Dynamics of shock waves in a liquid containing gas bubbles

Results are presented of a numerical solution of the Korteweg-de Vries-Burgers equation that describes the propagation and establishment process for a stationary structure to a shock wave in a gas-liquid medium. Data are obtained on the time for the establishment of a stationary structure of a shock wave, propagation velocity, and amplitude oscillations in the front of the shock wave. Experiments are discussed on the basis of the results obtained for the study of shock waves in a liquid containing gas bubbles.     

The nonlinear response of Cattaneo-type thermal loading of a laser pulse on a medium using the generalized thermoelastic model

The nonlinear thermoelastic responses of an elastic medium exposed to laser generated shortpulse heating are investigated in this article. The thermal wave propagation of generalized thermoelastic medium under the impact of thermal loading with energy dissipation is the focus of this research. To model the thermal boundary condition(in the form of thermal conduction),generalized Cattaneo model(GCM) is employed. In the reference configuration, a nonlinear coupled Lord-Shulman-type generalized thermoelasticity formulation using finite strain theory(FST) is developed and the temperature dependency of the thermal conductivity is considered to derive the equations. In order to solve the time-dependent and nonlinear equations, Newmark's numerical time integration technique and an updated finite element algorithm is applied and to ensure achieving accurate continuity of the results, the Hermitian elements are used instead of Lagrangian's. The numerical responses for different factors such as input heat flux and nonlinear terms are expressed graphically and their impacts on the system's reaction are discussed in detail.The results of the study are presented for Green–Lindsay model and the findings are compared with Lord-Shulman model especially with regards to heat wave propagation. It is shown that the nature of the laser's thermal shock and its geometry are particularly determinative in the final stage of deformation. The research also concluded that employing FST leads to achieving more accuracy in terms of elastic deformations; however, the thermally nonlinear analysis does not change the results markedly. For this reason, the nonlinear theory of deformation is required in laser related reviews, while it is reasonable to ignore the temperature changes compared to the reference temperature in deriving governing equations.     

Uniaxial wave propagation in a nonlinear thermoviscoelastic medium with temperature dependent properties

The problem of finite wave propagation in a nonlinearly thermoviscoelastic thin rod whose viscoelastic properties are temperature dependent is considered. The rod is subjected to mechanical or thermal time-dependent loading. The coupled equations of motion and heat conduction are based on a constitutive theory of nonisothermal nonlinear viscoelasticity which is described by single-integral terms only. This theory is reformulated here for the uniaxial motion of a compressible rubbery material. The solution of the field equations is obtained by a numerical procedure which is developed for the present case and is able to handle successfully shock waves whenever they built up in the nonlinear material.     

The influence of magnetic field upon the collapse of a cylindrical shock wave

In the present paper, the problem of propagation of collapsing cylindrical shock wave in an ideal gas permeated by a transverse magnetic field with infinite electrical conductivity is investigated. Here it is assumed that the medium ahead of the shock front is uniform and at rest. Also, its counter pressure concerning the motion of the wave front is neglected. This problem admits a self similar solution of second kind. The similarity exponent has been computed by solving a nonlinear eigenvalue problem and integrating numerically the self-similar equations for various values of adiabatic heat exponent and Cowling number. Numerical computations have been performed to determine the flow field behind the shock wave. The influence of magnetic field strength and adiabatic heat exponent on the flow parameters for various cases is presented.     

On the Interphase Heat Transfer Effect on Nonlinear Wave Propagation in a Gas-Liquid Mixture

The effect of interphase heat transfer on shock wave propagation is investigated. A multiwave nonlinear equation which in the limiting case of the absence of heat transfer decomposes into two classic generalizations of the Boussinesq equations is derived. Quasi-isothermal and quasi-adiabatic propagation regimes for which the heat transfer is fairly intense are considered. For both regimes, nonlinear equations describing the wave propagation are obtained. The equation describing the first regime is investigated in detail. Exact analytic solutions of this equation are given and used to study the shock wave structures and the solitary wave behavior. Formulas for the dependence of the heat transfer rate on the equilibrium-mixture parameters are obtained.     

Analytical Solution for Isothermal Flow in a Shock Tube Containing Rigid Granular Material

Analytical solution of shock wave propagation in pure gas in a shock tube is usually addressed in gas dynamics. However, such a solution for granular media is complex due to the inclusion of parameters relating to particles configuration within the medium, which affect the balance equations. In this article, an analytical solution for isothermal shock wave propagation in an isotropic homogenous rigid granular material is presented, and a closed-form solution is obtained for the case of weak shock waves. Fluid mass and momentum equations are first written in wave and (mathematical) non-conservation forms. Afterwards by redefining the sound speed of the gas flowing inside the pores, an analytical solution is obtained using the classical method of characteristics, followed by Taylor’s series expansion based on the assumption of weak flow which finally led to explicit functions for velocity, density and pressure. The solution enables plotting gas velocity, density and pressure variations in the porous medium, which is of high interest in the design of granular shock isolators.     

