Mikhail V. Averiyanov

Nonlinear propagation of spark-generated N-waves in air: Modeling and measurements using acoustical and optical methods

The propagation of nonlinear spherically diverging N-waves in homogeneous air is studied experimentally and theoretically. A spark source is used to generate high amplitude (1.4 kPa) short duration (40 μs) N-waves; acoustic measurements are performed using microphones (3 mm diameter, 150 kHz bandwidth). Numerical modeling with the generalized Burgers equation is used to reveal the relative effects of acoustic nonlinearity, thermoviscous absorption, and oxygen and nitrogen relaxation on the wave propagation. The results of modeling are in a good agreement with the measurements in respect to the wave amplitude and duration. However, the measured rise time of the front shock is ten times longer than the calculated one, which is attributed to the limited bandwidth of the microphone. To better resolve the shock thickness, a focused shadowgraphy technique is used. The recorded optical shadowgrams are compared with shadow patterns predicted by geometrical optics and scalar diffraction model of light propagation. It is shown that the geometrical optics approximation results in overestimation of the shock rise time, while the diffraction model allows to correctly resolve the shock width. A combination of microphone measurements and focused optical shadowgraphy is therefore a reliable way of studying evolution of spark-generated shock waves in air.

Nonlinear and diffraction effects in propagation of N-waves in randomly inhomogeneous moving media.

Finite amplitude acoustic wave propagation through atmospheric turbulence is modeled using a Khokhlov-Zabolotskaya-Kuznetsov (KZK)-type equation. The equation accounts for the combined effects of nonlinearity, diffraction, absorption, and vectorial inhomogeneities of the medium. A numerical algorithm is developed which uses a shock capturing scheme to reduce the number of temporal grid points. The inhomogeneous medium is modeled using random Fourier modes technique. Propagation of N-waves through the medium produces regions of focusing and defocusing that is consistent with geometrical ray theory. However, differences up to ten wavelengths are observed in the locations of fist foci. Nonlinear effects are shown to enhance local focusing, increase the maximum peak pressure (up to 60%), and decrease the shock rise time (about 30 times). Although the peak pressure increases and the rise time decreases in focal regions, statistical analysis across the entire wavefront at a distance 120 wavelengths from the source indicates that turbulence: decreases the mean time-of-flight by 15% of a pulse duration, decreases the mean peak pressure by 6%, and increases the mean rise time by almost 100%. The peak pressure and the arrival time are primarily governed by large scale inhomogeneities, while the rise time is also sensitive to small scales.

Random focusing of nonlinear acoustic N-waves in fully developed turbulence: Laboratory scale experiment

A laboratory experiment was conducted to study the propagation of short duration (25 μs) and high amplitude (1000 Pa) acoustic N-waves in turbulent flow. Turbulent flows with a root-mean-square value of the fluctuating velocity up to 4 m/s were generated using a bidimensional nozzle (140 × 1600 mm(2)). Energy spectra of velocity fluctuations were measured and found in good agreement with the modified von Kármán spectrum for fully developed turbulence. Spherical N-waves were generated by an electric spark source. Distorted waves were measured by four 3 mm diameter microphones placed beyond the turbulent jet. The presence of turbulence resulted in random focusing of the pulse; more than a threefold increase of peak pressures was occasionally observed. Statistics of the acoustic field parameters were evaluated as functions of the propagation distance and the level of turbulence fluctuations. It is shown that random inhomogeneities decrease the mean peak positive pressure up to 30% at 2 m from the source, double the mean rise time, and cause the arrival time about 0.3% earlier than that for corresponding conditions in still air. Probability distributions of the pressure amplitude possess autosimilarity properties with respect to the level of turbulence fluctuations.

Modeling of nonlinear shock wave propagation and thermal effects in high‐intensity focused ultrasound fields.

Numerical simulations based on the Khokhlov–Zabolotskaya‐type equation are currently used to characterize therapeutic high‐intensity focused ultrasound fields in water and to predict bioeffects in tissue. Here results from three different algorithms that differ in calculating the nonlinear term in the equation are presented. Shock capturing schemes of Godunov type, exact implicit solution with further extrapolation of the waveform over a uniform temporal grid, and direct modeling in the frequency domain are tested. In the case of weak nonlinearity, all schemes give essentially the same solution. However, at high peak pressures around 50 MPa and strong shocks developed in the focal region, the predictions of acoustic variables and heat deposition become sensitive to the algorithm employed. The parameters of the schemes, such as number of harmonics or temporal samples and the inclusion of artificial absorption that provides consistent results, are discussed. It is shown that the spectral and Godunov‐type ap...

Comparison of time and frequency domain approaches to simulate propagation of weak shocks.

Simulations of acoustic fields using finite‐difference methods are performed either in time or frequency domains. A method of fractional steps with an operator splitting procedure is frequently applied to solve nonlinear equations of the evolution type. Either time or frequency domain solvers can be used to calculate different terms in the equation over a propagation grid step. In this work, several algorithms that have been used to simulate quadratic nonlinear term in the Burgers or KZK‐type evolution equations are applied to model the propagation of weak shocks. Shock capturing schemes of Godunov type, exact analytic solution with further extrapolation of the waveform over a uniform temporal grid, time‐domain conservative schemes, direct modeling in the frequency domain, and asymptotic spectral approach are compared. The parameters of the schemes that would provide the results of the same accuracy, an artificial absorption necessary for stability of the schemes, resolution of shocks, and internal viscos...

