Rayleigh-BeМЃnard convection : structures and dynamics

RAYLEIGH-BENARD CONVECTION STRUCTURES AND DYNAMICS

ADVANCED SERIES IN NONLINEAR DYNAMICS Editor-in-Chief: R. S. MacKay (Univ. Warwick) Published Vol. 1

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RAYLEIGH-BENARD CONVECTION STRUCTURES AND DYNAMICS

A. V. Gelling Institute ol Nuclear Physics M. V Lomonosov Moscow State University

World Scientific Singapore • New Jersey 'London • Hong Kong

Published by World Scientific Publishing Co. Pte. Ltd. P O Box 128, Farrer Road, Singapore 912805 USA office: Suite IB, 1060 Main Street, River Edge, NJ 07661 UK office: 57 Shelton Street, Covent Garden, London WC2H 9HE

Library of Congress Cataloging-in-Publication Data Gelling, A. V. (Alexander V.) Rayleigh-Benard convection : structures and dynamics / A V. Getling. p. cm - (Advanced series in nonlinear dynamics ; v. 11) Includes bibliographical references and index. ISBN 9810226578 1. Rayleigh-Benard convection. I. Title. II. Series. QC3302G48 1998 536'.25-dc21 97-52073 CIP

British Library Cataloguing-in-Publication Data A catalogue record for this book is available from the British Library. First published 1998 Reprinted 2001

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Printed in Singapore

To the memory of

Su brain any an Ch an drasekh ar, my outstanding predecessor, the pioneer of systematizing our knowledge of convection

This page is inttnsionally intentionallyFcfl leftblink blank Thiipagtis

CONTENTS

1. Introduction 2. Basic Concepts 2.1. The Boussinesq Approximation 2.2. The Rayleigh-Benard Problem 2.3. Linear Analysis 2.4. Nonlinear Regimes and Bifurcations 2.5. Planforms of Convection Cells 3. Investigation Tools 3.1. Experiment 3.2. Theoretical Approaches: A. Expansions in Small Amplitude (Small Reduced Rayleigh Number) 3.3. Theoretical Approaches: B. Two-Dimensional Models of ThreeDimensional Convection 3.3.1. Amplitude Equations 3.3.2. Manneville's "Microscopic" Equations 3.3.3. Model Equations 3.3.4. The Lyapunov Functional 3.3.5. The Pomeau Manneville Phase Equation 3.3.6. The Cross-Newell Phase Equation 3.4. Theoretical Approaches: C. Numerical Simulation 4. Basic Types of Convective-Flow Structures 4.1. Two-Dimensional Rolls and Three-Dimensional Cells 4.1.1. Thermocapillary Effect 4.1.2. Temperature Dependence of Viscosity 4.1.3. Temperature Dependence of Other Material Parameters of the Fluid 4.1.4. Asymmetry of Boundary Conditions 4.1.5. Curvature of the Unperturbed Temperature Profile vii

1 9 10 13 16 20 23 27 27 30 33 36 41 43 45 47 49 54 59 59 61 62 70 73 74

viii

CONTENTS

4.1.6. Finite Thermal Conductivity of Horizontal Boundaries 4.1.7. Deformation of the Free Surface 4.1.8. Time-Dependent Heating 4.1.9. The Presence of Suspended Solid Particles 4.1.10. Tertiary Flows 4.1.11. Hexagonal Cells in a Vertically Symmetric Layer 4.2. Patterns of Quasi-Two-Dimensional Rolls 4.3. Convection Textures. Roll-Pattern Defects 4.3.1. Dislocations 4.3.2. Disclinations 4.3.3. Grain Boundaries 5. Convection Regimes 5.1. Regime Diagram 5.2. Phase Turbulence 5.3. Spiral-Defect Chaos 6. Selection of the Wavenumbers of Convection Rolls 6.1. Wavenumbers in Experiments with Random Initial Disturbances .. 6.2. Searches for Universal Selection Criteria 6.3. Stability of Two-Dimensional Roll Flows 6.3.1. Theoretical Results 6.3.2. Experimental Results 6.4. Lyapunov Functional and Selection 6.5. "Selection Mechanisms" 6.5.1. Grain-Boundary Motion 6.5.2. Spatial Ramp of Parameters 6.5.3. Motion and Equilibrium of Dislocations 6.5.4. Axisymmetric Flows 6.5.5. Convection-Front Propagation 6.5.6. Convection-Front Propagation Combined with Roll Relaxation 6.5.7. Relaxation of Rolls in Contact with a Disordered Flow 6.5.8. The Effect of Sidewalls 6.5.9. The Preferred Wavenumber and Realized Wavenumbers (Discussion and Conclusion) 7. Peculiarities of Stratification and Vertical Structure of Convection 7.1. Effects of Strong Temperature Dependence of Viscosity 7.2. Penetrative Convection 7.3. Small-Scale Motions in a Globally Unstable Layer 7.3.1. A layer with a Piecewise-Linear Unperturbed Temperature Profile

