Uniform estimates on the velocity in Rayleigh–Bénard convection

Uniform estimates on the velocity in Rayleigh–Bénard convection

Manuel Núñez

Citation: Journal of Mathematical Physics46, 033102 (2005); doi: 10.1063/1.1855400

View online: https://doi.org/10.1063/1.1855400

View Table of Contents: http://aip.scitation.org/toc/jmp/46/3

Uniform estimates on the velocity in Rayleigh–Bénard

convection

Manuel Núñeza兲

Departamento de Análisis Matemático, Universidad de Valladolid, 47005 Valladolid, Spain 共Received 17 March 2004; accepted 23 November 2004; published online 8 February 2005兲

The kinetic energy of a fluid located between two plates at different temperatures is easily bounded by classical inequalities. However, experiments and numerical simulations indicate that when the convection is turbulent, the volume of the do-mains in which the speed is large, is rather small. This could imply that the maxi-mum of the speed, in contrast with its quadratic mean, does not admit an a priori upper bound. It is proved that, provided the pressure remains bounded, a uniform estimate for the speed maximum does indeed exist, and that it depends on the maxima of certain ratios between temperature, pressure, and velocity. © 2005 American Institute of Physics. 关DOI: 10.1063/1.1855400兴

I. INTRODUCTION

The study of thermal convection of a fluid powered by the difference of temperature between two plates, known as Rayleigh–Bénard convection, has been an extensively studied subject for a long time. Computer modeling and physical experiments have produced an enormous wealth of information: for recent reviews, see Refs. 1 and 2. Perhaps unavoidably, there has not been a comparable volume of rigorous studies, if we except the study of the stability of different patterns 共Ref. 3, pp. 23–95兲. It is well known that when the difference of temperature between the top and bottom plates exceeds a certain amount, usually measured in terms of the Rayleigh constant R, convection sets in. Near the onset, convection cells occur; with increasing R, and depending also on the ratio between viscosity and thermal diffusivity共the Prandtl number兲more complex patterns appear and bifurcations to chaotic states may occur. The same may be said of the temperature: for a colorful illustration, starting with regular rolls, see e.g., Ref. 4.

The standard mathematical model of Reyleigh–Bénard convection is given by the Boussinesq approximation to the equations of motion, which we repeat here for convenience. We will consider a d-dimensional domain U of the form⍀⫻关0 , h兴, and as usual we will assume that the tempera-ture is constant at the lower and upper lids, T = T0at⍀⫻兵0其 and T = Th⬍T0 at⍀⫻兵h其. The rest

of the boundary conditions will be discussed later. Let us denote by v the fluid velocity, T the temperature,␯the kinematic viscosity,␬the thermal diffusivity, and␲the kinetic pressure. Then the nondimensionalized Boussinesq approximation共see, e.g., Ref. 5兲to the equations of motion is

⳵v

⳵t =␯⌬v − v · ⵜv − ⵜ␲+共T − Th兲ed, 共1兲

⳵T

⳵t =␬⌬T − v · ⵜT, 共2兲

a兲_{Electronic mail: mnjmhd@am.uva.es}

46, 033102-1

ⵜ· v = 0. 共3兲

eddenotes the vertical unit vector. Traditionally the difference␪of the actual temperature with the

linear one between the lids共associated to pure heat conduction兲is used,

␪= T − T0+␤xd,

共4兲 ␤=T0− Th

h .

␲is also changed to␲+␤x_d2/ 2 −共T0− Th兲xd共which we denote again by␲兲. The final system is

⳵v

⳵t =␯⌬v − v · ⵜv − ⵜ␲+␪ed, 共5兲

⳵␪

⳵t =␬⌬␪− v · ⵜ␪+␤vd, 共6兲

ⵜ· v = 0. 共7兲

Boundary conditions are usually the following ones: the upper and lower plates are taken either rigid, where we assume a no-slip condition and v vanishes there, or stress free, in which casevd= 0

and the vertical derivatives of the remaining components of the velocity are also zero,⳵dvi= 0. The

lateral walls are assumed rigid and conducting, so that v and␪ vanish there.

