DMD analysis of coherent structures in a
turbulent forced jet
S.S. Abdurakipov, V.M. Dulin, D.M. Markovich
Institute of Thermophysics, Novosibirsk, Russia
Novosibirsk State University, Novosibirsk, Russia
IT SB RAS
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Outline

Motivation and Objectives

Experimental setup and apparatus

Data post-processing

Results

Conclusions
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Motivation and Objectives

Motivation




It is well recognized (Crow & Champagne, 1971; Longmire & Eaton, 1992; Swanson &
Richards, 1997) that dynamics of large-scale vortices are crucial for heat and mass
transfer process in turbulent shear flows, such as jets. It is also well known that periodical
forcing is an effective method to control formation of vortices these flows. However, as it
was shown in Broze & Hussain, 1996, vortex formation in forced jets can be rather
distinctive (depending on forcing frequency and amplitude), since dynamics of the
nonlinear open system is strongly affected by a feedback from downstream events of
vortex roll-ups and pairings. Particle Image Velocimery has now became a standard
technique to measured spatial distributions of instantaneous velocity and is straight
forward to investigate properties of coherent structures.
A widely used technique for identifying coherent structures from velocity data is Proper
Orthogonal Decomposition (POD) (Lumley 1967; Sirovich 1987; Holmes, Lumley &
Berkooz 1996). The method determines the most energetic structures by diagonalizing the
covariance matrix of snapshots. However, the important information about temporal
evolution of coherent structures is ignored.
Dynamic Mode Decomposition (DMD) method for identifying coherent structures was
recently developed based on Koopman analysis of nonlinear dynamical system by Schmid
(2010). In the DMD method the snapshots are assumed to be generated by a linear
dynamical system, which implies that the extracted basis is characterized by growth rate
and frequency content of the snapshots.
Task

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The aim of the present work is to investigate dynamics of coherent structures in a
forced jet flow by applying Dynamic Mode Decomposition to high-repetition PIV
data.
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Experimental setup and apparatus

Open aerodynamic rig:

Air jet at Re = 4800, U0 = 5.0 m/s
 Contraction nozzle d = 15 mm. Free stream
turbulence <3%
 Periodical forcing. St = 0.5, St = 1.0.
 LDV probe for the nozzle exit: urms = 10% of
U0.

High-repetition PIV measurements (1kHz):

PCO 1200HS CMOS camera in double shutter
mode. Image pair with 40s shift. 900 s
between the pairs.
 Double-head Nd:YLF Pegasus PIV laser (2x10
mJ at 1.1 kHz)
 Synchronizing processor

Processing (600 image pairs)

Background removal from images
 Iterative cross-correlation algorithm with
continuous window shifting (Scarano, 2002).
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LDV
probe
Laser
CMOS Camera
Dumper
Loudspakers
Flow
seeder
Flowmeters
Air
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Data post-processing
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Dynamic Mode Decomposition (DMD)



V1N  u1 ,u2 ,u3 ,...,uN 
Ensembles of PIV velocity fields
Decomposition of data ensemble into
,where u i corresponds to ti
N 1
N 1
k 1
k 1

u( x, ti )   bk (ti )k ( x)   e
k
r

 iik ti
k ( x)
Can be done by finding global Eigen modes of A
~
ui 1  e At ui  Aui

According to Schmid (2010) eigen values of A are approximated by following the modified
Arnoldi method.
u N  a1u1  a2u2  aN 1u N 1  r  V1N -1a  r

The least-squares problem for minimization or ||r||:
V2N  AV1N 1   Au1 ,Au 2 ,Au3 ,...,Au N-1 
 N
N -1
V2  u 2 ,u3 ,...,u N   u 2 ,u3 ,...,V1 a  r


Is solved by QR decomposition
Eigenvalue problem of S:
Sy k  k y k ,

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V1N 1  QR  S  R1QT V2N or
Y   y1 ,...,y N 1 
DMD basis for a small time-shift :
  1 ,...,N 1  ,
 AV1N 1  V1N 1S  r  eT
  V1N -1Y
k  1
k 
a  R1QT u N
0
0
 
e  . 
.
. 
1 
 
Ln(k )
 kr  iik
2t
(Schmid , JFM, 2010)
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Application to synthetic database

