Investigation of a Single-Point Nonlinearity Indicator in the Propagation of High-Amplitude Jet Noise

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Cody Hodges
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1 2th AIAA/CEAS Aeracustics Cnference (27th AIAA Aeracustics Cnference) 8 - May 26, Cambridge, Massachusetts AIAA Investigatin f a Single-Pint Nnlinearity Indicatr in the Pragatin f High-Amlitude Jet Nise Lauren E. Falc * and Anthny A. Atchley The Pennsylvania State University, University Park, PA, 682 Kent L. Gee Brigham Yung University, Prv, UT, 8462 There is evidence t suggest that nnlinearity is imrtant in the ragatin f highamlitude jet nise [Gee et al., AIAA J. 4(6), 98-4 (25)]. Tyically, the wer sectral density (PSD) is used t assess the imact f jet nise n the surrunding envirnment, but such an assessment requires multile measurement lcatins t bserve the nnlinear evlutin f the PSD. The difference in the PSDs measured at different lcatins deends n a cmbinatin f surce level, nzzle diameter, and ragatin distance. As a result, full scale measurements have t be extended ver large distances, and mdel scale measurements require high measurement bandwidths. These cnstraints cmlicate the measurement and make it difficult t bserve nnlinear effects using the PSD. Here a different technique fr determining the imrtance f nnlinearity is investigated. The imaginary art f the crss-sectral density f the ressure and the square f the ressure, als called the quadsectral density (QSD), is related t the rate f nnlinear change f the PSD. Thus, the extent t which the PSD is evlving nnlinearly can be determined at a single measurement lcatin. In the absence f absrtin, energy is cnserved, and the integratin f the rduct f the QSD with frequency ver all frequencies must be zer; when nnlinearity is resent, the value f the QSD can be nnzer fr many frequencies. Because nnlinearity tends t transfer energy frm lw frequencies t high frequencies and the QSD is sitive at frequency cmnents that are lsing energy, using the integral f the QSD ver its sitive values as a nnlinearity indicatr eliminates the need fr high bandwidth measurements. Exerimental measurements were taken in a lane wave tube with a wrking length f 9.55 m in which bundary layer lsses dminate ver atmsheric absrtin. Exerimental and numerical results shw that the rati f the integral f the QSD ver the frequencies fr which it is sitive (Q s ) t the integral ver the frequencies fr which it is negative (Q neg ) is clse t ne. Als, because the QSD is thirdrder in ressure, nrmalizing its integral by the cube f the rms ressure yields a quantity that is easily cmared acrss exerimental cnditins. Results fr bth eridic and bradband signals are resented and the racticality f using the QSD as a single-int indicatr f nnlinearity addressed. Nmenclature c = equilibrium sund seed d = exerimental jet nzzle diameter d = full-scale jet nzzle diameter = acustic ressure Q 2 = imaginary art f the crss-sectral density f the square f the ressure and the ressure (als QSD) QSD neg = all values f the QSD which are negative QSD s = all values f the QSD which are sitive Q neg = summatin ver frequency f (*QSD neg ) * Ph.D. Candidate, Grad. Prg. in Acustics, 27 Alied Science Bldg., Student Member AIAA. Prfessr f Acustics and Head, Grad. Prg. in Acustics, 27 Alied Science Bldg., Member AIAA. Assistant Prfessr, Deartment f Physics and Astrnmy, N9 ESC, Member AIAA. Cyright 26 by the authrs. Published by the, Inc., with ermissin.

