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
3Aerospace & 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
1000
10000
Re
10
-310
-2friction 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 12American 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
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 12American 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
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”)
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
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 12American 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 0.1Re = 500
Re = 1500
Re = 3000
Re = 5000
Results:
Expected value of polymer stretch conditioned on (R
A
, Q
A
) for Re = 5000
R
a/<Q
w>
3/2Q
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 4R
a/<Q
w>
3/2Q
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 0.65JPDF R
Avs Q
A Monday, 26 November 12American 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 3Re = 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
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 12American Physical Society, 65th Annual Fall DFD Meeting, November 18 - 20, 2012, San Diego, California