Thermoviscous Fluid Research Articles

Numerical model for nonlinear standing waves and weak shocks in thermoviscous fluids.

Nonlinear standing waves in a one-dimensional tube are studied numerically by using a finite-difference algorithm. The numerical code models the acoustic field in resonators for homogeneous, thermoviscous fluids. Calculations are performed exclusively in the time domain, and all harmonic components are obtained by one resolution. The fully nonlinear differential equation is written in Lagrangian coordinates. It is solved without truncation. Effects of absorption are included. Displacement and pressure wave forms are calculated at different locations and results are shown for different excitation levels and tube lengths. Amplitude distributions along the resonator axis for every harmonic component are also evaluated. Simulations are performed for amplitudes ranging from linear to strongly nonlinear and weak shock. A very good concordance with classic experimental and analytical results is obtained.

Finite element simulation of nonlinear wave propagation in thermoviscous fluids including dissipation.

A recently developed finite element method (FEM) for the numerical simulation of nonlinear sound wave propagation in thermoviscous fluids is presented. Based on the nonlinear wave equation as derived by Kuznetsov, typical effects associated with nonlinear acoustics, such as generation of higher harmonics and dissipation resulting from the propagation of a finite amplitude wave through a thermoviscous medium, are covered. An efficient time-stepping algorithm based on a modification of the standard Newmark method is used for solving the non-linear semidiscrete equation system. The method is verified by comparison with the well-known Fubini and Fay solutions for plane wave problems, where good agreement is found. As a practical application, a high intensity focused ultrasound (HIFU) source is considered. Impedance simulations of the piezoelectric transducer and the complete HIFU source loaded with air and water are performed and compared with measured data. Measurements of radiated low and high amplitude pressure pulses are compared with corresponding simulation results. The obtained good agreement demonstrates validity and applicability of the nonlinear FEM.

A High-Order Finite-Difference Algorithm for the Analysis of Standing Acoustic Waves of Finite but Moderate Amplitude

A numerical formulation for modelling standing acoustic waves of finite but moderate amplitude is presented. A thermoviscous fluid contained in a one-dimensional tube with rigid walls is considered. The fluid is initially at rest and is excited by means of a harmonic piston. A second-order wave equation for viscous and homogeneous fluids is used. The perturbation method is employed. The numerical simulation is carried out by a multi-time-step, implicit, six-point finite-difference scheme of high order in the time domain. Displacement and pressure waveforms and distributions are presented. Numerical results are validated by comparison with an analytical model. The numerical scheme is illustrated with several examples.

Amplitude degradation of time-reversed pulses in nonlinear absorbing thermoviscous fluids

The linear wave equation in a lossless medium is time reversible, i.e., every solution p(x, t) has a temporal mirror solution p(x,− t). Analysis shows that time reversal also holds for the lossless nonlinear wave equation. In both cases, time-reversal invariance is violated when losses are present. For nonlinear propagation loses cannot normally be ignored; they are necessary to prevent the occurrence of multivalued waveforms. Further analysis of the nonlinear wave equation shows that amplification of a time-reversed pulse at the array elements also leads to a violation of time reversal even for lossless nonlinear acoustics. Numerical simulations are used to illustrate the effect of nonlinearity on the ability of a time-reversal system to effectively focus on a target in an absorbing fluid medium. We consider both the amplitude and arrival time of retrodirected pulses. The numerical results confirm that both shock generation (with the accompanying absorption) and amplification at the array, adversely affect the ability of a time-reversal system to form strong retrodirective sound fields.

On some nonlinear effects in ultrasonic fields

Nonlinear effects associated with intense sound fields in fluids are considered theoretically. Special attention is directed to the study of higher effects that cannot be described within the standard propagation models of nonlinear acoustics (the KZK and Burgers equations). The analysis is based on the fundamental equations of motion for a thermoviscous fluid, for which thermal equations of state exist. Model equations are derived and used to analyze nonlinear sources for generation of flow and heat, and other changes in the ambient state of the fluid. Fluctuations in the coefficients of viscosity and thermal conductivity caused by the sound field, are accounted for. Also considered are nonlinear effects induced in the fluid by flexural vibrations. The intensity and absorption of finite amplitude sound waves are calculated, and related to the sources for generation of higher order effects.

