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Studying the Topology and Dynamics of Elasto-inertial Channel Flow Turbulence Using the Invariants of the Velocity Gradient Tensor and Dynamic Mode Decomposition

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(1)

American Physical Society

Studying the Topology and Dynamics of

Elasto-inertial Channel Flow Turbulence Using

the Invariants of the Velocity Gradient Tensor

and Dynamic Mode Decomposition

J. Soria

1,5

, Y. Dubief

2

, V. Terrapon

3

& I. Moreno-Bermejo

4

1

Laboratory for Turbulence Research in Aerospace and Combustion,

Dept. of Mechanical and Aerospace Engineering, Monash University, Melbourne, Australia

2

School of Engineering, University of Vermont, Burlington, VT 05405, USA

3

Aerospace & Mechanical Engineering Department, University of Liège, Belgium

4

Center for Turbulence Research, Stanford University, CA, USA

5

Dept. of Aeronautical Engineering, King Abdulaziz University, Jeddah, Kingdom of Saudi Arabia

Vermont Advanced Computing Center

National Institutes of Health.

Marie Curie FP7 Career Integration Grant

Australian Research Council

(2)

1000

10000

Re

10

-3

10

-2

friction factor

Wi=100

Wi=700

MDR

Turbulent

Laminar

Introduction

Excitation of extensional sheet flow and elliptical

pressure redistribution of energy

Increase of extensional viscosity in

sheets

Formation of sheets of C

@

t

C + (u

· r) C

C

· (ru) + (ru)

t

· C

T

r

2

p = 2Q +

1

Re

r · (r · T)

Monday, 26 November 12

(3)

American Physical Society, 65th Annual Fall DFD Meeting, November 18 - 20, 2012, San Diego, California

Flow Topology

Chong et al. (1990) generalised the idea of critical point

theory by attaching the origin of a non-rotating,

translating coordinate system to every fluid particle

in this reference frame the flow at the origin is a critical

point

topological character of the flow pattern of the fluid

particle is governed by A

ij

= (VGT)

the topological character is Galilean Invariant

VGT has characteristic equation

λ

i

are the eigenvalues of A

ij

, P

A

, Q

A

and R

A

are the

(4)

Flow Topology

(Chong et al. 1990, Soria et al. 1994)

incompressible flows, invariants of VGT A

ij

:

local topology dependents only on

Q

A

and R

A

D

A

is the discriminant of A

ij

:

A A A A Monday, 26 November 12

(5)

American Physical Society, 65th Annual Fall DFD Meeting, November 18 - 20, 2012, San Diego, California

Flow Topology

(Chong et al. 1990, Soria et al. 1994)

A

ij

can be split:

S

ij

- rate-of-strain tensor

(symmetric ∴ real eigenvalues)

3 corresponding invariants (P

S

, Q

S

, R

S

)

α

1

, α

2

, α

3

are eigenvalues = principal strain rates s.t. α

1

≤ α

2

≤ α

3

W

ij

- rate-of-rotation tensor

(skew-symmetric ∴ complex eigenvalues)

3 corresponding invariants (P

W

, Q

W

, R

W

)

P

S

= P

W

= R

W

= 0

Q

S

is negative definite

Q

w

is positive definite

sgn(R

S

) = sgn(α

2

)

Truesdell (1954) introduced kinematic vorticity number

local measure of rotational strength to rate of irrotational stretching

of fluid element:

κ = ∞ (solid body rotation), κ = 0 (irrotational stretching)

A

ij

= S

ij

+ W

ij

(6)

JPDF of Q

W

- -Q

S

and its relationship to turbulence structure

(Perry & Chong (1994))

372

_ Q ,

1 . q . q .

Dissipation

- . ° ° - °

:.- :"

:: .-:

• . , . ~ . - . . . . , . ° . * ° - . - ° - ° o • . - . - - . . . - o O . - . - . ° . ° , -

: : : : : : : : : : : : : : : : : : : : : : :

, . - , ° . ~ ° - ,

: . : . : - : . : -

: - : . : - : . :

o . . . . o • - . - o - . .

: . : . : . : . : . : .

: - : . : . : . : . : .

: - : - i - : - : - :

- . , . - . - . - . - • , ° - . , ° o . - • • . . . . .