2 Q s = summatin ver frequency f (*QSD s ) r = radial ragatin distance S = wer sectral density (als PSD) x = linear ragatin distance = generalized atmsheric and bundary-layer absrtin cefficient = cefficient f nnlinearity = equilibrium density = angular frequency = angular frequency f the surce I. Intrductin N the study f the ragatin f jet nise, the wer sectral density (PSD) is tyically the quantity used t I assess imact n the surrunding cmmunity. There is evidence t suggest that a crrect assessment requires knwledge f whether the ragatin is linear r nnlinear. If the ragatin is nnlinear, then the nnlinearity must be accurately accunted fr in any redictin mdel. It has been shwn that, under sme cnditins, a linear mdel des nt accurately redict the ragatin f jet nise, esecially at higher frequencies. The imrtance f nnlinearity is usually determined by examining the evlutin f the PSD with ragatin distance. This methd, which requires measurements at multile lcatins, has inherent cmlicatins fr bth full-scale and mdel-scale jet measurements. In full-scale exeriments, many effects influence the ragatin f the nise, including wind and temerature gradients, grund imedance, and the satial extent and directivity f the surce. The cmlexity f this envirnment calls fr measurements (acustic and meterlgical) at many lcatins s that these effects can be quantified and selectively remved during analysis. In mdel-scale measurements, the scaling-u f frequency and the cnstraints f wrking in an anechic chamber f finite dimensin ften make it difficult t detect nnlinear effects by examining the PSD. Nnlinear effects are cumulative, s maximizing the value f r/d f a measurement (where r is radial distance frm the surce and d is jet diameter) is imrtant t their detectin. Because mdel-scale measurements are usually cnducted in an enclsed sace, r is limited. Furthermre, decreasing d increases the frequency bandwidth needed fr the measurements. Frequency bandwidth well abve the eak frequency (ne t tw decades) is generally necessary t see significant differences between measurements and linear redictins and therefre t detect the resence f nnlinearity. Peak frequencies are usually arund a few hundred Hertz in full-scale measurements; in mdel-scale measurements, they are usually n the rder f a few kilhertz 2. Frequency is scaled as d /d (where d is the fullscale jet diameter), meaning that a greater bandwidth is required fr a smaller jet. While it is ssible t recrd data at a samle rate f rughly 2 khz (thereby btaining a Nyquist frequency f khz), amlitude and hase calibratins are nt readily available fr mst micrhnes at these high frequencies. A tyical /8 micrhne (Brüel & Kjær tye 48) has a flat amlitude resnse t abut 4 khz; a similar /4 micrhne (Brüel & Kjær tye 498) is reliable nly t 7 khz. Phase resnses are nt secified fr these micrhnes. This restrictin, cmbined with the limitatin n r sed by the measurement sace, makes it very difficult t see nnlinearity in mdel-scale measurements f the PSD. In light f these factrs (meterlgy, measurement sace, and frequency bandwidth), it wuld be beneficial t be able t determine the resence r imrtance f nnlinearity with a measurement at a single lcatin. This aer exlres the use f a single-int nnlinearity indicatr based n the wrk f Mrfey and Hwell. II. Thery and Derivatin Mrfey and Hwell derive an exressin cntaining a quantity that has the tential t serve as a single-int nnlinearity indicatr. They begin with the Burgers Equatin, a secnd-rder arablic wave equatin that includes the effects f bth nnlinearity and absrtin. After sme maniulatin, including transfrmatin t the frequency dmain and ensemble-averaging, they btain an equatin similar t d dx 2αx β 2αx ( e S ) ω e Q 2 =, () where x is distance frm the surce, a generalized absrtin cefficient, S the PSD, the cefficient f nnlinearity, and Q 2 the imaginary art f the crss sectral density (als knwn as the quadsectral density, r QSD) f the square f the ressure and the ressure. Equatin () has been mdified slightly frm its riginal frm ρ c 2

