Nonlinear Spectral Analysis
in Aeroacoustics
Dr. K. Srinivasan
Department of Mechanical Engineering
Indian Institute of Technology Madras
Acknowledgement
• Funding agencies:
– AFOSR (Dr. John Schmisseur)
– ISRO (ISRO-IITM Cell)
• Co-researchers:
–
–
–
–
–
–
–
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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
2
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
3
Scenarios in resonant acoustics
Free-jet Resonance:
Screech
Jet interaction
with solid devices
(a) Impingement
(b) Hole tone,
Ring tones
(c) Resonance tube
Hartmann Whistle
(d) Edge tone
(e) Cavity tones
4
Screech
Raman, Prog. Aero. Sci., vol. 34, 1998
5
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.
6
Y
7
Hartmann Whistle
Hartmann Whistle
Jet Nozzle
Hartmann
Tube
Flow
Direction
Tube Length
Adjustment
Spacing
Adjustment
9
Relevant Parameters
L
s
• Tube Length (L)
• Spacing (S)
• Nozzle Pressure Ratio (NPR)
10
New Frequency Prediction Model
• Dimensionless numbers involving frequency
P0
RT0
2 
 2 2
2 2
0 f S
f S
• Linearity used in
developing a
frequency model
c0
f 
S k1L  k2 S 
2
Dimensionless no 2 vs L/s 6bar
s23
s28
s32
s35
s39
s42
70
60
50
40
30
20
10
0.5
1
1.5
L/S
2
2.5
11
Resemblance with Helmholtz
resonator
Spill-over
A
Area of neck
Volume V
L
length of neck
c
A
f 
2 VL
Shock Cell(s)
Tube Volume
AL
Spacing S
f 
c
2
A
ALS
c0
f 
S k1L  k2 S 
12
Evidence of Non-linearity
•Highly coherent spectral
components summed.
Mic 1
Mic 2
•Intense modulation
(quadratic nonlinearities)
•Lissajous show complex patterns. Similar with 2 Piezos.
13
Twin jet Coupling
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
15
Resonant coupling induced damage
(Berndt, 1984)
16
Twin jet coupling
• Aerodynamic, acoustic and stealth advantages derived
from nozzles of complex geometry.
• Acoustic fatigue damage observed in earlier aircraft.
17
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.
18
Modal Interactions in
twin jets
19
Illustration of earlier results
(a) Single jet modes
Nozzle
Bevel Angle = 300
Jet Flow Direction
Microphone
(b) V-shaped configuration: Twin jet coupling modes
Spanwise
symmetric
coupling
mode
Spanwise
antisymmetric
coupling
mode
(c) Twin jet: Arrowhead-shaped configuration
No coupling
20
Experimental Setup
Parameters
• Stagnation Pressure:
26 psig to 40 psig, in
steps of 1 psi
• Mach No. Range:
1.3  Mj  1.5
Measured Quantities
• Stagnation Pressure
• Sound Pressure signals
• Nozzle spacings:
7.3  s/h  7.9
s
h
21
Signal Conditioning & DAq
Stagnation Pressure
Mic+Preamp.
+ Pow. Supp.
Anti-alias
filter
1 – 100 kHz
1 – 100 kHz
NI Board
Sampling rate:200 kSa/s
Sampling time: 1.024 s
Interface
22
Outline of the Method
•
•
•
•
Spectra
Frequency locking
Phase locking
Phase angle
substantiated by
high phase
coherence.
• Observations for
different geometric
and flow
parameters
23
Time series Analyses
Mic 2 Power, (Log units)
• 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
Mic 1 Power, (Log units)
Mach No. 1.4
24
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: Yi
Yi
Xi
Time Series: Xi
25
X-Y phase plots & non-linearity
• Phase plots (time domain) also pointed out
to non-linearity at some Mach numbers.
1.3
Fuzziness
1.37
Curvature
1.33
1.4
1.5
X and Y axes: Acoustic Pressures. Range: -2000 Pa to 2000 Pa for all plots
26
Time-Localization Studies
• To gain additional knowledge, phase plots
within a data set were plotted: x-x phase plots
x(t)
ti
ti+t
ti+2t
t
x-x
Window 1 Window 2
x(w1)
Phase plot
x(w2)
27
Cross Spec, x-x, y-y, and x-y plots
A
C
B
D
Note: x-x and y-y plots are dissimilar, but x-y plots look similar
28
Cross Spec, x-x, y-y, and x-y plots
A
C
B
D
Note: x-x, y-y, and x-y plots change within the time series.
29
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 nonlinearities.
30
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');
31
Higher order spectral methods
• Tool Employed – Cross Bispectrum.
T


1
Byxx ( f1 , f 2 )     lim  y (t ) x (t   1 ) x (t   2 ) dt exp 2i ( f1  1  f 2  2 )d 1 d 2
T  T
  
0

 
• Description: In two time series signals,
Quantifies the relationship between a pair of
frequencies in the spectrum.
• x-Bispectrum: (k )
(k )
( k )*
( k )*
SYXX ( f1 , f 2 )  Y
• Ensemble Average:
• x-Bicoherence:
1
SYXX ( f1 , f 2 ) 
M
b ( f1 , f 2 ) 
2
c
( f1  f 2 ) X
( f1 ) X
M
S
k 1
(k )
YXX
SYXX ( f1 , f 2 )
 1

M
2  1
Y ( k ) ( f1  f 2 ) 

k 1
 M
M
( f2 )
( f1 , f 2 )
2
M
X
k 1
(k )
2
( f1 ) X ( k ) ( f 2 )  32

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 nonlinear interactions.
• Thomas (2003): Book chapter on HOS
tools applicable to shear flows.
33
Demonstration
• Two sinusoids generated:
f (t )  sin 1t  sin 2t
Spectra
(a)
g (t )  sin 1t sin(2t   )
Cross-Bicoherence
(b)
34
Interpreting results from CBC spectra
Sum Int. Region
Diff. Int. Region



