Laurette TUCKERMAN Rayleigh-Bénard Convection and Lorenz Model

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1 Laurette TUCKERMAN Rayleigh-Bénard Convection and Lorenz Model

2 Rayleigh-Bénard Convection

3 Rayleigh-Bénard Convection Boussinesq Approximation Calculation and subtraction of the basic state Non-dimensionalisation Boundary Conditions Linear stability analysis Lorenz Model Inclusion of nonlinear interactions Seek bifurcations

4 Boussinesq Approximation µ (viscosity diffusivity of momentum),κ(diffusivity of temperature), ρ (density) constant except in buoyancy force. Valid for T 0 T 1 not too large. Governing equations: Boundary conditions: ρ(t) = ρ 0 [1 α(t T 0 )] U = 0 ρ 0 [ t + (U )]U = µ U P gρ(t)e z [ t + (U )]T = κ T advection diffusion buoyancy U = 0 at z = 0,d T = T 0,1 at z = 0,d

5 Calculation and subtraction of base state Conductive solution: (U,T,P ) Motionless: U = 0 uniform temperature gradient: T = T 0 (T 0 T 1 ) z [ d density: ρ(t ) = ρ α(t 0 T 1 ) z ] d Hydrostatic pressure counterbalances buoyancy force: P = g dz ρ(t ) [ ] = P 0 gρ 0 z + α(t 0 T 1 ) z2 2d

6 Write: Buoyancy: T = T + ˆT P = P + ˆP ρ(t + ˆT) = ρ 0 (1 α(t + ˆT T 0 )) = ρ 0 (1 α(t T 0 )) ρ 0 αˆt = ρ(t ) ρ 0 αˆt P gρ(t)e z = P gρ(t ) ˆP + gρ 0 αˆte z = ˆP + gρ 0 αˆte z Advection of temperature: (U )T = (U )T + (U )ˆT ( = (U ) T 0 (T 0 T 1 ) z ) + (U )ˆT d = T 0 T 1 U e z + (U )ˆT d

7 Governing equations: ρ 0 [ t + (U )]U = ˆP + gρ 0 αˆte z + µ U U = 0 [ t + (U )] ˆT = T 0 T 1 U e z + κ ˆT d Homogeneous boundary conditions: U = 0 at z = 0,d ˆT = 0 at z = 0,d

8 Scales: z = d z, t = d2 Equations : Non-dimensionalization κ t, U = κ dū, ˆT = µκ d 3 gρ 0 α T, ˆP = µκ d 2 P κ 2 ρ 0 [ t + (Ū ) ] Ū = µκ P + µκ Te d 3 d 3 d 3 z + µκ Ū d 3 κ Ū = 0 d 2 µκ 2 [ t + (Ū ) ] T = κ 0 T 1 Ū e d 5 z + gρ 0 α dt µκ2 d d 5 gρ 0 α T Dividing through, we obtain: [ t + (Ū ) ] Ū = µ [ P + Te z + Ū ] ρ 0 κ [ t + (Ū ) ] T = (T 0 T 1 )d 3 gρ 0 α Ū e z + T κµ

9 Non-dimensional parameters: the Prandtl number: Pr µ ρ 0 κ momentum diffusivity / thermal diffusivity the Rayleigh number: Ra (T 0 T 1 )d 3 gρ 0 α κµ non-dimensional measure of thermal gradient

10 Boundary conditions Horizontal direction: periodicity2π/q Vertical direction: at z = 0, 1 T = 0 z=0,1 perfectly conducting plates w = 0 z=0,1 impenetrable plates Rigid boundaries at z = 0,1: u z=0,1 = v z=0,1 = 0 zero tangential velocity Incompressibility x u + y v + z w = 0 = z w = ( x u + y v) u z=0,1 = v z=0,1 = 0 = x u z=0,1 = y v z=0,1 = 0 = z w z=0,1 = 0

11 Free surfaces atz = 0,1 to simplify calculations: [ z u + x w] z=0,1 = [ z v + y w] z=0,1 = 0 zero tangential stress w z=0,1 = 0 = x w z=0,1 = y w z=0,1 = 0 = z u z=0,1 = z v z=0,1 = 0 = x z u z=0,1 = y z v z=0,1 = 0 = zz w z=0,1 = z ( x u + y v) z=0,1 = 0 Not realistic, but allows trigonometric functionssin(kπz)

12 Two-dimensional case U = ψ e y = No-penetration boundary condition: { u = z ψ w = x ψ } = U = 0 0 = w = x ψ = Horizontal flux: 1 z=0 dz u(x,z) = 1 z=0 { ψ = ψ1 atz = 1 ψ = ψ 0 atz = 0 dz z ψ(x,z) = ψ(x,z)] 1 z=0 = ψ 0 ψ 1 Arbitrary constant= ψ 0 = 0 Zero flux = ψ 1 = 0 Stress-free: 0 = z u = 2 zz ψ Rigid: 0 = u = zψ atz = 0,1

