Rough solutions of the Stochastic Navier-Stokes Equation

Transcription

1 variable Error in solutions of the Equation Björn Center for Complex and Nonlinear Science Department of Mathematics, UC Santa Barbara and University of Iceland, Reykjavík RICAM, Linz, December 2016 Co-authors, Gregory Bewley, Cornell, Michael Sinhuber, Göttingen, John Kaminsky and Shahab Karmini, UC Santa Barbara

2 Outline variable Error in variable 4 5 Error in 6 7

3 Outline variable Error in variable 4 5 Error in 6 7

4 Deterministic Equations variable Error in A general incompressible fluid flow satisfies the Equation u t + u u = ν u p u(x,0) = u 0 (x) with the incompressibility condition u = 0, Eliminating the pressure using the incompressibility condition gives u t + u u = ν u + 1 trace( u) 2 u(x,0) = u 0 (x) turbulence is quantified by the dimensionless Taylor-Reynolds number Re λ = Uλ ν

5 Reynolds Decomposition variable Error in velocity is written as U + u, pressure as P + p U describes the large scale flow, u describes the small scale turbulence This is the classical Reynolds decomposition (RANS) U t + U U = ν U P x j R ij last term the eddy viscosity, where R ij = u i u j is the Reynolds stress, describes how the small scale influence the large ones. Closure problem: compute R ij.

6 A Closure variable Error in Large scale flow Small scale flow U t + U U = ν U P x j R ij U(x,0) = U o (x). u t + u u = ν u + 1 trace( u) 2 + Noise u(x,0) = u 0 (x). What is the form of the Noise? It will contain both additive noise and multiplicative u noise.

7 with Turbulent Noise variable Error in Adding the two types of additive noise and the multiplicative noise we get the stochastic equations describing fully developed turbulence du = (ν u (U + u) u u U + 1 tr( u) 2 )dt 2 + c 1 k dbt k e k (x) + d k k 1/3 dt e k (x) k 0 k 0 + u( M k 0 u(x,0) = u 0 (x) R h k Nk (dt,dz)) (1) Each Fourier component e k = e 2πik x comes with its own Brownian motion bt k and deterministic bound k 1/3 dt

8 Outline variable Error in variable 4 5 Error in 6 7

9 Obukhov ory variable In 1941 Kolmogorov and Obukhov [7, 6, 9] proposed a statistical theory of turbulence structure functions of the velocity differences of a turbulent fluid, should scale with the distance (lag variable) l between them, to the power p/3 E( u(x,t) u(x + l,t) p ) = S p = C p l p/3 Error in A. Kolmogorov A. Obukhov

10 Obukhov Refinded Similarity with She- Intermittency Corrections variable Error in Obukhov 41 theory was criticized by Landau for including universal constants C p and later for not including the influence of the intermittency In 1962 Kolmogorov and Obukhov [8, 10] proposed a refined similarity hypothesis S p = C p < ε p/3 > l p/3 = C p l ζ p (2) l is the lag and ε a mean energy dissipation rate scaling exponents ζ p = p 3 + τ p include the She- intermittency corrections τ p = 2p 9 + 2(1 (2/3)p/3 ) and the C p are not universal but depend on the large flow structure

11 Solution of the Proof of Obukhov refined hypothesis variable Error in We solve (1) using the Feynmann-Kac formula, and Girsanov s orem solution is u = e Kt e t 0 Uds e t 0 dq M t u 0 + k 0 t 0 e K (t s) e (t s) 0 Udr e t s dq M t s (c 1/2 k dbs k + d k k 1/3 ds)e k (x) K is the operator K = ν + 1 tr( u ) M t is the Martingale M t = e { t 0 (U+u)(B s,s) db s 1 t0 2 (U+u)(B s,s) 2 ds} Using M t as an integrating factor eliminates the inertial terms from the equation (1)

12 Feynmann-Kac formula computation of the intermittency corrections variable Error in Feynmann-Kac formula gives the exponential of a sum of terms of the form t t t dq k = ln(1 + h k )N k (dt,dz) h k m k (dt,dz), s 0 R by a computation similar to the one that produces the geometric Lévy process [1, 2], m k the Lévy measure. form of the processes e t 0 R ln(1+h k )N k (dt,dz) t 0 R h k m k (dt,dz) = e Nk t lnβ+γln k = k γ β Nk t was found by She and [11], for h k = β 1 It was pointed out by She and Waymire [12] and by Dubrulle [5] that they are log-poisson processes. 0 R

13 KOSL of the Structure Functions, higher order Re λ 16,000 Comparison of ory and Experiments variable Error in Figure: exponents of the structure functions as a function of order, theory or Obukov-She- scaling (red), experiments (disks), dns simulations (circles), from [4], and experiments (X), from [11]. Kolomogorov-Obukhov 41 scaling is also shown as a blue line for comparion.