Effect of heat transfer on the plane-channel poiseuille flow of a thermo-viscous fluid

A steady-state plane channel flow of viscous incompressible fluid with no-slip and heat transfer boundary conditions is considered. The flow is induced by a fixed pressure difference and the fluid viscosity depends on the temperature in accordance with a power law. It is shown numerically that the dependence of the Peclet number on the nondimensional pressure difference is not single-valued. An investigation of the solution’s dependence on the Biot number shows that for Biot numbers greater than unity the velocity profile has a point of inflection.

FDTD simulation of finite-amplitude pressure and temperature fields for biomedical ultrasound.

Full wave simulations provide a valuable tool for studying the spatial and temporal nature of an acoustic field. One method for producing such simulations is the finite-difference time-domain (FDTD) method. This method uses discrete differences to approximate derivatives in the governing partial differential equations. We used the FDTD method to model the propagation of finite-amplitude sound in a homogeneous thermoviscous fluid. The calculated acoustic pressure field was then used to compute the transient temperature rise in the fluid; the heating results from absorption of acoustic energy by the fluid. As an example, the transient temperature field was calculated in biological tissue in response to a pulse of focused ultrasound. Enhanced heating of the tissue from finite-amplitude effects was observed. The excess heating was attributed to the nonlinear generation of higher-frequency harmonics which are absorbed more readily than the fundamental. The effect of nonlinear distortion on temperature rise in tissue was observed to range from negligible at 1 MPa source pressure to an 80% increase in temperature elevation at 10 MPa source pressure.

Higher order effects in finite-amplitude sound fields

Nonlinear effects associated with intense sound fields in fluids are considered theoretically. Special attention is directed to the study of higher-order effects that cannot be described within the standard propagation models of nonlinear acoustics (the KZK and Burgers equations). The analysis is based on the fundamental equations of motion for a homogeneous, thermoviscous fluid, for which thermal equations of state exist. Model equations are derived and used to analyze nonlinear sources for generation of flow and heat, and other changes in the ambient state of the fluid. Fluctuations in the coefficients of viscosity and thermal conductivity caused by the sound field are accounted for. Also considered are nonlinear effects induced in the fluid by surface waves in the transducer. The intensity and absorption of finite-amplitude sound waves are calculated, and related to the sources for generation of higher-order effects.

Coupled thermal-acoustic simulation results with temperature-dependent tissue parameters for therapeutic ultrasound

Recently the authors used direct simulations of the transient acoustic pressure field to calculate heat energy deposition and, thus, temperature fields in two-dimensional tissue like media [J. Acoust. Soc. Am. 102, 3172(A) (1997)]. It is known that acoustic-induced temperature rises will alter the properties of the medium in hyperthermia situations. In this presentation the simulations are extended to simultaneously solve the wave propagation and tissue heating problems so that the effect of temperature-dependent tissue parameters can be accounted for directly. In other words, there is a continuous feedback of temperature on the sound propagation parameters, which in turn affects the thermal energy deposition. The simulations are second order accurate in time, fourth order in space full wave calculations using the finite-difference time-domain technique, and allow for finite-amplitude wave propagation in an inhomogeneous, thermoviscous fluid. The model allows for spatially and temporally varying sound speed, density, attenuation coefficient, and nonlinearity parameter, as well as variable thermal properties and perfusion. Acoustic pressure, thermal dose, and tissue temperature are calculated, and conclusions are made regarding qualitative and quantitative aspects of focusing behavior contrasted to the case where the propagation and thermal parameters are kept constant. [Work sponsored by ONR and DARPA.]