Let us state some classical results, since T satisfies 共2兲, which is a scalar parabolic equation without terms in T, it also satisfies the maximum and minimum principles共see, e.g., Ref. 6兲. That means that T lies always between T0and Th, which makes excellent physical sense. Therefore␪is

uniformly bounded. By multiplying 共5兲by v, integrating in U and making use of the boundary conditions, one gets

1

2 d

dt

冕

U

v2_{dV +␯}

冕

U

兩ⵜv兩2dV =

冕

U

␪vddV艋储␪储⬁Vol共U兲2储vd储2, 共8兲

where Vol共U兲denotes the volume共area in dimension two兲of U. Since with our boundary condi-tions a Poincaré inequality holds, there exists a positive constant c such that

c

冕

U

兩v2兩dV艋

冕

U

兩ⵜv兩2dV. Thus, by standard inequalities,

1

2 d

dt储v储2

2

+ ␯c 2 储v储2

2_艋 1

2␯2c2储␪储⬁

2

Vol共U兲, 共9兲

which implies that储v储2 is bounded for all time.

As in many other turbulent situations, modeling of the chaotic phase of convection shows a tendency of the flow to concentrate the velocity in regions of small volume.7Thus the bounded-ness of the kinetic energy does not provide an a priori bound upon the maximum of the speed. It is true that physically it seems obvious that this maximum cannot grow without limit, but never-theless it is interesting to obtain rigorous estimates in terms of the main magnitudes of the

problem. Our only hypothesis will be the boundedness of the pressure ␲ 关or, to be specific, of ␲/共1 +v兲兴.

II. ANALYSIS OF THE MOMENTS OF THE VELOCITY

Let us start with the momentum equation

⳵v

⳵t =␯⌬v − v · ⵜv − ⵜ␲+␪ed, 共10兲 and for p = 1 , 2…, let

F_p=

冕

U

vp_{dV, 共11兲}

wherev=兩v兩 represents the modulus of v. F_pis a function of time. Sincev2_{= v · v,}

⳵v2p

⳵t = 2pv

2p−2_{v ·}⳵v

⳵t.

Therefore

1

2pF ˙

2p=

冕

U

v2p−2v ·⳵v

⳵tdV,

and taking into account the momentum equation,

1

2pF ˙

2p= −

冕

U

v2p−2_{共v · ⵜv兲· v dV +␯}

冕

U

v2p−2_{v ·⌬v dV +}

冕

U

共␪v2p−2_v

d−v2p−2v · ⵜ␲兲dV.

共12兲

In the first place,

冕

U

v2p−2共v · ⵜv兲· v dV =

冕

U

1

2pv · ⵜv

2p_{dV = 0. 共13兲}

As for the dissipative term,

冕

U

v2p−2v ·⌬v dV =

冕

U

兺

j

共ⵜ·共vjv2p−2ⵜvj兲− ⵜvj· ⵜ共v2p−2vj兲兲dV

=1 2

冕

_⳵U

v2p−2⳵v

2

⳵nd␴−

冕

U

冉

v2p−2兩ⵜv兩2+p − 1 2 v

2p−4_兩ⵜ

v2兩2

冊

dV. 共14兲

It is understood that the last term 共multiplied by p − 1兲 vanishes when p = 1; there are never negative powers ofv. As for the boundary integral, it also vanishes, since⳵v2_{/⳵n vanishes in all}

冕

U

共␪v2p−2vd−v2p−2v · ⵜ␲兲dV. 共15兲

Since

−

冕

U

v2p−2_{v · ⵜ␲dV = −}

冕

⳵U

v2p−2_{␲v · n d␴+}

冕

U

共p − 1兲␲v2p−4_{v · ⵜv}2_{dV, 共16兲}

and again the boundary integral vanishes, the term in共16兲is

冕

U

共v2p−2␪vd+共p − 1兲␲v

2p−4_{v · ⵜ}

v2兲dV, 共17兲

with the same meaning as before when p = 1. We may bound共17兲in several ways. We first choose

