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)

l Degrees of freedom 3 ⎛ L⎞ 9 N = ⎜ ⎟ ∝ Re 4 ⎝ η⎠ 1 ν3 4 η = ( ε ) The scales of turbulence

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

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 Parametrization: LES Filters ˆ f (x) = ∫ f (x' )G(x − x' )dx' ⇔ f (k) = fˆ (k)Gˆ (k)

Filter Function Fourier transform

sin k x x' 1 if ki ≤ kc Cut-off spectral 3 [ c( i − i )] Πi =1 0 otherwise π(xi − x' i ) { 2 2 3 ⎛ Δ k ⎞ ⎛ 6 ⎞ 2 ⎛ 6 x − x' ⎞ exp⎜ − ⎟ exp⎜ ⎟ ⎜ ⎟ Gaussian ⎜ 2 ⎟ ⎜ − 2 ⎟ ⎝ 24 ⎠ ⎝ πΔ ⎠ ⎝ Δ ⎠ ⎡ 1 ⎤ 1 sin⎢ Δ ki ⎥ 1 3 if x − x' ≤ Δ ⎣ ⎦ Δ i i 3 2 Top-hat 2 Πi =1 1 0 otherwise Δ ki { 2 LES Parametrization: LES

Viscosity Filters: f = ∫ f (x')G(x − x')dx' f ≠ f Backscatter Filtered part

τij = ui uj − ui u j + ui u' j + u' i u j +u' i u' j

Issue: Reynolds stress parametrization LES Parametrization: LES Global models

Smagorinski Spectral

1 ˆ 2 ˆ τ = τ δ − 2ν S i kjτ ij = νT (k)k ui ij 3 kk ij T ij 1 E(kc ,t) ⎡ ⎛ 3.03kc ⎞ ⎤ 2 2 ν (k) = ⎢ 0.441+15.2 exp⎜ − ⎟ ⎥ νT = Cs Δ 2S pqS pq T 3 ( )( ) Ko kc ⎣ ⎝ k ⎠ ⎦

1 ∞ 1 2 Sij = (∂i u j + ∂ j ui ) u = E(k)dk 2 2 ∫0

Cs = 0.1 − 0.2 Ko =1.3 −1.8

Lesieur, « Turbulence in Fluids », Kluwer, 1990 Parametrization: LES Gradient Model

2 τ ij = C(Δx) Sik S jk

C= adjustable constant

LES: gradient Test via laboratory experiment       



 Control parameters  Reynolds number = Rotation number = Turbulence Asymmetry

intensity f1 − f2 ϑ = f1 + f2 Calibration of C using angular momentum conservation

1.5

Lz(z) 1 Kp

0.5

0 −0.5 0 0.5 z

Kuzzay, Faranda and B. Dubrulle , Phys Fluids (2015) Check LES vs Experiment

(d) (b) 0.5

z 0 0.2 T −0.5 −0.5 0 0.5 ⇧ x T 0.15

0.1

(b) 0.05 0.5

z 0

0 −0.5

−0.5 0 0.5 x 104 105 106 Re Kuzzay, Faranda and B. Dubrulle , Phys Fluids (2015) Check LES vs Experiment (d) (c)

0.5 0.5

0 z 0 z

−0.5 (b) −0.5

−0.5 0 0.5 −0.5 0 0.5 x T x 0.2 ⇧T 0.15

0.1 (a)

0.05 0.5

0 z 0

−0.05 −0.5 −1 −0.5 0 0.5 1 −0.5 0 0.5 x

✓ Kuzzay, Faranda and B. Dubrulle , Phys Fluids (2015) Why planes do ﬂight…

1 2 2 2 ∂t (δuℓ ) +ε = ∇ℓ (νT ∇ℓ (δuℓ ) )+νΔℓ (δuℓ ) 2 F , , u2 Kolmogorov Idea: νT = (ε ℓ δ ℓ ) (1−3α )/2 On dimensional ground α u2 1+α νT = ε (δ ℓ ) ℓ

Mixing length,smagorinski: α =1/ 3 Gradient Model: α = 0 Why planes do ﬂight…

One scale integration: ℓ 2 ε = (νT +ν)∇ℓ (δuℓ ) d

Two scale integration:

1−α (εℓ) 2 3(1−α)/2 = K (δuℓ ) d (1−α)

So except if you take alpha=1, you get Kolmogorov spectrum!!! But burners do not burn….

Kolmogorov self-similarity 3 2,5 1/3 2 δul = u(x + l) − u(x) ∝ l 1,5 n n / 3 δul ∝ l (p) zeta 1 0,5 ⎛ δu ⎞ 0 uP u, l F⎜ ⎟ -0,5 δ (δ ) = ⎜ 1 ⎟ 0 2 4 6 8 ⎝ l 3 ⎠

u P(u) Transverse velocity increments

0,1

0,01

0,001

0,0001

10-5

10-6

10-7 0 20 40 60 80 100 (u / l 0.35 )

uP(u) Frisch Turbulence, Cambridge University Press 1991 uP(u) uP(u) uP(u) uP(u) uP(u) uP(u) References

Lesieur, « Turbulence in Fluids », Kluwer, 1990 Frisch Turbulence, Cambridge University Press 1991 Frisch in « Lecture notes on turbulence » World Scientiﬁc, 1989 Dubrulle & Frisch, Phys. Rev A, vol 43, p5355 (1991) J-P. Laval, B. Dubrulle and S. Nazarenko, Phys. Fluid 13 1995-2012 (2001). B. Dubrulle, J-P. Laval, S. Nazarenko and O. Zaboronski, J. Fluid Mech. 520 1-21 (2004) J-P. Laval , B. Dubrulle and J.C. McWilliams , Phys. Fluids 15 (5) : 1327-1339 (2003) J-P. Laval and B. Dubrulle , Eur. Phys. J B 49 471-481 (2006).

G. Parisi, U. Frisch in Turbulence and Predictability in Geophysical Fluid Dynamics, Proceed. Intern. School. of Physics “E. Fermi”. eds. Ghil M, Benzi R, Parisi G. (1985) 84.