Dberg Number Research Articles

Reduced-Order Comparison of Simulated and Measured Coalescing Mach Waves near Supersonic Jets

Prior measurements of the sound field produced by a laboratory-scale, Mach 3 jet flow (Baars and Tinney, Journal of Sound and Vibration, Vol. 333, No. 12, 2014, pp. 2539–2553; Fiévet et al., AIAA Journal, Vol. 54, No. 1, 2016, pp. 254–265) suggest that acoustic waveforms steepen early on in their development. This explained the discrepancy between theoretical predictions, based on effective Gol’dberg numbers, that shocks should not form, and observations of steepened Mach waves close to laboratory-scale jets. The present work continues studying this phenomenon by exploring coalescence processes that occur when neighboring waveforms intersect, forming larger-amplitude waveforms with increased cumulative nonlinear distortion. A numerical model based on the Khokhlov–Zabolotskaya–Kuznetsov (KZK) equation is developed to show that coalescence-induced steepening is sensitive to the intersection angle between adjacent waveforms, waveform duration, and cylindrical spreading effects. High frame-rate schlieren images of sound waves propagating from the post-potential core region of a laboratory-scale Mach 3 jet are then captured along an angle following the ridge of most intense noise to study the development and evolution of coalescence. A shock detection algorithm isolates shock-like events, which are tracked using a translating coordinate system and decomposed using proper orthogonal decomposition. Reduced-order reconstructions of both schlieren images and the KZK model identify common patterns that characterize the shock coalescence process.

Examination of Mach wave coalescence in a Mach 3 jet flow

Prior acoustic measurements of the sound field produced by a laboratory-scale, Mach 3 jet flow [Baars et al., AIAA (2013); Fiévet et al., AIAA (2016)] showed a discrepancy between the theoretical prediction that shocks should not form, based on effective Gol’dberg numbers, and the apparent observation of steepened waves. This suggests an additional mechanism relating to wave coalescence is responsible for increased wave steepening. To better understand this new mechanism, high framerate schlieren images of sound waves propagating from the post potential core of the same Mach 3 jet are studied. A shock detection algorithm is developed to isolate shock-like features in the images and along the propagation path, which are then used to determine occurrences of coalescence-induced steepening in the flow. Numerical models of coalescing waves using the Khokhlov–Zabolotskaya–Kuznetzov (KZK) equation are then leveraged to determine identifiable characteristics of coalescence events to improve detection. POD-based reduced-order representations of the isolated events are then studied in a Lagrangian frame. Ensemble averages of tracked events are used to identify common patterns for coalescence, which is believed to contribute to “crackle” noise in full-scale jet flows. [WAW is supported by the ARL:UT Chester M. McKinney Graduate Fellowship in Acoustics.]

Effects of acoustic nonlinearity on pulse-echo attenuation coefficient estimation from tissue-mimicking phantoms.

The ultrasonic attenuation coefficient (ACE) can be used to classify tissue state. Pulse-echo spectral-based attenuation estimation techniques, such as the spectral-log-difference method (SLD), account for beam diffraction effects using a reference phantom having a sound speed close to the sound speed of the sample. Methods like SLD assume linear propagation of ultrasound and do not account for potential acoustic nonlinear distortion of the backscattered power spectra in both sample and reference. In this study, the ACE of a sample was computed and compared using the SLD with two independent references (high attenuating and low attenuating phantoms but with similar B/A values) and over several pressure levels. Both numerical and physical tissue-mimicking phantoms were used in the study. The results indicated that the biases in ACE increased when using a reference having low attenuation, whereas the high attenuating reference produced more consistent ACE. Furthermore, increments in ACE vs input pressure were correlated to the log-ratio of Gol'dberg numbers between the sample and reference (R2=0.979 in simulations and R2=0.734 in experiments). Therefore, the results suggest that to reduce bias in ACE using spectral-based methods, both the sound speed and the Gol'dberg number of the reference phantom should be matched to the sample.

Comparison between the Gol’dberg number and the Morfey-Howell Q/S for finite-amplitude plane waves

In this paper we compare the two most popular indicators of nonlinearity, namely, the Gol’dberg number [Sov. Phys. Acoust. 2, 346-350 (1956)] and the Morfey-Howell Q/S [AIAA J. 19, 986-992 (1981)] for the case of an initially-sinusoidal, finite-amplitude plane wave in a thermoviscous medium. The Gol’dberg number Γ, defined as the ratio of the absorption length at the fundamental frequency to the shock formation distance, is a parameter that single-handedly dictates the entire history of the wave [for example, weak (Γ < 1) and strong (Γ >> 1) waves]. However, it fails to quantify the strength of nonlinearity that each individual harmonic “sees” at any given point. In other words, not every harmonic sees the apparent strength of nonlinearity according to Γ. The Q/S, on the other hand, is defined for each harmonic as a function of propagation distance, and hence keeps track of the ever-changing strength of nonlinearity during the course of propagation. One interesting ramification of this is that using Q/S it is possible to pinpoint the distance at which the value of nonlinearity relative to absorption actually becomes Γ for the fundamental component.

