5.3 Nonlinear stability of Rayleigh-Bénard convection

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1 Nonliner stbility of Ryleigh-Bénrd convection In Chpter 1, we sw tht liner stbility only tells us whether system is stble or unstble to infinitesimlly-smll perturbtions, nd tht there re cses in which system cn be unstble to finite-mplitude perturbtions even if it is linerly stble. In the previous sections, we studied the liner stbility of Ryleigh-Bénrd convection. The next nturl step is to determine whether the stbility of this system is well-described by liner theory, or whether finite mplitude instbilities for Ryleigh numbers below R c re possible. There re two different wys of doing this. The first is to perform wekly nonliner nlysis of the stbility of the system close to the criticl Ryleigh number R c. The second is to study the energy stbility of the problem. These two methods ber mny similrities with stndrd tools of dynmicl systems theory: the first is relted to norml forms, nd the second to Lypunov stbility. We now study both in turn, strting with the ltter. Energy stbility is somewht cruder, but more generl tool thn wekly nonliner theory it tells us bout globl stbility in generl without giving us ny informtion bout wht the ctul nonliner dynmics of the system re. To gin insight on this more specific problem, wekly nonliner theory is the wy to go. As this will become evidently cler, both techniques require equl mounts of inspirtion nd perspirtion to work hold on to your socks! Non-dimensionl equtions Before we begin, we recst the governing equtions in non-dimensionl form. This is stndrd step in most pplied mthemticl studies, nd cn ctully provide quite interesting insight into the problem t hnd. In wht follows, we rescle time, spce, nd temperture perturbtions using these new units: [l] = H, [t] = H2 κ T nd [T ] = T (5.1) The unit lengthscle is the only nturl lengthscle of the system the seprtion between the two pltes. The choice of the unit time is less obvious; here, we choose the therml diffusion time cross H. The unit temperture, gin, is pretty strightforwrd, nd is the temperture difference between the two pltes. Finlly, the unit velocity becomes [v] = [l] [t] = κ T H (5.2) We will worry bout non-dimensionlizing p shortly. We now crete the non-dimensionl vribles T = T ˆT, u = κ T H û, etc... nd note tht spce is lso rescled, so tht = 1 H ˆ. The non-dimensionl

2 5.3. NONLINEAR STABILITY OF RAYLEIGH-BÉNARD CONVECTION119 momentum eqution cn be derived with the following steps: κ 2 T û H 3 ˆt + κ2 T H 3 û ˆ û = 1 ˆ p + αg T ˆT ez + ν κ T Hρ m H 2 H ˆ 2 û û ˆt + û ˆ û = H2 κ 2 T ρ ˆ p + m αg T H3 κ 2 T ˆT e z + ν κ T ˆ 2û (5.3) We recognize some of the importnt non-dimensionl numbers introduced erlier: the Ryleigh number nd the Prndtl number. Also note tht this clcultion suggests tht good non-dimensionliztion for p would be such tht so finlly, Similr steps led to H 2 κ 2 T ρ p = ˆp (5.4) m û ˆt + û ˆ û = ˆ ˆp + RPr ˆT e z + Pr ˆ 2 û (5.5) ˆ û = 0 ˆT ˆt + û ˆ ˆT ŵ = ˆ 2 ˆT (5.6) At this point, we see tht Ryleigh-Bénrd convection only ever needs to be chrcterized by 2 non-dimensionl numbers : R nd Pr. This mens tht two completely different systems (sy, with different plte seprtion, different temperture offset, different viscosity, different therml diffusivity, etc.. ) cn ctully behve exctly the sme wy s long s their Ryleigh nd Prndtl numbers re exctly the sme. This rther surprising result is the min reson why it does mke sense, for instnce, to do erodynmic studies of irplnes using smller-scle models. From here on, we drop the hts, but remember tht ll the quntities re now non-dimensionl Energy stbility of Ryleigh Bénrd convection Lypunov stbility in dynmicl systems Lypunov stbility theory is n excellent wy of proving whether stedy stte is globlly stble insted of being just linerly stble. To see how it works, it is best to strt with simple exmple bsed on 2D dynmicl system. Consider the following system: f = f + 4g ġ = f g 3 (5.7)

