Computing Internal and External Flows Undergoing Instability and

Bifurcations

V. V. S. N. Vijay, Neelu Singh and T. K. Sengupta

High Performance Computing Laboratory

Department of Aerospace Engineering

I. I. T. Kanpur, Kanpur 208016

INDIA

Organization of The Presentation

Instability, Bifurcation and Multi-Modal Dynamics. Hopf- bifurcation and Landau-Stuart-Eckhaus (LSE) Equation. Numerical Methods: Aliasing Error, Dispersion Error. Evidence of Multiple Hopf-Bifurcations for Flow Past a

Cylinder. Multiple Bifurcations in Lid-Driven Cavity Flow. Dynamical System Approach - Proper Orthogonal

Decomposition (POD). Free Stream Turbulence (FST) : Effects and Its Modeling. Conclusions.

Bifurcation and Instabilities

Flow Past a Circular Cylinder, Re = 60 Flow Inside a Square Cavity, Re = 8500

Landau Equation and Hopf-Bifurcation

If disturbance velocity is given by

(1) And if its amplitude is given by

(2)

then, linearized evolution equation for amplitude is

(3)

Landau (1944) proposed a corresponding nonlinear evolution equation

(4)

tit ireconsttA .

22

2 Adt

Adr

422

2 AlAdt

Adrr

Landau Equation and Hopf-Bifurcation

Above and near Recr,

(5) For Eq. (4), an equilibrium exists (using Eq. (5)) that is given by

(6)

Variation of |Ae| beyond Recr is parabolic and is symptomatic of Hopf-bifurcation.

termsorderhigherK crr ReRe

rcre lkA /ReRe2||

Re

|Ae|

Re = Recr

0

Analysis of Numerical Methods

Non-dimensional effectiveness of convection terms

Dissipation Discretization

Error Dynamics for Convection Discretization

Evidences of Multiple Modes in Flow Past a Cylinder

Present Computations, Re = 60 Expt. by Strykowski (1986), Re = 49

Presence of multiple modes is evident in both the cases.

Multiple Hopf-Bifurcations and Its Modeling in Flow past a Cylinder

cre kkkkA ReRe;|| 44

33

221

84.615-0.217.6-9.5825.7133 -250

63.868-7.2105-5124.480 -133

51.934136-67492.67.6951.93 – 80.0

Recrk4 x 109k3 x 108k2 x 106k1 x 104Re

Strykowski’s Data (1986)

Further Evidences of Multiple Bifurcations in Flow Past a Cylinder

Vortex shedding began at Re = 65 in

Homann’s experiment. The first bifurcation was suppressed by

use of highly viscous oil as working media. Provansal et. al (1987) also have reported

different critical Reynolds number in

the same tunnel.

Results due to Homann (1936)

6911 cm10.0

630.8 cm12.5

580.6 cm 16.7

RecrDL/D

Multiple Bifurcations in Lid-Driven Cavity Flow

Proper Orthogonal Decomposition (POD) of DNS Data

A spatio-temporal fluid dynamical system can be analyzed by POD. We project the space-time dependent field into Fourier space with definitive

energy/ enstrophyenergy/ enstrophy content. Disturbance vorticity is represented as an ensemble of M snapshots

The covariance matrix as given below

The eigenvalues and eigenvectors of [R] provide the enstrophy content and

the POD modes, respectively.

M

mmm rtatr

1

' )()(,

NjitrtrM

RM

mmjmiij ,,2,1,,','

1

1

Flow Past a Circular Cylinder : Cumulative Enstrophy Distribution for Different Time Range

Eigen-modes

Cum

ulat

ive

Ens

trop

hyC

onte

nt

10 20 30

0.4

0.5

0.6

0.7

0.8

0.9

1

t = 200 - 430t = 350 - 430

[1]

{2}

{5}

{6}

[5]

{1}

[2]

[3]

[6]

t

'(0.504

4D,0)

