Averaging vs Chaos in Turbulent Transport?

Transcription

1 Averaging vs Chaos in Turbulent Transport? Houman Owhadi CALTECH, Applied and Computational Mathematics, Control and Dynamical Systems. Averaging vs Chaos in Turbulent Transport? p. 1/6

2 The Model PDE in R d tt + v. T = κ T κ > 0. v (C(R d )) d, div(v) = 0. Averaging vs Chaos in Turbulent Transport? p. 2/6

3 The Model PDE in R d tt + v. T = κ T κ > 0. v (C(R d )) d, div(v) = 0. The stream Matrix v = div(γ) Γ (C 1 (R d )) d d, Γ i,j = Γ j,i. v i = d j=1 jγ i,j. Averaging vs Chaos in Turbulent Transport? p. 2/6

4 Motivations: turbulent flows M. Rutgers Averaging vs Chaos in Turbulent Transport? p. 3/6

5 The multi scale decomposition Γ := n=0 γ n E n ( x ρ n ). Averaging vs Chaos in Turbulent Transport? p. 4/6

6 The multi scale decomposition Γ := n=0 γ n E n ( x ρ n ). Spatial scale ρ R +, ; 2 ρ < Averaging vs Chaos in Turbulent Transport? p. 4/6

7 The multi scale decomposition Γ := n=0 γ n E n ( x ρ n ). Eddies geometrical parameters T d := Torus of Dimension d and side 1. n, E n (C 1 (T d )) d d ; E n;i,j = E n;j,i ; E n (0) = 0. sup n N sup m,i,j {1,...,d} m E n;i,j 1 Averaging vs Chaos in Turbulent Transport? p. 4/6

8 The multi scale decomposition Γ := n=0 γ n E n ( x ρ n ). Circulation rates γ R +, ; 1 < γ < ρ Averaging vs Chaos in Turbulent Transport? p. 4/6

9 The Spectrum is not Kolmogorov v(l) velocity of the eddies of size l E(k) The kinetic energy distribution in the Fourier modes Kolmogorov Our Model v(l) l 1 3 E(k) k 5 3 v(l) l ln γ ln ρ 1 Kolmogorov γ = ρ 4 3 E(k) k 1 2ln γ ln ρ Our model γ < ρ Averaging vs Chaos in Turbulent Transport? p. 5/6

10 ρ n One scale. Stream lines of the flow γ n E n ( x ρ n ) Averaging vs Chaos in Turbulent Transport? p. 6/6

11 ρ n+1 ρ n Two scales. Stream lines of γ n E n ( x ρ n ) and γ n+1 E n+1 ( x ρ n+1 ). Averaging vs Chaos in Turbulent Transport? p. 7/6

12 ρ n+1 ρ n Two scales. sum Stream lines of γ n E n ( x ρ n ) + γ n+1 E n+1 ( x ρ n+1 ). Averaging vs Chaos in Turbulent Transport? p. 8/6

13 ρ n+2 ρ n+1 ρ n Three scales. sum γ n E n ( x ρ n ) + γ n+1 E n+1 ( x ρ n+1 ) + γ n+2 E n+2 ( x ρ n+2 ) Averaging vs Chaos in Turbulent Transport? p. 9/6

14 ρ n+2 ρ n+1 ρ n sum γ n E n ( x ρ n ) + γ n+1 E n+1 ( x ρ n+1 ) + γ n+2 E n+2 ( x ρ n+2 ) Modification of the ratio ρ Averaging vs Chaos in Turbulent Transport? p. 10/6

15 ρ n+2 ρ n+1 ρ n sum γ n E n ( x ρ n ) + γ n+1 E n+1 ( x ρ n+1 ) + γ n+2 E n+2 ( x ρ n+2 ) Modification of the ratio ρ Averaging vs Chaos in Turbulent Transport? p. 11/6

16 The circulation rates γ k γ n ρ n sum γ n E n ( x ρ n ) + γ n+1 E n+1 ( x ρ n+1 ) + γ n+2 E n+2 ( x ρ n+2 ) Averaging vs Chaos in Turbulent Transport? p. 12/6

17 The circulation rates γ k γ n ρ n sum γ n E n ( x ρ n ) + γ n+1 E n+1 ( x ρ n+1 ) + γ n+2 E n+2 ( x ρ n+2 ) Averaging vs Chaos in Turbulent Transport? p. 13/6

18 E k : Geometry of the eddies sum γ n E n ( x ρ n ) + γ n+1 E n+1 ( x ρ n+1 ) + γ n+2 E n+2 ( x ρ n+2 ) Averaging vs Chaos in Turbulent Transport? p. 14/6

