Non linear Spectral analysis by panniuniu

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									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:
   –   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,   (c) Resonance tube   (d) Edge tone   (e) Cavity tones
                   Ring tones
                                                                                            4
                                    Hartmann Whistle
            Screech




                                         5
Raman, Prog. Aero. Sci., vol. 34, 1998
            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
                         Tube Length
              Hartmann   Adjustment
              Tube



  Flow                       Spacing
  Direction                  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
                  0 f S
                      2 2
                            f S
• Linearity used in               Dimensionless no 2 vs L/s 6bar

  developing a         2                                                s23
                                                                         s28
                       70
  frequency model      60
                                                                         s32
                                                                         s35
                                                                         s39

          c0           50                                                s42


f                     40


    S k1L  k2 S    30

                       20

                       10
                            0.5     1         1.5         2        2.5
                                              L/S
                                                                               11
               Resemblance with Helmholtz
                       resonator
     A                                               Spill-over
Area of neck

                                     Shock Cell(s)     Tube Volume
                          Volume V
                                                           AL


              L
        length of neck
                                      Spacing S

              c           A
         f                                 f 
                                                 c    A
             2          VL                     2   ALS
                                                    c0
                                          f 
                                              S k1L  k2 S 



                                                                     12
    Evidence of Non-linearity
•Highly coherent spectral       Mic 1
                                                     Mic 2
components summed.
•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          Spanwise
                                                      symmetric         antisymmetric
                                                      coupling          coupling
                                                      mode              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
                           • Nozzle spacings:
Measured Quantities          7.3  s/h  7.9
                                      s
• Stagnation Pressure
                                h
• Sound Pressure signals                              21
   Signal Conditioning & DAq

Stagnation Pressure

Mic+Preamp.           Anti-alias     NI Board
+ Pow. Supp.          filter
                                    Sampling rate:200 kSa/s
  1 – 100 kHz         1 – 100 kHz
                                    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
• 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.
Mic 2 Power, (Log units)




                           Mach No. 1.33               Mach No. 1.4




                            Mic 1 Power, (Log units)                  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




                                                                              26
 X and Y axes: Acoustic Pressures. Range: -2000 Pa to 2000 Pa for all plots
       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

           Window 1 Window 2                 x-x
                                   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 non-
  linearities.
                                                  30
            Drawbacks of SOS




• SOS cannot discern
  between linearly
                       t = [0:1e-5:1]';
  superposed and       x = 0.5*(sin(2*pi*3000*t)+sin(2*pi*13000*t));
                       y = sin(2*pi*5000*t).*sin(2*pi*8000*t);
  quadratically        [p f] = spectrum(x,y,1024,[],[],100000);
                       semilogy(f,p(:,1),f,p(:,2));
  modulated signals.   xlabel ('Frequency (Hz)');
                       ylabel ('PSD (1^2/Hz)');
• So, use restricted   legend('3k+13k','5k*8k');

  to linear systems.                                              31
           Higher order spectral methods
    • Tool Employed – Cross Bispectrum.
                  
                                 T
                                                                  
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
                                1
                           T  T
                             0                               
    • Description: In two time series signals,
      Quantifies the relationship between a pair of
      frequencies in the spectrum.
    • x-Bispectrum: S ( k ) ( f , f )  Y ( k ) ( f  f ) X ( k )* ( f ) X ( k )* ( f                                                        )
                                       YXX       1      2                      1         2                           1                   2

    • Ensemble Average:                                                   1              M
                                                      SYXX ( f1 , f 2 ) 
                                                                          M
                                                                                       S
                                                                                        k 1
                                                                                                (k )
                                                                                                YXX       ( f1 , f 2 )

    • x-Bicoherence:                       b ( f1 , f 2 ) 
                                             2                                        SYXX ( f1 , f 2 )
                                                                                                          2


                                                               1                         2  1                                           2
                                             c                      M                              M
                                                              
                                                              M
                                                                     Y ( k ) ( f1  f 2 )  M
                                                                    k 1                    
                                                                                                   X
                                                                                                   k 1
                                                                                                              (k )
                                                                                                                     ( 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 non-
  linear interactions.
• Thomas (2003): Book chapter on HOS
  tools applicable to shear flows.

