Rayleigh-B enard convection: a priori estimate on the Boussinesq - PowerPoint PPT Presentation

Rayleigh-B´ enard convection: a priori estimate on the Boussinesq system that carry physical meaning C. Doering & M. Westdickenberg, & C. Seis, & C. Nobili, & A. Choffrut Max Planck Institute for Mathematics in the Sciences, Leipzig

Rayleigh-B´ enard convection in the turbulent regime

Rayleigh-B´ enard convection Temperature advection and diffusion Buoyancy, acceleration, and viscosity T = 0 u = 0 z = 1 1 Pr ( ∂ t u + u · ∇ u ) + ∇ p � 0 � ∂ t T + u · ∇ T − △ T = 0 = −△ u + RaT 1 ∇ · u = 0 z = 0 T = 1 u = 0 Rayleigh number Ra = α g δT h 3 Prandtl number Pr = ν , ν κ κ ... in Boussinesq approximation

Beyond the Rayleigh-B´ enard instability Ra < Ra ∗ : pure diffusion, T = 1 − z , u = 0 Ra > Ra ∗ : unstable to convection rolls Ra ≫ 1: steady → periodic → “turbulent” Schlieren picture for Ra ∼ 10 9 L. Kadanoff

Efficiency of heat transport: the Nusselt number Diffusion and convection � vertical heat flux � 0 � heat flux q = Tu − ∇ T, vertical heat flux = q · 1 � 0 � Nusselt number = space-time average of q · 1 z = 1 z Nu = lim t ↑∞ � t � � 0 � 1 1 dx dt ′ (0 , L ) d − 1 × (0 , 1) q · y L d − 1 1 t 0 z = 0 � �� � L

Scaling of Nu in Ra and Pr ... Similarity law: Nu = f ( Ra, Pr, L ) Grossmann & Lohse ’00 Ahlers et al ’05 ... in theory and experiment

First part: Upper bounds on Nu at Pr = ∞ Limitations of the background field method Second part: Upper bounds on Nu at large Pr Re only matters in boundary layer

Malkus’ marginal stability argument... T = 0 , u = 0 z = 1 ∂ t T + u · ∇ T − △ T = 0 T = 0 , u = 0 ˆ z = H ˆ � 0 � −△ u + ∇ p = RaT 1 u · ˆ ∇ T − ˆ ∂ ˆ t T + ˆ △ T = 0 ∇ · u = 0 z = 0 � 0 � T = 1 , u = 0 − ˆ u + ˆ △ ˆ ∇ ˆ p = T 1 x = Ra − 1 x, t = Ra − 2 3 ˆ 3 ˆ t ˆ ∇ · ˆ u = 0 1 1 3 � u = Ra 3 ˆ u, Nu = Ra Nu z = 0 ˆ T = 1 , u = 0 ˆ .. as a rescaling

Nusselt number Nu independent of height H ≫ 1 1 � 3 for Ra ≫ 1 Nu ∼ Ra ⇐ ⇒ Nu ∼ 1 for H ≫ 1 NUSSELT−ZAHL BLAU: H = 250, Nu = 0.0648, Varianz = 3.61e−04 ROT: H = 500, Nu = 0.0584, Varianz = 2.15e−04 SCHWARZ: H = 1000, Nu = 0.0545, Varianz = 1.19e−04 0.15 Simulation 0.1 (M. Zimmermann) : Nu(t) 0.05 0 100 200 300 400 500 600 700 Zeit

Upper bounds on Nusselt via a priori estimates Nu � ln 2 / 3 H , • Constantin & Doering (’99): Stokes maximal regularity in L ∞ Nu � ln 1 / 3 H , • Doering & O. & Reznikoff (’06): background temperature field method Nu � ln 1 / 15 H , • O. & Seis (’11): background temperature field method • Nobili & O. (’16) exponent 1/15 optimal for background temperature field method Nu � ln 2 / 3 ln H , • O. & Seis (’11): Stokes maximal regularity in L ∞ and background temperature field method

Two insights from rigorous results • Optimal background temperature profile is non-monotone stable for H 0 � log − 1 15 H 1 15 H yields Nu � log • Optimal background temperature profile has no physical meaning

Notations � t � 1 0 dt ′ 1 Time & horizontal average: �·� := lim (0 , L ) d − 1 dy L d − 1 t t ↑∞ z = H z Temperat. fluctuations: θ := T − � T � y � 0 � z = 0 � �� � Vertical velocity: w := u · 1 L θ ↔ w : Stokes & no-slip b. c. � plate & clamped b. c. △ 2 w = − △ y θ, w = ∂ z w = 0 for z = 0 , H .