3 120 It hs n obvious fixed point t f = g = 0. Linerizing round it, we find tht smll perturbtions stisfy so f = f + 4g ġ = f (5.8) f = f 4f (5.9) This suggests tht f e λt with λ 2 + λ + 4 = 0. This hs solutions so tht λ = 1 ± = 1 2 ± i 15 2 ( f(t) = e t/2 cos( 15t/2) + b sin( ) 15t/2) (5.10) (5.11) where nd b re two integrtion constnts, nd similrly for g. This implies tht, linerly speking t lest, the origin is stble spirl. But this is only true for initil conditions close to the origin. Do ll possibly initil conditions lwys end up decying to 0 s well? To nswer this question, let s construct Lypunov function E(t). By definition it hs to be strictly positive, must be equl to zero t the fixed point (here, f = g = 0), nd hs to stisfy /dt < 0 except t the fixed point, where it must be 0. Let s try: E(t) = f 2 + γ 2 g 2 (5.12) where γ 2 remins to be determined (but is positive). By construction, we see tht E is indeed positive everywhere except t f = g = 0 where it is 0. Furthermore, dt = 2(f f + γ 2 gġ) = 2( f 2 + 4gf γ 2 gf γ 2 g 4 ) (5.13) If we tke γ 2 = 4, then the term in fg conveniently vnishes, nd we re left with dt = 2f 2 8g 4 (5.14) which is clerly negtive, except t the fixed point. Wht does this buy us? Well, we see tht given ny initil condition f 0, g 0, the dynmicl system will evolve in time following (5.8). However, s we hve just demonstrted, this lso mens tht E will evolve in time ccording to (5.14), nd will therefore lwys decrese since /dt < 0. As t, E 0 (since E hs to be positive), which then necessrily implies tht both f nd g must lso be going to 0. To summrize, it is possible to prove tht fixed point is globlly stble provided we cn find sclr function E of the dependent vribles, tht stisfies: E is strictly positive, except t the fixed point where it must be 0

4 5.3. NONLINEAR STABILITY OF RAYLEIGH-BÉNARD CONVECTION121 /dt is strictly negtive, except t the fixed point where it must be 0 A nice feture of this method is tht it very esily generlizes to systems with ny number of dimensions, nd cn be used in fluid dynmics to prove the globl stbility of stedy stte. The energy stbility criterion We now ttempt to crete Lypunov function to study the stbility properties of Ryleigh-Bénrd convection. Since E hs to be sclr function, nd yet hs to cpture the dynmics of the whole fluid system, it is best to crete is s n integrl over domin D, where we tke D to be the spce between the pltes. Since this hs infinite horizontl extent, we then reduce it to some portion of the horizontl plne, nd require periodicity in x (recll tht we re considering here 2D problem only). Hence E = stuff (5.15) where denotes the sptil integrl over D. For resons tht will be pprent shortly, it is lso best to mke E qudrtic in the dependent vribles, rther thn, sy, qurtic, or higher-order. The simplest qudrtic, positive definite integrl we know is the one tht is bsed, for instnce, on the kinetic energy of the fluid. Dotting the momentum eqution with u, nd integrting over domin, we get 1 2 t u u u 2 = u p + RPrwT + Pru 2 u (5.16) Since u = 0, we hve u u 2 = (u u 2 ) nd u p = (pu) (5.17) Furthermore, since the boundry conditions re w = 0 on the top nd bottom boundry, nd periodic in x, the integrl over the domin of these divergences re ll zero. Finlly, using integrtion by prts nd the sme properties of the boundry conditions, we hve (using Einstein s convention of repeted indices) u 2 u = u i jj u i = ( j u i ) 2 = u 2 (5.18) The kinetic energy eqution then becomes 1 2 t u 2 = RPr wt Pr u 2 (5.19) This sttes tht the totl kinetic energy in the domin chnges s results of the conversion of potentil energy (first terms on the RHS) or viscous dissiption (second term on the RHS). While viscous dissiption is lwys negtive, the first term cn be positive (nd must be, for instbility to occur!). If tht is the cse, u 2 cn either increse or decy depending on which of the two terms, energy injection or energy dissiption, is the lrgest.

