VIEWS: 12 PAGES: 60 POSTED ON: 10/27/2011 Public Domain
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+2t 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 2i ( 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+2f) (2f1+f) (2f1+2f) f f1 f1+f 2f1 2f1+f fs -f (f1-f) (f1) (2f1-f) (2f1-2f) 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