Tables of integrals

The growth of a vapor bubble in a superheated liquid is controlled by three factors: the inertia of the liquid, the surface tension, and the vapor pressure. As the bubble grows, evaporation takes place at the bubble boundary, and the temperature and vapor pressure in the bubble are thereby decreased. The heat inflow requirement of evaporation, however, depends on the rate of bubble growth, so that the dynamic problem is linked with a heat diffusion problem. Since the heat diffusion problem has been solved, a quantitative formulation of the dynamic problem can be given. A solution for the radius of the vapor bubble as a function of time is obtained which is valid for sufficiently large radius. This asymptotic solution covers the range of physical interest since the radius at which it becomes valid is near the lower limit of experimental observation. It shows the strong effect of heat diffusion on the rate of bubble growth. Comparison of the predicted radius‐time behavior is made with experimental observations in superheated water, and very good agreement is found.

The growth of vapor bubbles in superheated liquids. report no. 26-6

The simulation of nonlinear ultrasound propagation through tissue realistic media has a wide range of practical applications. However, this is a computationally difficult problem due to the large size of the computational domain compared to the acoustic wavelength. Here, the k-space pseudospectral method is used to reduce the number of grid points required per wavelength for accurate simulations. The model is based on coupled first-order acoustic equations valid for nonlinear wave propagation in heterogeneous media with power law absorption. These are derived from the equations of fluid mechanics and include a pressure-density relation that incorporates the effects of nonlinearity, power law absorption, and medium heterogeneities. The additional terms accounting for convective nonlinearity and power law absorption are expressed as spatial gradients making them efficient to numerically encode. The governing equations are then discretized using a k-space pseudospectral technique in which the spatial gradients are computed using the Fourier-collocation method. This increases the accuracy of the gradient calculation and thus relaxes the requirement for dense computational grids compared to conventional finite difference methods. The accuracy and utility of the developed model is demonstrated via several numerical experiments, including the 3D simulation of the beam pattern from a clinical ultrasound probe.

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Modeling nonlinear ultrasound propagation in heterogeneous media with power law absorption using a k-space pseudospectral method

流体力学用語集 非線形音響学(Nonlinear acoustics)

Hamilton-Jacobi equations are frequently encountered in applications,e.g.,in control theory and differential games.Hamilton-Jacobi equations are closely related to hyperbolic conservation laws-in one space dimension the former is simply the integrated version of the latter.Similarity also exists for the multidimensional cases,and this is helpful in designing difference approximations.In this paper central weighted essentially non-oscillatory (CWENO) schemes for Hamilton-Jacobi equations are investigated,which yield uniform high-order accuracy in smooth regions and sharply resolve discontinuities in the derivatives.The schemes are numerically tested on a variety of one-dimensional problems,including a problem related to control optimization.High-order accuracy in smooth regions,high resolution of discontinuities in the derivatives,and convergence to viscosity solutions are observed.

High-order essentially non-oscillatory schemes for Hamilton-Jacobi equations

Weak shock wave focusing at fold caustics is described by the mixed type elliptic/hyperbolic nonlinear Tricomi equation. This paper presents a new and original numerical method for solving this equation, using a potential formulation and an “exact” numerical solver for handling nonlinearities. Validation tests demonstrate quantitatively the efficiency of the algorithm, which is able to handle complex waveforms as may come out from “optimized” aircraft designed to minimize sonic booms. It provides a real alternative to the approximate method of the hodograph transform. This motivated the application to evaluate the ground track focusing of sonic boom for an accelerating aircraft, by coupling CFD Euler simulations performed around the mock-up on an adaptated mesh grid, atmospheric propagation modeling, and the Tricomi algorithm. The chosen configuration is the European Eurosup mock-up. Convergence of the focused boom at the ground level as a function of the matching distance is investigated to demonstrate the efficiency of the numerical process. As a conclusion, it is indicated how the present work may pave the way towards a study on sonic superboom (focused boom) mitigation.

/pdf/numerical-simulation-of-shock-wave-focusing-at-fold-caustics-1jw7clp98m.pdf

Numerical simulation of shock wave focusing at fold caustics, with application to sonic boom.