19 E k : Geometry of the eddies sum γ n E n ( x ρ n ) + γ n+1 E n+1 ( x ρ n+1 ) + γ n+2 E n+2 ( x ρ n+2 ) Averaging vs Chaos in Turbulent Transport? p. 15/6

20 A simple example of the multi-scale flow d = 2, for all n, E n (x 1, x 2 ) = ( ) 0 h(x 1, x 2 ) h(x 1, x 2 ) 0 with h(x 1, x 2 ) := cos(2πx 1 ) sin(2πx 2 ) Averaging vs Chaos in Turbulent Transport? p. 16/6

21 h 2 0(x, y) = 2 k=0 γk h( x ρ k, y ρ k ) with ρ = 2.9, γ = 1.25 Averaging vs Chaos in Turbulent Transport? p. 17/6

22 Another example 2 h 2 0(x, y) = with ρ = 3, γ = 1.1 and k=0 γ k h( x ρ k, y ρ k) h(x, y) =2 sin(2πx + 3 cos(2πy 3 sin(2πx + 1))) sin(2πy + 3 cos(2πx 3 sin(2πy + 1))) Averaging vs Chaos in Turbulent Transport? p. 18/6

23 Another example Averaging vs Chaos in Turbulent Transport? p. 18/6

24 Motivations: turbulent flows Averaging vs Chaos in Turbulent Transport? p. 19/6

25 Two opposed descriptions? The transport is Superdiffusive, Highly mixing, self-averaging (Kolmogorov 41, Richardson 26, Obukhov 41) Averaging vs Chaos in Turbulent Transport? p. 20/6

26 Two opposed descriptions? High density gradients, coherent patterns, sensitive to the geometry of the flow, (Poincaré 08, Landau-Lifshitz 42-85, Ruelle-Takens 71) Averaging vs Chaos in Turbulent Transport? p. 20/6

27 Questions for our model The transport is Superdiffusive or not? Sensitive to the particular geometry of the flow or not? Averaging vs Chaos in Turbulent Transport? p. 21/6

28 Our results: outline We can define a parameter λ R + from the characteristics ( γ, (E n ) n N ) of our model, inverse of a local Peclet (Reynolds) number if λ > 0 superdiffusive + highly mixing + self-averaging Averaging vs Chaos in Turbulent Transport? p. 22/6

29 Our results: outline We can define a parameter λ R + from the characteristics ( γ, (E n ) n N ) of our model, inverse of a local Peclet (Reynolds) number if λ > 0 superdiffusive + highly mixing + self-averaging if λ = 0 self-averaging collapses, highly sensitive, high gradients Averaging vs Chaos in Turbulent Transport? p. 22/6

30 Results SDE { dy t = 2κdω t + v(y t )dt y 0 = x ω standard BM in R d. Averaging vs Chaos in Turbulent Transport? p. 23/6

31 Results SDE { dy t = 2κdω t + v(y t )dt y 0 = x ω standard BM in R d. Exit Time τ(r) := inf{t > 0 : y t r} Averaging vs Chaos in Turbulent Transport? p. 23/6

32 Initial Distribution m r (dx) := dx B(0,r) dx1 B(0,r) Averaging vs Chaos in Turbulent Transport? p. 24/6

33 Initial Distribution m r (dx) := dx B(0,r) dx1 B(0,r) Mean Exit Time E mr [ τ(r) ] = 1 Vol ( B(0, r) ) B(0,r) E x [ τ(r) ] dx Averaging vs Chaos in Turbulent Transport? p. 24/6

34 One Point Fast Motion Theorem If λ > 0 then C(d, 1/λ, 1/ lnγ) < such that for ρ > Cγ one has lim sup r 1 ( ln r ln [ ] ) E mr τ(r) < 2 Averaging vs Chaos in Turbulent Transport? p. 25/6

35 One Point Fast Motion Theorem For λ > 0, ρ > Cγ and r > ρ E mr [ τ(r) ] = r 2 ν(r) ν(r) = ln γ ( ) 1 + ǫ(r) ln ρ Averaging vs Chaos in Turbulent Transport? p. 25/6

36 One Point Fast Motion Theorem For λ > 0, ρ > Cγ and r > ρ E mr [ τ(r) ] = r 2 ν(r) ν(r) = ln γ ( ) 1 + ǫ(r) ln ρ ǫ(r) 0.5C γ ρ < 0.5 Averaging vs Chaos in Turbulent Transport? p. 25/6