Two representations of Nusselt number w = ∂ z w = 0 z = H Recall definition z △ 2 w = −△ y θ � H 1 Nu = 0 � Tw − ∂ z T � dz y H z = 0 w = ∂ z w = 0 Get in addition z Nu = � θw � − d dz � T � for all z , z ′ � H 0 �|∇ T | 2 � dz Nu =

More flexible representation of Nusselt number τ 1 τ ( z ) with τ ( z = 0) = 1 , τ ( z = H ) = 0 z H Decompose T = τ + θ ; keep relation between θ & w Average Nu = − ∂ z � T � + � w θ � w. r. t. − dτ dz : � H � H dτ dτ Nu = dz ∂ z � T � dz − dz � w θ � dz 0 0 � H 0 �|∇ T | 2 � dz : Combine with Nu = �� H � � H � H dz ) 2 dz − 0 ( dτ dτ 0 �|∇ θ | 2 � dz + 2 Nu = dz � w θ � dz 0

Optimal bound through saddle point problem � � H �� H �� � H Nu ≤ � dz ) 2 dz − 0 ( dτ dτ 0 �|∇ θ | 2 � dz + 2 Nu := min max dz � w θ � dz τ ( θ,w ) 0 τ 1 min over all τ with τ = 1 , 0 for z = 0 , H z w = ∂ z w = 0 H △ 2 w = −△ y θ max over all ( θ, w ) with w = ∂ z w = 0 Hopf’43, Nicolaenko&Scheurer&Temam’85, Constantin&Doering’92

Plus I: A marginal stability criterion � H dz ) 2 dz 0 ( dτ � Nu = min τ min over all τ with τ = 1 , 0 for z = 0 , H with � H � H dτ 0 �|∇ θ | 2 � dz + 2 dz � w θ � dz ≥ 0 0 for all ( θ, w ) with △ 2 w = −△ y θ and θ = w = ∂ z w = 0 at z = 0 , H ... captures transition H ∗ to convection rolls

Plus II: Amenable to horizontal Fourier transform ... � H dz ) 2 dz 0 ( dτ � Nu = min τ min over all τ with τ = 1 , 0 for z = 0 , H with � H � � 2+ � 1 � 2 dz ( k 2 − d 2 dz ( k 2 − d 2 d dz 2 ) 2 w dz 2 ) 2 w k 0 � H dz w ( k 2 − d 2 dτ dz 2 ) 2 w dz ≥ 0 + 2 0 for all ( k, w ( z ) ) with and ( k 2 − d 2 dz 2 ) 2 w = w = dw dz = 0 at z = 0 , H Busse, Howard, Chan ’71, Ierley&Kerswell&Plasting ’05 ... monitor bifurcations in optimal k as H ↑ ∞

Stability of logarithmic temperature profile τ dτ Linear profile: τ = az + b , dz = a � H � H 0 a � ( ∂ z w ) 2 � dz 0 a � w θ � dz ≥ 2 light heavy z 1 b dτ Log profile: τ = ln z − a , dz = z − a Lemma 1 (Doering & O. & Reznikoff ’06) τ � H � H 1 z − a � ( ∂ z w ) 2 � dz 1 z − a � w θ � dz ≥ 2 0 0 for all ( θ, w ) with △ 2 w = −△ y θ z and w = ∂ z w = 0 at z = 0 , H