3 t aly t lane waves. The left-hand side f Eq. () is the satial derivative f the absrtin-crrected PSD. The right-hand side can be viewed as a surce term fr changes in the PSD that are due t nnlinearity. Thus, the QSD culd be a valuable tl in determining the resence r imrtance f nnlinearity in the ragatin f a wave. A nrmalizatin f this quantity, ften referred t as Q/S r the Mrfey-Hwell nnlinearity indicatr, has recently been used by several researchers,2,4 in the analysis f high-amlitude nise. Hwever, its hysical meaning is nt well understd and has nt yet been thrughly investigated. One f the signatures f nnlinearity in sund ragatin is the transfer f energy frm lwerfrequency t higher-frequency sectral cmnents. A ragating sinusid with sufficient amlitude will generate sectral cmnents at harmnics f the fundamental frequency; as the ragatin distance increases, mre f the higher harmnics becme imrtant. These harmnics gain energy at the exense f the fundamental. Thus, the left-hand side f Eq. () fr an riginally sinusidal wave shuld be negative at the fundamental frequency and sitive fr the harmnic frequencies. Figure shws this quantity fr an initially sinusidal wave at tw different ragatin distances. These data were taken in a 5.2-cm diameter lane wave tube with a fiberglass anechic terminatin and driven by tw JBL 242H drivers. The micrhnes used were /4 Brüel & Kjær tye 498 munted with the diahragm arximately flush with the inner wall f the tube. The micrhne lcatins were. m,.25 m, 6.4 m, and 9.55 m frm the surce, resectively. Fr a mre cmlete descritin f the exerimental aaratus, lease see Falc et al. 5 Analysis is simlified by temrarily assuming lssless ragatin f lane waves. This simlificatin aids in the understanding f the rblem; lsses can be added later. Rewriting Eq. () fr a lssless case ( = ) gives d dx Integrating bth sides ver frequency yields d dx S (Pa 2 /(Hz*m)) β = ω Q 2. (2) ρ c β S dω = ρ c 5-5 ωq dω d(e 2αx S )/dx. m, 47.6 db, σ =..25 m, 45.9 db, σ = Frequency (khz) Figure. The left-hand side f Eq. () fr measured wavefrms at tw different ragatin distances. 2. () The integral n the left-hand side f Eq. () is the mean-square ressure f the signal, which is rrtinal t energy. In the absence f lsses and shcks, the ttal energy f the wave shuld nt change with ragatin distance. It fllws that ω dω Q 2 =. (4) If we define QSD s as all values f the QSD which are sitive (frequencies at which energy is being lst) and QSD neg as all values f the QSD which are negative (frequencies at which energy is being gained), cnservatin f energy and Eq. (4) dictate that

4 ωqsd s ( ω) dω = ωqsd ( ω) dω. (5) Fr the linear case, bth f these integrals shuld equal zer. Fr the nnlinear case, bth integrals shuld have nnzer values equal in magnitude. Once shcks have frmed, hwever, Eq. (5) aarently n lnger hlds. Even in the absence f exlicit atmsheric and bundary-layer lsses, the wave will exerience lsses at the shcks which will decrease its ttal energy 6. Thus, the magnitude f the left-hand side f Eq. (5) shuld be larger than that f the right-had side when shcks are resent. As a result, the rati f these integrals can be used as a simle test fr the resence f shcks (and therefre f nnlinearity). Given the restrictins n r/d mentined abve in the cntext f mdel-scale jet measurements, the ability t detect nnlinearity in the re-shck regin (at smaller values f r/d) and using a limited bandwidth wuld be articularly beneficial. Either integral frm Eq. (5) culd