                                                33
                  Demonstration
  • Two sinusoids generated:
      f (t )  sin 1t  sin 2t   g (t )  sin 1t sin(2t   )

        Spectra                    Cross-Bicoherence




(a)                                (b)                              34
Interpreting results from CBC spectra
                                 Plot shows CBC
         Sum Int. Region          contours
                                • X and Y axes:
            Diff. Int. Region     Frequencies
                                 interacting
                                  non-linearly.
                               • Resultant
                                  frequencies read
                                  from the plot.
                                • Strength quantified
                                  by CBC value (color)
                               • , - participating
                                         frequencies.
                                •  - Resultant
                                       frequency      35
    Influence of Phase on CBC
      f (t )  sin 1t  sin  2 t                       g (t )  sin 1t sin(2t   )
                                                                                             Phase
• To examine the effect of                                1

  phase (), on the cross-                               0.9
                                                         0.8
  bicoherence, various 



                                     Cross-Bicoherence
                                                         0.7
                                                         0.6
  used.                                                  0.5

• The resultant plot showed                              0.4
                                                         0.3
  that CBC is insensitive to                             0.2
                                                         0.1
  small phase differences,                                0

  but declines sharply for                                     0    1           2           3
                                                                    Phase Difference (radians)
                                                                                                     4


  large phase differences
  (  /2 and greater).
                                                                                                         36
 Effect of Magnitude of Non-Linear part
   f (t )  ½(sin 1t  sin 2t   )              0    0.05
     g (t )  A f (t )  B sin 1t sin(2t   )   A+B=1

• 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.                                              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:     Arrowhead-shaped:
     Coupled        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

             (2f1)
 f1


             (f1+f)    (f1+2f)   (2f1+f) (2f1+2f)
 f

             f1        f1+f         2f1      2f1+f      fs
-f
             (f1-f)    (f1)       (2f1-f)   (2f1-2f)
                  Cluster 1             Cluster 2




  -fs                                                          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                             C




s/h = 7.5       s/h = 7.7               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
                                       1   bc2 ( f i , f j )  n
  I c ,n    (i, j ),    (i , j ) 
         i 1 j 1                     0   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 (threshold 0.3)
   variation with Mach number
                                  120
   Interaction Density (Ic,0.3)



                                  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, 1 mm
                                                                     7.5       V-shaped, 27.7
                                                                                           mm
                                    V-shaped, 3 7.9
                                                mm                   7.3
                                                          Arrowhead, 0mm       Single jet

                                                                                                   51
  Interaction density (threshold 0.4)
     variation with Mach number
                                     80
     Interaction Density (I c,0.4)




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




                                                                            Avg. interaction density
   Avg. Interaction density




                              70                                                                       50
                              60
                                                                                                       40
                              50
                                                                                                       30




                                                                                      (Ic,0.3)
             (Ic,0.3)




                              40
                              30                                                                       20
                              20
                                                                                                       10
                              10
                               0                                                                        0
                                1.29 1.32 1.35 1.38 1.41 1.44 1.47 1.50                                     7.3        7.5            7.7       7.9
                                             Mach number                                                          Inter-nozzle spacing (s/h )




                              50
                                                                      (b)                              25
                                                                                                                                                (d)
  Avg. interaction density




                                                                            Avg. interaction density
                              40                                                                       20

                              30                                                                       15
            (Ic,0.4)




                                                                                      (Ic,0.4)
                              20                                                                       10

                              10                                                                       5

                              0                                                                        0
                               1.29 1.32 1.35 1.38 1.41 1.44 1.47 1.50                                      7.3       7.5            7.7        7.9
                                             Mach num ber                                                         Inter-nozzle spacing (s/h )
                                                                                                                                                      53
Significance of Interaction Density Metric
         N    M
                                             1            bc2 ( f i , f j )  n
  I c ,n    (i, j ),          (i , j ) 
         i 1 j 1                           0            bc2 ( f i , f j )  n




                                                                                    54
              Physics of Fluids, vol.17, Art.096103, 2005
Significance of   Mic 1
                  Jet 1
                             Mic 3
                             Twin jet
                                               Mic 2
                                               Jet 2     Mic 1        Mic 2


  Interaction
Density contd…
                                         α = yaw angle


                                                           Jet flow direction
                          Jet flow direction




                                                                                55
CBC spectra of                  Mic 1
                                         Mic 2

Hartmann Whistle Data

            Spacing 30mm, length 40 mm




            Spacing 40mm, length 30 mm




                                                 56
    Interaction Density                           Mic 1

          vs NPR                                              Mic 2




                     Interaction Density vs NPR
         60
         50
         40
Ic,0.3




         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

								
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