100 200 300 400

-0.8

-0.4

0

0.4

0.8

Re = 60

Eigen-modes

Cum

ulat

ive

Ens

trop

hyC

onte

nt

10 20 30

0.4

0.5

0.6

0.7

0.8

0.9

1

t = 200 - 430t = 350 - 430

[1]

{2}

{5}

{6}

[5]

{1}

[2]

[3]

[6]

t

'(0.504

4D,0)

100 200 300 400

-0.8

-0.4

0

0.4

0.8

Re = 60

Flow in LDC : Cumulative Enstrophy Distribution for Different Time Range

Leading Eigenvalues For Flow Past a Cylinder Obtained by POD of Enstrophy for Re = 60

----

0.0026040.021043

0.0026040.021151

0.0367050.04038

0.0371150.04224

0.2809160.17497

0.2832520.17554

--0.48416

2.4149351.61711

2.4975581.66274

Eigenvalues (t = 351-430)Eigenvalues ( t = 200-430)

Leading Eigenvalues for Lid-Driven Cavity Flow Obtained by POD of Enstrophy for Re = 8500

----

0.00004470.0002115

0.00005480.0002225

0.0002899--

0.00032810.0003878

0.0012810.0020369

0.0193140.013615

0.0235610.016495

Eigenvalues (t = 300-500)Eigenvalues (t = 200-500)

Relating LSE Equation with POD Modes

The LSE modes are related to the POD modes by,

For example, two leading LSE modes are governed by,

These two constitute stiff-differential equations that can be reformulated as

where

2

2112

2

1111111 AAAAA

dt

Ad

2

2222

2

1221222 AAAAA

dt

Ad

2

22

2

11

1AA

dt

Yd

Y

22122211112121 ;;; AAY

l

ljjjjjjj taitaA 212212 ;)]()([

)]()([2

1212 xixf jj

jj

Amplitude Functions of POD Modes for Flow Past a Cylinder for Re = 60

Amplitude Functions of POD Modes for Flow in a Square-Cavity for Re = 8500

Leading Eigenfunctions of POD Modes for Flow Past a Cylinder for Re = 60 and t : 200-430

Min = -19.85Max = 16.92

Mode 2 (2)

Min = -7.22Max = 7.23

Mode 3 (3)

Min = -13.93Max = 20.01

Mode 1 (1)

Leading Eigenfunctions of POD Modes for Flow Past a Cylinder for Re = 60 and t : 200-430

(contd.)

Min = -5.58Max = 5.58

Mode 5 (5) Min = -5.62Max = 5.61

Mode 6 (6)

Min = -2.50Max = 2.29

Mode 7 (7) Min = -2.36Max = 2.44

Mode 8 (8)

Multi-Modal Feature in the Wake

Vorticity variation and FFT of

vorticity data shown at different

streamline positions

( 1.5D, 3D, 6D and 10D )

Note the presence of additional

modes seen in FFT of data at

6D and 10D.

t

Vor

ticity

100 150 200 250 300 350 400

-1

-0.5

0

0.5

1

(d)

FrequencyA

mpl

itude

(lo

gsc

ale)

0 0.5 1 1.5 2-2

-1

0

1

2

10 D

Vor

ticity

100 150 200 250 300 350 400

-1

-0.5

0

0.5

1

(c)

Am

plitu

de(

log

scal

e)

0 0.5 1 1.5 2-2

-1

0

1

2

6 D

Vor

ticity

100 150 200 250 300 350 400

-1

-0.5

0

0.5

1

(b)

Am

plitu

de(

log

scal

e)

0 0.5 1 1.5 2-2

-1

0

1

2

3 D

Vor

ticity

100 150 200 250 300 350 400

-1

-0.5

0

0.5

1

(a)

Am

plitu

de(

log

scal

e)

0 0.5 1 1.5 2 2.5-2

-1

0

1

2

1.5 D

Vortex Topology Inside a Lid-Driven Square Cavity

Leading Eigenfunctions of POD Modes for Flow in a Square-Cavity for Re = 8500

Higher Eigenfunctions for LDC Flow for Re = 8500

Different Critical Reynolds Number Reported in Different Facilities for Flow Past a

