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22. Comparison of nonlinearly-and linearly-predicted waveform segments at 1000 m with and without dispersion.
Source publication
The role of nonlinearity in the propagation of noise radiated from
high-performance jet aircraft has not been a well-understood phenomenon
in the past. To address the problem of finite-amplitude noise
propagation, a hybrid time-frequency domain model has been developed to
numerically solve the generalized Mendousse-Burgers equation, which is a
para...
Citations
... The forced Burgers' equation is considered in [61] for studying various errors encountered in the simulations of atmospheric flow. Propagation of high-intensity noise in the atmosphere can be simulated using Burgers' equation [62], and this is the key idea used for the prediction of propagation of noise from jet aircraft and helicopters [63][64][65]. A recent study [66] describes Burgers' equation as a mathematical model for one-dimensional groundwater recharge by spreading. ...
Even if numerical simulation of the Burgers’ equation is well documented in the literature, a detailed literature survey indicates that gaps still exist for comparative discussion regarding the physical and mathematical significance of the Burgers’ equation. Recently, an increasing interest has been developed within the scientific community, for studying non-linear convective–diffusive partial differential equations partly due to the tremendous improvement in computational capacity. Burgers’ equation whose exact solution is well known, is one of the famous non-linear partial differential equations which is suitable for the analysis of various important areas. A brief historical review of not only the mathematical, but also the physical significance of the solution of Burgers’ equation is presented, emphasising current research strategies, and the challenges that remain regarding the accuracy, stability and convergence of various schemes are discussed. One of the objectives of this paper is to discuss the recent developments in mathematical modelling of Burgers’ equation and thus open doors for improvement. No claim is made that the content of the paper is new. However, it is a sincere effort to outline the physical and mathematical importance of Burgers’ equation in the most simplified ways. We throw some light on the plethora of challenges which need to be overcome in the research areas and give motivation for the next breakthrough to take place in a numerical simulation of ordinary / partial differential equations.
... For a simple test case, an initially sinusoidal waveform with amplitude similar to the model-scale jet noise was numerically propagated in air using a hybrid time-frequency domain algorithm for the GBE. 35 Unlike the jet noise case, which exhibits range and frequency-dependent geometric spreading in the near field, spherical spreading (m ¼ 1) was assumed at all distances. For ease of subsequent comparison, the distance was scaled with respect to a jet nozzle diameter (D j )-equal to 3.5 cm-and the atmospheric conditions were taken to be the same as in the experiment discussed in Sec. ...
A single-point, frequency-domain nonlinearity indicator is calculated and analyzed for noise from a model-scale jet at Mach 0.85, Mach 1.8, and Mach 2.0. The nonlinearity indicator, νN, has been previously derived from an ensemble-averaged, frequency-domain version of the generalized Burgers equation (GBE) from Reichman, Gee, Neilsen, and Miller [J. Acoust. Soc. Am. 139, 2505–2513 (2016)]. The indicator gives the spatial rate of change due to nonlinear processes in sound pressure level (SPL) spectrum, and two other indicators from the GBE—νS and να—give the same quantity due to geometric spreading and absorption, respectively. Trends with frequency, angle, distance, and jet condition—supported both by spectral analysis and by calculation of the GBE-derived indicators—reveal a concentration of nonlinear effects along radials close to the plume with large overall SPLs. The calculated indicators for nonlinearity and absorption effects far from the source combine to give the same decay predicted by nonlinear theory for monofrequency sources. Trends in the νN indicator are compared with trends observed for other indicators such as pressure-derivative skewness and bicoherence, revealing both the qualitative and quantitative advantages of the νN indicator.
... For plane waves in a lossless medium (BBF), nonlinear losses at the shock cause the waveform to decay in amplitude, and after shock formation N is asymptotically negative for all frequencies [see Eqs. (25) and (26) in Ref. 17]. However, shocks in a lossy medium eventually thicken due to absorption, after which the waveform decay is slower than linearly predicted. ...
... Since there is no known analytical solution to the GBE with thermoviscous absorption and spherical spreading, a numerical solution is used. 25 To compare with the analysis from Sec. 3.2, a Gol'dberg number of 30 is desired. However, due to divergence there is much less nonlinear steepening for a spherical wave than for a plane wave of the same initial amplitude. ...