Time-reversal invariance of nonlinear acoustic wave propagation is experimentally investigated. Reversibility is studied for propagation shorter or longer than shock formation distance. In the first case, time-reversal invariance holds and a sinusoid distorted by nonlinearities during forward propagation progressively recovers its initial shape after the time-reversal operation. In the second case, reversibility is broken locally at the shock front as a time-reversal operation transforms a stable compression shock into an unstable expansion shock. Achieving experimentally the time-reversal process with a time-reversal mirror made of reversible piezoelectric transducers for very broadband signals, would require transducers with huge bandwidths. To date, such transducers remain unavailable. In order to overcome this technical limitation, we restricted ourselves in this study to one-dimensional (1D) propagation, for which an experimental ersatz of a time-reversal mirror can be used. Indeed, in a 1D case, the time-reversal operation applied on a plane wave can be mimicked for an antisymmetric wave form by a reflection of the plane wave onto a pressure-release interface.

/pdf/breaking-of-time-reversal-invariance-in-nonlinear-acoustics-40wdx10rzk.pdf

Breaking of time reversal invariance in nonlinear acoustics.

Spherical caps are widely used to produce efficiently focused ultrasonic fields. Due to the curvature of the source surface, no exact analytical expression of the radiated field is known. The approximate model of Williams and O’Neil [J. Acoust. Soc. Am. 17, 219‐227 (1946); 21, 516–526 (1949)] is expected to be limited to high‐frequency, slightly focusing sources. In order to determine more precisely the limits of that model, and to dispose of an efficient tool for describing precisely the focused field, a new numerical method is presented. It is based on two expansions of the pressure field into spherical harmonics in two conveniently chosen domains. The expansions are then matched. Validity of the method is assessed by making a comparison with the exact analytical solution for the planar case. Then, the sound field radiated by a sharply focusing radiator is investigated. Comparison of the results of the numerical procedure with the approximate model indicates a much wider domain of validity of the model than generally thought. More precise conditions for the applicability of the model are derived, that follows from simple considerations of geometrical acoustics.

Continuous field radiated by a geometrically focused transducer: Numerical investigation and comparison with an approximate model

A numerical method is presented to simulate the focusing of sonic booms. In the vicinity of caustics, the pressure satisfies the nonlinear Tricomi equation. To solve this equation, an iterative algorithm, based on an unsteady version of the equation, is used. The algorithm is a modification of the pseudospectral code used for solving the Zabolotskaya-Khokhlov (KZ) equation. In the linear case, the code is validated by comparison with analytical solutions. For an incoming N wave, the shape of the outgoing wave observed in flight tests is recovered. In the nonlinear case, no analytical solution is known to compare with the numerical output. Validation of the numerical scheme is completed by means of four different tests. First, comparisons with the linear case shows that the numerical solutions behave as expected from the physics. Second, the solution after convergence is proved to be independent of the initial guess. Third, the maximal signal amplitude is proved to converge while increasing the discretization. Finally, the numerical scheme is checked against the nonlinear Guiraud's scaling law. In the last part, an application to the focusing of Concorde sonic boom in acceleration is presented, and ways of reducing focused booms are discussed.

Numerical Simulation of Sonic Boom Focusing

Numerical simulation of nonlinear acoustics and shock waves in a weakly heterogeneous and lossless medium is considered. The wave equation is formulated so as to separate homogeneous diffraction, heterogeneous effects, and nonlinearities. A numerical method called heterogeneous one-way approximation for resolution of diffraction (HOWARD) is developed, that solves the homogeneous part of the equation in the spectral domain (both in time and space) through a one-way approximation neglecting backscattering. A second-order parabolic approximation is performed but only on the small, heterogeneous part. So the resulting equation is more precise than the usual standard or wide-angle parabolic approximation. It has the same dispersion equation as the exact wave equation for all forward propagating waves, including evanescent waves. Finally, nonlinear terms are treated through an analytical, shock-fitting method. Several validation tests are performed through comparisons with analytical solutions in the linear case and outputs of the standard or wide-angle parabolic approximation in the nonlinear case. Numerical convergence tests and physical analysis are finally performed in the fully heterogeneous and nonlinear case of shock wave focusing through an acoustical lens.

François Coulouvrat

Papers

Numerical simulation of shock wave focusing at fold caustics, with application to sonic boom.

Breaking of time reversal invariance in nonlinear acoustics.

Continuous field radiated by a geometrically focused transducer: Numerical investigation and comparison with an approximate model

Numerical Simulation of Sonic Boom Focusing

Acoustic shock wave propagation in a heterogeneous medium: a numerical simulation beyond the parabolic approximation.