14 KOSL of the Structure Functions, low order Re λ 16,000 variable Error in Figure: exponents of the structure functions as a function of order ( 1, 2], theory or Obukov-She- scaling (red), experiments (disks), dns simulations (circles), from [4]. Obukov 41 scaling is also shown as a blue line for comparion.

15 Computation of the structure functions Can we do better than Obukhov? variable Error in scaling scaling of the structure functions is where S p C p x y ζ p, ζ p = p 3 + τ p = p 9 + 2(1 (2/3)p/3 ) p 3 being the Kolmogorov scaling and τ p the intermittency corrections. scaling of the structure functions is consistent with Kolmogorov s 4/5 law. Let S p denote structure function without the absolute value, then to leading order, were ε = de dt S 3 = 4 ε x y 5 is the energy dissipation

16 Can we compute the Reynolds number dependence of the structure functions? John Kaminsky in his Ph.D. thesis variable Error in S 1 (x,y,t) = 2 C k Z 3 \{0} d k (1 e λ k t ) k ζ 1 + 4π 2 ν C k ζ sin(πk (x y)). We get a stationary state as t, and for x y small, S 1 (x,y,t) 2πζ 1 C k Z 3 \{0} d k 1 + 4π2 ν C k 4 3 where ζ 1 = 1/3 + τ Similarly, S 2 (x,y,t) = 4 { C 2 C c k(1 e 2λ k t ) 2 + k Z 3 k ζ 2 + 4π 2 ν C k ζ d 2 k (1 e λ k t ) k ζ 2 + 8π 2 ν C k ζ π4 ν 2 k ζ 2+ 8 C 2 3 where ζ 2 = 2/3 + τ x y ζ 1. } sin 2 (πk (x y)),

17 Higher order structure functions third and pth structure functions are: variable Error in S 3 (x,y,t) = 8 C 3 [{ C 2 c k d k (1 e 2λ k t )(1 e λ k t ) k Z 3 k ζ 3 + 8π 2 ν C k ζ π4 ν 2 k ζ 3+ 8 C 2 3 d k 3 (1 e λ k t ) 3 }] k ζ π 2 ν C k ζ π4 ν 2 C 2 k ζ π6 ν 3 C 3 k ζ 3+4 sin 3 (πk (x y)). S p (x,y,t) = 2p C p A p sin p [πk (x y)], k 0 A p = 2 p 2 Γ( p+1 2 )σp k 1F 1 ( 1 2 p, 1 2, 1 2 ( M k σ k ) 2 ), k ζ p + p k π 2 ν C k ζ p O(ν 2 ) and M k = d k (1 e λ k t ), and σ k = ( C 2 c k(1 e 2λ k t )). +

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19 wind generating homogeneous turbulence Comparison of the new theory and experiments variable Error in comes from the Max Planck Institute for Dynamical and Self-Orgranization, in Göttingen, Germany (E. Bodenschatz). It was generated by the variable density turbulence (VDTT). pressurized gases circulate in the VDTT in an upright, closed loop. At the upstream end of two test sections, the free stream is disturbed mechanically. in the current paper is generated by a fixed grid, but the gas stream can also be disturbed by an active grid resulting in even higher Reynolds number turbulence. In the wake of the grid the resulting turbulence evolves down the length of the without the middle region being substantially influenced by the walls of the.

20 Variable Density Turbulent Tunnel (VDTT) variable Error in test sections are about 8 meters long so the turbulence to evolve through at least one eddy turnover time, around 1 second. This means that the turbulence can be observed over the time that it takes the energy to cascade all the way from the large eddies to the dissipate scale, see [3]. Measurements were taken from Taylor Reynolds Numbers 110, 264, 508, 1000, and One might think that the system length is the square root of the cross sectional value of the A, but the relevant system length is the grid size D of the grid.

