Significance of Localized Vortical Disturbances in Wall-Bounded and

Free Shear Flows Jacob CohenJacob Cohen11, Ilia Shukhman, Ilia Shukhman22

Michael KarpMichael Karp11 and Jimmy Philip and Jimmy Philip11

1. Faculty of Aerospace Engineering, Technion, Haifa, Israel1. Faculty of Aerospace Engineering, Technion, Haifa, Israel2. Institute of Solar-Terrestrial Physics, Russian Academy of Sciences, 2. Institute of Solar-Terrestrial Physics, Russian Academy of Sciences,

Siberian Department, Irkutsk 664033, P.O. Box 4026, RussiaSiberian Department, Irkutsk 664033, 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 xy

x

Re 1667, 0.29injv

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

2/3o Re~v

5

1~ Reov

Scaling in Pipe Flow

Side view Front view

Similar scaling law was also obtainedfor 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 Eigenmodesfor n=1, ω=0

Re=3000

Ben-Dov, Levinski & Cohen, (PoF) 2003

7

8

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 resultsUniform shear flow (T=5)

DNS

p

Plane Poiseuille flow (T=4.8) HPIV

ωy

2δ

Optimal hairpin 4545o o inclinationinclination60 < z60 < z++ < 115 < 115

- disturbance length scale

2 2/1/2 3( ) xp e

13

4545o o inclination inclination

p

Toroidal Disturbance:

Optimal Hairpin

60 < z+ < 60 < z+ < 115115

Localized Disturbance

Vortical disturbance

Base flow

∆δ

δ∆

1 >>

15

To predict the interaction between a localized vortical disturbance and various linear shear

flows

0

20

0

( ) ( ) (| | )

( ) 0

ii i j

j x

i

VV x V x x O x

x

V x

Disturbance: 3D, finite-amplitude, localized, viscous.

Base flow:

Objective of the model

16

1 12 2( , ,0) , 0,0,V y x

Couette, σ=Ω Hyperbolic, σ2 > Ω2

(Stagnation flow Ω=0)Elliptic, σ2 < Ω2

(Solid-body rotationσ=0)

Base Flow – Linear Shear

[1/s]=[1/s]=-80 [1/s]=-80; =0 =0; [1/s]=-80

V1V2

17

The disturbance vorticity equation:

Theoretical-based model

N Non linear terms

V V v v vt

tottot vVV

vVtottot

18

Theoretical Model – cont.

31 2 1

1

32 1 2

2

33 3

3

1

2

3

{ }

{ }

{ }

1

2

1

2

v

x

v

x

v

x

N

N

N

M

M

M

where

2 11 2

1 1

2{

2} x x

t x x

M

ii iN v v

19

1. Fourier transform

Theoretical Model - continuation

k

Following Shukhman, 2006

3 3

31ˆ ˆ;

2ik r ik r

j j j jr e d r k e d k

2 2 21 2 1 3 1 2 3 1 1

2 2 22 1 2 3 1 2 3 2 2

1 1 2 2 3 3

1 ˆˆ ˆˆ ( )21 ˆˆ ˆˆ ( )2

ˆ ˆ ˆ

{

0

ˆ }

ˆ { }

0

ik v k k k N

ik v k k k N

v k k k

M

M

where 1 22 1

ˆ { }1 1

ˆ2 2i k k

t k k

M

20

Theoretical Model – cont.

2. Lagrangian variables

1 22 1

31 2

1 2 3

1 1ˆ2 2

k kt k k

dkdk dkd

dt t dt k dt k dt k

M

1/21 1 2

1/22 2 1

3 3

( ) cosh( ) sinh( ),

( ) cosh( ) sinh( ),

( ) ,

k t q t q t

k t q t q t

k t q

2 21( ); ;

2( 0)i isign k t q

where

21

Theoretical Model – cont.

1 1 2 2 11 22 1 12

2 1 2 2 12 21 2 22

3 1 1 2 2 3

1 ˆ2

1 ˆ2

[ ( , ) ( , ) ( , ) ( , )]

k k q k qd qq k q N q

dt k

k k q k qd qq k q N q

dt kk q t q t k q t q t q

ttqktq jj ,,ˆ, 1

2

ˆ[ ]ˆ( ) ;

kv k i

k

2 2 2 21 2 3k k k k 2

' ' 3 '( ) ( ) ( )k

A B A k B k k d k

3

22

The equations are of the form:

Euler’s method:

Theoretical Model – summary

, ,jj

d qF q q t

dt

1 2

nn n j

j j

d qt O t

dt

Inverse Fourier Transform:

1 1 3, ;n n ik rj j q t e d k

0k r q r

3 3d k d q

23

The Gaussian disturbance:

Initial Disturbance

22

31

1 2 3

0, , e ,

, ,

r

t r F F

*

*

2T t

representative length

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

-4 -3 -2 -1 0 1 2 3 4-4

-3

-2

-1

0

1

2

3

4

x/

Isosurface of 0.7*wmax at T=2 , nonlinear case (=1)

y/

T=2

-2 -1.5 -1 -0.5 0 0.5 1 1.5 2

x 10-3

0

5

10

15

20

25

30Vorticity magnitude along X=-Y principal axis, T=1

x=-y [m]

[

1/se

c]

Numerical

Fluent

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=2T=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

CVPHairpins

=

Thank You