 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
35
frequency
Influence of Phase on CBC
f (t )  sin 1t  sin 2 t
g (t )  sin 1t sin(2t   )
Phase
1
0.9
Cross-Bicoherence
• To examine the effect of
phase (), on the crossbicoherence, various 
used.
• The resultant plot showed
that CBC is insensitive to
small phase differences,
but declines sharply for
large phase differences
(  /2 and greater).
0.8
0.7
0.6
0.5
0.4
0.3
0.2
0.1
0
0
1
2
3
4
Phase Difference (radians)
36
Effect of Magnitude of Non-Linear part
f (t )  ½(sin 1t  sin 2t   )
g (t )  A f (t )  B sin 1t sin(2t   )
• Nonlinear part
systematically varied.
• The resultant spectra
of g(t) and crossbicoherence between
f(t) and g(t) examined.
• Note that the crossbispectrum looks
similar. Only the
magnitudes differ.
0    0.05
A+B=1
37
How do SOS and HOS compare in
their respective tasks?
A = 0.5, B = 0.5
A = 0.95, B = 0.05
A = 0.9, B = 0.1
A = 0.99, B = 0.01
38
A = 0.995, B = 0.005
HOS is more robust; detects even
very small magnitudes of non-linearity
39
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.
40
Results: Coupled and Uncoupled Jets
V-shaped:
Coupled
Arrowhead-shaped:
Did not Couple
41
Single and Twin Jets
• Single jets show lesser non-linearity than twin jets
in terms of number and strength of interactions.
42
Spectra at Mach numbers in the
symmetric coupling range
Interaction Clusters
Mj = 1.3
Mj = 1.33
43
As Mach number increases…
Mj = 1.4, Mode Switching
Mj = 1.46, Antisymmetric
44
Clustering illustrated
fs/2
f1
f
(2f1)
(f1+f)
f1
-f
-fs
(f1+2f)
f1+f
(f1)
(f1-f)
Cluster 1
(2f1+f) (2f1+2f)
2f1
2f1+f
fs
(2f1-2f)
(2f1-f)
Cluster 2
45
Close-up view of a cluster
46
Effect of inter-nozzle spacing
Mj = 1.32 (symmetric)
s/h = 7.3
s/h = 7.5
s/h = 7.7
More dots (NL interactions) as s/h increases
47
Effect of inter-nozzle spacing
Mj = 1.46 (antisymmetric)
A
B
s/h = 7.5
s/h = 7.7
C
s/h = 7.9
48
Closer look at the straightly
aligned interactions
49
NL Interaction Quantification
• Based on number of interactions
– Interaction Density: Number of peaks in the
CBC spectrum above a certain (interaction
threshold) value.
N
M
I c ,n    (i, j ),
i 1 j 1
 1
 (i , j ) 
 0
bc2 ( f i , f j )  n
bc2 ( f i , f j )  n
– Threshold values of 0.3, and 0.4 used.
– Interaction density variation with parameters
of the study.
50
Interaction Density (Ic,0.3)
Interaction density (threshold 0.3)
variation with Mach number
120
100
80
60
40
20
0
1.28
1.31
1.34
1.37
1.4
1.43
1.46
1.49
1.52
Fully Expanded Mach Number (Mj)
V-shaped, 07.3
mm
V-shaped, 3 7.9
mm
V-shaped, 17.5
mm
7.3
Arrowhead, 0mm
V-shaped, 27.7
mm
Single jet
51
Interaction density (threshold 0.4)
variation with Mach number
Interaction Density (I c,0.4)
80
70
60
50
40
30
20
10
0
1.28
1.31
1.34
1.37
1.40
1.43
1.46
1.49
1.52
Fully Expanded Mach Number (M j )
Moderate increase
around symmetric
Peak at coupling-transition
Mach number
52
Average Interaction density metric
(a)
70
60
50
40
30
20
10
0
1.29 1.32 1.35 1.38 1.41 1.44 1.47 1.50
Avg. interaction density
(Ic,0.3)
Avg. Interaction density
(Ic,0.3)
•Interaction density averaged over all Mach numbers for
a particular spacing, and vice-versa.
(c)
50
40
30
20
10
0
7.3
(b)
50
40
30
20
10
0
1.29 1.32 1.35 1.38 1.41 1.44 1.47 1.50
Mach num ber
7.7
7.9
Inter-nozzle spacing (s/h )
Avg. interaction density
(Ic,0.4)
Avg. interaction density
(Ic,0.4)
Mach number
7.5
(d)
25
20
15
10
5
0
7.3
7.5
7.7
7.9
Inter-nozzle spacing (s/h )
53
Significance of Interaction Density Metric
N
M
I c ,n    (i, j ),
i 1 j 1
 1
 (i , j ) 
 0
Physics of Fluids, vol.17, Art.096103, 2005
bc2 ( f i , f j )  n
bc2 ( f i , f j )  n
54
Significance of
Interaction
Density contd…
Mic 1
Jet 1
Mic 3
Twin jet
Mic 2
Jet 2
Mic 1
Mic 2
α = yaw angle
Jet flow direction
Jet flow direction
55
CBC spectra of
Hartmann Whistle Data
Mic 1
Mic 2
Spacing 30mm, length 40 mm
Spacing 40mm, length 30 mm
56
Interaction Density
vs NPR
Mic 1
Mic 2
Interaction Density vs NPR
60
50
Ic,0.3
40
30
20
10
0
4
6
8
NPR
s30d40
s40d30
s45d35
57
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.
58
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.
59
60