Effective Gol'dberg number for diverging waves.

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.

Piecewise-Spreading Regime Model for Calculating Effective Gol’dberg Numbers for Supersonic Jet Noise

A method for calculating the effective Gol’dberg number for diverging waveforms is presented, which leverages known features of a high-speed jet and its associated sound field. The approach employs a ray tube situated along the Mach wave angle where the sound field is not only most intense, but advances from undergoing cylindrical decay to spherical decay. Unlike other efforts, a “piecewise-spreading regime” model is employed, which yields, separately, effective Gol’dberg numbers for the cylindrically and spherically spreading regions in the far field. The new approach is applied to a plethora of experimental databases, encompassing both laboratory- and full-scale jet noise studies. The findings demonstrate how cumulative nonlinear distortion is expected to form in the acoustic near field of laboratory-scale round jets where pressure amplitudes decay cylindrically; waveform distortion is not expected in the acoustic far field where waveform amplitudes diverge spherically. On the other hand, where full-scale jet studies are concerned, effective Gol’dberg number calculations demonstrate how cumulative waveform distortion is significant in both the cylindrical- and spherical-spreading regimes. The laboratory-scale studies also reveal a pronounced sensitivity to humidity conditions, relative to the full-scale counterpart.

Effect of Frequency-Dependent Attenuation on Predicted Histotripsy Waveforms in Tissue-Mimicking Phantoms

Tissue-mimicking phantoms are employed for the assessment of shocked histotripsy pulses in vitro. These broadband shock waves are critical for tissue ablation and are influenced by the frequency-dependent attenuation of the medium. The density, sound speed and attenuation spectra (2–25 MHz) were measured for phantoms that mimic key histotripsy targets. The influence of non-linear propagation relative to the attenuation was described in terms of Gol'dberg number. An expression was derived to estimate the bandwidth of shocked histotripsy pulses for power law-dependent attenuation. The expression is independent of the fundamental frequency of the histotripsy pulse for linear frequency-dependent attenuation.

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

The spatial evolution of acoustic waveforms produced by a laboratory-scale Mach 3 jet are investigated using both and . pressure field microphones located along rays emanating from the postpotential core where the peak sound emission is found to occur. The measurements are acquired in a fully anechoic chamber, where ground or other large surface reflections are minimal. Various statistical metrics are examined along the peak emission path, where they are shown to undergo rapid changes within 2 m from the source region. An experimentally validated wave-packet model is then used to confirm the location where the pressure amplitude along the peak emission path transitions from cylindrical to spherical decay. Various source amplitudes, provided by the wave-packet model, are then used to estimate shock formation distance and Gol’dberg numbers for diverging waves. The findings suggest that cumulative nonlinear distortion is likely to occur at laboratory scale near the jet flow, where the waveform amplitude decays cylindrically, but less likely to occur farther from the jet flow, where the waveform amplitude decays spherically. Direct inspection of the raw time series reveals how steepened waveforms are generated by roguelike waves that form from the constructive interference of waves from neighboring sources as opposed to classical cumulative nonlinear distortion.

On cumulative nonlinear acoustic waveform distortions from high-speed jets

AbstractA 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$\Lambda $for diverging acoustic waves. The latter properly accounts for spherical spreading, whereas the classical Gol’dberg number$\Gamma $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.

Effective Gol’dberg number for diverging waves

Interest in characterizing nonlinearity in jet noise has motivated consideration of an effective Gol’dberg number for diverging waves [Baars and Tinney, Bull. Am. Phys. Soc. 57, 17 (2012)]. Fenlon [J. Acoust. Soc. Am. 50, 1299 (1971)] developed expressions for the minimum value of Γ, the Gol’dberg number as defined for plane waves, for which shock formation occurs in diverging spherical and cylindrical waves. The conditions were deduced from a generalized Khokhlov solution and depend on the ratio xsh/r0, where r0 is source radius, and xsh the plane-wave shock formation distance for Γ=∞. Alternatively, by taking the ratio of the nonlinear and thermoviscous terms in Fenlon’s Eq. (2), it is proposed here that effective Gol’dberg numbers may be identified for spherical and cylindrical waves: Λ=Γexp(-πxsh/2r0) and Λ=Γ/(1 + πxsh/4r0), respectively. For a given value of Λ, the diverging waves achieve approximately the same degree of nonlinear distortion as a plane wave for which the value of Γ is the same. Conversely, to achieve the same degree of nonlinear distortion as a plane wave with a given value of Γ, the value of Γ for, e.g., a spherical wave must be larger by a factor of exp(πxsh/2r0). Extensions to other spreading laws are presented.