Acoustic streaming at high Reynolds numbers due to focused piston radiation in real fluids

In an earlier presentation [Hamilton et al., J. Acoust. Soc. Am. 97, 3376(A) (1995)], a theoretical model for acoustic streaming at high Reynolds numbers (based on streaming velocities) in thermoviscous fluids was used for analytical and numerical predictions based on the assumption of weakly nonlinear focused Gaussian beams. Here, numerical calculations of streaming are presented for radiation from circular pistons, both focused and unfocused, and at amplitudes for which shock formation occurs. The forcing function for the streaming equations is evaluated with time-domain numerical solutions of the nonlinear parabolic wave equation. Effects of thermoviscous dissipation on the acoustic waveforms, and in particular on shock structures, are taken into account explicitly. The calculations provide a consistent description of acoustic streaming generated in real fluids by sound beams at moderate amplitudes, either with or without shocks. Because the shocks are not considered to be discontinuous features of the solutions, uniformly valid predictions of streaming are obtained in the transition region encompassing the birth and death of shocks amid the competing effects of dissipation and diffraction. The model accommodates both continuous and pulsed sources. Comparisons are made with experiments reported in the literature. [Work supported by the Office of Naval Research.]

Time-domain modeling of finite-amplitude sound in relaxing fluids

A time-domain computer algorithm that solves an augmented Burgers equation is described. The algorithm is a modification of the time-domain code developed by Lee and Hamilton [J. Acoust. Soc. Am. 97, 906–917 (1995)] for pulsed finite-amplitude sound beams in homogeneous, thermoviscous fluids. In the present paper, effects of nonlinearity, absorption and dispersion (both thermoviscous and relaxational), geometrical spreading, and inhomogeneity of the medium are taken into account. The novel feature of the code is that effects of absorption and dispersion due to multiple relaxation phenomena are included with calculations performed exclusively in the time domain. Numerical results are compared with an analytic solution for a plane step shock in a monorelaxing fluid, and with frequency-domain calculations for a plane harmonic wave in a thermoviscous, monorelaxing fluid. The algorithm is also used to solve an augmented KZK equation that accounts for nonlinearity, thermoviscous absorption, relaxation, and diffraction in directive sound beams. Calculations are presented which demonstrate the effect of relaxation on the propagation of a pulsed, diffracting, finite-amplitude sound beam.

Propagation of finite‐amplitude broadband noise.

Burgers’ equation is used to predict the effect of nonlinearity on the power spectral density of plane broadband noise traveling in a nondispersive thermoviscous fluid. The source signal is assumed to be stationary Gaussian noise, which, because of nonlinear propagation distortion, becomes non‐Gaussian as it travels. As opposed to time‐domain methods, the method presented here is based directly on the power spectral density of the signal, not the signal itself. The Burgers equation is transformed into an unclosed set of linear equations that describe the evolution of the joint moments of the signal. A method for solving the system of equations is presented. Only the evolution of appropriately selected joint moments needs to be calculated in order to predict the evolution of the power spectral density of the signal. The results are in good agreement with a time‐domain code [Cleveland et al., J. Acoust. Soc. Am. 98, 2865(A) (1995)]. The method can be also applied when the source condition is a stationary, ergodic, and Gaussian stochastic process. [Work supported by NASA.]

Theoretical modeling of the acoustic pressure field produced by commercial lithotripters

A theoretical model for the acoustic field produced by commercial lithotripters based on the KZK equation is presented. The KZK equation has been used previously to model high-intensity sound beams in thermoviscous fluids. Both electrohydraulic and piezoelectric lithotripters are considered. To model the acoustic field reflected from ellipsoidal reflectors found in electrohydraulic litho- tripters, geometrical acoustics are used to define a directivity function at the mouth of the reflector. An equivalent focused source with a shading function defined by the directivity function is then assumed as the boundary condition for the KZK equation. A code that solves this equation entirely in the time domain is used to obtain results for the acoustic pressure. The numerical results are compared with previous experiments [Coleman etal., Ultrasound Med. Biol. 13(10), 651–657 (1987)] and good agreement is found. Results for propagation in both water and tissue are presented. Positive pressures of 40–100 MPa and negative pressures of 4–10 MPa are predicted in the focal region. A reflector with a pressure release surface is also considered. In this case the field close to the source is an inverted replica of that of the rigid reflector. However, strong distortion due to nonlinearity affects the rest of field. Negative pressures of 20–30 MPa are predicted in the focal region for this type of reflector. [Work supported by NIH.]

Nonlinear analysis of the generalized thermo-magnetodynamic problem

The results reported in this paper were obtained in the course of an investigation of a simple model problem of global geodynamics and magnetic field of the planet Uranus. The problem leads to an initial-boundary value problem for a coupled system of equations of magnetodynamics of an incompressible, electrically conducting, thermo-viscous and ferromagnetic Bingham's fluid. The variational formulation of the problem and the existence and uniqueness theorem will be given.

