Eddy-diffusivity fields in turbulent transport: explicit expressions

Transcription

1 DICCA University of Genova Italy Eddy-diffusivity fields in turbulent transport: explicit expressions Andrea Mazzino Lagrangian transport: from complex flows to complex fluids Lecce 7-10 March, 2016

2 Outline passive scalar turbulence

3 Outline E() - Large-scale transport (asymptotic) IR limit 5/3 0 L E() Θ - Pre-asymptotic transport New explicit expressions for the eddy diff. 5/3 0 L E() Θ - Toward real cases 5/3 0 L

4 Large-scale (asymptotic) transport Van Gogh s turbulence

5 Large-scale (asymptotic) transport = 2π Van Gogh s turbulence

6 Large-scale (asymptotic) transport E() 5/3 0

7 Large-scale (asymptotic) transport E() 5/3 IR limit Focus: scalar transport in the IR limit L / = 1 0 L

8 Large-scale (asymptotic) transport E() IR limit 5/3 0 L E() e i Lx e i x scalar initial condition 0 L

9 Large-scale (asymptotic) transport E() IR limit 5/3 0 L E() How does the initial condition evolve? e i Lx scalar initial condition e i x 0 L

10 Large-scale (asymptotic) transport Evolution: t θ(x,t)+v θ(x,t)=d 0 2 θ(x,t) t =0 t = t + dt θ(x, 0) e ±i Lx v(x, 0) e ±i x v θ(x, 0) e ±ilx e ±ix = e ±i (1+)x E() e i Lx e i x scalar initial condition 0 L

11 Large-scale (asymptotic) transport Evolution: t θ(x,t)+v θ(x,t)=d 0 2 θ(x,t) t =0 t = t + dt θ(x, 0) e ±i Lx v(x, 0) e ±i x v θ(x, 0) e ±ilx e ±ix = e ±i (1+)x E() e i Lx e i x scalar initial condition 0 L

12 Large-scale (asymptotic) transport Evolution: t θ(x,t)+v θ(x,t)=d 0 2 θ(x,t) next time step: θ(x,dt) e ±i (1±)x v(x,dt) e ±i x v θ(x, 0) e ±i Lx E() e i Lx e i x scalar initial condition 0 L

13 Large-scale (asymptotic) transport Evolution: t θ(x,t)+v θ(x,t)=d 0 2 θ(x,t) next time step: θ(x,dt) e ±i (1±)x v(x,dt) e ±i x v θ(x, 0) e ±i Lx This the origin of the eddy diffusivity E() beating 0 L

14 0 L Large-scale (asymptotic) transport E() θ t θ(x,t)+v θ(x,t)=d 0 2 θ(x,t) full-scale description L 0 E() t Θ(x,t)=D eddy αβ α β Θ(x,t) Θ IR limit Proof via multiple-scale expansion (Biferale et al., 1995)

15 Lac of scale separation no scale separation E() Θ 5/3 no effective equation 0 L we need an intermediate step

16 Pre-asymptotic transport we create a spectral E() gap such that Θ = L / 1 5/3 0 L

17 Pre-asymptotic transport we create a spectral E() gap such that Θ = L / 1 5/3 0 L the renormalization process occurs w.r.t the eddy diffusivity is now a field: D αβ (x, 2 t)

18 Pre-asymptotic transport E() Θ 5/3 via formal multiple-scale expansion: 0 L t Θ + U E Θ = i j (D ij (x,t)θ) U E i U i + j D ij A.M. (1997); A.M., Musacchio, Vulpiani (2005); Cencini, A.M., Musacchio, Vulpiani (2006);

19 Pre-asymptotic transport E() Θ 5/3 via formal multiple-scale expansion: 0 L t Θ + U E Θ = i j (D ij (x,t)θ) or in Lagrangian terms dx(t) dt = U E (x(t),t)+σ(x(t),t)η(t) D ij (x,t)= 1 2 (σ ip(x,t)σ jp (x,t))

20 Pre-asymptotic transport D ij (x,t) from an auxiliary differential problem E() Θ 5/3 expensive! 0 L need for explicit expressions for the eddy diffusivity field

21 Explicit expressions for D ij (x,t) Already done in some perturbative limits and simple flows: A.M., Musacchio, Vulpiani (2005): Cencini, A.M., Musacchio, Vulpiani (2006): U/u 1 U/u 1 E() here: u turbulence such to mimic ideal U 0 L 5/3 u random, homogeneous, stationary and scaling

22 Explicit expressions for D ij (x,t) E() U 0 L u 5/3 random, homogeneous, stationary and scaling there are many ways to do that but only one allows us to determine D ij (x,t) analytically:...by exploiting parallel flows...

