Studying the Topology and Dynamics of Elasto-inertial ... · 5Dept. of Aeronautical Engineering, King Abdulaziz University, Jeddah, Kingdom of Saudi Arabia Vermont Advanced Computing

American Physical Society65th Annual Fall DFD MeetingNovember 18 - 20, 2012San Diego, California

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 41Laboratory for Turbulence Research in Aerospace and Combustion,

Dept. of Mechanical and Aerospace Engineering, Monash University, Melbourne, Australia2School of Engineering, University of Vermont, Burlington, VT 05405, USA

3Aerospace & Mechanical Engineering Department, University of Liège, Belgium4Center for Turbulence Research, Stanford University, CA, USA

5Dept. 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

Part of this research was conducted during the 2012 CTR Summer Program at Stanford

Monday, 26 November 12

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

1000 10000Re

10-3

10-2

frict

ion

fact

or

Wi=100Wi=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@tC+ (u ·r)C

C · (ru) + (ru)t ·C�Tr2p = 2Q+1� �

Rer · (r ·T)



Flow TopologyChong et al. (1990) generalised the idea of critical point theory by attaching the origin of a non-rotating, translating coordinate system to every fluid particlein this reference frame the flow at the origin is a critical pointtopological character of the flow pattern of the fluid particle is governed by Aij = (VGT)the topological character is Galilean InvariantVGT has characteristic equation

λi are the eigenvalues of Aij, PA, QA and RA are the 1st, 2nd and 3rd tensor invariants



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

incompressible flows, invariants of VGT Aij:

local topology dependents only on QA and RA

DA is the discriminant of Aij:

A

A

A A



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

Aij can be split:Sij - rate-of-strain tensor (symmetric ∴ real eigenvalues)

3 corresponding invariants (PS, QS, RS)α1, α2, α3 are eigenvalues = principal strain rates s.t. α1 ≤ α2 ≤ α3

Wij - rate-of-rotation tensor (skew-symmetric ∴ complex eigenvalues)

3 corresponding invariants (PW, QW, RW)PS = PW = RW = 0QS is negative definiteQw is positive definitesgn(RS) = sgn(α2)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)

Aij = Sij + Wij

?



JPDF of QW - -QS 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~, a = 7 w i j w i j ,-~ 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”)



Results:Conditional volume integrals when DA > |DA (given)|

Enstrophy due to focal regions

10−15 10−10 10−5 10010−2

10−1

100

Da/<Qw>3

∫Qw

( Da/<

Qw

>3 > D

a/<Q

w>3 (g

iven

)) dV

Re = 500Re = 1500Re = 3000Re = 5000



Results:Conditional volume integrals when DA > |DA (given)|

“Dissipation” of mechanical energy due to focal regions

10−15 10−10 10−5 10010−2

10−1

100

Da/<Qw>3

∫ −Q

s( Da/<

Qw

>3 > D

a/<Q

w>3 (g

iven

)) dV

Re = 500Re = 1500Re = 3000Re = 5000



Results:JPDF RA - QA

Ra/<Qw>3/2

Qa/<

Qw>

−2 −1.5 −1 −0.5 0 0.5 1 1.5 2x 10−3

−0.04

−0.03

−0.02

−0.01

0

0.01

0.02

0.03

0.04

Ra/<Qw>3/2

Qa/<

Qw>

−6 −4 −2 0 2 4 6x 10−3

−0.06

−0.04

−0.02

0

0.02

0.04

0.06

Ra/<Qw>3/2

Qa/<

Qw>

−0.01 −0.005 0 0.005 0.01

−0.1

−0.05

0

0.05

0.1

Ra/<Qw>3/2

Qa/<

Qw>

−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



Results:Expected value of polymer stretch conditioned on (RA, QA) for Re = 5000

Ra/<Qw>3/2

Qa/<

Qw>

−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

Ra/<Qw>3/2

Qa/<

Qw>

−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

0.65JPDF RA vs QA



Results:JPDF QW - -QS

Qw/<Qw>

−Qs/<Qw>

0 0.5 1 1.5 2 2.5 30

0.5

1

1.5

2

2.5

3

Qw/<Qw>

−Qs/<Qw>

0 0.5 1 1.5 2 2.5 30

0.5

1

1.5

2

2.5

3

Qw/<Qw>

−Qs/<Qw>

0 0.5 1 1.5 2 2.5 30

0.5

1

1.5

2

2.5

3

Qw/<Qw>

−Qs/<Qw>

0 0.5 1 1.5 2 2.5 30

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~, a = 7 w i j w i j ,-~ 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



Results:JPDF Σ - QW

Re = 500 Re = 1500

Re = 3000 Re = 5000

σ/<Qw>1/2

Qw/<Q

w>

−0.02 −0.015 −0.01 −0.005 0 0.005 0.01 0.015 0.020

0.05

0.1

0.15

0.2

0.25

0.3

0.35

0.4

0.45

0.5

σ/<Qw>1/2

Qw/<Q

w>

−0.1 −0.05 0 0.050

0.5

1

1.5

σ/<Qw>1/2

Qw/<Q

w>

−0.1 −0.05 0 0.05 0.10

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

σ/<Qw>1/2

Qw/<Q

w>

−0.1 −0.05 0 0.05 0.10

0.2

0.4

0.6

0.8

1

1.2



Summaryin 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 energywhile in the EIT regime (Re = 5000) they occupy ~64% of the volume containing ~63% of the enstrophy and “dissipate” ~64% of the mechanical energyduring the transition form laminar to the EIT regime, the JPDF of RA v. QA evolves from a somewhat symmetric shape around the 2-D flow axis (RA = 0) to the more tear-drop shape but which is different to that found in Newtonian turbulent flowsthroughout the transition form laminar to the EIT regime the dominant structure of the flow is sheet like as evidenced by the JPDF of Qw v. -Qs

polymer stretch in the EIT regime exhibits minima which are UFC topology and lie along the null discriminant which represents axisymmetric contraction topology


Studying the Topology and Dynamics of Elasto-inertial ... · 5Dept. of Aeronautical Engineering, King Abdulaziz University, Jeddah, Kingdom of Saudi Arabia Vermont Advanced Computing

Documents