Dr. K. SrinivasanDepartment of Mechanical Engineering

Indian Institute of Technology Madras

Nonlinear Spectral Analysis in Aeroacoustics

2

Acknowledgement• Funding agencies:

– AFOSR (Dr. John Schmisseur)– ISRO (ISRO-IITM Cell)

• Co-researchers:– Prof. Ganesh Raman, IIT, Chicago– Prof. David Williams, IIT Chicago– Prof. K. Ramamurthi, IIT Madras– Prof. T. Sundararajan, IIT Madras– Dr. Byung Hun-Kim, IIT, Chicago– Dr. Praveen Panickar, IIT, Chicago– Mr. Rahul Joshi, IIT Chicago– Mr. S. Narayanan, IIT Madras– Mr. P. Bhave, IIT Madras

3

Roadmap of the talk

• Examples of nonlinearity in aeroacoustics– Twin jet coupling– Hartmann whistle

• Twin jet coupling: Results from linear spectral analysis

• Motivation to use nonlinear spectral analysis • Results from nonlinear spectra• Interaction density metric• Universality of interaction density metric• Conclusions

4

Scenarios in resonant acoustics

(a) Impingement (b) Hole tone,Ring tones

(c) Resonance tube (d) Edge tone (e) Cavity tones

Jet interaction with solid devices

Free-jet Resonance: Screech

Hartmann Whistle

5

Screech

Raman, Prog. Aero. Sci., vol. 34, 1998

6

Other complications

• Non-axisymmetric geometry

• Spanwise oblique geometry and shock structure,

From: Raman, G., Physics of Fluids, vol. 11, No. 3, 1999, pp. 692 – 709.

7

Y

Hartmann Whistle

9

Hartmann Whistle

Hartmann Tube

Jet Nozzle

Flow Direction

Tube Length Adjustment

Spacing Adjustment

10

Relevant Parameters

s

L

• Tube Length (L)

• Spacing (S)

• Nozzle Pressure Ratio (NPR)

11

New Frequency Prediction Model

• Dimensionless numbers involving frequency

• Linearity used indeveloping a frequency model

220

220

02 Sf

RT

Sf

P

Dimensionless no 2 vs L/s 6bar

10

20

30

40

50

60

70

0.5 1 1.5 2 2.5

L/S

s23

s28

s32

s35

s39

s42

2

SkLkS

cf

21

0

12

Resemblance with Helmholtz resonator

ALS

Acf

2

Volume V

Llength of neck

AArea of neck

Tube Volume AL

Shock Cell(s)

Spill-over

Spacing S

VL

Acf

2

SkLkS

cf

21

0

13

Evidence of Non-linearityMic 1

Mic 2•Highly coherent spectral components summed.

•Intense modulation (quadratic nonlinearities)

•Lissajous show complex patterns. Similar with 2 Piezos.

Twin jet Coupling

15

Literature on twin jet coupling

• Berndt (1984) found enhanced dynamic pressure in a twin jet nacelle.

• Tam, Seiner (1987) Twin plume screech.• Morris (1990) instability analysis of twin jet.• Wlezien (1987) Parameters influencing interactions.• Shaw (1990) Methods to suppress twinjet screech.• Raman, Taghavi (1996, 98) coupling modes, relation

to shock cell spacing, etc.

• Panickar, Srinivasan, and Raman (2004) Twin jets from two single beveled nozzles.

• Joshi, Panickar, Srinivasan, and Raman (2006) Nozzle orientation effects and non-linearity

16

Resonant coupling induced damage (Berndt, 1984)

17

Twin jet coupling

• Aerodynamic, acoustic and stealth advantages derived from nozzles of complex geometry.

• Acoustic fatigue damage observed in earlier aircraft.

18

Our earlier work

• Panickar et al.(2004) concluded the following from their experiments:

– Single beveled jet - symm, antisymm and oblique modes.

– Twin jet - only spanwise symmetric and antisymmetric modes during coupling.

– A simple change to the configuration eliminated coupling between the jets.

Journal of Sound and Vibration, vol.278, pp.155-179, 2004.

19

Modal Interactions in twin jets

20

Illustration of earlier results(a) Single jet modes

(b) V-shaped configuration: Twin jet coupling modes

(c) Twin jet: Arrowhead-shaped configuration

Jet Flow Direction

Bevel Angle = 300

Nozzle

Microphone

No coupling

Spanwise symmetric coupling mode

Spanwise antisymmetric coupling mode

21

Experimental SetupParameters

• Stagnation Pressure: 26 psig to 40 psig, in steps of 1 psi

• Mach No. Range: 1.3 Mj 1.5

• Nozzle spacings: 7.3 s/h 7.9Measured Quantities

• Stagnation Pressure• Sound Pressure signals

s

h

22

Signal Conditioning & DAq

Mic+Preamp. + Pow. Supp.

Anti-alias filter

1 – 100 kHz

NI Board

1 – 100 kHzSampling rate:200 kSa/s

Sampling time: 1.024 s

Interface

Stagnation Pressure

23

Outline of the Method

• Spectra• Frequency locking• Phase locking• Phase angle

substantiated by high phase coherence.