37 One Point Fast Motion Theorem For λ > 0, ρ > Cγ and r > ρ E mr [ τ(r) ] = r 2 ν(r) ν(r) = lnγ lnρ ( 1 + ǫ(r) ) 0 < lnγ lnρ Averaging vs Chaos in Turbulent Transport? p. 25/6

38 One Point Fast Motion Theorem For λ > 0, ρ > Cγ and r > ρ E mr [ τ(r) ] = r 2 ν(r) ν(r) = ln γ ( ) 1 + ǫ(r) ln ρ Shear flow models: Avellaneda-Majda (91), Glimm-Zhang (92), Gaudron (00), Komorowski-Fannjiang (01), Ben Arous-Owhadi (01). Non shear flow, Kraichnan, Gaussian, annealed, one particle: Piterbarg 97, Komorowski-Olla 02, Fannjiang 02 Averaging vs Chaos in Turbulent Transport? p. 25/6

39 Fast Transport as an almost sure event Fast transport event H(r) := { τ(r) r 2 δ} with δ = 0.9 ln γ ln ρ Averaging vs Chaos in Turbulent Transport? p. 26/6

40 Fast Transport as an almost sure event Fast transport event with Observe that δ > 0 H(r) := { τ(r) r 2 δ} δ = 0.9 ln γ ln ρ Averaging vs Chaos in Turbulent Transport? p. 26/6

41 Fast Transport as an almost sure event Fast transport event with H(r) := { τ(r) r 2 δ} δ = 0.9 ln γ ln ρ Theorem If λ > 0 then for ρ > Cγ one has lim r P m r [ H(r) ] = 1 Averaging vs Chaos in Turbulent Transport? p. 26/6

42 Super-Diffusive two-points motion z t second passive tracer dz t = 2κd ω t + v(z t )dt. ω t standard BM independent of ω t. B(0, r, l) := { (y, z) R d R d : y z < r and y 2 + z 2 < l 2} τ(r, l) := inf{t > 0 : (y t, z t ) / B(0, r, l)} Averaging vs Chaos in Turbulent Transport? p. 27/6

43 Super-Diffusive two-points motion m r,l (dy dz) := dy dz (y,z) B(0,r,l) dy dz1 B(0,r,l). Theorem If λ > 0 then for ρ > Cγ one has lim sup r lim l 1 ( ln r ln [ ] ) E mr,l τ(r, l) < 2 lim E [ ] m r,l τ(r, l) = r 2 ν(r) l Averaging vs Chaos in Turbulent Transport? p. 27/6

44 Super-Diffusive two-points motion m r,l (dy dz) := dy dz (y,z) B(0,r,l) dy dz1 B(0,r,l). Theorem If λ > 0 then for ρ > Cγ one has lim lim P { } m r r,l τ(r, l) r 2 δ = 1 l Averaging vs Chaos in Turbulent Transport? p. 27/6

45 Strong Self Averaging property of the flow If λ > 0 The particular geometry of the eddies E n has no influence on the transport Averaging vs Chaos in Turbulent Transport? p. 28/6

46 Strong Self Averaging property of the flow If λ > 0 The particular geometry of the eddies E n has no influence on the transport The transport depends only on the power law of the velocity field. lnγ lnρ Averaging vs Chaos in Turbulent Transport? p. 28/6

47 What is λ? Renormalization procedure. Averaging vs Chaos in Turbulent Transport? p. 29/6

48 A reminder on homogenization M := {positive definite symmetric constant d d matrices} Averaging vs Chaos in Turbulent Transport? p. 30/6

49 A reminder on homogenization M := {positive definite symmetric constant d d matrices} S(T d ) := {skew symmetric d d matrices with coefficients in L (T d )}. Averaging vs Chaos in Turbulent Transport? p. 30/6

50 A reminder on homogenization M := {positive definite symmetric constant d d matrices} S(T d ) := {skew symmetric d d matrices with coefficients in L (T d )}. Let (a, E) M S(T d ) Averaging vs Chaos in Turbulent Transport? p. 30/6

51 A reminder on homogenization Operator { div (a + E( x ǫ )) u ǫ(x) = f(x) u ǫ = 0 in Ω in Ω Averaging vs Chaos in Turbulent Transport? p. 31/6

52 A reminder on homogenization Operator { div (a + E( x ǫ )) u ǫ(x) = f(x) u ǫ = 0 in Ω in Ω then σ(a, E) M such that as ǫ 0, u ǫ u 0 Averaging vs Chaos in Turbulent Transport? p. 31/6