74 76 77 79 79 83 85 96 96 97 98 99 99 101 110 119 121 124 125 125 137 139 140 141 144 147 152 156 161 172 175 178 191 191 194 199 199

ix

CONTENTS 7.3.2. 7.4. Astro7.4.1. 7.4.2. 8. Conclusion References Subject Index

A Layer with Radiative Energy Transfer and Geophysical Applications Solar Convection Zone The Earth's Mantle

204 208 208 211 215 219 239

CHAPTER 1

INTRODUCTION

Convection due to nonuniform heating is, without overstatement, the most widespread type of fluid motion in the Universe. It plays also an important role in various engineering setups. All this is sufficient to warrant steadfast and intent interest of investigators in convection. But of late, this interest has also been powerfully stimulated by other circumstances. Convection problems are a rich source of material for the development of new ideas concerning the relation between order and chaos in flows and between simplicity and complexity in the structure and behaviour of hydrodynamic objects. Convective flows can form more or less ordered spatial structures, and their investigation substantially contributes to understanding the general properties of pattern-forming systems, which are the main subject for study in synergetics, an actively developed branch of modern science. Convection in a plane horizontal fluid layer heated from below, or RayleighBenard convection," is the type of convection considered most frequently. In this phenomenon, spatial and temporal effects are largely decoupled because of the lack of intense streams determined by external conditions. For this reason, both the experimental and theoretical treatment of Rayleigh-Benard convection proves to be especially fruitful. °It is sometimes called Benard-Rayleigh convection as well as Benard convection or Rayleigh convection. It is pointless to omit the name of either of these pioneers of the comprehensive investigation of convection, an experimenter and oo the equation for the vertical component fiz of vorticity acquires the form Aft z = 0.

(2.25)

If we combine this equation with the boundary conditions Qz = 0

on a rigid boundary,

—r-- = 0 Oz

on a free boundary,

(2.26)

which immediately follow from Eqs. (2.13) and (2.14), we find that throughout the layer flz = 0

if at least one boundary is rigid,

ftj = const

if both boundaries are free.

(2.27)

As a rule, we shall not be interested in the case of the uniform rotation of the layer as a whole ( 0 2 = const). Therefore, we assume as a general property

16

2. BASIC

CONCEPTS

of convection in the infinite-Prandtl-number limit that the vertical vorticity component vanishes. More precisely, a state with fi2 = 0 ensues after the completion of fast transients. The vertical-heat-diffusion time r v is an important characteristic time of convection processes. We use this quantity as the unit of measurement of time, and the statement that fiz tends to a constant (zero) value with P —> oo is based on the implication that there are no fast variations in £lz, which could make the left-hand side of Eq. (2.24) finite. In other words, it is assumed t h a t d/dt = 0 ( 1 ) . If initially Qz ^ 0, transient processes nullifying the effect of initial conditions take, in general, times as short as 0 ( P _ 1 ) . It is clear that the idealized conditions of an infinite layer can only be ap­ proximated more or less closely, rather than reproduced, in a laboratory. We shall see in the subsequent chapters that the presence of container sidewalls, which bound a finite region of the fluid layer, can strongly affect the structure and evolution of convection in this region. An important parameter is thus the aspect ratio