5 122 A similr evolution eqution for nother positive definite functionl cn be constructed by considering the therml energy eqution insted, nd multiplying it by T. Integrting over the sme domin D, using the sme trick to get rid of the divergence, nd integrting the therml diffusion term by prts, we get 1 2 t T 2 = wt T 2 (5.20) Agin, we see tht T 2 cn either increse or decy depending on the reltive sizes of the first nd second term on the RHS. We cn now construct very generl qudrtic Lypunov functionl s E(u, T ) = (1/2) u 2 + γ 2 T 2, where γ 2 is n rbitrry positive constnt. The evolution eqution for E is then E t = (RPr + γ2 ) wt Pr u 2 γ 2 T 2 (5.21) If we cn somehow prove tht, for ll non-zero functions u, w nd T (stisfying u = 0) the RHS of this eqution is strictly negtive except t the fixed point, then E must strictly decrese with time. Since E 0, the only possible evolution of this system drives E towrds 0, so tht E 0 s t. In other words, ll perturbtions must decy, nd the system is globlly stble. Given tht this proof uses n energy-like functionl to show globl stbility, the criterion derived is often clled energy stbility. In order to determine when /dt < 0, it is sufficient to show tht (RPr + γ 2 ) wt is smller thn the dissiption term D = Pr u 2 + γ 2 T 2 for ll possible functions u, w, T (stisfying u = 0). To do tht, we now fix the totl dissiption, nd mximize (RPr + γ 2 ) wt, subject to the constrints D = D 0 (where D 0 is known), nd u = 0. Energy stbility would then simply require tht this mximum vlue be smller thn D 0. In order to mximize (RPr + γ 2 ) wt subject to these condition, we introduce the Lgrnge multipliers Λ 1 nd Λ 2, nd mximize insted S = (RPr + γ 2 ) wt + Λ 1 (Pr u 2 + γ 2 T 2 D 0 ) + Λ 2 u (5.22) over ll functions u, w, T, nd Λ 2. Note how ech Lgrnge multiplier is ssocited with one of the constrints. While Λ 1 is constnt, becuse we re trying to impose D = D 0 globlly, Λ 2 is function becuse we wnt to enforce u t every point in the domin D. We re now simply left to mximize S. Optimiztion using Euler-Lgrnge equtions. Let s recll how one my go bout mximizing functionl (rther thn function). Consider the much simpler functionl, sy, S(f) = L(f, f; x)dx (5.23)

6 5.3. NONLINEAR STABILITY OF RAYLEIGH-BÉNARD CONVECTION123 where f = df/dx nd where f is subject to simple conditions such s f() = f nd f(b) = f b. Stting tht f is the function tht mximizes S is equivlent to sying tht infinitesiml vritions in f result in zero chnge in S, t lest t first order. Indeed, ner the mximum x mx of norml single-vrible function g(x), g(x) = g(x mx )+0.5(x x mx ) 2 g (x mx ) g(x) g(x mx ) 0+O((x x mx ) 2 ) (5.24) The sme is true for S, so if f(x) = f mx (x) + δf(x), where f mx (x) is the function which mximizes S, nd δf(x) is smll perturbtion round it, then we expect tht δs = S(f mx + δf) S(f mx ) 0 (5.25) This condition is the one tht effectively yields f mx. Indeed, let s evlute δs: δs = which defines δl. Since then [ L(f mx + δf, f mx + δf; x) L(f mx, f ] mx ; x) dx δs = δl = δf f + δ f f [ δf f + δ f f Finlly, note tht δ f = d(δf)/dx so, using integrtion by prts, [ ] b dδf f dx dx = f δf δldx (5.26) (5.27) ] dx (5.28) δf d dx f dx (5.29) Since f hs to stisfy the boundry conditions, we cnnot perturb it t x = nd x = b. This mens tht δf() = δf(b) = 0, so the integrted term is equl to 0. This leves us with: δs = [ δf f δf d ] dx f dx = δf [ f d ] dx f dx = 0 (5.30) For this to be true for ny possible perturbing function δf(x), the term in the squre brckets hve to be zero. In other words, the function f mx stisfies the eqution f d dx f = 0 (5.31) (with the boundry condition f() = f, nd f(b) = f b ). This eqution is clled n Euler-Lgrnge eqution.

7 124 Note tht this method cn esily be generlized when L is functionl of mny dependent vribles {f i } i=1..i nd when the integrl is in mny dimensions {x j } j=1..j. For ech f i, we hve f i j x j ( f i / x j ) = 0 (5.32) Condition for energy stbility. We now use Euler-Lgrnge s equtions to mximize S given by (5.22). Using the nottion of the previous section, S is given by S = Ldxdz (5.33) where L is the functionl L = (RPr + γ 2 )wt + Λ 1 (Pr u 2 + γ 2 T 2 D 0 ) + Λ 2 u (5.34) where, recll, Λ 1 is constnt while Λ 2 is function of x nd z. Since we hve two independent vribles, we hve to clculte q x ( q/ x) z ( q/ z) = 0 (5.35) where q is either u, w, T or Λ 2. Let s work first with the derivtive with respect to Λ 2, which is the simplest one since L does not depend on ny derivtives of Λ 2. We simply hve Λ 2 = u = 0 (5.36) which recovers the incompressibility constrint. This suggests tht, s usul, we cn represent u by using strem function with u = φ/ z nd w = φ/ x. Similrly, the derivtive with respect to Λ 1 lso just recovers the constrint D = D 0. Let s now work with the derivtive with respect to T. We hve while since T = (RPr + γ2 )w (5.37) ( T/ x) = T 2γ2 Λ 1 x T 2 = (5.38) ( ) 2 ( ) 2 T T + (5.39) x z nd similrly for the derivtive with respect to T/ z. Putting these together using (5.35), we then get (RPr + γ 2 )w 2γ 2 Λ 1 2 T = 0 (5.40)