空化可压缩流动空穴溃灭激波特性研究

为深入研究空化可压缩流动中空泡/空泡团溃灭过程中激波产生、传播及其与空穴相互作用规律,本文采用数值模拟方法对空化可压缩流动空穴溃灭激波特性展开了研究.数值计算基于OpenFOAM开源程序,综合考虑蒸汽相和液相的压缩性,通过在原无相变两相可压缩求解器的控制方程中耦合模拟空化汽液相间质量交换的源项,实现了对空化流动的非定常可压缩计算.利用上述考虑汽/液相可压缩性的空化流动求解器,对周期性云状空化流动进行了数值模拟,并重点研究了空穴溃灭激波特性.结果表明:上述数值计算方法可以准确捕捉到空穴非定常演化过程及大尺度脱落空泡云团溃灭激波现象,大尺度脱落空泡云团溃灭过程分为3个阶段:(1) U型空泡团形成; (2) U型空泡团头部溃灭; (3) U型空泡团腿部溃灭.在U 型空泡团腿部溃灭瞬间,观察到激波产生,并向上游和下游传播,向上游传播的激波与空穴相互作用,导致水翼吸力面新生的附着型片状空穴回缩,直至完全溃灭.并且空穴溃灭激波存在回弹现象, 抑制了下一周期的空化发展.      

Effect of a shock wave on heat propagation

The nature of the propagation of a thermal wave produced by a powerful explosion was described in a number of papers, for example, [1–6]. It was shown by a numerical method [4] that a shock wave is present together with the thermal wave. In this paper, the effect of a homothermal shock wave on heat propagation is evaluated by an approximate method.     

Submodels of model of nonlinear diffusion with non-stationary absorption

We study the model, describing a nonlinear diffusion process (or a heat propagation process) in an inhomogeneous medium with non-stationary absorption (or source). We found tree submodels of the original model of the nonlinear diffusion process (or the heat propagation process), having different symmetry properties. We found all invariant submodels. All essentially distinct invariant solutions describing these invariant submodels are found either explicitly, or their search is reduced to the solution of the nonlinear integral equations. For example, we obtained the invariant solution describing the nonlinear diffusion process (or the heat distribution process) with two fixed "black holes", and the invariant solution describing the nonlinear diffusion process (or the heat distribution process) with the fixed "black hole" and the moving "black hole". The presence of the arbitrary constants in the integral equations, that determine these solutions provides a new opportunities for analytical and numerical study of the boundary value problems for the received submodels, and, thus, for the original model of the nonlinear diffusion process (or the heat distribution process). For the received invariant submodels we are studied diffusion processes (or heat distribution process) for which at the initial moment of the time at a fixed point are specified or a concentration (a temperature) and its gradient, or a concentration (a temperature) and its rate of change. Solving of boundary value problems describing these processes are reduced to the solving of nonlinear integral equations. We are established the existence and uniqueness of solutions of these boundary value problems under some additional conditions. The obtained results can be used to study the diffusion of substances, diffusion of conduction electrons and other particles, diffusion of physical fields, propagation of heat in inhomogeneous medium.     

Growth and decay of weak shock waves in magnetogasdynamics

The purpose of the present study is to investigate the problem of the propagation of weak shock waves in an inviscid, electrically conducting fluid under the influence of a magnetic field. The analysis assumes the following two cases: (1) a planar flow with a uniform transverse magnetic field and (2) cylindrically symmetric flow with a uniform axial or varying azimuthal magnetic field. A system of two coupled nonlinear transport equations, governing the strength of a shock wave and the first-order discontinuity induced behind it, are derived that admit a solution that agrees with the classical decay laws for a weak shock. An analytic expression for the determination of the shock formation distance is obtained. How the magnetic field strength, whether axial or azimuthal, influences the shock formation is also assessed.     

