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DMD analysis of coherent structures in a DMD analysis of coherent structures in a turbulent forced jetturbulent forced jet

S.S. Abdurakipov, V.M. Dulin, D.M. MarkovichS.S. Abdurakipov, V.M. Dulin, D.M. Markovich

Institute of ThermophysicsInstitute of Thermophysics, , NovosibirskNovosibirsk, , RussiaRussia

Novosibirsk State University, Novosibirsk, RussiaNovosibirsk State University, Novosibirsk, Russia

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Outline

Motivation and ObjectivesMotivation and Objectives

Experimental setup and apparatusExperimental setup and apparatus

Data post-processing Data post-processing

ResultsResults

ConclusionsConclusions

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Motivation and Objectives

MotivationMotivation 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.

TaskTask 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: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 PIVHigh-repetition PIV measurements (1kHz):measurements (1kHz): PCO 1200HS CMOS camera in double shutter

mode. Image pair with 40s 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).

CMOS Camera

Laser

Flowmeters

Air

Loudspakers

Flow

seeder

Dumper

LDV

probe

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Data post-processingData post-processing

Ensembles of PIV velocity fields ,where corresponds to

Decomposition of data ensemble into

Can be done by finding global Eigen modes of A

According to Schmid (2010) eigen values of A are approximated by following the modified Arnoldi method.

The least-squares problem for minimization or ||r||:

Is solved by QR decomposition Eigenvalue problem of S:

DMD basis for a small time-shift :

Dynamic Mode Decomposition (DMD)

iitA

i Ae uuu

~

1

1 1 11 2 or N T N T

NV QR S R Q V R Q a u

kS ,k ky y

1kki

kr

kk i

t

Ln

2

)(

YV 1-N1 1 1,..., ,N

11 N,...,Y yy

(Schmid , JFM, 2010) 6IT SB RAS

1 1

1 1

( , ) ( ) ( ) ( )k kr i i

N Ni t

i k i k kk k

x t b t x e x

u

-11 2 1 1 1

NN N Na a a V 1 2u u u u r a r

N1 2 3 N, , ,...,V 1u u u u

N 12 1 1 2 3 N-1

N -12 2 3 N 2 3 1

, , ,...,

, ,..., , ,...,

N

N

V AV A A A A

V V

u u u u

u u u u u a r

1 11 1N N TAV V S r e

00...1

e

iu it

Comparison of DMD versus Proper Orthogonal Decomposition method (Sirovich, 1987)

Where

and

Synthetic data:

2

2

-( )

-( )

cos( )

cos( / 3 / 2)

ytb

ytb

e kx t

e kx t

u

b = 0.01, k = π/b,

ω =2πf = 20π, σ = 0.4π,

dt = T/Nt, T = 2π/ω,

40% noise

0 4 8 12 16 200

0.1

0.2

0.3

0.4

0.5

i

l/еili

j1

j2

j3

j4

-0.05 -0.03 -0.01 0.01 0.03 0.05-0.01

0

0.01 j1

-0.05 -0.03 -0.01 0.01 0.03 0.05-0.01

0

0.01 j2

-0.05 -0.03 -0.01 0.01 0.03 0.05-0.01

0

0.01 j3

-0.05 -0.03 -0.01 0.01 0.03 0.05-0.01

0

0.01 j4

-0.05 -0.03 -0.01 0.01 0.03 0.05-0.01

0

0.01-0.05 -0.03 -0.01 0.01 0.03 0.05

-0.01

0

0.01

-0.05 -0.03 -0.01 0.01 0.03 0.05-0.01

0

0.01

-0.05 -0.03 -0.01 0.01 0.03 0.05-0.01

0

0.01

1

( ) ( )N

k kn n

n

x a xj

u

.( ), ( )n m mnspatx xj j

POD DMD

i

Hz

2maxk

i ik

al *

.( ), ( )k k

i ispat

a u x xj

Application to synthetic database

10 Hz10 Hz

5 Hz5 Hz

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Study of a forced turbulent jetStudy of a forced turbulent jet

Averaged flow structure

0 0 .5 10

1

2

3

4

0 0 .5 10

1

2

3

4

0 0 .5 10

1

2

3

4

0 0 .5 10

1

2

3

4

0 0 .5 10

1

2

3

4

0 0 .5 10

1

2

3

4Unfoced flow 00.52, 0.1rmsSt u U 01.0, 0.1rmsSt u U

z

d

z

d

z

d

r d r d r d r d r d r d

0

0 .0 1

0 .0 2

0 .0 3

0 .0 4

0 .0 5

0 .0 6

0 .10 .20 .30 .40 .50 .60 .70 .80 .91

0

U

U

2

20U

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

0 200 400 6000

200

400

600

800

1000300

150

||||

[Hz]

4

3

115

2

35

1

DMD spectra

Unfoced flow 00.52, 0.1rmsSt u U 01.0, 0.1rmsSt u U

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

0 200 400 6000

400

800

1200

1600

2000

||||

[Hz]