A frequency-domain nonlinearity indicator has previously been characterized for two analytical solutions to the generalized Burgers equation (GBE) [Reichman, Gee, Neilsen, and Miller, J. Acoust. Soc. Am. 139, 2505-2513 (2016)], including an analytical, asymptotic expression for the Blackstock Bridging Function. This letter gives similar old-age analytical expressions of the indicator for the Mendousse solution and a computational solution to the GBE with spherical spreading. The indicator can be used to characterize the cumulative nonlinearity of a waveform with a single-point measurement, with suggested application to noise waveforms as well.
... The spectrum peaks at a centre frequency of f 0 = 1.2 kHz. The signal is projected to several observer positions using the numerical algorithm presented by Gee (2005). The algorithm is an extended version of the seminal work presented by Pestorius & Blackstock (1974). ...
For some time now it has been theorized that spatially evolving instability waves in the irrotational near-field of jet flows couple both linearly and nonlinearly to generate far-field sound [Sandham and Salgado, Philos. Trans. R. Soc. Am. 366 (2008); Suponitsky, J. Fluid Mech. 658 (2010)]. An exhaustive effort at The University of Texas of Austin was initiated in 2008 to better understand this phenomenon, which included the development of a unique analysis technique for quantifying their coherence [Baars et al., AIAA Paper 2010–1292 (2010); Baars and Tinney, Phys. Fluids 26, 055112 (2014)]. Simulated data have shown this technique to be effective, albeit, insurmountable failures arise when exercised on real laboratory measurements. The question that we seek to address is how might jet flows manifest nonlinearities? Both subsonic and supersonic jet flows are considered with simulated and measured data sets encompassing near-field and far-field pressure signals. The focus then turns to considering nonlinearities in the form of cumulative distortions, and the conditions required for them to be realized in a laboratory scale facility [Baars, et al., J. Fluid Mech. 749 (2014)].
... The spectrum peaks at a centre frequency of f 0 = 1.2 kHz. The signal is projected to several observer positions using the numerical algorithm presented by Gee (2005). The algorithm is an extended version of the seminal work presented by Pestorius & Blackstock (1974). ...
A model is proposed for predicting the presence of cumulative nonlinear distortions in the acoustic waveforms produced by high-speed jet flows. The model relies on the conventional definition of the acoustic shock formation distance and employs an effective Gol'dberg number A for diverging acoustic waves. The latter properly accounts for spherical spreading, whereas the classical Gol'dberg number F is restricted to plane wave applications. Scaling laws are then derived to account for the effects imposed by jet exit conditions of practical interest and includes Mach number, temperature ratio, Strouhal number and an absolute observer distance relative to a broadband Gaussian source. Surveys of the acoustic pressure produced by a laboratory-scale, shock-free and unheated Mach 3 jet are used to support findings of the model. Acoustic waveforms are acquired on a two-dimensional grid extending out to 145 nozzle diameters from the jet exit plane. Various statistical metrics are employed to examine the degree of local and cumulative nonlinearity in the measured waveforms and their temporal derivatives. This includes a wave steepening factor (WSF), skewness, kurtosis and the normalized quadrature spectral density. The analysed data are shown to collapse reasonably well along rays emanating from the post-potential-core region of the jet. An application of the generalized Burgers equation is used to demonstrate the effect of cumulative nonlinear distortion on an arbitrary acoustic waveform produced by a high-convective-Mach-number supersonic jet. It is advocated that cumulative nonlinear distortion effects during far-field sound propagation are too subtle in this range-restricted environment and over the region covered, which may be true for other laboratory-scale jet noise facilities.
... In strict accordance with the standard, a divergence of 61.5 dB in L 1=3 x ð Þ from spherical spreading levels is allowed for frequencies of 6.3 kHz or more. 5 Air absorption, not covered by the standard, is accounted for here 12,13 prior to averaging the pressures using temperature and humidity information provided by an Oregon Scientific weather station, model BAR388HGA. ...
Qualifying an anechoic chamber for frequencies that extend into the ultrasonic range is necessary for research work involving airborne ultrasonic sound. The ANSI S12.55/ISO 3745 standard which covers anechoic chamber qualification does not extend into the ultrasonic frequency range, nor have issues pertinent to this frequency range been fully discussed in the literature. An increasing number of technologies employ ultrasound; hence the need for an ultrasonic anechoic chamber. This paper will specifically discuss the need to account for atmospheric absorption and issues pertaining to source transducer directivity by presenting some results for qualification of a chamber at Brigham Young University.