53 A reminder on homogenization Operator { div (a + E( x ǫ )) u ǫ(x) = f(x) u ǫ = 0 in Ω in Ω then σ(a, E) M such that as ǫ 0, u ǫ u 0 { div σ(a, E) u 0 = f u 0 = 0 in Ω in Ω Averaging vs Chaos in Turbulent Transport? p. 31/6

54 A reminder on homogenization SDE dx t = a 1 2.dωt + div E(x t )dt, Averaging vs Chaos in Turbulent Transport? p. 32/6

55 A reminder on homogenization SDE As ǫ 0, dx t = a 1 2.dωt + div E(x t )dt, ǫx t/ǫ 2 B t B t Brownian Motion with covariance matrix 2σ(a, E). Averaging vs Chaos in Turbulent Transport? p. 32/6

56 A reminder on homogenization Literature. 2σ(a, E): Effective diffusivity σ(a, E): Effective conductivity - Eddy viscosity. Dispersion matrix. Averaging vs Chaos in Turbulent Transport? p. 33/6

57 A reminder on homogenization Effective conductivity as a mapping σ : M S(T d ) M (a, E) σ(a, E) Averaging vs Chaos in Turbulent Transport? p. 34/6

58 A reminder on homogenization Effective conductivity as a mapping σ : M S(T d ) M (a, E) σ(a, E) for all (a, E) M S(T d ) a σ(a, E) a + t E(x)a 1 E(x)dx T d Averaging vs Chaos in Turbulent Transport? p. 34/6

59 What is λ? Renormalization sequence (A n ) n N For all n N, A n M A 0 = κ γ I d and A n+1 = 1 γ σ(a n, E n ) Averaging vs Chaos in Turbulent Transport? p. 35/6

60 What is λ? Renormalization sequence (A n ) n N For all n N, A n M A 0 = κ γ I d and A n+1 = 1 γ σ(a n, E n ) (A n ) n N does not depend on ρ Averaging vs Chaos in Turbulent Transport? p. 35/6

61 What is λ? Renormalization sequence (A n ) n N For all n N, A n M A 0 = κ γ I d and A n+1 = 1 γ σ(a n, E n ) (A n ) n N does not depend on ρ λ := lim inf n λ min(a n ) Averaging vs Chaos in Turbulent Transport? p. 35/6

62 Interpretation of A n Γ n 1 (x) := n 1 k=0 γ k E k ( x ρ k) Averaging vs Chaos in Turbulent Transport? p. 36/6

63 Interpretation of A n Γ n 1 (x) := n 1 k=0 γ k E k ( x ρ k) Assume ρ N. Then Γ n 1 periodic. div(κi d + Γ n 1 ) Homogen div σ(κi d, Γ n 1 ) Averaging vs Chaos in Turbulent Transport? p. 36/6

64 Interpretation of A n Γ n 1 (x) := n 1 k=0 γ k E k ( x ρ k) Assume ρ N. Then Γ n 1 periodic. div(κi d + Γ n 1 ) Homogen div σ(κi d, Γ n 1 ) lim ρ σ(κi d, Γ n 1 ) = γ n A n Averaging vs Chaos in Turbulent Transport? p. 36/6

65 Interpretation of A n Magnitude of the velocity vector field at scale n: V n γn ρ n Averaging vs Chaos in Turbulent Transport? p. 36/6

66 Interpretation of A n Magnitude of the velocity vector field at scale n: Local Peclet tensor V n γn ρ n Pe n := V n ρ n( lim ρ σ(κi d, Γ n 1 ) ) 1 Averaging vs Chaos in Turbulent Transport? p. 36/6

67 Interpretation of A n Magnitude of the velocity vector field at scale n: Local Reynolds tensor V n γn ρ n Re n := V n ρ n( lim ρ σ(κi d, Γ n 1 ) ) 1 Averaging vs Chaos in Turbulent Transport? p. 36/6

68 Interpretation of A n Magnitude of the velocity vector field at scale n: Local Reynolds tensor V n γn ρ n Re n := V n ρ n( lim ρ σ(κi d, Γ n 1 ) ) 1 A n = (Pe n ) 1 = (Re n ) 1 Averaging vs Chaos in Turbulent Transport? p. 36/6

69 When is λ > 0? λ + := lim sup n λ max (A n ) µ := lim sup n λ max (A n ) λ min (A n ) Averaging vs Chaos in Turbulent Transport? p. 37/6

70 When is λ > 0? λ + := lim sup n λ max (A n ) Theorem µ := lim sup n λ max (A n ) λ min (A n ) λ + C d λ and µ C d (λ ) 2 Averaging vs Chaos in Turbulent Transport? p. 37/6