Unsteady waves in a liquid containing vapor bubbles

Unsteady wave processes in vapor-liquid media containing bubbles are investigated taking into account the unsteady interphase heat and mass transfer. A single velocity model of the medium with two pressures is used for this, which takes into account the radial inertia of the liquid with a change in volume of the medium and the temperature distribution in it [1]. The system of original differential equations of the model is converted into a form suitable for carrying out numerical integration. The basic principles governing the evolution of unsteady waves are studied. The determining influence of the interphase heat and mass transfer on the wave behavior is demonstrated. It is found that the time and distance at which the waves reach a steady configuration in a vapor-liquid bubble medium are considerably less than the correponding characteristics in a gas-liquid medium. The results of the calculation are compared with experimental data. The propagation of acoustic disturbances in a liquid with vapor bubbles was studied theoretically in [2]. The evolution of waves of small but finite amplitude propagating in one direction in a bubbling vapor-liquid medium is investigated in [3, 4] on the basis of the generalization of the Burgers-Korteweg-de Vries equation obtained by the authors. An experimental investigation of shock waves in such a medium is reported in [5, 6], and the structure of steady shock waves is discussed [7].Translated from Izvestiya Akademii Nauk SSSR, Hekhanika Zhidkosti i Gaza, No. 5, pp. 117–125, September–October, 1984.     

Supersonic low-concentration dusty-gas flow past a plane cylinder in the presence of an oblique shock wave interacting with the bow shock

The motion of an inertial dispersed admixture near a plane cylinder immersed in a steady-state hypersonic dusty flow in the presence of an oblique shock wave interacting with the bow shock is considered. It is assumed that the free-stream particle mass concentration is small and the particles do not affect the carrier flow. The III and IV shock wave interaction regimes are considered. The gas flow parameters in the shock layer are calculated from the numerical solution of the full Navier-Stokes equations for the perfect gas. A TVD second-order finite-difference scheme constructed on the basis of a finite volume method is used. For calculating the dispersed-phase parameters, including the concentration, the full Lagrangian method is used. On a wide range of variation of the particle inertia parameters, the patterns of the particle trajectories, velocity, concentration, and temperature in the shock layer are studied. The possibility of aerodynamic focusing of the particles behind the shock wave intersection point and the formation of narrow beams with a high particle concentration is revealed. These beams impinge on the cylinder surface and result in a sharp increase in the local heat fluxes. The maximal possible increase in the heat fluxes caused by the particles colliding with the cylinder surface is estimated for the flows with and without the incident oblique shock wave.     

颗粒材料中致密波结构研究

采用一维两相流模型与相应颗粒构形应力函数,研究了致密波的形成及其结构．用简化两相流模型系统地讨论致密波对有关因素的依赖关系．分析指出：小于基体材料音速的致密波仅能在非理想颗粒材料中存在,从波前到波后,所有状态物理量光滑过渡．大于基体材料音速的致密波,波头可能存在间断．应力函数与致密粘性确定后,致密波速度决定致密波结构、宽度、终态压实度．采用一维两相流模型模拟了活塞驱动颗粒床形成致密波这一动态过程．用线方法（MOL）对该方程组求数值解．计算表明,经过短暂的非稳态过程,颗粒床中形成一稳态致密波．分析了活塞速度与初始孔隙率对致密波结构的影响,并对简化两相流模型与两相流模型的计算结果进行了对比．     

Thermal shielding of a sphere from the action of a strong shock wave

The time-dependent interaction of an incident shock wave with a sphere is considered in the presence of a heat supply region ahead of the body. The reflected shock configuration and the flow pattern are numerically investigated. The efficiencies of heat shields of different shapes are compared with respect to the longitudinal force acting on the sphere.     

A numerical study of weak shock wave propagation in a reactive bubbly liquid

A numerical study is presented on the response of a weakly shock compressed liquid column that contains reactive gas bubbles. Both the liquid and gas are considered compressible. Compressibility of the liquid allows calculation of shock and rarefaction waves in the pure liquid as well as in the gas/liquid mixture. A microscopic model for local bubble collapse is coupled with a macroscopic model of wave propagation through the gas/liquid mixture. In the particular cases presented here, the characteristic times of propagation of the shock wave and bubble collapse are of the same order of magnitude. Consequently, the coupling between various phenomena is very strong. The present model based on fundamental principles of two-phase fluid mechanics takes into account the coupling of localized bubble oscillations. This model is composed of a microscopic one in the scale of a bubble size, and a macroscopic one which is based on the mixture theory. The liquid under study is water, and the gas is a reactive mixture of argon, hydrogen and oxygen. Received 18 December 1995 / Accepted 2 June 1996     

Scattering of plane monochromatic waves from a heterogeneous inclusion of arbitrary shape in a poroelastic medium: An efficient numerical solution

