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Turbulence Modelling Turbulence Modelling B. Dubrulle CNRS SPEC, CEA Saclay Astrophysical Flows Disks/Galaxies Re ≈1013 Planetary Atmosphere 9 Re =10 8 Stars Re =10 Navier-Stokes Equations: ∂tu + u • ∇ u = − ∇ p + ν Δ u + f LU Control Parameter: Re = ν 2 Re =10 11 Re =107 Re =10 Re =1012 Re =1021 Laboratory Flows Grid Turbulence Taylor-Couette Von Karman 5 3 Re =10 6 Re =10 Re =10 Similarity Principle:same boundray conditions, same Reynolds->same behaviour Joule’s experiment (1840) Application of 772.24 foot pound force (4.1550 J.cal−1) results in elevation of temperature of a pound of water by one degree Fahrenheit Fluid good converters of Mechanical enery into heat Aside about Joule’s experiment NEQFLUIDS2016: Classical and Quantum Fluids Out of Equilibrium 13-16 Jul 2016 Grand Hotel de Paris, Villard de Lans (France) Robert&Collins Lessiveuse: n.m. Traduction: washing machine Ils utilisent le club de cartes comme leur lessiveuse. ->They use the card club as their washing machine. Aside about Joule’s experiment 1840 1991 Robert&Collins Von Karman Flow: n.m. Traduction: french washing machine They use their french washing machine as experimental set up-> Ils utilisent leur lessiveuse comme experience. Douady S, Couder Y and Brachet M E 1991 Direct observation of the intermittency of intense vorticity filaments in turbulence Phys. Rev. Lett. 67 983 A modern version: VK flow Work measured by Torques applied at Shafts Heat flux measured By keeping T constant Rousset et al, RSI 85, 103908 (2014); Zeroth law of turbulence SuperFluid! Non-dimensional Energy dissipation per unit mass is independent of viscosity!!!!! Ravelet PhD, Saint-Michel PhD, VKS collaboration, SHREK collaboration Observation: power-law spectrum Energy cascade Saclay team: Debue, Kuzzay, Faranda, Saw, Daviaud, Dubrulle et al, (2016) Observation of Fluid Movements Creation of structures finer and finer till dissipation by viscosity-> scale hierachy Scale hierarchy Scale hierarchy Leonard de Vinci Richardson’s Cascade (Jonathan Swift) Big whirls have litle whirls, Which feed on teir velocit, And litle whirls have lesser whirls , And so on t viscosit.” Lewis Fry Richardson 1881-1953 Kolmogorov Cascade (1941) Big whirls have litle whirls, Which feed on teir velocit, And litle whirls have lesser whirls , And so on t viscosit.” Lewis Fry Richardson Andrey Kolmogorov 1903-1987 Kolmogorov Theory (1) NSE+homogeneity 1 2 1 3 2 ∂t (δuℓ ) +ε = − ∇ℓ (δuℓ ) +νΔℓ (δuℓ ) 2 4 δuℓ = u(x + ℓ) − u(x) Karman Howarth equation Kolmogorov Theory (2) KH equation + self-similarity+stationarity 1 2 1 3 2 ∂t (δuℓ ) +ε = − ∇ℓ (δuℓ ) +νΔℓ (δuℓ ) 2 4 3 4 (δuℓ ) ∝− εℓ 3 2/3 −5/3 2 2/3 E k = C ε k (δuℓ ) ∝(εℓ) ( ) Interpretation:The energy cascade Work Energy cascade Heat Dissipative scale Injection scale (viscosity) Turbulence phenomenology Robust result Kolmogorov spectrum Constant energy transfer du 2 u3 = ε ≅ = cte Interpretation (Kolmogorov 1941) dt l Energy cascade 1 3 L ⇒ u ∝ (εl) N l Degrees of freedom 3 ⎛ L⎞ 9 N = ⎜ ⎟ ∝ Re 4 ⎝ η⎠ 1 ν3 4 η = ( ε ) The scales of turbulence η L Example: the sun Dissipation Solar Giant Granule scale spot Convective cell 0.1 km 103 km 3⋅104 km 2⋅105 km 3 N = (10 6 ) = 