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CS-Turbulence Interaction

1

Fazle Hussain, D. S. Pradeep & Eric Stout

University of Houston

Funded by NSF

KTH Workshop April 30, 2010

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4:56 EDT

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Trailing vortex:

A simple example of

coherent structure

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(a)

(e)

(d) (c) (b)

Motivation:

Circular jet

Plane jet

Mixing layer

Cylinder wake

Sheared vortex column

Strained vortex with CD

JFM ‘01

vortex column in turbulence

PRE ’93 JFM ‘06

Core Dynamics (CD) Fl.Dyn.Res. ‘94

generic CS

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CS-turbulence interaction: Idealized flow

Idealizations:

•No interaction with other CS no pairing or reconnection

•No background shear no elliptic instability

•Rectilinear, cylindrical CS no self-induced motion

•Random, fine-scale fluctuations homog., isotrop. k – sep.

Flow evolution using DNS initialized with 3-D vort. from lin. analysis Pseudo-spectral method (Rennich & Lele ’97; Pradeep & Hussain ’04) periodic in z, pot. flow @ r → ∞

DNS Re ≡ Γ/ν

1k – 20k

Oseen-Lamb vortex

r u

θ v z

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Numerical simulation method:

Triply-periodic “unbounded” flow

(Rennich-Lele ’97) Pradeep & H. ‘04

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Comparison:

time Unbounded BC

Periodic BC

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−10

−8

−6

−4

0 5 10 15 20

tag2

tag1

DNS Theory (a)

time Energy

Growth of q-vortex instability mode: Bending wave Re = 105

(DNS)

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Vortex-Turbulence Interaction

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LS

Organization of turbulence:

(a) (b)

(c)

(d) d1

d2 d3 d4

d5

FS

T = 0 T = 10

T =30 T = 120

dipoles meridional plane |ω| contours

Γ/ν = 12.5k

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spiral threads dipoles

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Transport effect of threads:

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centrifugal instability

Re = 5000

Self- limiting

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Instability Driven Intensification:

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18 0

0.4 0.8

0 2 4 6

T=0 50 100 200 300 400 500

(a)

V/v1

r/r1

0.001 0.01 0.1

0 4 8

T=0

100 200

300 400

500

u/v^

1

r/r1

(b)

• surprisingly little effect of turbulence on vortex decay

More rapid decay possible?

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Turb. statistics:

16

8

4 12

0

0 1 2 3

u’2

r

(a)

x 10−4

16

0 4 8 12

0 1 2 3

v’2

(b)

r

x 10−4

16

12

8

4

0

1 2 3

0

(c)

w’2

r

x 10−4

2 4

0

−2

−4

0 1 2 3

u’v’

x 10−5

r

(d) u

w

T = 15

r r

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q’

r

q’

q’

q’

r

r

r

Re=3000

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Other mechanisms?

• Thread/Vortex wave resonance

• Transient growth

Centrifugal instability is self-limiting Mechanisms of core perturbation growth:

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0 0.05 0.1 0.15

0 0.5 1 1.5

.. . .

. .

(b) .

c

w

c

r

λ γ=0.05

γ=0.1 γ=0.2

R

3

R

1

R

2

c

w

c

r

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(a) (b) (c) (d) (e) (f) (g) (h)

γ = 0.1 Re = 2000

t = 0 t = 5 t = 10 t = 15 t = 20 t = 25 t = 30 t = 35

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TRANSIENT GROWTH

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A rudimentary example:

TG: Temporary growth followed by decay

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ψ

+uv

-uv 2-D mechanism:

dV r

V r dt uv

dE ~ ∫ ( / )'

Rigid Rotating Rod

. .

A

B

(b)

ω

z

.

.

(d)

A’

ω

z B’

+u +v

.

r θ (b)

-u -v

+u-v

-u +v

+ tilt

(c)

- tilt

u v

+uv ψ

-uv

-ve strain

only shearing, no stretching

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z(z B) − ω z(z Aω)

. . . .

t

t t

t t

1 2

3 4

(e)

u<0 v>0 u>0

v<0

(d) 4

zA zB

t

u<0 v=0u>0 v=0

(c) t3

zA zB

V z x y r

θ

zA zB

u=0v>0 u=0

v<0

Unperturbed vortex surface

Perturbed vortex surface

(a) t1

u>0 v>0

u<0 v<0

Meridional streamline Vortex line

(b) t

2

zA zB

30

Rotation only

. .