Non-monotone Ansatz for background temp. profile τ 1 2 (1 + 1 z λ ln H − z ) 1 λ ≈ ln H δ = boundary layer width H z ���� ���� δ δ Theorem 1 (Doering & O. & Reznikoff ’06, O. & Seis ’11) For δ ≪ ln − 1 15 H have � H � H 0 �|∇ θ | 2 � dz + 2 dτ dz � w θ � dz ≥ 0 0 for all ( θ, w ) with △ 2 w = −△ y θ and θ = w = ∂ z w = 0 at z = 0 , H 1 � 15 H Hence Nu ≤ Nu � ln

ln − 1 15 is optimal for background field method Theorem 2 (Nobili & O. ’16) � H dτ Suppose that dz dz = 1 and 0 � H � H 0 �|∇ θ | 2 � dz + 2 dτ dz � w θ � dz ≥ 0 0 for all ( θ, w ) with △ 2 w = −△ y θ and θ = w = ∂ z w = 0 at z = 0 , H . � H 1 dz ) 2 dz � ln 0 ( dτ 15 H . Then 1 � 15 H . In particular Nu ∼ ln

Background field method not optimal Theorem 1 (Doering & O. & Reznikoff ’06, O. & Seis ’11) 1 � 15 H We have for H ≫ 1 : Nu � ln Theorem 2 (Nobili & O. ’16) 1 � 15 H . We have for H ≫ 1 : Nu � ln Theorem 3 (O. & Seis ’11) 1 3 ln H We have for H ≫ 1 : Nu � ln � In particular Nu ≪ Nu .

Theorem 2: Characterization of stable profiles Lemma 2 (Nobili & O. ’16) Suppose that � H dτ dz � w θ � dz ≥ 0 0 for all ( θ, w ) with △ 2 w = −△ y θ and w = ∂ z w = 0 at z = 0 , H . Then τ dτ • dz ≥ 0, � ˜ � δ H dz dz � (ln ˜ dτ H dτ • δ ) dz dz δ δ z 2 δ ˜ 2 δ H for all δ ≪ ˜ H .

Proof of Lemma 2: non-negativity Horizontal Fourier transform y � k : � H dz w ( k 2 − d 2 dτ dz 2 ) 2 w dz ≥ 0 0 dz = ( k 2 − d 2 for all k and all w ( z ) with w = dw dz 2 ) 2 w = 0 for z = 0 , H . � H dz w 2 dz ≥ 0 dτ dτ Limit k ↑ ∞ : = dz ≥ 0 ∀ w ⇒ 0 � H dz w d 4 dτ Limit k ↓ 0: ∀ w dz 4 w dz ≥ 0 0

Proof of Lemma 2: approximate logarithmic growth � H dz w d 4 dτ Have: ∀ w dz 4 w dz ≥ 0 0 Change of variables s = ln z , w = z 2 v : large scales w d 4 ds = − dv 2 dz 4 w = v ( d ds + 2)( d ds + 1) d ds ( d − 2 v dv ds − 1) v ≈ ds , dτ dz dz = dτ ds ds � ln H ds ( − dv 2 d 2 τ dτ ds 2 = d ds z dτ Get: ∀ v ds ) ds � 0 = ⇒ dz � 0 0

Two insights from rigorous results • Optimal background temperature profile is non-monotone stable for H 0 � log − 1 15 H 1 15 H yields Nu � log • Optimal background temperature profile has no physical meaning

Second part: Bounds on Nu at large but finite Pr Re only matters in thermal boundary layer

Recall Boussinesq approximation ... u = 0 T = 0 z = 1 1 Pr ( ∂ t u + u · ∇ u ) � 0 � ∂ t T + u · ∇ T − △ T = 0 −△ u + ∇ p = RaT 1 ∇ · u = 0 z = 0 T = 1 u = 0 Rayleigh number Ra = α g δT h 3 Prandtl number Pr = ν , ν κ κ ... at finite Prandtl number

Theorem 4 (Choffrut&Nobili&0. ’15)   1 1   3 for Pr ≥ ( Ra ln Ra )   ( Ra ln Ra ) 3 Nu � 1 1   ( Ra ln Ra   ) for Pr ≤ ( Ra ln Ra ) 2 3 Pr 1 2 for all Pr Constantin & Doering ’96: Nu � Ra 1 2 3 ln 3 Ra for Pr ≫ Ra X. Wang ’07: Nu � Ra Wang’s regime Pr ≫ Ra means Re ≪ 1 in bulk; 1 3 means Re � 1 our regime Pr � Ra in thermal boundary layer