71 When is λ > 0? λ + := lim sup n λ max (A n ) Theorem µ := lim sup n λ max (A n ) λ min (A n ) λ + C d λ and µ C d (λ ) 2 λ > 0 λ + < and µ < Averaging vs Chaos in Turbulent Transport? p. 37/6

72 When is λ > 0? for ζ > 0 V (ζ) := lim inf n λ min ( σ(ζid, E n ) ) ζ Averaging vs Chaos in Turbulent Transport? p. 38/6

73 When is λ > 0? for ζ > 0 V (ζ) := lim inf n λ min ( σ(ζid, E n ) ) ζ V is decreasing and V 1 thus is well defined. V (0) := lim ζ 0 V (ζ) Averaging vs Chaos in Turbulent Transport? p. 38/6

74 When is λ > 0? for ζ > 0 V (ζ) := lim inf n λ min ( σ(ζid, E n ) ) ζ Theorem If µ < and γ < V (0) then λ > 0 and C 1 λ λ + C 2. Averaging vs Chaos in Turbulent Transport? p. 38/6

75 Idea of the Proof. Ball B(0, r) r Averaging vs Chaos in Turbulent Transport? p. 39/6

76 Idea of the Proof. Ball B(0, r) r We want to compute E[τ(0, r)] Averaging vs Chaos in Turbulent Transport? p. 39/6

77 Idea of the Proof. Ball B(0, r) r We want to compute E[τ(0, r)] Main feature of the flow: Infinite number of scales 0, 1,..., Averaging vs Chaos in Turbulent Transport? p. 39/6

78 Separation between scales. ρ n(r) Scale n(r) = [lnr/ ln ρ] ρ n(r) r < ρ n(r)+1 r ρ n(r)+1 Averaging vs Chaos in Turbulent Transport? p. 40/6

79 Separation between scales. Scale n(r) = [lnr/ ln ρ] ρ n(r) r ρ n(r)+1 ρ n(r) r < ρ n(r)+1 Small scales 0,..., n(r) 1 Averaging vs Chaos in Turbulent Transport? p. 40/6

80 Separation between scales. Scale n(r) = [lnr/ ln ρ] ρ n(r) r ρ n(r)+1 ρ n(r) r < ρ n(r)+1 Small scales 0,..., n(r) 1 Intermediate scales n(r), n(r) + 1 Averaging vs Chaos in Turbulent Transport? p. 40/6

81 Separation between scales. Scale n(r) = [lnr/ ln ρ] ρ n(r) r ρ n(r)+1 ρ n(r) r < ρ n(r)+1 Small scales 0,..., n(r) 1 Intermediate scales n(r), n(r) + 1 Large scales n(r) + 2,..., Averaging vs Chaos in Turbulent Transport? p. 40/6

82 All Scales. Medium with infinite number of scales 0, 1,...,. Averaging vs Chaos in Turbulent Transport? p. 41/6

83 All Scales. Medium with infinite number of scales 0, 1,...,. Transport of a drop of dye? Averaging vs Chaos in Turbulent Transport? p. 41/6

84 Transport by Large Scales Large scales Averaging vs Chaos in Turbulent Transport? p. 42/6

85 Transport by Large Scales Large scales Their influence on the transport of the drop of dye is Averaging vs Chaos in Turbulent Transport? p. 42/6

86 Transport by Large Scales Large scales Their influence on the transport of the drop of dye is negligible Averaging vs Chaos in Turbulent Transport? p. 42/6

87 Transport by Small Scales Small Scales homogenized. Averaging vs Chaos in Turbulent Transport? p. 43/6

88 Transport by Small Scales Transport = Diffusion with Effective diffusivity σ(κi d, Γ n(r) ) γ n(r) A n(r) Exit Time: τ D (0, r) r 2 γ n(r) λ(a n(r) ) Averaging vs Chaos in Turbulent Transport? p. 43/6

89 Transport by Small Scales Transport by mixing, density gradients smoothed. Averaging vs Chaos in Turbulent Transport? p. 43/6

90 Transport by Intermediate Scales Intermediate Scales: not homogenized, not negligible. Averaging vs Chaos in Turbulent Transport? p. 44/6

91 Transport by Intermediate Scales Transport by convection through particular geometry. Exit time τ C (0, r) r V n(r) r2 γ n(r) Averaging vs Chaos in Turbulent Transport? p. 44/6

92 Transport by Intermediate Scales Transport by advection, density gradients increased. Averaging vs Chaos in Turbulent Transport? p. 44/6

93 Local Peclet number If λ > 0 Pe(r) := τ C(r) τ D (r) ( λ(a n(r) ) ) 1 Averaging vs Chaos in Turbulent Transport? p. 45/6