Full‐diffraction and parabolic axisymmetric numerical models to characterize nonlinear ultrasound fields of two‐dimensional therapeutic arrays.

Numerical modeling has been shown to be an effective tool to characterize nonlinear pressure fields for single‐element HIFU transducers, but it has not yet been applied for the much more complex three‐dimensional (3‐D) fields generated by therapeutic phased arrays. In this work, two approaches are presented to simulate nonlinear effects in the field of a 256‐element focused array. A new full‐diffraction approach includes rigorous 3‐D simulations of the nonlinear wave equation with a boundary condition given at the elements of the array. A second simpler approach is based on the KZK model and a focused piston source as the boundary condition. The effective aperture and initial pressure of the piston source are set by matching linear simulations of the two models in the focal region. It is shown that as output power is increased, agreement in the focal waveforms of the two simulations, even when shocks were present, is maintained up to very high power outputs of the array. These results demonstrate the feas...

Random focusing of nonlinear N‐waves in fully developed turbulence: Laboratory scale experiment and theoretical analysis.

The high‐amplitude shock wave generated by a supersonic aircraft propagates through the atmosphere toward the ground and generates an acoustic field with non‐uniform pressure distributions strongly influenced by atmospheric turbulence. Recent numerical simulations based on generalized KZK‐type equation including the effects of moving inhomogeneous media will be discussed. Formation of multiple focusing and defocusing zones is predicted. Nonlinear effects are significant not only in the random focusing zones but also in shadow zones of lower‐pressure levels due to scattering of high frequencies from the areas of focusing. A statistical analysis is performed, and the results are compared to experimental data obtained in the controlled laboratory scale experiments conducted in the ECL anechoic wind tunnel. A high‐power spark source is used to generate N‐waves. Correlation length scales and spectra of the turbulent velocity field are measured. Statistical distributions and mean values for peak positive pressure and shock arrival time are obtained and found to be in a good agreement with modeling. In focusing areas, waveforms with amplitudes more than four times higher than those measured without turbulence are observed. Pressure amplitude probability density distributions are shown to possess autosimilarity properties when changing the intensity of turbulence. [Work supported by RFBR, French Government.]

Nonlinear Propagation of Spark-generated N-waves in Relaxing Atmosphere: Laboratory-Scaled Experiments and Theoretical Study

Propagation of nonlinear spherically diverging N-waves in homogeneous atmosphere was studied experimentally and theoretically. Theoretical study was held on the basis of numerical solutions to the modified Burgers equation the relative effects of nonlinear, dissipation and relaxation phenomena on N-wave parameters, such as amplitude, duration and rise time were investigated. It is shown that under the experimental conditions, the duration of pulse increases mainly due to nonlinear propagation, whereas the amplitude and rise time depend on the combined effects of nonlinearity, dissipation and relaxation. Experimental study was done in the laboratory-scaled environment. Calibration of the amplitude and duration of the experimental waveforms was performed based on nonlinear lengthening of the propagating pulse. The frequency response of the measuring system was obtained. The results of numerical modelling showed a good agreement with the experimental data.

Mechanisms of nonlinear saturation in focused acoustic beams of periodic waves and single pulses

Physical mechanisms leading to saturation of various acoustic field parameters in nonlinear focused beams of periodic waves and single pulses were investigated numerically. A numerical algorithm based on the KZK equation was used in the simulations. Propagation of an initially harmonic wave and a single pulse (one period of a sine wave) emitted by a focused transducer with Gaussian apodization was modeled. It was shown that in periodic fields, saturation of the peak positive pressure is mainly due to the effect of nonlinear absorption at the shock front. In acoustic fields of single pulses the main mechanism of saturation is the nonlinear refraction. Maximum pressure in the periodic field, achieved at the focus, was found to be higher than that of the single pulse. The total energy of the beam of the periodic wave, however, decreases much faster with the distance from the source as compared to the single pulse focusing. These nonlinear propagation effects propose a possibility to use pulsed beams for more effective delivery of the wave energy to the focal region, and periodic waves—to achieve higher pressure values of at the focus. [Work supported by EB007643, NIH DK43881, DK075090, and RFBR 09-02-01530.]

Measurements of spark‐generated N‐waves in air using a combination of acoustical and optical methods.

Accurate measurement of broadband acoustic signals in air, particularly N‐waves, remains a challenge. Bandwidth of existing microphones typically does not exceed 150 kHz, which results in significant overestimations of the shock rise time. To better resolve the shock thickness, it is proposed to use a focused optical shadowgraphy technique. The approach is tested experimentally. A spark source is used to generate high amplitude N‐waves in air. Acoustic measurements are performed using conventional microphones (3 mm diameter), and optical shadowgrams are made using a collimated light beam from a 20‐ns flash source. The results of modeling based on the generalized Burgers equation are in a good agreement with the microphone measurements in respect to the wave peak pressure and duration. However, the measured rise time of the front shock is ten times longer than the calculated one, which is attributed to the limited bandwidth of the microphone. The recorded optical shadowgrams in the vicinity of the shock fr...