13 Two-dimensional case Temperature equation: t T + U T = RaU e z + T U T = u x T + w z T = z ψ x T + x ψ z T J[ψ,T] t T + J[ψ,T] = Ra x ψ + T

14 Velocity equation t U + (U )U = Pr[ P + Te z + U] Takee y : e y t U = e y t ψe y = t ψ e y P = 0 e y Te z = x T e y U = e y ψe y = 2 ψ t ψ e y (U )U = Pr[ x T + 2 ψ] f = f f

15 e y (U )U = z (U )u x (U )w = z (u x u + w z u) x (u x w + w z w) = z u x u + z w z u x u x w x w z w + u xz u + w zz u u xx w w xz w = z u ( x u + z w) x w ( x u + z w) + u x ( z u x w) + w z ( z u x w) = ( z ψ) x ( zz ψ xx ψ) + ( x ψ) z ( zz ψ xx ψ) = ( z ψ) x ( ψ) ( x ψ) z ( ψ) = J[ψ, ψ] t ψ + J[ψ, ψ] = Pr[ x T + 2 ψ]

16 Linear stability analysis Linearized equations: t ψ = Pr[ x T + 2 ψ] t T = Ra x ψ + T Solutions: ψ(x,z,t) = ˆψ sinqx sinkπz e λt q R, k Z +, λ C T(x,z,t) = ˆT cosqx sinkπz e λt functions scalars γ 2 q 2 + (kπ) 2 λγ 2 ˆψ = Pr[ qˆt + γ 4 ˆψ] λˆt = Ra q ˆψ γ 2ˆT

17 [ ˆψ λ ˆT ] = [ Pr γ 2 Pr q/γ 2 Ra q γ 2 ][ ˆψ ˆT ] Steady Bifurcation: λ = 0 Pr γ 4 Pr Ra q2 γ 2 = 0 Ra = γ6 q = (q2 + (kπ) 2 ) 3 2 q 2 Ra c (q,k)

18 Convection Threshold Conductive state unstable at (q,k) for Ra > Ra c (q,k)

19 Conductive state stable if min Ra < q R Ra c (q,k) k Z + 0 = Ra c(q,k) q = q2 3(q 2 + (kπ) 2 ) 2 2q 2q(q 2 + (kπ) 2 ) 3 q 4 = 2(q2 + (kπ) 2 ) 2 q 3 (3q 2 (q 2 + (kπ) 2 ) = q 2 = (kπ)2 2 ( Ra c q = kπ ),k = (kπ)2 /2 + (kπ) 2 ) 3 2 (kπ) 2 /2 Ra c Ra c (q = π ),k = 1 2 = 27 4 (kπ)4 = 27 4 (π)4 = 657.5

20 Rigid Boundaries Calculation follows the same principle, but more complicated. Boundaries damp perturbations= higher threshold q c = l c = π/q c = rolls circular Ra c q c l c stress-free boundaries 27 4 π4 π = rigid boundaries 1700 π 1

21 Lorenz Model: including nonlinear interactions J [ψ, ψ] = J[ψ, γ 2 ψ] = x ψ z ( γ 2 ψ) x ( γ 2 ψ) z ψ = 0 J[ψ,T] = ˆψˆT [ x (sinqxsinπz) z (cosqxsinπz) x (cosqx sinπz) z (sinqx sinπz)] = ˆψˆT qπ[cosqx sinπz cosqx cosπz + sinqx sinπz sinqx cosπz] +ˆψˆT qπ (cos 2 qx + sin 2 qx) sinπzcosπz = ˆψˆT qπ 2 sin2πz functions scalars

22 ψ(x,z,t) = ˆψ(t)sinqx sinπz T(x,z,t) = ˆT 1 (t)cosqx sinπz + ˆT 2 (t)sin2πz J[ψ,T 2 ] = ˆψˆT 2 [ x (sinqxsinπz) z (sin2πz) x (sin2πz) z (sinqx sinπz)] = ˆψˆT 2 q 2π cosqx sinπz cos2πz = ˆψˆT 2 q πcosqx (sinπz + sin3πz) Including ˆT 3 (t)cosqxsin3πz = new terms = Closure problem for nonlinear equations Lorenz (1963) proposed stopping att 2.