• Observations for different geometric and flow parameters

24

Time series Analyses

• presence of neighbouring jet in close proximity, and dissimilarities between the two jets.

• Parity plots of average spectra of the two channels in the frequency domain shows dissimilarities between jets, although they were frequency/phase locked.

Mach No. 1.33 Mach No. 1.4

Mic 1 Power, (Log units)

M

ic 2

Po

wer

, (L

og

un

its)

25

Phase plots of time series data• Time series data of acoustic pressure from a channel plotted against

the other: • X-Y phase plots

Time Series: Xi

Time Series: Yi

Xi

Yi

26

X-Y phase plots & non-linearity

• Phase plots (time domain) also pointed out to non-linearity at some Mach numbers.

1.3

1.33

1.37

1.4 1.5

Fuzziness

Curvature

X and Y axes: Acoustic Pressures. Range: -2000 Pa to 2000 Pa for all plots

27

Time-Localization Studies• To gain additional knowledge, phase plots

within a data set were plotted: x-x phase plots

t

x(t)

titi+t ti+2t

Window 1 Window 2

x(w1)

x(w2)

x-x Phase plot

28

Cross Spec, x-x, y-y, and x-y plots

A

B

C

D

Note: x-x and y-y plots are dissimilar, but x-y plots look similar

29

A

B

C

D

Note: x-x, y-y, and x-y plots change within the time series.

Cross Spec, x-x, y-y, and x-y plots

30

Further attempts to understand the non-linear behaviour

• Simulation of non-linear sinusoids to match their phase plots with experimental ones.– A Lissajou simulator for generating various

artificial phase plots. – These attempts were not much successful

and not an elegant approach to decipher the non-linearities.

• Conventional spectral analysis (SOS) does not reveal information about non-linearities.

31

Drawbacks of SOS

• SOS cannot discern between linearly superposed and quadratically modulated signals.

• So, use restricted to linear systems.

t = [0:1e-5:1]';x = 0.5*(sin(2*pi*3000*t)+sin(2*pi*13000*t));y = sin(2*pi*5000*t).*sin(2*pi*8000*t);[p f] = spectrum(x,y,1024,[],[],100000);semilogy(f,p(:,1),f,p(:,2));xlabel ('Frequency (Hz)');ylabel ('PSD (1^2/Hz)');legend('3k+13k','5k*8k');

32

Higher order spectral methods

• Tool Employed – Cross Bispectrum.

• Description: In two time series signals, Quantifies the relationship between a pair of frequencies in the spectrum.

• x-Bispectrum:

• Ensemble Average:

• x-Bicoherence:

)()()(),( 2)*(

1)*(

21)(

21)( fXfXffYffS kkkk

YXX

M

k

kYXXYXX ffS

MffS

121

)(21 ),(

1),(

M

k

kkM

k

k

YXXc

fXfXM

ffYM

ffSffb

1

2

2)(

1)(

1

2

21)(

2

2121

2

)()(1

)(1

),(),(

212211

0

2121 )(2exp)()()(1

lim),( ddffidttxtxtyT

ffBT

Tyxx

33

Use of HOS in shear flows

• Thomas and Chu (1991, 1993): Planar shear layers, traced the axial evolution of modes.

• Walker & Thomas (1997): Screeching rectangular jet, axial evolution of non-linear interactions.

• Thomas (2003): Book chapter on HOS tools applicable to shear flows.

34

Demonstration

• Two sinusoids generated:

Spectra Cross-Bicoherence

tttf 21 sinsin)( )sin(sin)( 21 tttg

(a) (b)

35

Interpreting results from CBC spectra

Plot shows CBC contours

• X and Y axes: Frequencies interacting non-linearly.

• Resultant frequencies read from the plot.

• Strength quantified by CBC value (color)

, - participating frequencies.

- Resultant frequency

Sum Int. Region

Diff. Int. Region

36

Influence of Phase on CBC

• To examine the effect of phase (), on the cross-bicoherence, various used.

• The resultant plot showed that CBC is insensitive to small phase differences, but declines sharply for large phase differences ( /2 and greater).

tttf 21 sinsin)( )sin(sin)( 21 tttg

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

0 1 2 3 4

Phase Difference (radians)

Cro

ss

-Bic

oh

ere

nc

e

Phase

37

Effect of Magnitude of Non-Linear part

• Nonlinear part systematically varied.

• The resultant spectra of g(t) and cross-bicoherence between f(t) and g(t) examined.

• Note that the cross-bispectrum looks similar. Only the magnitudes differ.

)sin(sin½)( 21 tttf)sin(sin)()( 21 ttBtfAtg A + B = 1

0 0.05

38

How do SOS and HOS compare in their respective tasks?

A = 0.5, B = 0.5

A = 0.9, B = 0.1

A = 0.95, B = 0.05

A = 0.99, B = 0.01

39

HOS is more robust; detects even very small magnitudes of non-linearity

A = 0.995, B = 0.005

40

How to use CBC

• Obtain the second order and third order spectra for the entire parametric space.

