Uniform estimates on the velocity in Rayleigh–Bénard convection

Uniform estimates on the velocity in Rayleigh–Bénard convection Manuel Núñez Citation: Journal of Mathematical Physics 46, 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 Published by the American Institute of Physics 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 domains in which the speed is large, is rather small. This could imply that the maximum 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 ad-dimensional domain Uof the form ⍀⫻关0,h兴, and as usual we will assume that the temperature is constant at the lower and upper lids, T=T0at ⍀⫻兵0其and T=Th⬍T0at ⍀⫻兵h其. The rest of the boundary conditions will be discussed later. Let us denote by vthe fluid velocity, Tthe 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: [email protected] JOURNAL OF MATHEMATICAL PHYSICS 46, 033102 共2005兲 46, 033102-10022-2488/2005/46共3兲/033102/9/$22.50 © 2005 American Institute of Physics 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 ␲ + ␤ xd 2/2−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 vvanishes there, or stress free, in which case vd=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 vand ␪ vanish there. Let us state some classical results, since Tsatisfies 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 Tlies always between T0and Th, which makes excellent physical sense. Therefore ␪ is uniformly bounded. By multiplying 共5兲by v, integrating in Uand making use of the boundary conditions, one gets 1 2 d dt 冕 U v2dV+ ␯ 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 conditions a Poincaré inequality holds, there exists a positive constant csuch 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⬁ 2Vol共U兲,共9兲 which implies that 储v储2is 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 boundedness 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 nevertheless it is interesting to obtain rigorous estimates in terms of the main magnitudes of the 033102-2 Manuel Núñez J. Math. Phys. 46, 033102 共2005兲 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 Fp= 冕 U vpdV,共11兲 where v=兩v兩represents the modulus of v.Fpis a function of time. Since v2=v·v, ⳵ v2p ⳵ t=2pv2p−2v· ⳵ 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·vdV+ ␯ 冕 U v2p−2v·⌬vdV+ 冕 U共 ␪ v2p−2vd−v2p−2v·ⵜ ␲ 兲dV. 共12兲 In the first place, 冕 U v2p−2共v·ⵜv兲·vdV= 冕 U 1 2pv·ⵜv2pdV=0. 共13兲 As for the dissipative term, 冕 U v2p−2v·⌬vdV= 冕 U兺 j共ⵜ·共vjv2p−2 ⵜvj兲−ⵜvj·ⵜ共v2p−2vj兲兲dV =1 2 冕 ⳵ U v2p−2 ⳵ v2 ⳵ nd ␴ − 冕 U 冉 v2p−2兩ⵜv兩2+p−1 2v2p−4兩ⵜv2兩2 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 of v. As for the boundary integral, it also vanishes, since ⳵ v2/ ⳵ nvanishes in all the boundary, including possible stress-free surfaces. The last term we must consider is 033102-3 Estimates on the convection velocity J. Math. Phys. 46, 033102 共2005兲 冕 U共 ␪ v2p−2vd−v2p−2v·ⵜ ␲ 兲dV.共15兲 Since − 冕 U v2p−2v·ⵜ ␲ dV=− 冕 ⳵ U v2p−2 ␲ v·nd ␴ + 冕 U共p−1兲 ␲ v2p−4v·ⵜv2dV,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 ␲ v2p−4v·ⵜv2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vdv2p−2 +vdv2p−1兲+共p−1兲 ␲ 1+v共v2p−4 +v2p−3兲v·ⵜv2dV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·ⵜv2兩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兩2dV− ␯ 共p−1兲 2 冕 U v2p−4兩ⵜv2兩2dV+ ␣ 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, 033102-4 Manuel Núñez J. Math. Phys. 46, 033102 共2005兲 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 U v2p−4兩ⵜv2兩2dV 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 U v2p−4兩ⵜv2兩2dV+ ␣ F2p−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 Uhas 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F2p⬍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 U v2p−4兩ⵜv2兩2dV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 F2p, 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 U v2p−4兩ⵜv2兩2dV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 Cdepending only on Usuch that 储f储2 d+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2 d+2 艋␭共储 ⵜf储2 d+储f储1 d兲储f储1 2,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 2v2p−2 ⵜv2,储ⵜf储2 2=p2 4 冕 U v2p−4兩ⵜv2兩2dV.共28兲 In the inequality 共27兲, there exist two alternatives. Either 033102-5 Estimates on the convection velocity J. Math. Phys. 46, 033102 共2005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2 2艌 冉 储f储2 d+2 −␭储f储1 d+2 ␭储f储1 2 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 F2p 共d+2兲/2 −␭Fp d+2 ␭Fp 2 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 2and 共b2兲 occurs, then F ˙2p艋0. Since alternative 共b1兲is included in 共B1兲F2p艋 ␥ pFp 2, 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 F2pis 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 033102-6 Manuel Núñez J. Math. Phys. 46, 033102 共2005兲 ␾ 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1/2n⌫2n−2共 ␶ 兲1/2n−1 ¯⌫2共 ␶ 兲1/4 ␾ 2共 ␶ 兲, ⌫2n−1共 ␶ 兲1/2n⌫2n−2共 ␶ 兲1/2n−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1/2n ¯.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2n−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 xholds, 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 ⌫2nsatisfies 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1/2n+1 艋23d/2共␭e兲1/2kd/4 冉 ␣ 共 ␶ 兲 ␯ + ␤ 共 ␶ 兲2 ␯ 2+1 冊 d/4,共46兲 and the same may be said of any finite product 033102-7 Estimates on the convection velocity J. Math. Phys. 46, 033102 共2005兲 兿 n=1 m ⌫2n共 ␶ 兲1/2n+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 Mthe 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 unnecessary, 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v2p−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 033102-8 Manuel Núñez J. Math. Phys. 46, 033102 共2005兲