冏

冕

U

v2p−2␪vd+共p − 1兲␲v2p−4v · ⵜv2dV

冏

=

冏

冕

U

␪ 1 +v共vdv

2p−2₊

vdv2p−1兲+共p − 1兲

␲ 1 +v共v

2p−4

+v2p−3_{兲v ·ⵜv}2_dV

冏

艋

冐

␪ 1 +v

冐

_⬁

冕

U

共v2p−2+v2p−1兲兩vd兩dV +共p − 1兲

冐

␲ 1 +v

冐

_⬁ ⫻

冕

U

共v2p−4_+v2p−3_{兲兩v · ⵜv}2_兩dV

艋

冐

␪ 1 +v

冐

_⬁

冕

U

共v2p−1+v2p兲dV +共p − 1兲

冐

␲ 1 +v

冐

_⬁ ⫻

冕

U

共v2p−3_+v2p−2_兲兩ⵜv2_{兩dV. 共18兲}

From now on we will denote

␣=

冐

␪ 1 +v

冐

_⬁,

␤=

冐

␲ 1 +v

冐

_⬁.

Notice that they are functions of t. Thus

1

2pF ˙

2p艋−␯

冕

U

v2p−2_兩ⵜv兩2_{dV −}␯共p − 1兲

2

冕

U

v2p−4_兩ⵜv2_兩2_{dV +␣}

冕

U

共v2p−1_+v2p_兲dV

+共p − 1兲␤

冕

U

共v2p−3+v2p−2兲兩ⵜv2兩dV. 共19兲

For our first estimate we will not make use of the first dissipative term. We have

␣

冕

U

共v2p−2+v2p−1兲dV艋␣共F2p−1+ F2p兲. 共20兲

As for the second term, by using Cauchy–Schwarz and Young’s inequalities,

共p − 1兲␤

冕

U

共vp−1+vp兲vp−2兩ⵜv2兩dV艋共p − 1兲␤

冉

冕

U

共vp−1+vp兲2dV

冊

1/2

冉

冕

U

v2p−4兩ⵜv2兩2dV

冊

1/2

艋共p − 1兲␤21 ␯

冕

U

共vp−1+vp兲2dV

+共p − 1兲␯ 4

冕

_Uv

2p−4_兩ⵜ

v2兩2dV

艋共p − 1兲␤22

␯共F2p−2+ F2p兲+

共p − 1兲␯

4

冕

U

v2p−4兩ⵜv2兩2dV.

共21兲 We have proved the recursive inequality

1

2pF ˙

2p艋−

␯共p − 1兲 4

冕

_Uv

2p−4_兩ⵜv2_兩2_{dV +␣F}

2p−1+␣F2p+

2共p − 1兲␤2

␯ F2p−2+

2共p − 1兲␤2

␯ F2p.

共22兲 We use now the fact that U has finite volume to bound all the Fkin terms of F2p,

F2p−2艋Vol共U兲1/pF2p 1−1/p

,

共23兲 F2p−1艋Vol共U兲1/2pF2p

1−1/2p

.

Let us begin studying a series of alternatives. It may happen

共A兲 F_2p⬍1. Otherwise,

共B兲 F2p−2艋Vol共U兲1/pF2p, F2p−1艋Vol共U兲1/2pF2p.

We will consider the consequences of alternative共B兲. We have

1

2pF ˙

2p+

␯共p − 1兲 4

冕

_Uv

2p−4_兩ⵜ

v2兩2dV艋

冉

␣共1 + m共U兲1/2p兲+2共p − 1兲␤

2

␯ 共1 + Vol共U兲

1/p_兲

冊

_F 2p,

共24兲

and if we call k = 1 + max兵Vol共U兲, 1其,

1

2pF ˙

2p+

␯共p − 1兲 4

冕

_Uv

2p−4_兩ⵜ

v2兩2dV艋k

冉

␣+2共p − 1兲␤

2

␯

冊

F2p. 共25兲

Let us now remember a particular case of the Gagliardo–Nirenberg inequality共Ref. 8, pp. 69 and 70兲. For any function f僆H1共U兲, there exists a constant C depending only on U such that