Nonlinear surface waves in soft, weakly compressible elastic media

Nonlinear surface waves in soft, weakly compressible elastic media are investigated theoretically, with a focus on propagation in tissue-like media. The model is obtained as a limiting case of the theory developed by Zabolotskaya [J. Acoust. Soc. Am. 91, 2569-2575 (1992)] for nonlinear surface waves in arbitrary isotropic elastic media, and it is consistent with the results obtained by Fu and Devenish [Q. J. Mech. Appl. Math. 49, 65-80 (1996)] for incompressible isotropic elastic media. In particular, the quadratic nonlinearity is found to be independent of the third-order elastic constants of the medium, and it is inversely proportional to the shear modulus. The Gol'dberg number characterizing the degree of waveform distortion due to quadratic nonlinearity is proportional to the square root of the shear modulus and inversely proportional to the shear viscosity. Simulations are presented for propagation in tissue-like media.

Moderately nonlinear ultrasound propagation in blood-mimicking fluid

In medical diagnostic ultrasound (US), higher than-in-water nonlinearity of body fluids and tissue usually does not produce strong nonlinearly distorted waves because of the high absorption. The relative influence of absorption and nonlinearity can be characterized by the Gol'dberg number Γ. There are two limiting cases in nonlinear acoustics: weak waves (Γ < 1) or strong waves (Γ >> 1). However, at diagnostic frequencies in tissue and body fluids, the nonlinear effects and effects of absorption more likely are comparable (Gol'dberg number Γ ≈ 1). The aim of this work was to study the nonlinear propagation of a moderately nonlinear US second harmonic signal in a blood-mimicking fluid. Quasilinear solutions to the KZK equation are presented, assuming radiation from a flat and geometrically focused circular Gaussian source. The solutions are expressed in a new simplified closed form and are in very good agreement with those of previous studies measuring and modeling Gaussian beams. The solutions also show good agreement with the measurements of the beams produced by commercially available transducers, even without special Gaussian shading. (E-mail: vince@bme.ri.ccf.org)

High amplitude, bifrequency experiments in porous rocks

Fluid-filled, porous, sedimentary rocks have the general characteristic that the sound speeds are strongly dependent on the state of stress and therefore, the third-order elastic coefficients (i.e., A, B, C) are generally much larger than for normal solids. However, the linear attenuation can also be very large such that in nonlinear wave propagation, the Gol’dberg number is small. Experiments studying harmonic generation in porous rocks have been made using an ultrasonic immersion system (i.e., water/solid/water) where a high amplitude (e.g., 500 kPa), bifrequency (0.95 and 1.05 MHz), tone burst is emitted and then sum and difference frequencies are detected after propagation through the water/solid/water system. A frequency domain numerical implementation of the KZK (Khokhlov–Zabolotskaya–Kuznetsov) nonlinear parabolic wave equation is used to describe the nonlinear propagation in the three layer system. Experiments on solids with well-known acoustic properties (linear and nonlinear) were used to validate the model. Measurements were then made on several sandstones and limestones and the model used to derive the nonlinear propagation parameter, β=f(A,B,C). Finally, these results are compared with independent measurements of A, B, C for these rocks.

Plane sound waves of finite amplitude for intermediate Gol’dberg numbers

Burgers equation provides a good description of the propagation of plane sound waves of finite amplitude for which nonlinear effects are important. Although an exact solution is known, simple asymptotic approximations are useful since they may provide starting points in the case of more general situations for which no exact solution is known. Approximation solutions of Burgers equation are sought by means of a “mixed asymptotic method” involving both the method of multiple scales and the method of strained coordinates. The main point is the introduction of a new slow variable binding the solution in the region before shock formation with the solution in the old age region. For an emitted monochromatic wave, the method yields a simple and complete description for Gol’dberg numbers up to 20, as it is shown by comparison to exact or other approximated solutions. For higher Gol’dberg numbers, the solution is valid up to the shock formation distance and it can be connected with Fay solution beyond.

Prediction of Finite-Amplitude Waveform Distortion with Dissipation and Spreading Loss

A new technique recently used for solution of the plane-wave Burgers' equation in a lossless medium has led to the development of an approximate analytical solution to Burgers' equation with dissipation for plane, cylindrical, and spherical sinusoidal waves. The solution is valid for the initial propagation zone prior to sawtooth formation for Gol'dberg numbers larger than 5 as indicated by comparison with a previous numerical solution of Burgers' equation for a plane wave. The approximate solution is expressed as a series that converges rapidly enough to permit calculations of harmonic levels by hand. For more viscous cases, where the approximate solution fails, the above technique leads easily to a numerical solution of Burgers' equation. By comparing the harmonic levels predicted by the numerical solution with empirical data taken on distorting spherical waves, it is concluded that numerical results tend to be below the measured values. It is felt that the numerical method can be of use in predicting distortion levels for practical underwater systems that exploit finite-amplitude effects.