23 Explicit expressions for D ij (x,t) n randomly oriented u = u(xx,t)n n

24 Explicit expressions for D ij (x,t) n randomly oriented u = u(x,t)n n

25 Explicit expressions for D ij (x,t) n randomly oriented u = u(x,t)n n

26 Explicit expressions for D ij (x,t) n randomly oriented u = u(x,t)n n

27 Explicit expressions for D ij (x,t) n randomly oriented u = u(x,t)n n

28 Explicit expressions for D ij (x,t) a few technical details: Ψ(x,t; x(0)) = da i dt = A i τ i + N i=1 A i (x(0),t) q i 2Bi τ i = α/(u rms q i ) Kaneda et al. (1999) sin[q i n(x(0)) x + θ i (x(0))] η i B i ε 2/3 q 5/3 τ i i ẋ(t) =v(x(t),t)+η(t) n D ij (x,t) via Taylor formula details in Boi, A.M., Lacorata (2016) JFM, under revision x(0)

29 Explicit expressions for D ij (x,t) D ij (x,t) via Taylor formula 2D case: D βγ = U 2 rms α 2 α 2 U 2 rms U 2 U β U γ +( U 2 δ βγ 2U β U γ ) α 2 α 2 U 2 rms U 2 +1 U 4 U 2 rms U N i=1 B i 2 τ i 3D case: D αβ = + α2 U 2 rms α U rms U U 2 δ αβ 3U α U β + U 4 δ αβ arctan (3 + α2 α U 4 α3 U rms Urms 3 U 2 rms U 5 1 U 2 )U α U β U 2 δ αβ N i=1 B i 2 τ i and the effective drift follows: U E i U i + j D ij

30 Range of validity U(x, 2 t) L = /L

31 Range of validity U(x, 2 t) L = /L

32 Range of validity U(x, 2 t) L = /L

33 Range of validity U(x, 2 t) L = /L E() Θ 0 L

34 Model performances in the absence of scale separation E() Θ U u observables to chec: M p (t) = x x(0) U 0 t p 0 L exact evolution dx(t) dt = U(x(t),t)+u(x(t),t) against dx(t) dt = U E (x(t),t)+σ(x(t),t)η(t)

35 Model performances in the absence of scale separation E() Θ different strategies for σ(x(t),t) U u dx(t) dt = U E (x(t),t)+σ(x(t),t)η(t) 0 L A: our closure B: only large scale σ(x(t),t)=0 U E = U C: our closure without effective velocity E() Θ U E = U D: naive closure σ from 5/3 0 L

36 Results (2D) mean square particle dispersion D naive < x(t)-x(0)-u 0 t 2 > C A E B exact our closures ẋ = U t E() Θ test in the absence of scale separation 0 L

37 Results (2D) 7 Pdf of particle displacement exact probability density function naive our closures x(t)-x(0)-u 0 t E() Θ test in the absence of scale separation 0 L

38 Application to drifter dispersion Surface current from 3 CODAR radars (April 23-30, 2012) TOSCA project Sampling time: 1/2 hour Spatial resolution: 500 m 15 drifters were released and their position recorded

39 Application to drifter dispersion Each drifter is thus followed for ~ 1 wee t= 1 wee t=0 In terms of 1 day windows (spaced by 1 hour): 24 x 6 = hour-long trajectories in total: 144 x 15 = hour-long trajectories

40 Application to drifter dispersion drifter initial positions two evolutions were computed: from radar dx(t) dt = U E (x(t),t)+σ(x(t),t)η(t)

41 Application to drifter dispersion drifter initial positions two evolutions were computed: from radar dx(t) dt = U E (x(t),t)+σ(x(t),t)η(t) our closure

42 Application to drifter dispersion Energy spectrum from radar E(ω) K41 ω Mean square particle dispersion was evaluated

43 Application to drifter dispersion our closure real traj. no closure x component our closure real traj. no closure time (hour) y component

44 Perspectives Toward dispersion of inertial particles 1 D αβ =lim t d d d (2 π) d t 0 t 0 dt 1 dt 2 e ı U [t 1 t 2 +τ (e t 2τ e t 1 τ )] e t t 1 τ τ ˇB αβ (, t 1 t 2 )(1 e t t 2 τ ) τ Stoes time ˇB αβ (, t ) Fourier transform of 2-point correlation function

45 Perspectives Toward dispersion of inertial particles 1 D αβ =lim t d d d (2 π) d t 0 t 0 dt 1 dt 2 e ı U [t 1 t 2 +τ (e t 2τ e t 1 τ )] e t t 1 τ τ ˇB αβ (, t 1 t 2 )(1 e t t 2 τ ) if U = U(X,T) E() Θ D αβ = D αβ (U, τ) U u 0 L

46 Perspectives Toward dispersion of inertial particles t Θ + U E Θ = i j [D ij (x,t; τ )Θ] E() Θ U 0 L u It now depends on inertia How does it wor???

47 Than you