储f储₂d+2艋C共储ⵜf储2+储f储1兲d储f储1 2

. 共26兲

Since共x + y兲d艋2d−1共xd+ yd兲, by taking␭= max兵2d−1C , 1其,

储f储₂d+2艋 ␭共储ⵜf储₂d+储f储₁d兲储f储₁2, 共27兲

with␭艌1. Notice that for f =vp_,

储f储2= F2p 1/2

, 储f储1= Fp, ⵜf =

p

2v

2p−2_ⵜ

v2, 储ⵜf储₂2= p

2

4

冕

_Uv

2p−4_兩ⵜ

v2兩2dV. 共28兲

共b1兲 储f储₂⬍ ␭1/共d+2兲储f储₁, 共29兲

or

共b2兲 储ⵜf储₂2艌

冉

储f储2

d+2_{−␭储f储} 1 d+2

␭储f储₁2

冊

2/d

. 共30兲

With our election of f, the first alternative means

F2p⬍ ␭2/共d+2兲Fp 2

. 共31兲

Alternative共b2兲, when taken into共25兲, yields

1

2pF ˙

2p艋−

␯共p − 1兲 p2

冉

F_2p共d+2兲/2−␭F_pd+2 ␭F_p2

冊

2/d

+ k

冉

␣+2共p − 1兲␤

2

␯

冊

F2p. 共32兲

Take now, for p艌2,

␥p=␭

冉

1 + kd/2

冉

p2 ␯共p − 1兲

冊

d/2

冉

␣+ 2共p − 1兲␤

2

␯

冊

d/2

冊

. 共33兲 Notice that␥p艌␭艌␭2/共d+2兲艌1. A short calculation will convince us that if F2p⬎␥pFp

2

and共b2兲 occurs, then F˙2p艋0. Since alternative共b1兲is included in

共B1兲 F_2p艋␥_pF2_p,

the remaining possibility is

共B2兲 F2p艋0.

Recall that all this assumes共B兲. Therefore the alternatives are共A兲F2p艋1,共B1兲or共B2兲.

III. UNIFORM ESTIMATES IN TIME

Let us consider a time interval关0 ,␶兴, and let

⌽p共␶兲= max兵1,max兵Fp共t兲:t僆关0,␶兴其其

共34兲 ⌫p共␶兲= max兵␥p共t兲:t僆关0,␶兴其.

For every t僆关0 ,␶兴where共A兲or 共B1兲occurs, certainly

F2p共t兲艋 ⌫p共␶兲⌽p共␶兲2. 共35兲

Now, if共35兲occurs for every t僆关0 ,␶兴, obviously

⌽2p共␶兲艋 ⌫p共␶兲⌽p共␶兲2. 共36兲

The other possibility is that for a certain t₁僆关0 ,␶兴,

F2p共t1兲⬎ ⌫p共␶兲⌽p共␶兲2, 共37兲

which implies that共B2兲holds, i.e., F˙2p共t1兲艋0. Let共t0, t1兴 be a maximal left interval where共37兲

occurs. We know that F2p is decreasing there. If t0⬎0, F2p共t0兲=⌫p共␶兲⌽p共␶兲2, which, since

F2p共t1兲艋F2p共t0兲, contradicts our hypothesis. The only possibility is that共35兲occurs nowhere, i.e.,

t0= 0. In that case, F2pis decreasing in关0 , t1兴and therefore F2p共t1兲艋F2p共0兲. Thus, in every case

⌽2p共␶兲艋max兵⌫p共␶兲⌽p共␶兲2,F2p共0兲其. 共38兲

Since储v共t兲储p= Fp共t兲1/p, if we denote

␾p共␶兲= max兵1,max兵储v共t兲储p:t僆关0,␶兴其其, 共39兲

the 2pth root of共38兲yields

␾2p共␶兲艋max兵⌫p共␶兲1/2p␾p共␶兲,储v共0兲储2p其. 共40兲

Hence

␾4共␶兲艋max兵⌫2共␶兲1/4␾2共␶兲,储v共0兲储4其,

␾8共␶兲艋max兵⌫4共␶兲1/8⌫2共␶兲1/4␾2共␶兲,⌫4共␶兲1/8储v共0兲储4,储v共0兲储8其¯,

共41兲 ␾2n共␶兲艋兵⌫₂n−1共␶兲1/2

n

⌫2n−2共␶兲1/2 n−1

¯⌫2共␶兲1/4␾2共␶兲,

⌫2n−1共␶兲1/2 n

⌫2n−2共␶兲1/2 n−1

¯⌫4共␶兲1/8储v共0兲储4…,储v共0兲储2n. Let us study the infinite product

⌫2共␶兲1/4⌫4共␶兲1/8¯⌫2n−1共␶兲1/2 n

¯. 共42兲 Recall that

␣共t兲=

冐

␪共t兲

1 +v共t兲

冐

_⬁, ␤共t兲=

冐

␲共t兲

1 +v共t兲

冐

_⬁. 共43兲

Let us denote by␣共␶兲,␤共␶兲 their respective maxima in关0 ,␶兴. By the expression in 共33兲,