'4

(2.28)

where L is the characteristic horizontal size of the region (for circular containers, L is traditionally put equal to the radius)." In describing the conditions of a particular experiment, the aspect ratios r r and Vy may also be needed, which are associated with the sizes Lz and Ly of the container in the x- and y-directions, respectively. In what follows, the role of the aspect ratio will be discussed repeatedly. Since we are interested primarily in the properties of convection little affected by sidewalls, the principal attention will be given to the case of large T values. If the horizontal extent L of the volume occupied by the fluid is finite, the characteristic time of horizontal diffusion of heat T^ — L2/x = T 2 r v is a parameter of great importance. In many cases, the times of large-scale processes are related, in one way or another, with this quantity. 2.3. Linear A n a l y s i s We assume v and 6 to be infinitesimal, linearize Eqs. (2.15) and (2.16) with respect to these variables, then apply the operator z curl curl to Eq. (2.15), and make use of Eq. (2.17). The system then reduces to two equations for vz and 0. On eliminating 8, we fix a horizontal wavevector k = {kx,ky.,0}, and seek vz in "Some authors define the aspect ratio as the reciprocal of the quantity indicated here.

2.3. LINEAR

ANALYSIS

17

the form of normal modes: vz ex

eXtw(x)f(z).

(2.29)

Here A is the growth rate, x = { x , y , 0 } , and io(x) is some spatially periodic solution of the two-dimensional Helmholtz equation Aw + k2w = 0, i.e., a linear combination N

w(x) = ^2

c e k

j ' -'x-

(2.30)

j=-iV

where t h e vectors k3 differ only in their orientation: |kf| = k; in addition, k - j = —kj and c_j = c" (the asterisk denotes complex conjugation; the last two equalities are necessary for w to be real). As a result, we obtain the following equation for / : (D

2

- k2 - A)(D 2 - k2 - i A ) ( D 2 - k2)f =

-Rk2f,

(2.31)

where D = d / d 2 . The transformed Eqs. (2.15) and (2.16) for the normal modes enable us to reduce Eqs. (2.22) and (2.23) to a set of conditions for the variable i>. (or / ) : / = D / = (D 2 - 2k2 - - ^ A ) D 2 / = 0

on a rigid boundary,

/ = D 2 / = D 4 / = 0 on a free boundary.

(2.32) (2.33)

Thus, Eq. (2.31) together with the boundary conditions (2.32) and/or (2.33) constitute an eigenvalue problem for the growth rates A and the eigenfunctions

/(*)•

If both layer boundaries are stress-free, this problem can be solved extremely simply and leads to the following explicit expression for the eigevalues A„ cor­ responding to the eigenfunctions fn = sin rc7rz (n = 1,2,...):

^

=

_£ + i „ V + *>)±/(^i)'(»V + * . ) . + ^ p .

(2.34)

It can be immediately seen from this expression that for any R > 0 both existing values of A„(/£, P, k) are real. One of them is always negative while the other one is positive if

R>RMS&±^1

(2.35)

and negative if R < Rn{k). In what follows we shall mean by a real \„ just this second value changing its sign at R = Rn{k).

18

2. BASIC

CONCEPTS

Fig. 1. The neutral-stability curve for a layer of motionless fluid. The region of stable states is below the curve and the region of unstable states is above the curve. If both layer surfaces are rigid or one surface is rigid while the other one is free, the calculations are more tedious, but the results are qualitatively the same (the eigenfunctions being different). In the case R < 0, if \R\ exceeds a certain value [depending on P and reaching its maximum (zero) at P = 1], the growth rate An has two conjugate complex values. Then the corresponding eigenfunctions describe decaying oscillations which are obviously associated with internal gravity waves. We see that infinitesimal perturbations with a given wavenumber k can grow (i.e., instability is possible) only provided that R > 0, and their growth is monotonic. As ReAi—the maximum of the real parts of the growth rates A n —passes through zero, increasing with /?, the corresponding imaginary part also becomes zero. The linear analysis thus indicates that convection sets in at a certain R as steady motion. In other words, a new steady state replaces the stable motionless state of the fluid. This property of Rayleigh-Benard convection is called the principle of exchange of stabilities. It can be shown [20, 3] that the validity of this principle, as well as other above-listed properties of A n , does not depend upon the boundary conditions. Each function 7?„(/c) has a minimum. The line R = Ri{k) in the plane (k, R) delimits the region where all infinitesimal perturbations decay and the region where the lowest perturbation mode n = 1 grows (Fig. 1). Obviously, if