170

340

1

2

0 200 400 6000

20

40

60

80

125

30

61

[Hz]

||||

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The instantaneous velocity and the most powerful DMD modes for St = 0.5

0 0.5 10

0.5

1

1.5

2

2.5

3

3.5

4

4.5

5 170 Hz

[Re ]

0 0.5 10

0.5

1

1.5

2

2.5

3

3.5

4

4.5

5 170 Hz

[Im ]

0 0.5 10

0.5

1

1.5

2

2.5

3

3.5

4

4.5

5

[Re ]

340 Hz

0 0.5 10

0.5

1

1.5

2

2.5

3

3.5

4

4.5

5

[Im ]

340 Hz

0 0.5 10

0.5

1

1.5

2

2.5

3

3.5

4

4.5

5z/d

00.10.20.30.40.50.60.70.80.911.11.21.3

x U0

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The instantaneous velocity and the most powerful DMD modes for St = 1.0

0 0.5 10

0.5

1

1.5

2

2.5

3

3.5

4

4.5

5z/d

40 Hz

[Re ]1

0 0.5 10

0.5

1

1.5

2

2.5

3

3.5

4

4.5

5z/d

40 Hz

[Im ]1

0 0.5 10

0.5

1

1.5

2

2.5

3

3.5

4

4.5

5

[Re ]2

z/d

r/d

120 Hz

0 0.5 10

0.5

1

1.5

2

2.5

3

3.5

4

4.5

5 150 Hzz/d

r/d

[Re ]3

0 0.5 10

0.5

1

1.5

2

2.5

3

3.5

4

4.5

5z/d

r/d

120 Hz

[Im ]

0 0.5 10

0.5

1

1.5

2

2.5

3

3.5

4

4.5

5 150 Hzz/d

r/d

[Im ]3

0 0.5 10

0.5

1

1.5

2

2.5

3

3.5

4

4.5

5z/d

r/d

300 Hz

[Re ]4

0 0.5 10

0.5

1

1.5

2

2.5

3

3.5

4

4.5

5z/d

r/d

300 Hz

[Im ]4

0 0.5 10

0.5

1

1.5

2

2.5

3

3.5

4

4.5

5

r/d

z/d

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Low-order reconstruction from most powerful DMD modes for St = 0.5

Low-order reconstruction of a sequence of seven velocity fields (time step 0.9 ms)

0 0.5 10

0.5

1

1.5

2

2.5

3

3.5

4

4.5

5 t = 0 msz/d

r/d 0 0.5 10

0.5

1

1.5

2

2.5

3

3.5

4

4.5

5 t = 0.9 msz/d

r/d 0 0.5 10

0.5

1

1.5

2

2.5

3

3.5

4

4.5

5 t = 1.8 msz/d

r/d 0 0.5 10

0.5

1

1.5

2

2.5

3

3.5

4

4.5

5 t = 2.7 msz/d

r/d 0 0.5 10

0.5

1

1.5

2

2.5

3

3.5

4

4.5

5 t = 3.6 msz/d

r/d 0 0.5 10

0.5

1

1.5

2

2.5

3

3.5

4

4.5

5 t = 4.5 msz/d

r/d 0 0.5 10

0.5

1

1.5

2

2.5

3

3.5

4

4.5

5t = 5.4 ms

z/d

r/d

x U0

02

x U0

-0.26-0.22-0.18-0.14-0.1-0.06-0.020.020.060.10.140.180.220.26

1 1 2 2*1 2( ) ( ) . .

k k k ki t i te x e x c c

u

*0r Uu

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Low-order reconstruction from most powerful DMD modes for St = 1.0

Low-dimensional reconstruction of a sequence of seven velocity fields (time step 0.9 ms)

Swirl strength criterion (Zhou and Adrian, 1999) for vortex detection

0 0.5 10

0.5

1

1.5

2

2.5

3

3.5

4

4.5

5z/d t = 0 ms

0 0.5 10

0.5

1

1.5

2

2.5

3

3.5

4

4.5

5z/d

t = 0.9 ms

0 0.5 10

0.5

1

1.5

2

2.5

3

3.5

4

4.5

5z/d

r/d

t = 1.8 ms

0 0.5 10

0.5

1

1.5

2

2.5

3

3.5

4

4.5

5z/d

r/d

t = 2.7 ms

0 0.5 10

0.5

1

1.5

2

2.5

3

3.5

4

4.5

5

r/d

z/d t = 3.6 ms

0 0.5 10

0.5

1

1.5

2

2.5

3

3.5

4

4.5

5

r/d

z/dt = 4.5 ms

02

x U0

0 0.5 10

0.5

1

1.5

2

2.5

3

3.5

4

4.5

5

r/d

-0.17-0.14-0.11-0.08-0.05-0.020.010.040.070.10.130.16

x U0t = 5.4 ms

4 4 3 3*4 3( ) ( ) . .

k k k ki t i te x e x c c

u

*0r Uu

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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!Thank you for your attention!

Acknowledgements:Acknowledgements: This work was fulfilled under foundation by Government of Russian Federation (Grant

No. 11.G34.31.0035, Supervisor Ak. V.E. Zakharov)