21 Outline variable Error in variable 4 5 Error in 6 7

22 variable Error in We have to fit the system size and bring the largest measurements into the range of the structure functions, r/η, where η is the Kolmogorov dissipative scale. largest eddies may be influenced by the system size and need to be modeled. large eddies should scale c k b 1 and d k a 1 for k small. small eddies should scale with k, c k k m and d k k m, for k large. constants C and D should measure the norm of u and the system length, for different Reynolds numbers.

23 Results of fits variable Taylor Reynolds Number a b D Table: fitted values for a, b, and D and C below. Error in Taylor Reynolds Number Second Third Fourth Sixth Eighth

24 Structure functions for Taylor-Reynolds number 110 variable Figure: Second Structure Function, Normal Scale and log-log scale, T-R 110 Error in Figure: Third and Fourth Structure function, log-log scale, T-R 110

25 Structure functions for Taylor-Reynolds number 110 and 1450 Figure: Sixth and Eighth Structure Function, T-R 110 variable Error in Figure: Second and Third Structure Function, T-R 1450

26 Structure functions for Taylor-Reynolds number 1450 variable Figure: Fourth and Sixth Structure Function, log-log scale, T-R 1450 Error in Figure: Eight Structure function, log-log scale, T-R 1450

27 Onsager s Observation variable Error in velocity u, lies in Sobolev space H s, where s = 11 6 when intermittency is not taken into account and s = when it is. This, in turn, implies that u lies in Sobolev space H s, where s = without intermittency and s = 18 with intermittency, now H s L p. This follows, by the Sobolev inequality, provided that or u p C u s, p. This is true for p = 2, p = 3, and p = 4, but does not hold for p = 6 and p = 8.

28 Divergence of the Sixth and Eight Structure Functions tell us how rough the fluid velocity is variable Error in Taylor Reynolds Number Second Third Fourth Sixth Eighth Table: fitted values for m, uncorrected structure functions. low value of m is due to the divergence of the sine series for the Sixth and the Eight Structure Functions. Fourth structure function sine series diverges with intermittency present.

29 Outline variable Error in variable 4 5 Error in 6 7

30 Do the Structure Functions with the Taylor-Reynolds number give a better fit? Figure: Second and Third Structure Function Error, with T-R 110 variable Error in Figure: Fourth and Sixth Structure Function Error, T-R 110

31 Structure Function error for Taylor-Reynolds number 1450 Figure: Second and Third Structure Function, log-log scale variable Error in Figure: Fourth and Sixth Structure function error, log-log scale

32 Outline variable Error in variable 4 5 Error in 6 7

33 Sobolev Function Spaces Shahab Karimi in his Ph.D. thesis variable Error in Let ū = Tn u dx and n = 2, or 3. We work with spaces of periodic functions Define Ḣ per = {u(x, ) u L 2 per (T n ), ū = 0, u = 0} V per = {u(x, ) u H 1 per (T n ), ū = 0, u = 0} V s = D(A s/2 ) = {u = c k e k k 2s c k 2 < }, k Z n 0 where A = P is the Stokes operator. H s -norm s on V s is equivalent to Vs. We have V 0 = Ḣper and V 1 = V per.

34 Equation variable Error in small scale stochastic equation (SNS) (1) for incompressible fluid on T n, n = 2,3, is du = (ν u (u. )u + p)dt + Ldt + dw (t) u = 0, ū = 0 u(x,0,ω) = u 0 (x,ω) u 0 is a random variable in L p (Ω;V α ), 1 p <. deterministic term L is the large deviation from mean noise W (t) above L = η k d k e ik x, k Z n 0 dw (t) = c 1/2 k db k t e ik x. k Z n 0

35 Integral Equation variable Error in A mild solution of the SNS equation (1), in the space V α, is a pair (u,τ), where τ is a strictly positive stopping time and u(. τ) L p (Ω;C([0,τ];V α )) is an F u 0 t -adapted process such that: t u(t) = S(t)u 0 S(t s)b(u s )ds + f (t) + W A (t), 0 a.s. for all t [0,τ], Bu = Pu u, S(t) = e At, where t t f (t) = S(t s)lds, W A (t) = S(t s)dw (s). 0 0 A mild solution (u,τ) in V α is unique if for any other mild solution (u,τ ), u(t τ τ ) = u (t τ τ ) almost surely.