Circular Cylinder

Different Recr have been reported :

Batchelor (1988) – 40

Landau & Lifshitz (1959) - 34

Reported by Critical Reynolds Number (Recr)

Kovasznay (1949) 40

Strykowski, Sreenivasan & Olinger (1990)

Between 45 - 46

Schumm, Berger & Monkewitz (1994)

46.7

Roshko (1954) 50

Kiya et al. (1982) 52

Tordella & Cancelli (1991) 53

Homann (1936) 65.3

Extrinsic or Intrinsic Dynamics ?D = 5mm, U= 17cm/s, Re = 53 D = 1.8mm, U= 46.9cm/s, Re = 53

3 4 5 6 7 80

1

2

3

Frequency

Amplitude

0 100 200 300 400

10-1

100

101

18 20 22 24 26 28 30 32 34

0.1

0.2

0.3

Frequency

Amplitude

0 100 200 300 400

10-1

100

101

Note: The input disturbance at the Strouhal number on the left frame is one order of magnitude larger as compared to the case on the right.

Modeling Free Stream Turbulence (FST)

Higher order statistics are modeled by the 1st-order moving-average time series model (Fuller,1978) with the low frequency noise added separately.

FST is modeled by the following equation: where the first two terms are given by Gaussian distribution and the

last term represents low frequency component of noise and u’ is the stream-wise disturbance component of velocity.

FFT of Experimental and Synthetic FST Data

Experimental Disturbance Velocity Synthetic Disturbance Velocity

Frequency

Am

plit

ud

e

0 50 100 150 200

10-3

10-2

10-1

100

101

Frequency0 50 100 150 200

10-3

10-2

10-1

100

101

Effect of FST on Results

Variation of lift and drag coefficient with time with and without FST for Re = 100

Comparison of saturation amplitude of lift coefficient (peak-to-peak) for the cases of

without and with FST (Tu = 0.06%)

Time

Cl

0 50 100 150 200 250

-0.4

-0.2

0

0.2

0.4No FSTFST=0.06%Tu

Re = 100

Time

Cd

0 50 100 150 200 250

-1.3

-1.2

-1.1

No FSTFST=0.06%Tu

Re = 100

Note: The noise causes early onset of asymmetry and very high frequency fluctuations in drag data.

Re

|Ae|

50 100 150 200 2500

0.1

0.2

0.3

0.4

0.5

0.6 Present Results (No FST) By DNSPresent Results (No FST) By Eq. 4.1Present Results (FST = 0.06% Tu)Norberg (2003)

Effect of Different Levels of FST

Dynamic range increases directly with Tu. Higher than the shown levels, could display completely different dynamics

that is also indicated by the early onset time implying separation bubbles

not allowed to form as in no-FST case.

Streamfunction Contours for Re = 50, With and Without FST

At this sub-critical Re, symmetric wake bubble is formed for no-FST case.

In the presence of FST, asymmetry builds up slowly from the tip of the bubble.

For higher FST levels, interface of the separation bubble can suffer Kelvin-Helmholtz instabilityKelvin-Helmholtz instability interrupting vortex street formation.

Conclusions Multiple Hopf-bifurcations have been shown computationally

as seen in Ae vs Re plots for both the cases.

For the cylinder, we have identified the second bifurcation at Re = 63.86, that is closer to value reported by Homann (1936) at Re = 65.3.

Multi-modal behaviour is explained with the help of POD analysis for both the flows. POD analysis is also used to explain the Landau-Stuart-Eckhaus Equation.

We have identified anomalous modes during the instability phase for both the flows from the POD amplitude functions. These modes do not satisfydo not satisfythe LSE Equations and need to be modeled differently.

For LDC flow, the higher modes which do not account for significant contribution to enstrophy are seen as localized spatio-temporal structures, e.g. the triangular vortex seen in the core.

With the help of results with FST and collective results in the literature, for flow past a cylinder we conclude that the onset of instability at first bifurcations is governed by extrinsic dynamicsextrinsic dynamics.

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