1018 Many degrees of freedom! Scale of turbulence L=5cm η=1mm L=10m η=0.06mm L=6000 km η=6mm Simulation of turbulence We have the basic equations… Can we simulate all these systeme? → → → ∇• u = 0 → → → → 1 → → ∂ u+ (u•∇)u = − ∇ p +νΔ u t ρ Computer Blue Gene Simulation of turbulence Simulation: 6 109 nodes --> 1 week of CPU on Blue Gene Storage: 108 nodes- -> 2Tb= 1 disk Simulation of turbulence 3 ! L $ N = # & "η % L=5cm η=1mm N=105 L=10m 16 η=0.06mm N=10 L=6000 27 km N=10 η=6mm Simulation of turbulence L=5cm η=1mm 1 mn of cpu Less than a disk L=10m 16 η=0.06mm N=10 L=6000 27 km N=10 η=6mm Simulation of turbulence L=5cm η=1mm 1 mn of cpu Lett than a disk L=10m 20 000 years of cpu η=0.06mm 100 billions disks L=6000 27 km N=10 η=6mm Simulation of turbulence L=5cm η=1mm 1 mn of cpu Less than a disk L=10m 20 000 years of cpu η=0.06mm 100 billions disks L=6000 1015 years of cpu km 1019 disks η=6mm Simulation of turbulence L=5cm η=1mm 1 mn of cpu Less than a disk L=10m 20 000 years of cpu η=0.06mm 100 billions disks L=6000 1015 years of cpu km 1019 disks η=6mm What can be done? Explosion Flux d’Energie E(k) (Viscosité Turbulente) Grandes Echelles Petites Echelles (Paramétrisées) 90 % des ressources k informatiques Building on Kolmogorov 1 2 1 3 2 ∂t (δuℓ ) +ε = − ∇ℓ (δuℓ ) +νΔℓ (δuℓ ) 2 4 Energy injection Viscous Energy transfer dissipation 1 2 2 2 ∂t (δuℓ ) +ε = ∇ℓ (νT ∇ℓ (δuℓ ) )+νΔℓ (δuℓ ) 2 Turbulent Viscosity What can be done? Explosion Flux d’Energie E(k) (Viscosité Turbulente) Grandes Echelles Petites Echelles (Paramétrisées) 90 % des ressources k informatiques Two ways to cut the scale space Fluctuations RANS: You keep the mean Mean Flow Flow Parametrize fluctuations Small scales LES: You keep the large scale Parametrize the small Large scale Mathematical translation 1 ∂ u + u ∇ u = − ∇ p + Δ u + f t i j j i i Re i i _ u = u + u' Spatial filter for LES _ Ensemble average for RANS 1 ∂ u + u ∇ u = − ∇ p + Δ u + f − ∇ τ t i j j i i Re i i j ij Reynolds stress τij = ui uj − ui u j + ui u' j + u' i u j +u' i u' j LES τij = +u' i u' j RANS Influence of decimated scales l 2 Typical time at scale l: δt ≈ ∝ l 3 u Decimated scales (small scales) vary very rapidly We may replace them by a noise with short time scale u = u + u' Dtu' i = Aiju' j +ξj ξi (x,t )ξ j (x' ,t' ) =κ ij (x, x')δ(t − t' ) Generalized Langevin equation Obukhov Model Simplest case No mean flow u = 0 Large isotropic friction A = −γδ , γ >>δt ij ij No spatial correlations κ ij(x,x') ∝γδij ! ! ! ! ⎛ 3 ⎞ ⎛ 3x2 3x •u u2 ⎞ P(x,u,t) = ⎜ 2 ⎟exp⎜− 3 − 2 − ⎟ ⎝ 2πεt ⎠ ⎝ εt εt εt ⎠ u ∝ εt Gaussian velocities x ∝ ε 2 / 3t 3/ 2 Richardson’s law u ∝ x1/ 3 Kolmogorov’s spectra LES: Langevin Influence of decimated scales: transport !• ! ! x = u + u' Leprovost&Dubrulle ! ! ! Ω = ∇ × u !