• Look for changes in gross features in the higher order spectra and establish a correspondence with earlier knowledge.

• Establish metrics from HOS to quantify non-linearity.

• If possible, trace the evolution of the spectra.

41

Results: Coupled and Uncoupled Jets

 

V-shaped: Coupled

Arrowhead-shaped: Did not Couple

42

Single and Twin Jets

• Single jets show lesser non-linearity than twin jets in terms of number and strength of interactions.

43

Spectra at Mach numbers in the symmetric coupling range

Mj = 1.3 Mj = 1.33

Interaction Clusters

44

As Mach number increases…

Mj = 1.4, Mode Switching Mj = 1.46, Antisymmetric

45

Clustering illustrated

f

-f

f1

(f1+f)

(f1-f)

f1+f

(f1+2f)

(f1)

f1

(2f1)

2f1

(2f1+f)

2f1+f

(2f1+2f)

(2f1-f) (2f1-2f)

Cluster 1 Cluster 2

fs/2

fs

-fs

46

Close-up view of a cluster

47

Effect of inter-nozzle spacing

s/h = 7.3 s/h = 7.5 s/h = 7.7

More dots (NL interactions) as s/h increases

Mj = 1.32 (symmetric)

48

A

B C

s/h = 7.5 s/h = 7.7 s/h = 7.9

Effect of inter-nozzle spacingMj = 1.46 (antisymmetric)

49

Closer look at the straightly aligned interactions

50

NL Interaction Quantification• Based on number of interactions

– Interaction Density: Number of peaks in the CBC spectrum above a certain (interaction threshold) value.

– Threshold values of 0.3, and 0.4 used. – Interaction density variation with parameters of

the study.

nffb

nffbjijiI

jic

jicN

i

M

jnc

),(

),(

0

1),(),,( 2

2

1 1,

51

Interaction density (threshold 0.3) variation with Mach number

0

20

40

60

80

100

120

1.28 1.31 1.34 1.37 1.4 1.43 1.46 1.49 1.52

Fully Expanded Mach Number (Mj)

Inte

ract

ion

Den

sity

(Ic,

0.3)

V-shaped, 0 mm V-shaped, 1 mm V-shaped, 2 mm

V-shaped, 3 mm Arrowhead, 0mm Single jet7.3 7.5 7.7

7.9 7.3

52

Interaction density (threshold 0.4) variation with Mach number

0

10

20

30

40

50

60

70

80

1.28 1.31 1.34 1.37 1.40 1.43 1.46 1.49 1.52

Fully Expanded Mach Number (M j)

Inte

ract

ion

Den

sity

(Ic,

0.4

)

Moderate increase around symmetric Peak at coupling-transition

Mach number

53

Average Interaction density metric

0

10

20

30

40

50

7.3 7.5 7.7 7.9

Inter-nozzle spacing (s/h )

Av

g. i

nte

rac

tio

n d

en

sit

y

(Ic,

0.3)

0

10

20

30

40

50

60

70

1.29 1.32 1.35 1.38 1.41 1.44 1.47 1.50

Mach number

Av

g. I

nte

rac

tio

n d

en

sit

y

(Ic,

0.3)

0

10

20

30

40

50

1.29 1.32 1.35 1.38 1.41 1.44 1.47 1.50

Mach number

Avg

. in

tera

ctio

n d

ensi

ty

(Ic

,0.4)

0

5

10

15

20

25

7.3 7.5 7.7 7.9

Inter-nozzle spacing (s/h )

Avg

. in

tera

ctio

n d

ensi

ty

(Ic

,0.4)

(a)

(b) (d)

(c)

•Interaction density averaged over all Mach numbers for a particular spacing, and vice-versa.

54Physics of Fluids, vol.17, Art.096103, 2005

Significance of Interaction Density Metric

nffb

nffbjijiI

jic

jicN

i

M

jnc

),(

),(

0

1),(),,( 2

2

1 1,

55

Mic 1 Mic 2

Jet flow direction

Jet flow direction

Mic 1Jet 1

Mic 2Jet 2

Mic 3Twin jet

α = yaw angle

Significance of Interaction Density contd…

56

Spacing 30mm, length 40 mm

Spacing 40mm, length 30 mm

Mic 1Mic 2CBC spectra of Hartmann

Whistle Data

57

Interaction Density vs NPR

Interaction Density vs NPR

0

10

20

30

40

50

60

4 6 8

NPR

Ic,0

.3

s30d40 s40d30 s45d35

Mic 1Mic 2

58

Conclusions

• Configurations that did not show conclusive linear coupling were found nonlinearly coupled. So, nonlinear coupling may be important in nozzle design.

• Nonlinearity in configs can be graded

• Two patterns of cross-bicoherence were observed, one that showed clustering, and another that showed a straight alignment.

59

Conclusions…• A new interaction density metric identified

and seems a relevant parameter to quantify non-linear coupling.

• The average interaction density peaks in the vicinity of mode jumps

• Therefore, higher order spectra could serve as useful tools in theoretical understanding as well as in practical situations.

60