94 Local Peclet number If λ > 0 Pe(r) := τ C(r) τ D (r) ( λ(a n(r) ) ) 1 λ > 0 Pe(r) < At every scale r, advection (irregularities, high gradients) is compensated by averaging (smoothing, dissipating). Averaging vs Chaos in Turbulent Transport? p. 45/6

95 If λ > 0 Local Peclet number Pe(r) := τ C(r) τ D (r) ( λ(a n(r) ) ) 1 λ > 0 Pe(r) < influence of the intermediate scales on the transport comparable to the influence of the small scales. τ(0, r) τ D (0, r) r2 γ n(r) Averaging vs Chaos in Turbulent Transport? p. 45/6

96 Local Peclet number n(r) lnr lnρ. If λ > 0 Pe(r) := τ C(r) τ D (r) ( λ(a n(r) ) ) 1 τ(0, r) r2 r2 ν γn(r) with ν = lnγ lnρ > 0. Averaging vs Chaos in Turbulent Transport? p. 45/6

97 If λ = 0 Averaging vs Chaos in Turbulent Transport? p. 46/6

98 If λ = 0 As r Pe(r) Averaging vs Chaos in Turbulent Transport? p. 46/6

99 As r If λ = 0 Pe(r) At every scale transport dominated by advection. Averaging vs Chaos in Turbulent Transport? p. 46/6

100 As r If λ = 0 Pe(r) At every scale transport dominated by advection. Collapse of the self-averaging property of the flow towards chaos. Averaging vs Chaos in Turbulent Transport? p. 46/6

101 As r If λ = 0 Pe(r) At every scale transport dominated by advection. Collapse of the self-averaging property of the flow towards chaos. The particular geometry of the eddies can not be neglected even if ρ is large. Averaging vs Chaos in Turbulent Transport? p. 46/6

102 Self-Similar Case Definition A n is self-similar and isotropic iff n N, E n = E ;. Averaging vs Chaos in Turbulent Transport? p. 47/6

103 Self-Similar Case Definition A n is self-similar and isotropic iff n N, E n = E ; ζ > 0, σ(ζi d, E) = λ(ζ)i d. Averaging vs Chaos in Turbulent Transport? p. 47/6

104 Self-Similar Case Definition A n is self-similar and isotropic iff n N, E n = E ; ζ > 0, σ(ζi d, E) = λ(ζ)i d. If A n is self-similar then it is a low order dynamical system. A n+1 = 1 γ σ(a n, E) Averaging vs Chaos in Turbulent Transport? p. 47/6

105 Self-Similar Case Definition A n is self-similar and isotropic iff n N, E n = E ; ζ > 0, σ(ζi d, E) = λ(ζ)i d. ( λ min σ(ζid, E) ) V (0) := lim ζ 0 ζ Averaging vs Chaos in Turbulent Transport? p. 47/6

106 Self-Similar Case Theorem If A n is self-similar and isotropic then If γ < V (0) then λ > 0 and lim n A n = ζ 0 I d where ζ 0 is the unique solution of V (ζ 0 ) = γ. Averaging vs Chaos in Turbulent Transport? p. 48/6

107 Self-Similar Case Theorem If A n is self-similar and isotropic then If γ < V (0) then λ > 0 and lim n A n = ζ 0 I d where ζ 0 is the unique solution of V (ζ 0 ) = γ. If γ = V (0) and (V (0) V (x))x p admits a non ero limit as x 0 with p > 0 then λ = 0 and lim n lnλ(a n ) lnn = 1 p. Averaging vs Chaos in Turbulent Transport? p. 48/6

108 Self-Similar Case Theorem If A n is self-similar and isotropic then If γ < V (0) then λ > 0 and lim n A n = ζ 0 I d where ζ 0 is the unique solution of V (ζ 0 ) = γ. If γ = V (0) and (V (0) V (x))x p admits a non ero limit as x 0 with p > 0 then λ = 0 and lim n lnλ(a n ) lnn = 1 p. If γ > V (0) then λ = 0 and 1 lim n n ln λ(a n) = ln ( V (0)) γ Averaging vs Chaos in Turbulent Transport? p. 48/6

109 Bifurcation Flow self similar and isotropic. for all n, E n = E. Shape of the eddy E over a period. 1 < V (0) <. Averaging vs Chaos in Turbulent Transport? p. 49/6