⌫2n共␶兲艋 ␭

冉

1 + kd/2

冉

22n ␯共2n− 1兲

冊

d/2

冉

␣共␶兲+2共2

n_{− 1兲}

␯ ␤共␶兲2

冊

d/2

冊

艋 ␭

冉

1 +

冉

22n+2k

冉

␣共␶兲 ␯ +

␤共␶兲2

␯2

冊冊

d/2

冊

艋 ␭

冉

1 +

冉

22n+2k

冉

␣共␶兲 ␯ +

␤共␶兲2

␯2 + 1

冊冊

d/2

冊

. 共44兲

The addition of 1 to the parentheses is intended to ensure that

x =

冉

22n+2k

冉

␣共␶兲 ␯ +

␤共␶兲2

␯2 + 1

冊冊

d/2

艌1.

Since for those x, log共1 + x兲艋1 + log x holds,

log⌫2n共␶兲艋log␭+ 1 + d

2

冉

log k +共2n + 2兲log 2 + log

冉

␣共␶兲

␯ + ␤共␶兲2

␯2 + 1

冊冊

.

Hence the sum of the logarithms of⌫2n satisfies

兺

n=1

⬁ 1

2n+1log⌫2n共␶兲艋 1

2

冉

1 + log␭+ d

2log k

冊

+ d 2log 2

兺

_n=1

⬁ 2n + 2

2n+1 + d 4log

冉

␣共␶兲 ␯ +

␤共␶兲2

␯2 + 1

冊

.

共45兲

Thus, since the sum of the series兺共2n + 2兲/ 2n+1_{is 3,}

兿

n=1

⬁

⌫2n共␶兲1/2 n+1

艋23d/2共␭e兲1/2kd/4

冉

␣共␶兲 ␯ +

␤共␶兲2

␯2 + 1

冊

d/4

, 共46兲

兿

n=1 m

⌫2n共␶兲1/2 n+1

. 共47兲

Since, on the other hand,

储v共0兲储_p艋储v共0兲储_⬁Vol共U兲1/p艋k储v共0兲储_⬁, 共48兲 we find

␾2n共␶兲艋23d/2共␭e兲1/2kd/4

冉

␣共␶兲 ␯ +

␤共␶兲2

␯2 + 1

冊

d/4

max兵␾2共␶兲,k储v共0兲储_⬁其. 共49兲

It is well known that储v储_p→储v储_⬁as p→⬁, so that␾₂n共␶兲→max兵1 ,储v共t兲储_⬁: t僆关0 ,␶兴其. Also,␾₂共␶兲 represents the maximum of the kinetic energy in关0 ,␶兴, which we denote by E共␶兲. Calling M the universal constant written in共49兲, we obtain our main estimate

max兵储v共t兲储_⬁:t僆关0,␶兴其艋M

冉

1 ␯max关0,␶兴

冐

␪共t兲 1 +v共t兲

冐

_⬁+

1 ␯2max

关0,␶兴

冐

␲共t兲 1 +v共t兲

冐

_⬁

2

+ 1

冊

d/4

⫻max兵E共␶兲,k储v共0兲储_⬁其. 共50兲

IV. COMMENTS AND EXTENSIONS OF THE ESTIMATES

In principle it could look as if the estimates in terms of␪ and␲ divided by 1 +vare unnec-essary, since if vis bounded so are ␪ and␲. While there is no conceptual gain in taking these magnitudes divided by 1 +v, the estimate 共50兲 is in fact finer than one involving only 储␪储_⬁ and 储␲储_⬁. It can be far better if the regions where␪and/or␲are larger coincide with regions of high velocity. In particular, high temperature deviation propels the fluid faster, so it is likely that ␪/共1 +v兲is considerably smaller than ␪.