A,E. PERRY AND M,S. CHONG

" ' " ' " " ' " ' " ' " " "

Vortex

tubes

: : : : : : : : : : : : : : : : : : :

• . - - . - ° - . ° o . - . - . - - ° - - . - . - * . - . - . . ° o • • • • . * * . - . - . - . 1 - . - ° - . - , - , , . - , - , - °

¢

:2::::::::::::::::"

: : : : : : : : : : : : : : : : : : ] : : : . . - . - . - . - . - : - : - : - . . .

. .

• . - . ° . - . - . - . . ° - . * ° . o * . . . o . ° . ' . . . ' . . . - . . . - . o . . . ' . ° - . - . - . - . - . - . ° . ° . - o * . * . - . - - . - . ° • - . - . - . - . - . - - . - . . - * * . - - . " - ' . ' . ' . ' . ° - ' . ° - ° - ' - ' . ~ - ' - ' - ' - ' - ' - ' . ' - ' - ' - ' - ' - ' - ' - ' - ° . ' .

Q~,

= 7 w i j w i j

a

,-~

Enstropy density

Fig. 13.

Physical interpretation of various regions in the

- Q ~ vs. Q~,

plot•

-Q,

Fig. 14.

o " . , ° ~ ' ~ ° . x o

.

f -

Plot of -Q~ vs. Q~o for compressible mixing layer computed by Chen (1990).

Figure 15 shows a plot of

-Qs

versus Qw from some preliminary work on

turbulent boundary layers using the DNS data of Spalart. Although the plot has

poor resolution (the figure is a blow-up from another plot) it indicates that most

(for “Newtonian Fluid”)

(7)

American Physical Society, 65th Annual Fall DFD Meeting, November 18 - 20, 2012, San Diego, California

Results:

Conditional volume integrals when D

A

> |D

A

(given)|

Enstrophy due to focal regions

10

−15

10

−10

10

−5

10

0

10

−2

10

−1

10

0

D

a

/<Q

w

>

3

∫Q

w

( D

a

/<Q

w

>

3

> D

a

/<Q

w

>

3

(given)) dV

Re = 500

Re = 1500

Re = 3000

Re = 5000

(8)

Results:

Conditional volume integrals when D

A

> |D

A

(given)|

“Dissipation” of mechanical energy due to focal regions

10

−15

10

−10

10

−5

10

0

10

−2

10

−1

10

0

D

a

/<Q

w

>

3

∫ −

Q

s

( D

a

/<Q

w

>

3

> D

a

/<Q

w

>

3

(given)) dV

Re = 500

Re = 1500

Re = 3000

Re = 5000

Monday, 26 November 12

(9)

American Physical Society, 65th Annual Fall DFD Meeting, November 18 - 20, 2012, San Diego, California

Results:

JPDF R

A

- Q

A

R a/<Qw> 3/2 Q a /<Q w > −2 −1.5 −1 −0.5 0 0.5 1 1.5 2 x 10−3 −0.04 −0.03 −0.02 −0.01 0 0.01 0.02 0.03 0.04 R a/<Qw> 3/2 Q a /<Q w > −6 −4 −2 0 2 4 6 x 10−3 −0.06 −0.04 −0.02 0 0.02 0.04 0.06 Ra/<Qw>3/2 Q a /<Q w > −0.01 −0.005 0 0.005 0.01 −0.1 −0.05 0 0.05 0.1 Ra/<Qw>3/2 Q a /<Q w > −0.02 −0.01 0 0.01 0.02 −0.1 −0.05 0 0.05 0.1

Re = 500

Re = 1500

Re = 3000

Re = 5000

(10)

Results:

Expected value of polymer stretch conditioned on (R

A

, Q

A

) for Re = 5000

R

a

/<Q

w

>

3/2

Q

a

/<Q

w

>

−0.02

−0.01

0

0.01

0.02

−0.1

−0.05

0

0.05

0.1

0 0.5 1 1.5 2 2.5 3 3.5 4

R

a

/<Q

w

>

3/2

Q

a

/<Q

w

>

−0.02

−0.01

0

0.01

0.02

−0.1

−0.05

0

0.05

0.1

0.4 0.45 0.5 0.55 0.6 0.65

JPDF R

A

vs Q

A Monday, 26 November 12

(11)

American Physical Society, 65th Annual Fall DFD Meeting, November 18 - 20, 2012, San Diego, California

Results:

JPDF Q

W

- -Q

S

Qw/<Qw> − Q s /<Q w > 0 0.5 1 1.5 2 2.5 3 0 0.5 1 1.5 2 2.5 3 Qw/<Qw> − Q s /<Q w > 0 0.5 1 1.5 2 2.5 3 0 0.5 1 1.5 2 2.5 3 Qw/<Qw> − Q s /<Q w > 0 0.5 1 1.5 2 2.5 3 0 0.5 1 1.5 2 2.5 3 Qw/<Qw> − Q s /<Q w > 0 0.5 1 1.5 2 2.5 3 0 0.5 1 1.5 2 2.5 3

Re = 500

Re = 1500

Re = 3000

Re = 5000

372 _ Q , 1 . q . q . Dissipation - . ° ° - ° :.- :" :: .-: • . , . ~ . - . . . . , . ° . * ° - . - ° - ° o • . - . - - . . . - o O . - . - . ° . ° , - : : : : : : : : : : : : : : : : : : : : : : : , . - , ° . ~ ° - , : . : . : - : . : - : - : . : - : . : o....o • - . - o - . . : . : . : . : . : . : . : - : . : . : . : . : . : - : - i - : - : - : - . , . - . - . - . - • , ° - . , ° o . - • • . ....

A,E. PERRY AND M,S. CHONG

" ' " ' " " ' " ' " ' " " " Vortex tubes : : : : : : : : : : : : : : : : : : : • . - - . - ° - . ° o . - . - . - - ° - - . - . - * . - . - . . ° o • • • • . * * . - . - . - . 1 - . - ° - . - , - , , . - , - , - ° ¢ :2::::::::::::::::" : : : : : : : : : : : : : : : : : : ] : : : . . - . - . - . - . - : - : - : - . . . . . • . - . ° . - . - . - . . ° - . * ° . o * . . . o . ° . ' . . . ' . . . - . . . - . o . . . ' . ° - . - . - . - . - . - . ° . ° . - o * . * . - . - - . - . ° • - . - . - . - . - . - - . - . . - * * . - - . " - ' . ' . ' . ' . ° - ' . ° - ° - ' - ' . ~ - ' - ' - ' - ' - ' - ' . ' - ' - ' - ' - ' - ' - ' - ' - ° . ' . Q~, = 7 w i j w i j a ,-~ Enstropy density

Fig. 13. Physical interpretation of various regions in the - Q ~ vs. Q~, plot•

-Q,

Fig. 14.

o " . , ° ~ ' ~ ° . x o

.

f -

Plot of -Q~ vs. Q~o for compressible mixing layer computed by Chen (1990). Figure 15 shows a plot of -Qs versus Qw from some preliminary work on turbulent boundary layers using the DNS data of Spalart. Although the plot has poor resolution (the figure is a blow-up from another plot) it indicates that most

(12)

Results:

JPDF Σ

- Q

W

Re = 500

Re = 1500

Re = 3000

Re = 5000

σ/<Qw>1/2 Q w /<Q w > −0.02 −0.015 −0.01 −0.0050 0 0.005 0.01 0.015 0.02 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 0.5 σ/<Qw>1/2 Q w /<Q w > −0.1 −0.05 0 0.05 0 0.5 1 1.5 σ/<Qw>1/2 Q w /<Q w > −0.1 −0.05 0 0.05 0.1 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 σ/<Qw>1/2 Q w /<Q w > −0.1 −0.05 0 0.05 0.1 0 0.2 0.4 0.6 0.8 1 1.2 Monday, 26 November 12

(13)

American Physical Society, 65th Annual Fall DFD Meeting, November 18 - 20, 2012, San Diego, California

Summary

in the transition from laminar regime (Re = 500) focal regions

occupy ~57% of the volume containing ~56% of the enstrophy and

“dissipate” ~57% of the mechanical energy

while in the EIT regime (Re = 5000) they occupy ~64% of the

volume containing ~63% of the enstrophy and “dissipate” ~64% of

the mechanical energy

during the transition form laminar to the EIT regime, the JPDF of R

A

v. Q

A

evolves from a somewhat symmetric shape around the 2-D

flow axis (R

A

= 0) to the more tear-drop shape but which is different

to that found in Newtonian turbulent flows

throughout the transition form laminar to the EIT regime the

dominant structure of the flow is sheet like as evidenced by the

JPDF of Q

w

v. -Q

s

polymer stretch in the EIT regime exhibits minima which are UFC

topology and lie along the null discriminant which represents

Figure

Fig.  13.  Physical  interpretation  of  various  regions  in the  - Q ~   vs.  Q~,  plot•
Fig.  13.  Physical  interpretation  of  various  regions  in the - Q ~   vs.  Q~, plot•

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