110 Bifurcation γ < V (0) λ > 0 The flow is self averaging Averaging vs Chaos in Turbulent Transport? p. 49/6

111 Bifurcation γ V (0) λ = 0 The self averaging property collapses. Averaging vs Chaos in Turbulent Transport? p. 49/6

112 λ = 0, proof of the collapse Assume A n to be self-similar and isotropic. Spatial scales ρ n R n Γ(x) = n=0 γ n E( x R n ) ρ min := inf n N R n+1 R n 2 ρ min Averaging vs Chaos in Turbulent Transport? p. 50/6

113 λ = 0, proof of the collapse For y [0, 1] d σ(n, y) := σ(κi d, Γ n 1 + γ n E(y + x R n )) Averaging vs Chaos in Turbulent Transport? p. 50/6

114 λ = 0, proof of the collapse σ(n, y) := σ(κi d, Γ n 1 + γ n E(y + x R n )) Averaging paradigm Relative translation by y has little influence on σ(n, y) for all y, lim ρ min σ(n, y) = lim ρ min σ(n, 0) Averaging vs Chaos in Turbulent Transport? p. 50/6

115 λ = 0, proof of the collapse σ(n, y) := σ(κi d, Γ n 1 + γ n E(y + x R n )) σ(n, y, ρ) := R 1 R 0,..., R n 1 lim R n 2 ; Rn R =ρ n 1 σ(n, y) Averaging vs Chaos in Turbulent Transport? p. 50/6

116 λ = 0, proof of the collapse σ(n, y) := σ(κi d, Γ n 1 + γ n E(y + x R n )) σ(n, y, ρ) := R 1 R 0,..., R n 1 lim R n 2 ; Rn R =ρ n 1 σ(n, y) σ(n, y, ρ) = γ n 1 σ(a n 1, E(ρx) + γe(x + y)) Averaging vs Chaos in Turbulent Transport? p. 50/6

117 λ = 0, proof of the collapse σ(n, y) := σ(κi d, Γ n 1 + γ n E(y + x R n )) σ(n, y, ρ) := R 1 R 0,..., R n 1 lim R n 2 ; Rn R =ρ n 1 σ(n, y) σ(n, y, ρ) = γ n 1 σ(a n 1, E(ρx) + γe(x + y)) If λ > 0 then l (R d ) and y [0, 1] d, lim sup n t lσ(n, y, ρ)l t lσ(n, 0, ρ)l < 1 + C d(ρλ ) 1 2. Averaging vs Chaos in Turbulent Transport? p. 50/6

118 λ = 0, proof of the collapse σ(n, y) := σ(κi d, Γ n 1 + γ n E(y + x R n )) σ(n, y, ρ) := R 1 R 0,..., R n 1 lim R n 2 ; Rn R =ρ n 1 σ(n, y) σ(n, y, ρ) = γ n 1 σ(a n 1, E(ρx) + γe(x + y)) If λ = 0 then for any ρ > 1 there exists E and y [0, 1] d such that for any l (R d ), lim n t lσ(n, y, ρ)l t lσ(n, 0, ρ)l =. Averaging vs Chaos in Turbulent Transport? p. 50/6

119 Two scale flows S ρ E(x) = E(ρx) Θ y E(x) = E(x y) As ζ 0 σ(ζi d, S ρ E + E) C 1 ζi d. Averaging vs Chaos in Turbulent Transport? p. 51/6

120 Two scale flows S ρ E(x) = E(ρx) Θ y E(x) = E(x y) As ζ 0 σ(ζi d, S ρ E+Θ y E) C 2 ζ 1 2 Id Averaging vs Chaos in Turbulent Transport? p. 51/6

121 On the nature of Turbulence. r The flow at scale r is laminar. Viscosity κ, velocity V (r) = V 0. Averaging vs Chaos in Turbulent Transport? p. 52/6

122 On the nature of Turbulence. r r perturbation A small perturbation is introduced. Averaging vs Chaos in Turbulent Transport? p. 53/6

123 On the nature of Turbulence. r Transport Diffusion + Convection τ D (r): exit time of the perturbation by diffusion τ C (r): exit time of the perturbation by convection Averaging vs Chaos in Turbulent Transport? p. 54/6

124 On the nature of Turbulence. τ D (r) r2 κ τ C (r) r V 0 r Transport Diffusion + Convection Averaging vs Chaos in Turbulent Transport? p. 54/6

125 On the nature of Turbulence. r r τ D (r) < τ C (r) If τ D (r) < τ C (r) the perturbation exits by diffusion and is smoothed before going out of B(0, r). Averaging vs Chaos in Turbulent Transport? p. 55/6

126 On the nature of Turbulence. r r τ D (r) < τ C (r) the laminar flow is stable at the scale r Averaging vs Chaos in Turbulent Transport? p. 55/6