When the flow is chaotic, the temperature may become irregular. Therefore it is possible that some primitive of␪共e.g., a function⌰such that for some coordinate j,⳵_j⌰=␪兲may have a better behavior than the temperature deviation: portions where␪is positive may compensate with others where it is negative to obtain a smooth result. We will see that储v储_⬁may also be bounded in terms of⌰/共1 +v兲. The method follows the steps of the previous one: the term

冕

U

␪v2p−2vddV 共51兲

may be written as

冕

U

共⳵j⌰兲v2p−2vddV = −

冕

U

⌰⳵j共v2p−2vd兲dV = −

冕

U

⌰共v2p−2⳵jvd+ 2共p − 1兲v2p−4vdv ·⳵jv兲dV.

共52兲

This may be bounded by

共2p − 1兲

冕

U

冏

⌰ 1 +v

冏

共v

2p−2₊

v2p−1兲兩ⵜv兩dV艋共2p − 1兲

冐

⌰ 1 +v

冐

_⬁

冕

U

共v2p−2+v2p−1兲兩ⵜv兩dV. 共53兲

Using now the Cauchy–Schwarz inequality, the term is bounded by

共2p − 1兲2

冐

⌰ 1 +v

冐

_⬁

1

2␯

冕

_U共v

2p−2₊

v2p兲dV +␯

冕

U

v2p−2兩ⵜv兩2dV. 共54兲 The last term may now be cancelled with the first of the dissipative terms共which we did not use in our previous proof兲and we are left with

共2p − 1兲2

冐

⌰ 1 +v

冐

_⬁

2 ₁

2␯共F2p−2+ F2p兲. 共55兲

The rest of the proof is analogous to the previous one. We are left with a bound of the form

max兵储v共t兲储_⬁:t僆关0,␶兴其艋M

冉

1 ␯2max

关0,␶兴

冐

⌰共t兲 1 +v共t兲

冐

_⬁

2

+ 1 ␯2max

关0,␶兴

冐

␲共t兲 1 +v共t兲

冐

_⬁

2

+ 1

冊

d/4

⫻max兵E共␶兲,k储v共0兲储_⬁其. 共56兲 Notice, however, that now a factor of the form 1 /␯2 appears before the maximum norm of ⌰/共1 +v兲, and this is squared, while before we had only 1 /␯and the power of␪/共1 +v兲was one. This may be important when the viscosity is low.

V. CONCLUSIONS

While the kinetic energy of the flow in Rayleigh–Bénard convection may be easily bounded by classical inequalities, the maximum of the velocity is harder to handle. As soon as the flow becomes chaotic small islands or filaments of high velocity are observed, which makes compatible the boundedness of the square mean of the velocity共the kinetic energy兲with the existence of large velocity peaks. It is proved here that, provided the pressure remains uniformly bounded, so does the velocity, and its maximum may be estimated by the maxima of the temperature deviation and pressure divided by one plus the velocity modulus. Other estimates may be made in terms of certain means of the temperature, which may be considerably smaller than the temperature itself if its distribution is irregular. The bounds depend on certain powers, depending on the dimension, of the previously mentioned magnitudes and the flow viscosity.

1

A. V. Getling, Rayleigh-Bénard Convection: Structures and Dynamics共World Scientific, Singapore, River Edge, London, 1998兲.

2

E. Bodenschatz, W. Pesch, and G. Ahlers, Annu. Rev. Fluid Mech. 32, 709共2000兲.

3

F. H. Busse, “Fundamentals of thermal convection,” in Mantle Convection: Plate Techtonics and Global Dynamics, edited by V. R. Peltier共Gordon and Breach, Montreux, 1989兲.

4

C. H. Majunder, D. A. Yuen, E. O. Sevre, J. M. Boggs, and S. Y. Bergeron, Electron. Geosci. 7, s10069-002-0004-4

共2002兲.

5

F. H. Busse, Rep. Prog. Phys. 41, 1929共1978兲

6

A. Friedman, Partial Differential Equations of Parabolic Type共Prentice-Hall, Englewood Cliffs, NJ, 1964兲.

7

E. S. C. Ching, C. K. Leung, X.-L. Qiu, and P. Tong, Phys. Rev. E 68, 026307共2003兲.

8