R< Rc = min fl,(fc) = «i(A:c),

(2.36)

2.3. LINEAR

19

ANALYSIS

the motionless state of the fluid in the layer is stable with respect to infinitesimal perturbations. The quantities Rc and kc are termed, respectively, the critical Rayleigh number and the critical wavenumber. The critical (neutral) regime (R = Rc) corresponds to the onset of steady-state motion with an infinitesimal amplitude and with a unique wavenumber k = kc. If R > Rc (supercritical regime), the layer is convectively unstable, and those perturbations can grow which have wavenumbers lying between the two roots of the equation R = R\(k) or, in other words, between the abscissas of the points where the straight line R = const intersects the two branches of the neutral curve R — Ri(k). It is the existence of this wavenumber range that creates the problem of the selection of wavenumbers in supercritical regimes. We shall discuss this problem in Chapter 6. For two stress-free boundaries Rc = —7r" = 657.511, 4

kc = 4 = = 2.221; V2

(2.37)

kc = 3.117;

(2.38)

for two rigid boundaries /? c = 1707.762,

and for one rigid and one stress-free boundary Rc= T h e growth rate \i(R, in the lowest order

1100.657,

kc = 2.682.

(2.39)

P,k) can be expanded near R = Rc, k = kc to yield

-{e-e0(k-kc)2},

^ =

(2.40)

TO

where

R — Rc C=

(2.41)

-RT

is the reduced Rayleigh number (or relative supercriticality), istic time and length scales

ro=

(^L*.^'

/ t°

l

and the character­

cPRA1'2 [2Rcdk^)k=kc

(2.42)

are called, respectively, the relaxation time and the coherence length. In certain (specially stipulated) cases we shall denote by c certain different but similar in their physical meaning quantities. The scales r 0 and £ 0 , as calculated in a number of works—in particular, Refs. 30 and 31, are equal to

2. BASIC

CONCEPTS

£o = 0.3847

for two rigid boundaries,

(2.43)

*-&r

for two free boundaries.

(2.44)

20 _ 1 + 1.954P T

°~

38.44 P

'

2(1 + P) T0

~

3TT'P

'

In Ref. 30 an experimental verification of values given by Eq. (2.43) was under­ taken. For reference we write down, restricting ourselves to the lowest-order terms, the following expansions in k — kc for the case of two stress-free boundaries: Ri(k) -Rc

= I8z2(k

- kc)2 - uV2rr(k

^p_fM

kc)3,

(2.45)

+ ^_fcf.

(2.46)

-

In two-dimensional models of three-dimensional convection, which we discuss in Chapter 3, the quantities r 0 and £0 play the role of "similarity criteria" of a sort: a simplified model, to give the best description of reality, must be adjusted by taking its similar parameters to be equal to the actual relaxation time and coherence length. Also, expansions (2.45) and (2.46) frequently appear as approximate methods for analysing convection are developed. 2.4. N o n l i n e a r R e g i m e s and Bifurcations The behaviour of finite-amplitude perturbations under supercritical condi­ tions is generally rather complex. It constitutes the bulk of the content of the subsequent chapters. Now we outline only some principal features of finiteamplitude convection near the critical regime. Sorokin [32] considered a Boussinesq fluid filling a closed container of ar­ bitrary shape. The thermal boundary conditions were assumed to produce, if the fluid is motionless, a constant (unperturbed) temperature gradient 0g (here g = g/