Asymptotic stability of shearing flows of incompressible thermoviscous fluids

Asymptotic stability of shearing flows of incompressible thermoviscous fluids

Time-domain modeling of pulsed finite-amplitude sound beams

A time-domain algorithm that solves the Khokhlov–Zabolotskaya–Kuznetsov (KZK) nonlinear parabolic wave equation is described. The algorithm models the propagation of pulsed finite amplitude sound beams radiated from axisymmetric sources in homogeneous, thermoviscous fluids. Numerical results are presented for waveform distortion and shock formation in directive beams radiated by pulsed circular pistons. Waveforms are calculated through the shock region and out to far-field locations where they are dominated by the nonlinearly generated low-frequency components. Effects of pulse duration, frequency modulation, and noise are examined. Methods for including relaxation and focusing are described.

Nonlinear propagation through a fluid of waves originating from a biharmonic sound source

A sufficiently strong sound source generates in a thermoviscous fluid, due to nonlinearity, a frequency spectrum consisting of all multiples of the original frequencies and the sums and differences of these multiples. After a certain distance, a shock front is formed because of the energy transfer from lower to higher frequencies. In the case of two original frequencies as a source (the biharmonic case), the damping of high frequencies leaves us at a large distance from the source with primarily the difference frequency. The propagation of plane waves is described by the Burgers’ equation whose solution in the regions before and after the shock formation exhibits significantly different approximate analytical expressions. In this work, an analytical description of the total amplitude in the region after formation of shock in the case of a biharmonic sound source is found. This is a generalization of the well-known Khokhlov solution for a monochromatic (single frequency) source. This description is turned into a Fourier series which can be specialized into the classical Fay solution for a monochromatic source. From this Fourier series the behavior of the individual frequencies is obtained, in particular the difference frequency which is also examined.

Time harmonic acoustic Bloch wave propagation in periodic waveguides. Part I. Theory

The propagation of linear, dissipative, time-harmonic acoustic waves in a broad class of periodic waveguides is investigated theoretically. It is shown that a Floquet-like theorem is applicable to the system of partial differential equations and boundary conditions that describes the dynamics of a thermoviscous fluid in a periodic waveguide, and that the solution wave functions are therefore Bloch wave functions. Expressions for the parameters that characterize the Bloch wave functions are derived and the band structure of these functions of frequency is investigated. It is found that conditions of axial inversion symmetry (the waveguide may be isotropic or anisotropic) and reciprocity impose substantial restrictions on the allowed Bloch wave solutions.

Boundary layer theory and subduction

Numerical models of thermally activated convective flow in Earth's mantle do not resemble active plate tectonics because of their inability to model successfully the process of subduction, other than by the inclusion of artificial weak zones. Here we show, using a boundary layer argument, how the “ rigid lid ” style of convection favored by thermoviscous fluids leads to lithospheric stresses which may realistically exceed the yield stress and thus cause subduction to occur through the visoc‐plastic failure of lithospheric rock. An explicit criterion for the failure of the lid is given, which is sensitive to the internal viscosity ηa below the lid. For numbers appropriate to Earth's mantle, this criterion is approximately ηa > 1021 Pa s.

Self-demodulation of amplitude- and frequency-modulated pulses in a thermoviscous fluid

The self-demodulation of pulsed sound beams in a thermoviscous fluid is investigated experimentally and theoretically. Experiments were performed in glycerin at megahertz frequencies with amplitude- and frequency-modulated pulses. The theory is based on the Khokhlov–Zabolotskaya–Kuznetsov (KZK) nonlinear parabolic wave equation. Numerical results were obtained from an algorithm that solves the KZK equation in the time domain [Y.-S. Lee and M. F. Hamilton, Ultrasonics International 91 Conference Proceedings (Butterworth–Heinemann, Oxford, 1991), pp. 177–180]. A quasilinear analytic solution, which describes the main features of the waveform at all axial locations, is developed in the limit of strong absorption. Theory and experiment are in good agreement throughout the near- and far fields.