Comparison of DMD versus
Proper Orthogonal
Decomposition method (Sirovich,
1987)
N
u ( x)  
k
n 1


 
i   aik
and
k
aik  u k ( x), i* ( x)

2
spat .
ue
y
 t -( )2
b
e
i
10 Hz
5 Hz
10 Hz
5 Hz
 max
spat .
cos(kx  t ) 
cos(kx / 3  t / 2)  
b = 0.01, k = π/b,
ω =2πf = 20π, σ =
0.4π,
dt = T/Nt, T = 2π/ω,
40% noise
  mn
 Hz 
Synthetic data:
y
 t -( )2
b
DMD
ank n ( x)
n ( x),m ( x)
Where
POD
Study of a forced turbulent jet
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Averaged flow structure
4
z
d
z
d
2
2
2
1
1
1
1
1
r d
1
0.9
0.8
0.7
0.6
0.5
0.4
0.3
0.2
0.1
3
2
1
1
3
4
2
2
0.5
St  1.0, urms  0.1U 0
3
3
0
z
d
4
3
3
0
4 St  0.52, u4rms  0.1U 0
Unfoced4flow
r d
0
0
0.5
1
0
r d
0
0.5
1
0
r d
0
0.5
1
0
r d
0
0.5
1
0
r d
0
0.5
1
0.06
0.05
U
U0
0.04
2

0.03
U 02

0.02
0.01
0
According to Broze and Hussain (1996),
Forcing at St = 0.52 => Amplification of vortices at
frequency of excitation
Forcing at St = 1.0 => Stable pairing with modulation of
second harmonic
DMD spectra
Unfoced flow
St  0.52, urms  0.1U 0
St  1.0, urms  0.1U 0

For the unforced flow thee harmonics were detected: fundamental St = 0.4, pairing f0/2 and
double pairing f0/4

Forcing at St = 0.52 => A strong peak at f0 was observed along with 2f0

Forcing at St = 1.0 => Several harmonics corresponding to modulation of pairing:
fundamental f0, sub-harmonic f0/2 and f1 + f2 = f0/2
The instantaneous velocity and the most powerful
DMD modes for St = 0.5
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The instantaneous velocity and the most powerful DMD modes
for St = 1.0
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Low-order reconstruction from most powerful DMD modes for
St = 0.5
u*r U 0
Low-order reconstruction of a sequence of
seven velocity fields (time step 0.9 ms)
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
u e
*
k
1

i1k t

1 ( x)  e
k
2

i2k t
2 ( x)  c.c.
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Low-order reconstruction from most powerful DMD modes for
St = 1.0
u*r U 0
Low-dimensional reconstruction of a sequence of
4k i4k t
3k i3k t
*
u e
4 ( x)  e
3 ( x)  c.c.
seven velocity fields (time step 0.9 ms)
Swirl strength criterion (Zhou and Adrian, 1999)
for vortex detection
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Conclusions

High-repetition PIV system was applied for estimation of a set of
instantaneous velocity fields in a forced turbulent jet. Three regimes
were considered:

1. Unforced flow with formation of ring-like vortices, their stable pairing
and double pairing process;
 2. Flow forced at St = 0.52, corresponding to the amplification of vortices
at the frequency of excitation;
 3. Flow forced at St = 1.0, resulting in a stable pairing of the forced
vortices with modulation of their magnitude.

The set of the velocity fields measured in the forced jet was
processed by a DMD algorithm. The procedure provided information
about dominant frequencies of velocity fluctuations in different flow
regions and about scales of the corresponding coherent structures.

It is concluded that superposition of relevant DMD modes (i.e.,
tangent approximation) satisfactory described nonlinear process of
coherent structures interaction, viz., pairing of the vortices and
modulation of their magnitude.
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Thank you for your attention!

Acknowledgements:
 This work was fulfilled under foundation by Government of Russian Federation (Grant
No. 11.G34.31.0035, Supervisor Ak. V.E. Zakharov)
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