What is Turbulence? Fabian Waleffe. Depts of Mathematics and Engineering Physics University of Wisconsin, Madison

Transcription

1 What is Turbulence? Fabian Waleffe Depts of Mathematics and Engineering Physics University of Wisconsin, Madison

2 it s all around,... and inside us! Leonardo da Vinci (c. 1500) River flow, pipe flow, flow from a faucet,... Clouds, smoke,... Wakes behind boats, golf balls, bikers, cars, airplanes... Nuclear fusion, stars,... Blood flow in heart and large arteries (R 20, 000 in aorta!)

3 Reynolds 1883: turbulence in pipe flow

4 Navier-Stokes equations 3D velocity field v(x, t): v = 0 t v + (vv) + p = 1 R 2 v Control parameter: Reynolds number R non-dimensional velocity 1 R non-dimensional viscosity

5 Simple geometries, simple flow? Pipes, Channels Laminar solutions: v = (1 y 2 )ˆx plane Couette v = yˆx Linearly stable yet: R 2, 000 R 350

6 Turbulence in channel flow Front Top Side Green, M. A., Rowley, C. W. & Haller, G. Detection of Lagrangian coherent structures in three-dimensional turbulence, J. Fluid Mech., 572, 2007,

7 Classical statistical approach to Turbulence v(x, t) random ensemble average v v = 0 t v + ( v v ) + p = 1 R 2 v vv Closure problem: Reynolds stress vv =?!

8 vv modeling [Prandtl-von Karman] Turbulence = collisions of eddies? eddy viscosity: vv ν T ( v + v T ) ν T? mixing length, Smagorinsky,...

9 vv modeling [Prandtl-von Karman] Turbulence = collisions of eddies? eddy viscosity: vv ν T ( v + v T ) ν T? mixing length, Smagorinsky,... [Richardson-Kolmogorov] Turbulence = Cascade of energy from large to small scales? scale-similarity, inertial range, return-to-isotropy,... K-Epsilon ν T, Dynamic model, RANS, LES,...

10 vv modeling [Prandtl-von Karman] Turbulence = collisions of eddies? eddy viscosity: vv ν T ( v + v T ) ν T? mixing length, Smagorinsky,... [Richardson-Kolmogorov] Turbulence = Cascade of energy from large to small scales? Walls?! Ouch! scale-similarity, inertial range, return-to-isotropy,... K-Epsilon ν T, Dynamic model, RANS, LES,...

11 vv modeling [Prandtl-von Karman] Turbulence = collisions of eddies? eddy viscosity: vv ν T ( v + v T ) ν T? mixing length, Smagorinsky,... [Richardson-Kolmogorov] Turbulence = Cascade of energy from large to small scales? Walls?! Ouch! scale-similarity, inertial range, return-to-isotropy,... K-Epsilon ν T, Dynamic model, RANS, LES,... v random? coherent structures!

12 Turbulence is controlled by boundary conditions (with apologies to Kolmogorov and Clay Institute)

13 What are coherent structures?

14 What are coherent structures? How do they fit with classical models of turbulence?

15 What are coherent structures? How do they fit with classical models of turbulence?? About 50 years of a posteriori, qualitative studies

16 Streaks with z-spacing Kline, Reynolds, Schraub & Runstadler, JFM 1967 (diagram from Smith & Walker, 1997)

17 Near wall coherent structures John Kim,

18 Characteristic near-wall coherent structure Derek Stretch, CTR Stanford, 1990 Streaks + staggered quasi-streamwise vortices: why?

19 Self-Sustaining Process (SSP) advection of mean shear Streaks O(1) instability of U(y,z) Streamwise Rolls O(1/R) nonlinear self interaction Streak wave mode (3D) O(1/R) WKH 1993, HKW 1995, W 1995, 1997

20 SSP theory SSP method Construction of Exact Coherent States from SSP ( Full NSE, Newton s method) PRL 1998, JFM 2001, PoF 2003

21 Laminar Couette flow: u=0 & u= Wx Fr

22 SSP: Streamwise Rolls create Streaks Wx Fr

23 SSP: Rolls create Streaks Wx Fr

24 SSP: Rolls create Streaks Wx Fr

25 SSP in action: Subcritical Bifurcation from Streaks Wx Fr

26 SSP in action: Subcritical Bifurcation from Streaks Wx Fr

27 SSP in action: Subcritical Bifurcation from Streaks Wx Fr

28 SSP in action: Subcritical Bifurcation from Streaks Wx Fr

29 SSP in action: Subcritical Bifurcation from Streaks Wx Fr

30 SSP in action: Subcritical Bifurcation from Streaks Wx Fr

31 SSP in action: Subcritical Bifurcation from Streaks Wx Fr

32 SSP in action: Subcritical Bifurcation from Streaks Wx Fr

33 SSP in action: Subcritical Bifurcation from Streaks Wx Fr

34 SSP: Self-Sustained! 3D Lower branch Wx Fr

35 SSP: Self-Sustained! 3D Lower branch Wx Fr

36 SSP: Bifurcation from Streaks Wx Fr

37 SSP: Bifurcation from Streaks Wx Fr

38 SSP: Self-Sustained! 3D Upper branch Wx Fr

39 SSP: Self-Sustained! 3D Upper branch Wx Fr

40 Homotopy Free-Free Couette (FFC) Rigid-Free Poiseuille (RFP) µ = 0 1 BC : (1 µ) du dy + µu = 0 Flow : ( 1 U L (y) = y + µ 6 y 2 ) 2