127 On the nature of Turbulence. r r τ D (r) < τ C (r) τ D (r) < τ C (r) r2 κ < r V 0 Re = rv 0 κ < 1 Averaging vs Chaos in Turbulent Transport? p. 55/6

128 On the nature of Turbulence. r r τ D (r) > τ C (r) If τ D (r) > τ C (r) the perturbation exits by convection and propagates. Averaging vs Chaos in Turbulent Transport? p. 56/6

129 On the nature of Turbulence. r r τ D (r) > τ C (r) The flow is unstable at the scale r and fluctuates and this scale. Averaging vs Chaos in Turbulent Transport? p. 56/6

130 On the nature of Turbulence. r r τ D (r) > τ C (r) τ D (r) > τ C (r) r2 κ > r V 0 Re = rv 0 κ > 1 Averaging vs Chaos in Turbulent Transport? p. 56/6

131 On the nature of Turbulence. r/ρ r change of scale Let s look at the flow at the scale r/ρ Averaging vs Chaos in Turbulent Transport? p. 57/6

132 On the nature of Turbulence. r/ρ r change of scale The flow is laminar at this scale r/ρ with velocity V (r/ρ) = V 0 /γ Averaging vs Chaos in Turbulent Transport? p. 57/6

133 On the nature of Turbulence. r/ρ r change of scale A small perturbation is introduced at the scale r/ρ Averaging vs Chaos in Turbulent Transport? p. 58/6

134 On the nature of Turbulence. r r Iteration This self similar process is iterated, until the dissipation scale l is reached. Averaging vs Chaos in Turbulent Transport? p. 59/6

135 On the nature of Turbulence. r r Iteration At the dissipation scale l, τ C (l) τ D (l) Averaging vs Chaos in Turbulent Transport? p. 59/6

136 On the nature of Turbulence. r σ(r) rv (r) The multi-scale structure creates an effective viscosity σ(r) rv (r) Averaging vs Chaos in Turbulent Transport? p. 60/6

137 On the nature of Turbulence. r σ(r) rv (r) τ C (r) r/v (r) τ D (r) r 2 /σ(r) The multi-scale structure stabilizes the flow since σ(r) rv (r) τ D (r) τ C (r) Averaging vs Chaos in Turbulent Transport? p. 60/6

138 On the nature of Turbulence. r V (x) = V 0 (x/r) α We write γ = ρ 1+α Averaging vs Chaos in Turbulent Transport? p. 61/6

139 On the nature of Turbulence. r V (x) = V 0 (x/r) α ǫ(x) (V (x))3 x Kolmogorov α = 1/3 The energy dissipation ǫ(x) at scale x is σ(x)( V (x) x )2 (V (x))3 x Averaging vs Chaos in Turbulent Transport? p. 61/6

140 On the nature of Turbulence. r V (x) = V 0 (x/r) α ǫ(x) (V (x))3 x Kolmogorov α = 1/3 If α > 1/3 the larger eddies are dissipated before the smaller ones Averaging vs Chaos in Turbulent Transport? p. 61/6

141 On the nature of Turbulence. r V (x) = V 0 (x/r) α ǫ(x) (V (x))3 x Kolmogorov α = 1/3 If α < 1/3 the smaller eddies are dissipated before the larger ones Averaging vs Chaos in Turbulent Transport? p. 61/6

142 On the nature of Turbulence. r V (x) = V 0 (x/r) α ǫ(x) (V (x))3 x Kolmogorov α = 1/3 The relation σ(x) xv (x) is at the core of the Kolmogorov law Averaging vs Chaos in Turbulent Transport? p. 61/6

143 On the nature of Turbulence. r V (x) = V 0 (x/r) α ǫ(x) (V (x))3 x Kolmogorov α = 1/3 In the anisotropic case, the relation λ max (σ(x))λ min (σ(x)) x 2 (V (x)) 2 restores the isotropy of the flow. Averaging vs Chaos in Turbulent Transport? p. 61/6

144 On the nature of Turbulence. r V (x) = V 0 (x/r) α ǫ(x) (V (x))3 x Kolmogorov α = 1/3 At the dissipation scale, τ C (l) τ D (l) l/v (l) l 2 /κ r/l ( V 0r κ )3 4 Averaging vs Chaos in Turbulent Transport? p. 61/6

145 On the nature of Turbulence. r V (x) = V 0 (x/r) α ǫ(x) (V (x))3 x Kolmogorov α = 1/3 intermittency at the smaller scales Averaging vs Chaos in Turbulent Transport? p. 61/6