Jacob Cohen 1, Ilia Shukhman 2 Michael Karp 1 and Jimmy Philip 1 1. Faculty of Aerospace Engineering, Technion, Haifa, Israel 2. Institute of Solar-Terrestrial Physics, Russian Academy of Sciences, Siberian Department, Irkutsk , P.O. Box 4026, Russia Significance of Localized Vortical Disturbances in Wall-Bounded and Free Shear Flows Jacob Cohen 1, Ilia Shukhman 2 Michael Karp 1 and Jimmy Philip 1 1. Faculty of Aerospace Engineering, Technion, Haifa, Israel 2. Institute of Solar-Terrestrial Physics, Russian Academy of Sciences, Siberian Department, Irkutsk , P.O. Box 4026, Russia Technion – Israel Institute of Technology – Faculty of Aerospace Engineering

2 Streaks – Alternating high and low speed fluid in spanwise - direction: Counter Rotating Vortex Pair (CVP) – Streamwise vortex pair: Hairpins – Inclined pair of vortices (in streamwise dir.) connected by a short head (in spanwise dir.): Kim et al.,1971 Bakewell & Lumley,1967 Acarlar & Smith,1987 Coherent Structures in Turbulent Wall Bounded Shear Flows

3 y x y x Vertical Plane z x (b1) (b2) z x Horizontal Plane CVPs HAIRPINs Coherent Structures in Sub-Critical Plane Poiseuille flow

4 Scaling in Sub-Critical Channel Flow

5 Scaling in Pipe Flow Side viewFront view Similar scaling law was also obtained for Transition to Turbulence by Hof et. al, (PRL) 2003

6 Velocity components of the pair of least stable modes Transient Growth (spatial) : Pipe flow Spatial Eigenmodes for n=1, ω=0 Re=3000 Ben-Dov, Levinski & Cohen, (PoF) 2003

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9 Transient Growth (spatial) : plane Poiseuille flow Temporal Case: - two modes - many modes

10 Cross-sectional vector map for temporal case Pipe: pair of modes Pipe: many modes Channel: Pair of modes

11 Cohen, J. et al. AIAA,2006 Localized Disturbances + Linear Shear Mixing Layer / Stagnation Flow Pure Shear

12 Comparison with DNS results Uniform shear flow (T=5) DNS Plane Poiseuille flow (T=4.8) HPIV ωyωy 2δ 45 o inclination Optimal hairpin 45 o inclination 60 < z + < 115  - disturbance length scale

13 45 o inclination  Toroidal Disturbance: Optimal Hairpin 60 < z+ < 115

Localized Disturbance Vortical disturbance Base flow ∆ δ δ ∆ 1 >>

15 To predict the interaction between a localized vortical disturbance and various linear shear flows Disturbance: 3D, finite-amplitude, localized, viscous. Base flow: Objective of the model

16 Couette, σ=Ω Hyperbolic, σ 2 > Ω 2 (Stagnation flow Ω=0) Elliptic, σ 2 < Ω 2 (Solid-body rotation σ=0) Base Flow – Linear Shear  [1/s]=  [1/s]=-80  [1/s]=-80;  =0  =0;  [1/s]=-80 V1V1 V2V2

17 The disturbance vorticity equation: Theoretical-based model

18 Theoretical Model – cont. where

19 1.Fourier transform Theoretical Model - continuation Following Shukhman, 2006 where

20 Theoretical Model – cont. 2. Lagrangian variables where

21 Theoretical Model – cont

22 The equations are of the form: Euler’s method: Theoretical Model – summary Inverse Fourier Transform:

23 The Gaussian disturbance: Initial Disturbance

24 Linear case - the disturbance evolves to CVP Results – Stagnation flow T=0 T=2

25 Linear case – comparison to the analytical solution Results – Stagnation flow ( Leonard, 2000 ) T=1, Ω=0, σ=-80 1/sec

26 Non-Linear case – Vortex center moves Results – Stagnation flow T=2 T=1 x=-y principal axis

27 Linear case – comparison to the analytical solution Results – Couette flow T=1, Ω=σ=-40 1/sec

28 Non-Linear case – Generation of Hairpins Results – Couette flow T=2 T=1T=0

29 Linear case – rotation and splitting Results – Solid Body Rotation flow

30 Non-Linear case – two unsymmetrical parts Results – Solid Body Rotation flow

31 Conclusions An analytical based solution method has been developed The method can solve the interaction between a family of linear shear flows and any localized disturbance The solution is carried out using Lagrangian variables in Fourier space which is convenient and enables fast computations Localized Disturbance + Linear Shear Flow CVP Hairpins = Thank You