-1

0

1

0

1

2

-1

-0.5

0

0.5

1

u

Streamwise velocity, Re=1000 β=2 t=25

zβ/π

y

-1

0

1

0

1

2

-1

-0.5

0

0.5

1

u

Streamwise velocity, Re=1000 β=4 t=40

zβ/π

y

Sinuous Instability

Spanwise

Inflection

Secondary instability of Transient Growth in Couette Flow

Michael Karp and Jacob Cohen

Faculty of Aerospace Engineering,

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

Motivation• Linear Stability Theory (LST) is unable to predict transition

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 up 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

Flow Theoretical (LST) Experimental

Pipe Poiseuille ∞ (Stable) ~2000

Plane Poiseuille 5772 ~1000

Plane Couette ∞ (Stable) ~360

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

Table 1. Critical Reynolds numbers for transition, Theory vs. Experiment

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

utilize it to predict nonlinear transition to turbulence

Results• Optimal disturbances

• Even TG – Sinuous, max. spanwise shear (β≈4≈4≈4≈4)

• Odd TG – Sinuous, max. spanwise shear (β≈4≈4≈4≈4)

• Odd TG – Varicose, max. spanwise shear (β≈≈≈≈2)

• Secondary instability verified by obtaining transition in ‘Channelflow’ DNS (Gibson, 2012)

0

1

2 0

1

2

-1

-0.5

0

0.5

1

xα/πzβ/π

y

ω�

ω�

Even (2 vortices)

0

1

2 0

1

2

-1

-0.5

0

0.5

1

xα/πzβ/π

y

ω�

ω�ω

�

ω�

Odd (4 vortices)

Re=1000 β=4 α=2 t=40

z β/π

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

Re=1000 β=2 α=2 t=25

z β/π

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

Re=1000 β=4 α=2 t=20

z β/π

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

Secondary disturbance

(streamwise component)

Sinuous

Instability

Spanwise

Inflection

Varicose

Instability

Wall-normal

Inflection

Energy during transition

( ) ( )0

0 exp, ,d

t

d

t

iNt y z iA x

Mdωα τ τ

= − −

∫ɶu u

( ) ( ) ( )2 , ,, , ,, .ˆ . .,NLx L d

t y t x yy zt ze y z δεε= + + + +u u uu

Couette + 4 modes + nonlinear + secondary

AmplitudeEigenfunction Streamwise Eigenvalue Long time

wavenumber correction

Photo by Brooks Martner

0 50 100 150 2000

100

200

300

400

500

t

G(t

)

DNS (TG)

DNS

Analytical (TG)

Analytical

0 50 100 150 2000

50

100

150

200

250

300

350

400

t

G(t

)

DNS (TG)

DNS

Analytical (TG)

Analytical

0 50 100 150 2000

200

400

600

800

1000

tG

(t)

DNS (TG)

DNS

Analytical (TG)

Analytical

Photo by NASA/JPL

Isosurfaces of the Q definition during transition

0 1 0

5

10

15

-1

0

1

x

Qdef, t = 40

z

y

0 1 0

5

10

15

-1

0

1

x

Qdef, t = 60

z

y

0 1 0

5

10

15

-1

0

1

x

Qdef, t = 80

z

y

0 1 0

5

10

15

-1

0

1

x

Qdef, t = 20

z

y

0 1 0

5

10

15

-1

0

1

x

Qdef, t = 40

z

y

0 1 0

5

10

15

-1

0

1

x

Qdef, t = 60

z

y

0 2 0

5

10

15

-1

0

1

x

Qdef, t = 40

z

y

0 2 0

5

10

15

-1

0

1

x

Qdef, t = 30

z

y

0 2 0

5

10

15

-1

0

1

x

Qdef, t = 25

z

y