On cumulative nonlinear acoustic waveform distortions from high-speed jets

Abstract

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 Λ for diverging acoustic waves. The latter properly accounts for spherical spreading, whereas the classical Gol'dberg number Γ 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.

Coalescence in the Sound Field of a Laboratory-Scale Supersonic Jet

“…The placement of these microphones involved both a two-dimensional grid of points in the (x, r) plane as well as a line array along the peak emission path, which was found to emanate from a region centered around x∕D j 17.5 (based on a visual extrapolation of the data; see Sec. 3.1.1 of Baars et al [19]). Spectral-based metrics are constructed using 2 13 samples per block, thus resulting in a frequency resolution of δf 12.5 Hz.…”

“…However, no broadband shock tones or screech were observed in any of the acoustic spectra, and so these shocks were not expected to have any influence on the measurements reported here (or those acquired in 2011 with the 1∕4 in. microphones [17][18][19]). A common set of values for P ∞ , T ∞ , and T 0 were chosen by averaging over the test conditions reported in Table 1.…”

“…One of the underlying challenges in implementing the scaling model developed by Baars [18] and Baars et al [19], which predicts shock formation distances and the degree of cumulative nonlinear distortion in the acoustic waveform from high-speed jets, is that it requires one to know the size and amplitude of the source. This is a difficult proposition, even with the advent of more sophisticated numerical simulations at our disposal; consider the different source definitions found in the open literature [37][38][39].…”

“…microphones were used to map the acoustic field of an unheated supersonic jet (between x∕D j 5: 10: 145 and r∕D j 25: 10: 95) operating at a gas dynamic Mach number of 3 [17][18][19]. Contrary to that study, we will employ higher-fidelity instruments (1∕8 in.…”

Asymptotic behavior of a frequency-domain nonlinearity indicator for solutions to the generalized Burgers equation

“…The terms on the right-hand side of Eq. (1) show that the change in pressure over distance is related to the separate effects of divergence, absorption, and nonlinearity. If L n is defined as the SPL for the nth harmonic of a periodic signal with fundamental frequency x 1 , the time-domain GBE in Eq.…”