36 Maximal Mild local solutions variable Error in Definition A local mild solution (u,τ) in V α is maximal provided that: i) if (u,τ ) is a local mild solution in V α then τ τ a.s., ii) re exists a sequence {τ n } n of stopping times such that τ n τ and for all n N, (u,τ n ) is a local mild solution in V α. If τ = almost surely, then the solution is global. orem Suppose 1 α < 3, 1 p <, f C([0,T 0 ];V α ) and W A (.) L p (Ω ; C([0,T 0 ];V α )) for some fixed T 0 > 0, and u 0 L p (Ω;V α ). n SNS has a unique mild solution in V α.

37 Maximal and Global variable Error in orem ( Maximal Mild Solution) Given the assumptions of above orem, there exists a unique (up to null sets) maximal mild solution of SNS (1) in V α. orem τ sup u(t) α 1 + u(t) 2 α dt < 0 t<τ 0 (Global Existence) Let α 2, u 0 L p (Ω;V α ), d k e k V α 2, W A T >0 L p (Ω;C([0,T ];V α )), 1 < α < 3, n = 2. n SNS (1) has a unique global mild solution in V α.

38 Outline variable Error in variable 4 5 Error in 6 7

39 variable Error in stochastic closure theory for reproduces the statistical theory of K-O with the intermittency corrections of She-. We computed the dependence of the structure functions of turbulence on the Taylor-Reynolds number. Comparisons with from the VDTT are excellent. classical Prandtl wind experiment is finally explained. After 100 years! Very surprisingly the also determines the smoothness of the fluid velocity u. Error analysis favors the Taylor-Reynolds number corrections and confirms the roughness of solutions: α = 4/3 and α = 2, n = 2, α = 29/18, n = 3. unique global (rough) solutions in V α is proven, n = 2, and unique local (rough) solutions, n = 3.

40 Computation of the Eddy Viscosity (LES is similar) variable Error in With the stochastic closure, we can now compute the eddy viscosity R ij = u i u j, using the same method we used to compute the structure functions uu j = 2 t0 x j C e ( u+ u T )ds 2π[(k c 1/2 k )c 1/2 k + (2/C)(k d k )d k ] k>0 k ζ 2 K u (1 ζ 2)/2 e t 0 ( u+ u T )ds (1 ζ 2)/4 u S = 1 2 ( u + ut ) is the rate of strain tensor first (multiplicative) term is an exponential (dynamic) Smagorinsky term e 2 k

41 variable Error in B.. Obukhov statistical theory of turbulence. J. Nonlinear Sci., DOI /s z. B.. Obukhov ory of. Springer, New York, DOI / Eberhard Bodenschatz, Gregory P Bewley, Holger Nobach, Michael Sinhuber, and Haitao Xu. Variable density turbulence facility. Review of Scientific Instruments, 85(9):093908, S. Y. Chen, B. Dhruva, S. Kurien, K. R. Sreenivasan, and M. A. Taylor.

42 variable Error in Anomalous scaling of low-order structure functions of turbulent velocity. Journ. of Fluid Mech., 533: , B. Dubrulle. Intermittency in fully developed turbulence: in log-poisson statistics and generalized scale covariance. Phys. Rev. Letters, 73(7): , A. N. Kolmogorov. Dissipation of energy under locally istotropic turbulence. Dokl. Akad. Nauk SSSR, 32:16 18, A. N. Kolmogorov. local structure of turbulence in incompressible viscous fluid for very large reynolds number. Dokl. Akad. Nauk SSSR, 30:9 13, 1941.

43 variable Error in A. N. Kolmogorov. A refinement of previous hypotheses concerning the local structure of turbulence in a viscous incompressible fluid at high reynolds number. J. Fluid Mech., 13:82 85, A. M. Obukhov. On the distribution of energy in the spectrum of turbulent flow. Dokl. Akad. Nauk SSSR, 32:19, A. M. Obukhov. Some specific features of atmospheric turbulence. J. Fluid Mech., 13:77 81, Z-S She and E.. Universal scaling laws in fully developed turbulence. Phys. Rev. Letters, 72(3): , 1994.

44 Z-S She and E. Waymire. Quantized energy cascade and log-poisson statistics in fully developed turbulence. Phys. Rev. Letters, 74(2): , variable Error in