... = nonlinear coefficients, = 0 c 3 0 , kg 1 s 0 = density, kg m 3 = dimensionless propagation distance, x= x = retarded time, s = phase of the Fourier transform of the pressure, rad = phase of the Fourier transform of the pressure square, rad ! = angular velocity, rad=s Introduction A COUSTIC waves can propagate nonlinearly for highamplitude sound, causing a distortion of the waveform, eventually resulting in shocks, and then involving the dissipation of sound energy due to the shock wave. ...
... One-dimensional nonlinear propagation can be described with the Burgers equation [1], which includes nonlinear steepening and atmospheric absorption. Previously, a mixed method [2,3] had been developed to calculate the propagation of a periodic wave and broadband jet engine noise. This method predicts the nonlinear steepening in the time domain from the Earnshaw solution and the atmospheric absorption effects in the frequency domain. ...
... The real part of the complex coefficient 0 is associated with attenuation and can be obtained by the method presented by Bass et al. [7,8]. A formula for the imaginary part describing the dispersion effects and their variation with frequency is given in [3,4]. Finally, Eq. (3) can be expressed as two ordinary differential equations for the amplitude and phase [4], respectively: ...
An efficient numerical method to solve nonlinear sound propagation is presented. The frequency-domain Burgers equation, which includes nonlinear steepening and atmospheric absorption, is formulated in the form of the real and imaginary parts or the pressure. The new formulation effectively eliminates possible numerical issues associated with zero amplitude at higher frequencies occurring in a previous frequency-domain algorithm. In addition, to circumvent a high-frequency error that can occur in the truncated higher frequencies, a split algorithm is developed, in which the Burgers equation is solved below a cutoff frequency and a recursive analytic expression is used beyond the cutoff frequency. Finally, the Lanczos smoothing filter is incorporated to remove the Gibbs phenomenon. The new method is found to successfully eliminate high-frequency numerical oscillations and to provide excellent agreement with the exact solution for an initially sinusoidal signal with only a few harmonics. The new method is applied to a broad range of applications with a comparison to other methods to assess the robustness and numerical efficiency of the method. These include sonic boom, broadband supersonic jet engine noise, and helicopter high-speed impulsive noise. It is shown that the new method provides the fastest and most accurate predictions compared to the other methods for all the application problems.
... In general, the character of the spectra displayed in Figure 6 is quite like that observed by Gee et al. 1,8 in field experiments with both the F/A-18 and F-22 aircraft. A second series of our laboratory experiments on the jet noise in the proximity to a ground plane was conducted with the jet at a higher separation distance of h j / D = 4.9 (Fig. 6 b). ...
The influence of ground reflections on the measurement of aircraft engine exhaust noise is examined in this paper. A series of experiments were performed in the UCI aeroacoustic facility of supersonic jet noise with and without a ground plane in close proximity to the jet. Thus the effect of the ground plane on the radiated noise was isolated. Additionally, a computational model for the phenomenon was developed, which included the determination of a distribution of noise sources within the jet column. This distribution was developed from phased microphone array measurements incorporating an advanced beamforming algorithm. The developed model did a reasonably good job of predicting the effect of the reflecting ground surface. The most important exception is in the additional noise source caused by aerodynamic "scrubbing" of the turbulent jet on the surface far downstream of the jet exit. Guidance on how to alter the proposed model to account for porosity of the ground in field experiments over grassy or sandy terrain, is also given.
An effective Gol'dberg number is proposed for determining the degree of nonlinear distortion achieved in a diverging wave field. For values that are large compared with unity, the degree of nonlinear waveform distortion is virtually the same as that for a plane wave characterized by the traditional Gol'dberg number having the same numerical value. Expressions for the effective Gol'dberg number are proposed for spherical and cylindrical waves, Gaussian beams, and exponential horns.
The effects of nonlinearity on the power spectrum of jet noise can be directly compared with those of atmospheric absorption and geometric spreading through an ensemble-averaged, frequency-domain version of the generalized Burgers equation (GBE) [B. O. Reichman et al., J. Acoust. Soc. Am. 136, 2102 (2014)]. The rate of change in the sound pressure level due to the nonlinearity, in decibels per jet nozzle diameter, is calculated using a dimensionless form of the quadspectrum of the pressure and the squared-pressure waveforms. In this paper, this formulation is applied to atmospheric propagation of a spherically spreading, initial sinusoid and unheated model-scale supersonic (Mach 2.0) jet data. The rate of change in level due to nonlinearity is calculated and compared with estimated effects due to absorption and geometric spreading. Comparing these losses with the change predicted due to nonlinearity shows that absorption and nonlinearity are of similar magnitude in the geometric far field, where shocks are present, which causes the high-frequency spectral shape to remain unchanged.