be used as a nnlinearity indicatr in the re-shck regin, but since sitive values f Q 2 tend t ccur at lwer frequencies, using the left-hand side f Eq. (5) decreases the need fr a high-bandwidth measurement. neg III. Analysis Measurements were made using the lane wave tube described abve with tw JBL 2426H drivers as the sund surce. The drivers were sulied with a sinusidal signal ranging frm 5 Hz t khz in frequency, and the resulting sund waves had amlitudes frm 5 db t 45 db re 2Pa at the micrhne clsest t the surce. Wavefrms were catured at all fur micrhne lcatins. Numerical redictins fr the secnd, third, and furth micrhne lcatins were generated by using the measured wavefrm at the first micrhne as the inut t an Andersn-tye ragatin algrithm 7. Analytical redictins were made using the exerimental cnditins as inuts t Blackstck s slutin 8 fr the harmnic amlitudes f an initially sinusidal wave in the absence f absrtin. All redictins were generated with the same samle rate as the measurements. PSDs and QSDs were calculated by breaking the time signal int shrter recrds with a 5% verla, alying a Hanning windw t each recrd, taking a Furier transfrm, and averaging. Qneg /Qs..5 Q neg /Q s Q n eg /Q s..5 Q neg /Q s (a).99.e-5.e-4.e-.e-2.e-.e+.e+ A. Single-Frequency Surce Data The rati Q neg /Q s, where Q neg is the integral n the right-hand side f Eq. (5) and Q s is the integral n the lefthand side, was ltted as a functin f, r ragatin distance nrmalized by shck frmatin distance, given by xβω σ =. (6) ρ c 4.99.E-5.E-4.E-.E-2.E-.E+.E+ Figure 2. Rati f the integrals in Eq. (5) as a functin f nrmalized ragatin distance fr measured data (a) and analytical data (b). (b)

5 Figure 2 shws these lts fr the measured wavefrms and fr the analytically generated wavefrms. As exected accrding t Eq. (5), the value f the rati is clse t ne at all ragatin distances. Many f the data ints at smaller ragatin distances reresent lw-amlitude surce cnditins fr which the QSD is small. In these cases the signal-t-nise rati is als small, resulting in the deviatins frm unity at these ints. Nrmalized Q s Nrmalized Qs y = 254.8x r 2 = y = x r 2 =.9967 Qs/(Prms ) 5 Qs/(Prms ) (a).e+ 2.E- 4.E- 6.E- 8.E-.E+ Each value f Q s was nrmalized by the cube f the rms ressure f the signal and ltted as a functin f nrmalized distance, and a line was fitted t each lt. Results are shwn in Fig. fr the measured and analytical data; the numerically ragated data yields results s similar t thse fr the measurements that it is nt shwn here. The data in Fig. were generated using a 5 Hz surce and amlitudes ranging frm 5 db t 45 db re 2Pa. The equatins fr the tw lines are similar and r 2 >.97 fr bth cases. This result suggests that, fr an arbitrary single-frequency surce signal, knwledge f the QSD culd be used t btain the shck frmatin distance f the signal and therefre infrmatin abut the surce. B. Bradband Nise Data Bradband nise data were generated by using surce cnditins similar t thse f the measurements as inuts t the Andersn-tye algrithm. The frequency cntent f the surce was 5 Hz t Hz, the amlitude was between 5 db and 45 db re 2Pa, and the signal was saved at ragatin distances crresnding t the micrhne lcatins in the lane wave tube..e+ 2.E- 4.E- 6.E- 8.E-.E+ Figure. A nrmalizatin f Q s as a functin f nrmalized distance fr measured data (a) and analytical data (b) fr a 5 Hz surce. Q neg /Q s (b) Q s Nrmalizatin Qneg/Qs eff Qs/(Prms^) y = x +.48 r 2 = eff Pre-shck Pst-shck Figure 4. The rati Q neg /Q s fr numerically-figurgenerated bradband nise data. distance fr numerically-generated bradband 5. A nrmalizatin f Q s vs. nrmalized nise. 5