Scattering of plane longitudinal monochromatic waves from a heterogeneous inclusion of arbitrary shape in an infinite poroelastic medium is considered. Wave propagation in the medium is described by Biot’s equations of poroelasticity. The scattering problem is formulated in terms of the volume integral equations for displacements of the solid skeleton and fluid pressure in the pore space in the region occupied by the inclusion. An efficient numerical method is applied to solve these equations. In the method, Gaussian approximating functions are used for discretization of the problem. For regular node grids, the matrix of the discretized problem has Toeplitz’s properties, and the Fast Fourier Transform technique can be used for the calculation of matrix–vector products. The latter accelerates substantially the process of iterative solution of the discretized problem. For material parameters of typical sedimentary rocks, the system of differential equations of poroelasticity contains a differential operator with a small parameter. As the result, the wave field in the inclusion region is split up into a slowly changing part, and boundary layer functions concentrated near the inclusion interface. The method of matched asymptotic expansions is used for the numerical solution in this case. For a spherical inclusion, the results of the numerical and matched asymptotic expansion methods are compared with a semi-analytical series solution. For a non-spherical heterogeneous inclusion, an example of the numerical solution is presented.     

温度对冲击相变波传播的影响

冲击加载下,相界面的传播是一热力耦合过程。相变波阵面不仅是力学和物质间断面,也是温度界面。为考虑温度对相变波传播的影响,本文首先建立了相界面上的热传导方程和热力耦合的相变本构方程,然后采用一维特征线理论和有限差分数值计算相结合的方法,分析了温度界面和相变波的基本相互作用规律,进而给出了连续温度梯度下和绝热冲击下相变波传播规律。结果表明,温度对相变波传播的作用主要体现在两个方面,一方面是作为温度界面将与各类间断面相互作用,另一方面冲击相变波阵面后区域热力学状态变化影响卸载波结构。其原因在于相变方式（可逆、不可逆）和相变阈值应力具有强烈的温度相关性。     

Interaction Between a Time-Dependent Laminar Boundary Layer and an Inviscid Subsonic Flow Near a Region of Local Energy Supply

The results of numerical modeling of the time-dependent flows of a viscous heat-conducting gas occurring in the region of interaction between an external inviscid flow and a laminar boundary layer near a zone of local energy supply at high subcritical Reynolds numbers are presented. The solution of the Navier-Stokes equations is constructed on the basis of the method of matched asymptotic expansions. Numerical solutions of the nonlinear boundary-value problem describing the flow in the wall region of the boundary layer are given in similarity variables. It is shown that time- and space-localized energy supply results in the formation of a self-consistent flow disturbance, whose downstream propagation is accompanied by a disturbance amplitude growth during a short time interval, even after the energy supply has stopped. Calculations of the flows induced by two heat sources placed in tandem make it possible to conclude that the time lag for the second energy supply zone and the distance between the sources can be so chosen that superposition of the disturbances induced by the first and second sources leads, due to nonlinear effects, to a considerable increase in the amplitude of the total flow disturbance.__________Translated from Izvestiya Rossiiskoi Academii Nauk, Mekhanika Zhidkosti i Gaza, No. 3, 2005, pp. 64–75.Original Russian Text Copyright © 2005 by Kazakov.     

Propagation of longitudinal deformation wave along a lifting rope of variable length

The problem of motion of the rope of variable length consists of solving the boundary value problem with a variable boundary for the one-dimensional wave equation. A change of the rope length is caused by the force acting at the upper cross section of the rope. Studying the wave propagation process along the rope is based on the known integro-differential relation. The problem is reduced to solving two ordinary differential equations with a retarded argument that describe the variable length of the rope and the position of its lower end. The value of argument for functions involved in the right-hand side of the equations lag behind the argument value in the left-hand side of the equations by a time interval it takes for a propagation of the deformation wave throughout the current rope length. The problem is solved by using a technique of the sequential continuation of solution in the cyclic mode for each of the equation alternately. A computer realization of this technique presents no problem. A pilot computer program has been developed for solving the problem. Results of the numerical solution are presented in the case that the active force varies with time according to a piecewise linear relation.     

Numerical study of the lift force influence on two-phase shock tube boundary layer characteristics

The importance of the lift force acting on the dispersed phase in the boundary layer of a laminar gas-particle dilute mixture flow generated by a shock wave is investigated numerically. The particle phase is supposed to form a continuum and is described by an Eulerian approach. The ability of the Eulerian model to simulate particle flows and the importance of the two-way coupling are proven by comparison with experimental data as well as with the numerical results from schemes based on a Lagrangian approach. The models used for the lift force are discussed through comparisons between numerical and experimental results found in the literature. Some results about the formation of a dust cloud are numerically reproduced and show the major role of the lift force. Simulations of two-dimensional two-phase shock tube flows are also performed including the lift force effects. Although the wave propagation is weakly influenced by the lift force, the force modifies substantially the dynamics of the flow near the wall. Received 17 February 2000 / Accepted 30 November 2000