8 5.3. NONLINEAR STABILITY OF RAYLEIGH-BÉNARD CONVECTION125 Similrly, it cn be shown tht (RPr + γ 2 )T Λ 2 z 2PrΛ 1 2 w = 0 Λ 2 x 2PrΛ 1 2 u = 0 (5.41) We cn then eliminte Λ 2 between the two momentum-like equtions, to get nd finlly we cn eliminte, sy, T, to get (RPr + γ 2 ) T x = 2PrΛ 1 4 φ (5.42) (RPr + γ 2 ) 2 2 φ x 2 = 4Prγ2 Λ φ (5.43) This shows tht the solution φ tht mximizes the functionl S is the solution of liner eigenvlue problem, where the eigenvlue is Λ 1 (ll the other prmeters being known nd fixed). Since the solutions hve to stisfy the sme boundry conditions s the originl problem (i.e. periodic in x nd impermeble, stressfree in z, with T given on the boundries), they hve to be of the form φ(x, z) = ˆφe ikxx sin(nπz) ik x T (x, z) = 2PrΛ 1 (RPr + γ 2 ) (k2 x + n 2 π 2 ) 2 φ(x, z) (5.44) (where we implicitly men the rel prt of these quntities) with k 2 x(rpr + γ 2 ) 2 = 4Prγ 2 Λ 2 1(k 2 x + n 2 π 2 ) 3 (5.45) Let s now go bck the originl question, nd determine under which condition the mximum of (RPr + γ 2 ) wt is indeed smller thn D 0. First note tht by (5.40), for the optiml functions, (RPr + γ 2 ) wt = 2γ 2 Λ 1 T 2 T = 2γ 2 Λ 1 T 2 (5.46) We re then left to estimte the sign of Using (5.44), we hve while dt = ( 2Λ 1 1)γ 2 T 2 Pr u 2 (5.47) T 2 = (k 2 x + n 2 π 2 ) T 2 = ( u 2 2 φ = x 2 4Pr 2 Λ 2 1 k 2 x(rpr + γ 2 ) 2 (k2 x + n 2 π 2 ) 5 φ 2 (5.48) ) 2 ( 2 ) 2 φ x z ( 2 ) 2 φ z 2 = (k 2 x + n 2 π 2 ) 2 φ 2 (5.49)

9 126 so dt = [ ( 2Λ 1 1)γ 2 4PrΛ 2 ] 1 kx(rpr 2 + γ 2 ) 2 (k2 x + n 2 π 2 ) 3 1 Pr(kx 2 + n 2 π 2 ) 2 φ 2 We cn simplify this gretly using (5.45): (5.50) dt = (2Λ 1 + 2)Pr(k 2 x + n 2 π 2 ) 2 φ 2 (5.51) which is lwys negtive s long s 2Λ + 2 > 0, which implies Λ 1 > 1. Recll tht Λ 1 is the solution of (5.45), so k x (RPr + γ 2 ) Λ 1 = ± 2 (5.52) Prγ(kx 2 + n 2 π 2 ) 3/2 Note tht if Λ 1 > 0, energy stbility is lwys gurnteed becuse of (5.47). The intervl we need to worry bout is therefore 1 < Λ 1 < 0. The condition for the negtive root to be lrger thn 1 is equivlent to sying tht (RPr + γ 2 ) 2 4Prγ 2 < (k2 x + n 2 π 2 ) 3 k 2 x (5.53) This will lwys be true s long s the LHS of this inequlity is smller thn ny possible vlue tht the RHS my tke. As it turns out, we hve lredy worked out the minimum of this expression it s the sme s in liner theory! The minimum vlue, 27π 4 /4, is chieved for n = 1, nd for k 2 x = π 2 /2. Energy stbility is therefore gurnteed provided: (RPr + γ 2 ) 2 < 27Prπ 4 γ 2 (5.54) At this point, it is worth reclling tht we constructed not single Lypunov function, but n entire fmily of them ech corresponding to different vlue of γ. For ech Lypunov function, we get sufficient criterion for energy stbility s R < R c (γ) where 27Prπ 2 γ γ 2 R c (γ) = (5.55) Pr To find the mximum possible vlue of R below which it is possible to gurntee stbility, we simply hve to choose the γ tht mximizes the RHS of this lst inequlity. This occurs when γ = 27Prπ 2 /2. Putting everything together, we cn then prove the following result: if R < mx R c (γ) = 27π4 γ 4 (5.56) then the system is energy stble. Note how this criticl vlue is exctly the sme s the one we hd obtined for the liner stbility criterion. This rther remrkble result proves tht the criterion for liner stbility in Ryleigh-Bénrd convection is lso the criterion for globl stbility. This implies tht below R c = 27π 4 /4, it is not possible to destbilize the fluid however lrge the perturbtion is!