• ! ! ! ! ! ! Ω = (Ω•∇)u + (Ω•∇)u' ' ' βkl = ukul Stochastic computation ' ' αijk = ui∂ ku j ∂t Ωi + uk ∇k Ωi = Ωk∇k ui + ∇ k[βkl ∇l Ωi ] + 2α kil ∇kΩl Turbulent viscosity AKA effect Parametrization: RANS Issue: Reynolds stress parametrization τij = +u' i u' j = −αijk uk − βijkl∇k ul AKA effect Turbulent Viscosity Helicity effect 4 order tensor Influence on mean flow (breaks Galilean invariance) Can be « negative » (instabilities) Produces large scale-instabilities (cf dynamo effect) Sulem, Frisch, She Dubrulle&Frisch RANS Parametrization: RANS$ AKA effect Use to explain: Solar Granulation (Kishan, MNRAS, 1991) Galaxy Clustering (Kishan, MNRAS, 1993) Large-scale vortices in disks (Kitchatinov et al, A&A, 1994) Liitle (not?) used in general turbulence No general theory ! ! Analogy with dynamo: 1 u'•(∇ × u') α = ε ijk 3 τ ijk 3D isotropic RANS: AKA Parametrization: RANS$ Viscosity Not necessarily isotrop(cf shear flows) (Dubrulle&Frisch, Isotropic Case βijkl = νT δ jk δil Characteristic lenght Dimensional analysis νT = KV L Characteristic Constant velocity 1 Kolmogorov theory V = (εL) 3 4 1/3 3 νT = Kε L RANS: Viscosity Example 1: Mixing length Convection ! ! "!! 1 !!!" ! ! Fc ∂tu + u •∇u = − ∇p + βθ g +νT Δu Core ρ dr Radiative H Hp p = − Inertia = Buyancy d ln p Vc 2 vc 1 ≈ βgθ 3 L vc = (gβFcL) Fc = vcθ 1 4 3 3 νT = (gβFc) L L = α Hp 1 < α < 2 RANS: Viscosity Example 2: Smagorinski Viscosity written function of mean gradients 2 νT = (csΔ) ∇ j ui∇ j ui Mesh size Adjustable Constant RANS: Viscosity Example 3: RNG Viscosity in spectral space ˆ 2 νT Δu i → −ν T k uˆ i 1 ⎛ ε ⎞ 3 νˆ = K⎜ ⎟ T ⎝ k4 ⎠ Computed with renormalization group RANS: Viscosity Multi-scale computation L l l ε = <<1 L Main idea: use scale separation Method: a) Linearization equations b) development of operators and fields in power of ε c) Solve order by order d) Solvability conditions provide coefficient Example: Turbulent diffusivity Frisch in « Lecture notes on turbulence » World Scientific, 1989 Before we start… Is there a solution to ? Ax = y Solvability condition: y ∈Im(A) ⇔ ∀z ∈Ker(A+ ), y • z = 0 In space of periodical functions f •g = ∫ f (x,t)g(x,t)dxdt Ker(A+ ) = 1: x → 1 Solvability condition 1• y = y = ∫ y(x)dx =0 RANS: multi Turbulent diffusion (1) Basic equation: passive scalar ! 2 x ∂ Θ+ u •∇ Θ = κ∇ Θ Fast variables: T x x t ! ! u = 0 Slow variables X = εx T = ε 2t expansion ∇x ← ∇x + ε∇X 2 ∂t ← ∂t + ε ∂ T Θ(x, X,t,T) = Θ(0) + εΘ(1) + ε2Θ(2) + ... Frisch in « Lecture notes on turbulence » World Scientific, 1989 Turbulent diffusion (2) Order 0 (0) ! (0) 2 (0) Equation ∂tΘ + u∇xΘ = κ∇xΘ Solvability 0=0 (0) (0) (0) Solution Θ (x,t, X,T) = Θ (X,T) = Θ Order 1 Equation (1) ! (1) ! (0) 2 (1) ∂tΘ + u∇xΘ = −u∇X Θ +κ∇xΘ ! ! Solvability u = 0 (1) ! (0) Θ = χ∇X Θ Solution ! ! ! ! 2 ! Frisch in « Lecture notes on turbulence » World Scientific, 1989 ∂t χ + u∇x χ = −u +κ∇x χ Turbulent diffusion (3) Order 2 (2) ! (2) (0) Equation ∂tΘ + u∇xΘ +∂T Θ = ! (1) 2 (2) 2 (0) −u∇XΘ +κ∇xΘ +κ∇X Θ (1) +2κ∇x∇XΘ (0) 2 (0) Solvability ∂ Θ = κ∇ + D ∂ ∂ Θ T ( X ij Xi X j ) 1 ⎡ ⎤ Dij = − ⎣ ui χ j + uj χi ⎦ Turbulent Diffusivity 2 = κ ∂l χi∂i χl > 0 Frisch in « Lecture notes on turbulence » World Scientific, 1989 Two ways to cut the scale space Fluctuations RANS: You keep the mean Mean Flow Flow Parametrize fluctuations Small scales LES: You keep the large scale Parametrize the small Large scale Mathematical translation 1 ∂ u + u ∇ u = − ∇ p + Δ u + f t i j j i i Re i i _ u = u + u' Spatial filter for LES _ Ensemble average for RANS 1 ∂ u + u ∇ u = − ∇ p + Δ u + f − ∇ τ t i j j i i Re i i j ij Reynolds stress τij = ui uj − ui u j + ui u' j + u' i u j +u'
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