Patterns of Turbulence. Dwight Barkley and Laurette Tuckerman

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1 Patterns of Turbulence Dwight Barkley and Laurette Tuckerman

2 Plane Couette Flow Re = U gap/2 ν Experiments by Prigent and Dauchot Re400 Z (Spanwise) o Gap 2 Length 770 X (Streamwise)

3 Examples: Patterns & Bifurcations in Turbulent Flows Current understanding of Turbulent Patterns in Couette flow

JFM (1986) Very Large Aspect Ratio Taylor-Couette Experiment inner

4 Spiral Turbulence in Taylor-Couette Flow =Ri/Ro=0.88 z Coles JFM (1965) van Atta JFM (1966) Andereck et al. JFM (1986) Very Large Aspect Ratio Taylor-Couette Experiment inner outer cylinders approximately counter rotating Prigent & Dauchot PRL (2002) =Ri/Ro=0.983 Mirrors

5 Similar phenomenon seen in flow between rotating and stationary disks

6 Simple symmetry breaking in Highly Turbulent von Karman Flow

7 Gravity-Wave in Highly Turbulent Swirling Flow

8 Return to Couette flow Can we understand as pattern formation? + SO(2)xD 4 k

9 A little background - Streaks and Streamwise Vortices Turbulent Spot (from Saclay Group) z x streaks

10 Minimal Flow Unit Simulations Jimenez and Moin, JFM (1991) Poiseuille Flow Hamilton, Kim, Waleffe (HKW), JFM (1995) Plane Couette Flow Objective: Find minimum constrained domain which supports turbulence Periodic in x and z Z Reduce Reynolds number to near minimum for turbulence X Re=400 Reduce streamwise and spanwise dimensions to near minimal Streamwise x Spanwise ~ 6 x 4

11 Minimal Flow Unit Turbulent Spot (from Saclay Group) Mimimal Flow Unit L x L y L z 6 x 2 x 4 z x streaks

12 Computational Approach Start with Minimal Flow Unit (small domain supporting turbulent cycle of streamwise vortex pair) MFU Tilt Domain (maintain vortex pair spacing) L X Very Long (one direction only) L X =10 L Z = 120 Z (spanwise) 4 6 X (streamwise) z' ' ' MFU Gap = L y = 2 x' Simulate large length scales oblique to streamwise direction Study role of tilt & length in pattern selection

13 Computational Domains: Angles and Size Z x

Numerical Simulations Direct Numerical Simulations (Full DNS) of Incompressible Navier-Stokes Equations u t u u= p 1 R e 2 u u=0 Prism (Ron Henderson) P P

14 Numerical Simulations Direct Numerical Simulations (Full DNS) of Incompressible Navier-Stokes Equations u t u u= p 1 R e 2 u u=0 Prism (Ron Henderson) P P Spectral-Element Mesh (20 to 50 elements) (7x7 or 9x9 polynomial expansions) Fourier (M=256 to 2048) Long Parallel Typically 1-2 million grid points Up to 20 million

15 Bifurcation Diagram from Couette experiments Isolated (Spots) Periodic Intermittent } Uniform Turbulence 280? Simple Couette Re

16 Results

turbulence Steady Pattern Space-Time Plot 40 Time

17 Results For each domain: Start at Re = 500 Obtain turbulent flow Decrease Re in small steps Monitor turbulence Steady Pattern Space-Time Plot 40 Time History points Decreasing Reynolds Number Uniform Turbulence z'

18 Visualization P Turbulent Kinetic Energy K=uu c 2 v 2 w 2 Couette flow P P Band Streamwise P

19 Band Streamwise

20 Movie of Patterned Flow Turbulent Kinetic Energy Re = 350 Image shown every time unit over 300 time units

21 Long Space-Time Diagram 43,000 time units Three Types of Patterned States { Time { Turbulent flow T=500 { Laminar flow

Periodic and bounded from zero 0 0 Turbulent patch ~ constant size

22 Two Types of Steady Patterns Exponentially localized Re = 300 Re = 350 Localized State Periodic State turbulent spot turbulent bands Periodic and bounded from zero 0 0 Turbulent patch ~ constant size Turbulent regions split ~ constant wavelength Domain length slowly increased

23 Movie of Periodic State Streamwise velocity in z'-y plane y Re=350 Pattern wavelength z' ½ computational domain

24 Movie of Periodic State Streamwise vorticity in z'-y plane y Re=350 Pattern wavelength z' ½ computational domain

25 Movie of Isolated State Streamwise velocity in z'-y plane Re=300 y z' 75% computational domain

turbulence and steady patterns) Simulation of 40,000 time units at

26 Intermittency Found in Transition Region Between Uniform Turbulence and Steady Patterns Re = 410 (between uniform turbulence and steady patterns) Simulation of 40,000 time units at fixed parameters Flow never settles into a steady pattern but intermittently bubbles

27 Brief Survey of Other Patterned States

28 θ = 24 θ = 0 New band forms Periodic emission Remaining band moves Two bands lost Turbulence at very low Re Spatiotemporal intermittency

29 Quantitive Studies Parabolic neutral curve? Uniform Turbulence Re Re Patterns or

30 What is the transition Re for pattern onset? 43,000 time units Time Turbulent flow Scaling of Pattern Amplitude in Experiment Laminar flow Prigent et al Physica D

31 25,000 time units Averaged Spectra < w^ m > T Instantaneous spatial Fourier transform Average over T long compared with fluctuations, short compared with pattern time scale m=2 Laminar flow m=3 Time Turbulent flow uniform turbulence < w^ m > T m

32 Continuous Onset of Pattern < w^ 1 > T Patterns Domain Lz=40 High resolution 8,000 time units minimum at each Re Rec460 Uniform Turbulence Re We can get transition Re, but it is expensive (6 x 10 4 cpu hours).

33 Meanflows and Modeling

34 Mean and Fluctuating Fields Deviation from Couette uuu c u T u 2 T u T 2 Mean: RMS: Streamwise velocity mean: rms: y T=2000 time units Re=350 From Prigent et al. z' Center of turbulent region Mean flow the ~same in turbulent and laminar regions

75 Color: iso-surfaces of fluctuations (turbulent KE) Center of turbulent

35 Mean and Fluctuating Fields II Turbulent-Laminar Pattern at Re=350 T=8000 time units Meanflow in midplane Meanflow at y = ± 0.75 Color: iso-surfaces of fluctuations (turbulent KE) Center of turbulent region Blue: y=-0.75 Red: y=0.75 Black: direction of motion for top/bottom plates Center of turbulent region z x

36 Conclusions As Re is decreased or is increased Laminar Localized Periodic Intermittent Uniform Quantitative agreement with experiment All generated with only ONE extended direction Other direction ~ minimal flow unit Phenomenology Turbulent Patterns Properties of mean and fluctuating fields

Turbulent-laminar patterns in plane Couette flow

Turbulent-laminar patterns in plane Couette flow Dwight Barkley Mathematics Institute, University of Warwick, Coventry CV4 7AL, United Kingdom Laurette S. Tuckerman LIMSI-CNRS, BP 133, 91403 Orsay, France