. .

t t

t

t

t

1 2

3

4

uv

(f)

u

uv = 0

uv > 0 => E↑

uv < 0 => E↓

vort. => E osc.

strain => E ↑ ωθ = 0

ωθ = max.

Core dynamics m = 0

Unperturbed vortex surface

d

@ vort. Surface r ↓ => ν ↑ Γ = 2πr(V+ν) = const. r ↑ => ν ↓

Perturbed vortex surface

v Vortex Line

Meridional Streamline

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0 20 40

0 10 20

(a)

E(t)

t

β/α=0 β/α=0.05

β/α=0.1 β/α=0.2 2

r

V = α r + β

model flow

E ~2 t

pure strain

increasing vorticity

t ω

r

S ω

θ

=

Higher strain S i.e. α ↑

⇒ωθ gen.

⇒ uv ↑

⇒E ↑

inc. vort(β)

⇒ arrest E ↑ sooner

β/α = rotation/strain

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LIN. INVISCID TG

Strain: unbounded growth (lin. sense) eventually saturate at NL level

core vorticity: arrest growth & period of growth

→ core oscillation

VISCOSITY damps both

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Optimal gains:

1 10 10

1 2 3

(a)

k Gmax

Re = 500 1000

2500

5000

2

10 10 10

0 1 2

(a) Gmax

Re = 5000

10,000 20,000

k

3

2

0 2 4 6

0 2 4 6 8

(a)

r E

t=250

t=100

t=50 t=0 (E 10)x

0 4 8 12 16

0 1 2 3 4

(b)

E

r

t=90

t=60

t=30 t=0

(E 100)x

Energy evolution:

Bending m = 1 axi-sym m = 0

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Re effect on tilting/stretching

0 2 4

0 2 4 6

(a)

r ω r

Re=500

1000

5000 2500

−2 0 2 4

0 2 4

(b)

r ω r

10,000

Re=5000

20,000

m = 0 m = 1

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of optimal modes

NONLINEAR TRANSIENT GROWTH

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initial perturbation amplitude:

Case A Linear

B 0.6%

C 2%

D 6%

E

0 40 80 120

−6

−8

−4

−2 (a)

log(K)

t

A B

C D

z x

(a)

ω = 0.01θ

A

t = 90

(b) ω = −0.05θ

ω = −0.3

B

t = 70

(c) ω = 0.15θ

ω = 0.55θ

C

t = 50

(d) ω = 0.3θ

ω =0.6θ

D

t = 30 Structure at time of max. energy:

Re = 5000 m = 1

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High-amplitude perturbation evolution: Case D, m = 1

t = 0 t = 10 t = 20 t = 30 t = 100

Re = 5000

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Re-effect:

ω = 0.05 ω = 0.25 ω = 0.60 ω = 0.55 ω = 0.45

w = 0.05 ω = 0.25 ω = 0.65 ω = 0.95 ω = 0.65

Re = 2000

Re = 5000

Re = 10,000

t=0t=0 t=10 t=30 t=75 t=100

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Re-effect in time

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Regenerative Transient Growth? Re = 5000

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Regenerative Transient Growth?

t = 100 t = 120 t = 140 t = 160 t = 180

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Some conclusions

• Turbulence induces and amplifies core fluctuations – amplitudes exceeding those of external perturbations.

• Several potential mechanisms of core transition / accelerated vortex decay studied.

• Circulation overshoot => centrifugal instability: amplifies perturbations, but inherently self-limiting.

• Weak “threads” can resonate with vortex core dynamics waves, but not strong perturbations.

•Transient growth: orders-of-magnitude amplification

•Strongest transient growth for bending waves.

Further study

Nonlinear transient growth, regenerative transient growth, vortex breakup and turbulence self-sustenance

References

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