6 The same analysis was alied t the nise data as t the single-frequency surce data. An effective shck frmatin distance was calculated using the center frequency and the rms ressure f the surce 9. Because nise tends t steeen and shck at smaller values f than single-frequency surces, shck frmatin can haen fr eff <. Figure 4 cntains a lt f the rati Q neg /Q s fr the bradband nise data. The rati is nearly equal t fr all but the largest nrmalized ragatin distances. Based n the analysis abve, it is likely that shcks have frmed in these signals and that energy is being lst at the shcks rather than being transferred t ther frequency cmnents. The nrmalizatin f Q s mentined abve is shwn in Fig. 5. A line has been fitted t all but the last tw data ints because Fig. 4 indicates that shcks are likely resent in these signals. The crrelatin cefficient fr this line is very clse t ne, and the equatin f the line is similar t the equatins btained fr the single-frequency surce data. Its sle is larger, but this is t be exected as shck frmatin seems t ccur at smaller values f eff fr the nise data. This figure suggests that the receding analysis is equally valid fr single-frequency surce signals and bradband nise signals (and therefre tentially fr jet nise), given that an arriate nrmalized ragatin distance is used fr the nise signals. IV. Cnclusin The QSD has been shwn t rvide infrmatin abut nnlinearities in the ragatin f single-frequency surce signals and bradband nise signals. The value f the rati Q neg /Q s indicates whether shcks are resent in a signal. A nrmalizatin f Q s culd be used t btain the shck frmatin distance and surce cnditins f a signal f unknwn rigin. Because these methds f analysis are valid fr bradband nise, they culd be used in the evaluatin f jet nise rvided a suitable nrmalized ragatin distance is emlyed. Acknwledgments L. E. Falc wuld like t thank the Natinal Science Fundatin and the Office f Naval Research fr their surt f this wrk. K. L. Gee wuld like t thank the Strategic Envirnmental Research and Develment Prgram. References Gee, K. L., T. B. Gabrielsn, A. A. Atchley, and V. W. Sarrw, Preliminary Analysis f Nnlinearity in Military Jet Aircraft Nise Pragatin, AIAA Jurnal, Vl. 4, N. 6, 25, Petitjean, B. P., K. Viswanathan, and D. K. McLaughlin, Acustic Pressure Wavefrms Measured in High Seed Jet Nise Exeriencing Nnlinear Pragatin, 4 rd AIAA Aersace Sciences Meeting and Exhibit, 25, AIAA Mrfey, C. L., and G. P. Hwell, Nnlinear Pragatin f Aircraft Nise in the Atmshere, AIAA Jurnal, Vl. 9, N. 8, 98, McInerny, S. A., and S. M. Ölçmen, High-intensity Rcket Nise: Nnlinear Pragatin, Atmsheric Absrtin, and Characterizatin, J. Acust. Sc. Am., Vl. 7, N. 2, 25, Falc, L. E., K. L. Gee, A. A. Atchley, and V. W. Sarrw, Investigatin f a Single-Pint Nnlinearity Indicatr in One- Dimensinal Pragatin, Frum Acusticum Paer # 7, Budaest, Aug Rudnick, I., On the Attenuatin f a Reeated Sawtth Shck Wave, J. Acust. Sc. Am., Vl. 25, N. 5, 95,. 2,. 7 Gee, K. L., V. W. Sarrw, T. B. Gabrielsn, and A. A. Atchley, Nnlinear Mdeling f F/A-8E Nise Pragatin, th AIAA/CEAS Aeracustics Cnference, 25, AIAA Blackstck, D. T., Cnnectin Between the Fay and Fubini Slutins fr Plane Sund Waves f Finite Amlitude, J. Acust. Sc. Am., Vl. 9, N. 6, 966, Gurbatv, S. N. and O. V. Rudenk, Statistical henmena, in Nnlinear Acustics, M.F. Hamiltn and D.T. Blackstck, eds., , Academic Press, San Dieg: 998. Rudenk, O. V., Interactins f Intense Nise Waves, Sv. Phys. Us., Vl. 29, N. 7, 986,

Investigation of a Single-Point Nonlinearity Indicator in One-Dimensional Propagation. 2 Theory

Investigation of a Single-Point Nonlinearity Indicator in One-Dimensional Propagation. 2 Theory Investigatin f a Single-Pint Nnlinearity Indicatr in One-Dimensinal Prpagatin Lauren Falc, Kent Gee, Anthny Atchley, Victr Sparrw The Pennsylvania State University, Graduate Prgram in Acustics, University