41 FFC RFP

42 FFC RFP

43 FFC RFP

44 FFC RFP

45 FFC RFP

46 Poiseuille traveling wave!

47 Poiseuille traveling wave!

48 SSP, ECS: Generic structurally stable, dynamically unstable Plane Couette and Channel, free-slip, no-slip, any-slip! Pipe: Faisst & Eckhardt PRL 2003, Wedin & Kerswell JFM 2004, Hof et al. Science 2004, Pringle & Kerswell PRL 2007

49 Optimum Traveling Wave: 100 +! min R τ = 2h + = 44 for L + x = 274, L + z = 105 just right!

50 Out-of-the-blue-sky (saddle-node) DRAG R Lower branch does NOT bifurcate from laminar flow! RRC (α, γ) = (1, 2), (1.14, 2.5), up to R asymptotics

51 Vortex visualization ω x 2Q= 2 p = Ω ij Ω ij S ij S ij

52 Upper and Lower branches 0.6 max(q), (α, γ) = (1.14, 2.5).

53 Upper and Lower branches 0.6 max(q), (α, γ) = (1.14, 2.5).

54 Upper and Lower branches 0.6 max(q), (α, γ) = (1.14, 2.5).

55 Upper and Lower branches 0.6 max(q), (α, γ) = (1.14, 2.5).

56 Upper and Lower branches 0.6 max(q), (α, γ) = (1.14, 2.5).

57 Upper and Lower branches 0.6 max(q), (α, γ) = (1.14, 2.5).

58 Upper and Lower branches 0.6 max(q), (α, γ) = (1.14, 2.5).

59 Upper and Lower branches 0.6 max(q), (α, γ) = (1.14, 2.5).

60 Upper and Lower branches 0.6 max(q), (α, γ) = (1.14, 2.5).

61 Upper and Lower branches 0.6 max(q), (α, γ) = (1.14, 2.5).

62 Upper and Lower branches 0.6 max(q), (α, γ) = (1.14, 2.5).

63 Upper and Lower branches no-slip Couette 5 4 Dissipation Rate Energy Input Rate Steady State & turbulent (by Jue Wang & John Gibson) in RRC, R = 400, (α, γ)=(0.95,1.67)

64 Upper branches Turbulence (no-slip Couette) y 0 y U RMS solid: Turbulent (avg t=2000), dash: fixed point Mean and RMS velocity profiles

65 LB eigenvalues, (α, γ) = (1.14, 2.5), (1, 2), R = Only 1 unstable eig!

66 LB Transition (α, γ) = (1.14, 2.5), (1, 2), R = Dissipation Rate Dissipation Rate Energy Input Rate Energy Input Rate Dissipation Rate Dissipation Rate Energy Input Rate Energy Input Rate

67 Lower branch R = max(q), R = 1000, (α, γ) = (1.14, 2.5).

68 Two states of fluid flow Turbulent Laminar

69 Separatrix, transition threshold Turbulent Laminar

70 Unstable Coherent States! Turbulent Laminar

71 PCF data (R = 400) 5 4 Dissipation Rate Energy Input Rate Periodic solutions in HKW (1.14, 1.67) by Viswanath, JFM 2007 & Gibson (TBA)

72 Visualizing State Space ( 10 5 dof s) 0.1 u NB u LM a a a RRC, R=400, Gibson, Halcrow, Cvitanovic, JFM to appear arxiv.org/

73 Visualizing State Space (PCF, R=400) u UB 0.1 u LB τ z u UB u LM a 3 0 τ xz u LB τ x u UB τ xz u UB 0 a a 1 RRC, R=400, Gibson, Halcrow, Cvitanovic, JFM to appear arxiv.org/

74 Conclusions Turbulence is not the random collisions of fluid molecules

75 Conclusions Turbulence is not the random collisions of fluid molecules Turbulence is not a cascade of energy from large to small scales

76 Conclusions Turbulence is not the random collisions of fluid molecules Turbulence is not a cascade of energy from large to small scales not small scale random fluctuations but...

77 Conclusions Turbulence is not the random collisions of fluid molecules Turbulence is not a cascade of energy from large to small scales Multiscale Exact Coherent States (ECS) do the transport

78 Conclusions Turbulence is not the random collisions of fluid molecules Turbulence is not a cascade of energy from large to small scales Multiscale Exact Coherent States (ECS) do the transport ECS= 3D Traveling wave and periodic solutions of Navier-Stokes

79 Conclusions Turbulence is not the random collisions of fluid molecules Turbulence is not a cascade of energy from large to small scales Multiscale Exact Coherent States (ECS) do the transport ECS= 3D Traveling wave and periodic solutions of Navier-Stokes ECS: upper turbulence, lower transition

80 Conclusions Turbulence is not the random collisions of fluid molecules Turbulence is not a cascade of energy from large to small scales Multiscale Exact Coherent States (ECS) do the transport ECS= 3D Traveling wave and periodic solutions of Navier-Stokes ECS: upper turbulence, lower transition ECS unstable manifolds: low dimensional(?) control

81 Conclusions Turbulence is not the random collisions of fluid molecules Turbulence is not a cascade of energy from large to small scales Multiscale Exact Coherent States (ECS) do the transport ECS= 3D Traveling wave and periodic solutions of Navier-Stokes ECS: upper turbulence, lower transition ECS unstable manifolds: low dimensional(?) control Asymptotic theory of lower branch solutions? Proof?