23 Lorenz Model t ˆψ = Pr(qˆT 1 /γ 2 γ 2 ˆψ) tˆt 1 + qπ ˆψˆT 2 = Ra q ˆψ γ 2ˆT 1 tˆt 2 + qπ ˆψˆT 1 = (2π) 2ˆT 2 2 sinqxsinπz cosqxsinπz sin2πz Define: X πq 2γ 2 ˆψ, Y πq2 2γ 6 ˆT 1, Z πq2 2γ 6 ˆT 2, τ γ 2 t, r q2 4π2 γ6ra, b γ = 8 2 3, σ Pr Famous Lorenz Model: Ẋ = σ(y X) Ẏ = XZ + rx Y Ż = XY bz

24 σ = Pr (often set to 10, its value for water) r = Ra/Ra c Damping= σx, Y, bz Advection = XZ, XY Symmetry between (X, Y, Z) and( X, Y, Z)

25 Lorenz Model Pitchfork Bifurcation Steady states: 0 = σ(y X) = X = Y 0 = XZ + rx Y = X = 0 or Z = r 1 0 = XY bz = Z = 0 or X = Y = ± b(r 1) 0 0 0, b(r 1) b(r 1) r 1, b(r 1) b(r 1) r 1

26 Jacobian: Df = For (X,Y,Z) = (0,0,0): Eigenvalues: Df(0,0,0) = σ σ 0 r Z 1 X Y X b σ σ 0 r b λ 1 + λ 2 = Tr = σ 1 < 0 λ 1 λ 2 = Det = σ(1 r) λ 3 = b < 0

27 0 < r < 1 = λ 1,2,3 < 0 = stable node r > 1 = λ 1,3 < 0,λ 2 > 0 = saddle Pitchfork bifurcation at r = 1 creates X = Y = ± b(r 1), Z = r 1

28 Lorenz Model: Hopf Bifurcation For X = Y = ± b(r 1),Z = r 1, Df = σ σ b(r 1) ± b(r 1) ± b(r 1) b Eigenvalues: λ 3 + (σ + b + 1)λ 2 + (r + σ)bλ + 2bσ(r 1) = 0 Hopf bifurcation λ = iω : iω 3 (σ + b + 1)ω 2 + i(r + σ)bω + 2bσ(r 1) = 0

29 (σ + b + 1)ω 2 + 2bσ(r 1) = 0 ω 3 + (r + σ)bω = 0 2bσ(r 1) σ + b + 1 = ω2 = (r + σ)b 2bσ(r 1) = (r + σ)b(σ + b + 1) 2bσr 2bσ = rb(σ + b + 1) + σb(σ + b + 1) r = σ(σ + b + 3) σ b 1 = for σ = 10, b = 8/3 At r = 24.74, the two steady states undergo a Hopf bifurcation (shown to be subcritical) = unstable limit cycles exist for r < 24.74

30 Lorenz Model: Bifurcation Diagram

31 Lorenz Model: Strange Attractor for r = 28 Lorenz Model: Time Series for r = 28

32 Water wheel (Malkus) motion described by Lorenz model

33 Instabilities of straight rolls: Busse balloon cross-roll Rayleigh knot zig-zag Prandtl skew-varicose oscillatory wavenumber

34 skew-varicose instability cross-roll instability Continuum-type stability balloon in oscillated granulated layers, J. de Bruyn, C. Bizon, M.D. Shattuck, D. Goldman, J.B. Swift & H.L. Swinney, Phys. Rev. Lett

Matthews Nature 404 (2000) Phys. Rev. E.

35 Complex spatial patterns in convection Experimental spiral defect chaos Spherical harmonicl = 28 Egolf, Melnikov, Pesche, Ecke P. Matthews Nature 404 (2000) Phys. Rev. E. 67 (2003) Convection in cylindrical geometry. Bajaj et al. J. Stat. Mech. (2006)

36 Small containers: multiplicity of states cylindrical container with R = 2H two-tori torus mercedes four rolls pizza dipole two rolls three rolls CO asym three rolls experimental photographs by numerical simulations by Hof, Lucas, Mullin, Borońska, Tuckerman, Phys. Fluids (1999) Phys. Rev. E (2010)

toroidal convection cells moves radially inwards

37 Small containers: a SNIPER bifurcation in a cylindrical container with R = 5H Pattern of five toroidal convection cells moves radially inwards in time. From Tuckerman, Barkley, Phys. Rev. Lett. (1988).

38 Timeseries fast away from SNIPER slow near SNIPER Phase portraits pitchfork pitchfork saddle-nodes before SNIPER after

39 Geophysics Convection and plate tectonics Numerical simulation of convection in earth s mantle, showing plumes and thin boundary layers. By H. Schmeling, Wikimedia Commons.

Boulder School for Condensed Matter and Materials Physics. Laurette Tuckerman PMMH-ESPCI-CNRS

Boulder School for Condensed Matter and Materials Physics Laurette Tuckerman PMMH-ESPCI-CNRS laurette@pmmh.espci.fr Hydrodynamic Instabilities: Convection and Lorenz Model Web sites used in demonstrations