Stability of streaks in shear flows

Michael Karp and Jacob Cohen

Faculty of Aerospace Engineering

Technion - Israel Institute of Technology, Haifa 32000, Israel Research supported by the Israeli Science Foundation under Grant No. 1394/11

Research Aim • Study the secondary instability of TG in Couette flow and utilize it to

predict nonlinear transition to turbulence

References “Tracking stages of transition in Couette flow analytically” Karp, M.,

and Cohen, J., J. Fluid Mech., 748, 2014, pp 896 – 931.

Mathematical Method • Analytical approximation of linear TG using 4 modes

• Calculation of nonlinear interactions between the 4 modes

• Secondary stability analysis of the modified baseflow

• Long time correction of ud using solvability condition

Summary • Maximal growth is not essential for transition

• The role of the TG is to generate inflection points

• Optimal disturbances occur at maximal shear

• Most transition stages are captured analytically

Transient Growth (TG) • A mechanism where infinitesimal disturbances grow in a stable flow.

During this growth, the baseflow can be modified significantly and

instability may occur.

• Most efficient TG occurs for streamwise independent vortices

• Vortical structures (Isosurfaces of the Q def.) during transition

0

1

2 0

1

2

-1

-0.5

0

0.5

1

x/z/

y

Symmetric (Even, 2 vortices)

0

1

2 0

1

2

-1

-0.5

0

0.5

1

x/z/

y

Antisymmetric (Odd, 4 vortices)

0

0 expRe

, ,d

t

d

ti

A i xN

Mdt y z

u u

2

0 ,,, , ...ˆ ,, NLx L t y tyey z zz y U u u

Couette + 4 modes + nonlinear

Amplitude

Eigenfunction Streamwise Eigenvalue Long time

wavenumber correction

Photo by Brooks Martner

0 , , , ,, , ..., , , ,d dd t x yx y z zty z u U u u

TG baseflow + Secondary disturbance + correction term

Re=1000 =1.6 =1.2 t=43

z /

y

0 0.5 1 1.5-1

-0.8

-0.6

-0.4

-0.2

0

0.2

0.4

0.6

0.8

1Re=1000 =3.5 =1.5 t=36

z /

y

0 0.5 1 1.5-1

-0.8

-0.6

-0.4

-0.2

0

0.2

0.4

0.6

0.8

1Re=1000 =3.6 =1.8 t=61

z /

y

0 0.5 1 1.5-1

-0.8

-0.6

-0.4

-0.2

0

0.2

0.4

0.6

0.8

1

0 1 2 0

2

4

6

8

10

-1

0

1

x/

z/

y

0 1 2 0

2

4

6

8

10

-1

0

1

x/

z/

y

0 1 2 0

2

4

6

8

10

-1

0

1

x/

z/

y

0 1 2 0

2

4

6

8

10

-1

0

1

x/

z/

y

0 1 2 0

2

4

6

8

10

-1

0

1

x/

z/

y

0 1 2 0

2

4

6

8

10

-1

0

1

x/

z/

y

0 1 2 0

2

4

6

8

10

-1

0

1

x/

z/

y

0 1 2 0

2

4

6

8

10

-1

0

1

x/

z/

y

DNS

t=75

Theory

t=50 t=85 t=90

DNS

t=30

Theory

t=25 t=35 t=40

0 1 2 0

2

4

6

8

10

-1

0

1

x/

z/

y

0 1 2 0

2

4

6

8

10

-1

0

1

x/

z/

y

0 1 2 0

2

4

6

8

10

-1

0

1

x/

z/

y

0 1 2 0

2

4

6

8

10

-1

0

1

x/

z/

y

0 1 2 0

2

4

6

8

10

-1

0

1

x/

z/

y

0 1 2 0

2

4

6

8

10

-1

0

1

x/

z/

y

0 1 2 0

2

4

6

8

10

-1

0

1

x/

z/

y

0 1 2 0

2

4

6

8

10

-1

0

1

x/

z/

y

• Secondary instability verified by obtaining transition in

‘Channelflow’ DNS (Gibson, 2012)

(a) Even TG – Sinuous, max. spanwise shear (β=3.6)

(b) Odd TG – Sinuous, max. spanwise shear (β=3.5)

(c) Odd TG – Varicose, max. wall-normal shear (β=1.6)

Even TG (2 vortices) - sinuous

Odd TG (4 vortices) - varicose

0 50 100 150 2000

100

200

300

400

500

t

G(t

)

DNS (TG)

DNS

Analytical (TG)

Analytical

0 50 100 150 2000

500

1000

1500

2000

2500

3000

t

G(t

)

DNS (TG)

DNS

Analytical (TG)

Analytical

• Energy growth during transition

Even TG (2 vortices) - sinuous Odd TG (4 vortices) - varicose

Results • Formation of streamwise velocity streaks containing inflection points

Even TG (2 vortices) Odd TG (4 vortices)

• Secondary disturbance (streamwise component), legend below

(a) (b) (c)