential for bubbles of supercritical size, when the viscous friction effect is small. ABSTRACT

Key Words: cavity, bubbles, heat and mass transfer, two-phase flow, multivelocity effects

1. Introduction

We shall model such a flow in a plane-parallel statement, considering, for example, the flow over a body (Figure l a ) , the rear critical point of which is substituted by a finite stagnation zone (cavity) with smooth closure of jets. Such a cavity can be generated by means of artificial injection (ventilation) of vapor or gas into the cavity. Since the boundary of such a cavity is Taylor instable, the flow will be accompanied by intensive outflow of gas (or vapor) in form of bubbles and by the formation of a two-phase tail. The first example corresponds to flow over a convex cylinder according to

876


the Chaplygin-Kolscher scheme [GUR 651 with formation of finite stagnation zone behind the cylinder with pressure exceeding the pressure in a free stream. Here one can use the well-known solution for a circular cylinder [GUR 651; we, however, shall use a simpler solution for a flow over a plate (Figure l a ) [PRO 981. Though such a scheme can not be applied for description of flow over a plate (with singularities at points E and F), it is quite suitable for studying the flow in the cavity neighbourhood. The second example presents the collision of two symmetric jets with formation of the stagnation region [GUR 651. We shall suppose for simplicity, that the jet-leading channels are stretched up to infinity. In this case, while moving upwards along the stream, the flow becomes non-univalent, but we shall ignore this fact, since we consider the flow in the neighbourhood of jets' closure point B (Figure l b ) and in the outgoing jet. If the flow is reversed, then we arrive at the problem of "splitting" of the jet flowing onto the region with heightened pressure, which was considered in [YAK 871. Near the cavity the flow is highly inhomogeneous, and the bubbles fall into the region of high pressure gradients; as a result, the bubbles motion with respect to the liquid becomes essential. When the multi-velocity effects are taken into account, the model is more complicated: the medium becomes anisotropic [VOI 751. 2. The main relations and results.

2.1 Two-Phase liquid flow with low concentration of bubbles formulations.

To model for a rarefied bubble mixture [GAR 731, [VOI 751, which takes into account the two-velocity effects: dpu dpv -+-10,

ax

ay

F, 2 F1 = --nu 3

3

dnul

dnvl

ax

ay

-+--

+ Fz = 0,

-

0,

PI

PI

dlVl dV pl(- 3-) - 2 n ~ ~ ~ l-aV () , ~ F2 l = k.rrpla(V - V1), dt dt

Varia

877

Here p = p l ( l - a) is the mixture density under the neglection of vapor (gas) density in a bubble (pl=const is the liquid density), V(u, v) is the massaverage mixture velocity, which is equal to the mean liquid velocity under the neglection of bubbles mass, V l (ul ,vl) is the mean velocity of bubbles in an elementary volume, cx = 4/37rna3 is the volume concentration of bubbles (n is the numerical density of bubbles), a is the radius of bubbles, which are supposed to be spherical. For a steady motion, supposing the slipping-related stress tensor -rij to be a tensor function of the slipping vector W(W,, Wy) = V1-V one can write [VOI 751

Further, F1 is the hydrodynamic force acting on a bubble of variable radius moving with variable velocity [VOI 731, [YAK 731; F2 is the viscous drag force. In the first example, where the bubble size is small and the relative velocity is low (the Reynolds numbers of relative motion Re zz l ) , the k = 4 value was accepted for the viscous drag force F2 (the Adamar-Rybchinskii formula). For the second example, where larger bubbles are considered and the Reynolds numbers of relative motion reach the values of the order of lo3, the asymptotic Levich formula was applied [LEV 621 (k = 12). Further, ~1 is the liquid viscosity coefficient (the viscosity is taken into account in the bubbleslliquid interaction process only), pk is the gas pressure in a cavity, u = 1 - for the isothermal process and v = y (y is the Poisson adiabatic index) - for the adiabatic expansion-compression of gas, a is the bubble boundary velocity, C is the surface tension factor. We shall suppose that the bubbles only weakly disturb the liquid flow. We

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