Interaction of elementary waves for equations of potential flow

Interaction of elementary waves for equations of unsteady potential flow in gas dynamics is considered. Under the assumptions on weakness of strength of the elementary waves the structure of solutions has been given in various cases of interaction of simple wave with shock, or interaction between simple waves or shocks. Hence the complete results on interaction of weak elementary waves for second-order equation of potential flow are obtained. Project supported by the National Natural Science Foundation of China and the State Education Commission of China.     

管道内激波的不稳定性

本文将无限大激波阵面的激波不稳定性理论[1]推广到矩形截面管道内的激波不稳定性问题.首先,给出这个问题的数学提法,包括扰动方程与三类边界条件.其次,给出扰动方程的普遍解.上游和下游的普遍解分别含有5个待定常数.再次,在一类边界条件和一个假定下,证明了激波前扰动为0,激波后两个声扰动之一为0.边界条件是,X→±∞处扰动物理量为0.假定只讨论激波不稳定性问题,从而可先设ω=iγ,γ是不稳定性增长率,为正实数.另一类边界条件是管壁上法向速度扰动为0,它使波数只能取一组离散值.最后,用扰动激波上的5个守恒方程这一边界条件来决定激波后4个待定常数和扰动激波振幅这个未知量时,导出了色散关系.结果表明,正实数γ确是存在.不稳定激波有两种模式,一种模式为γ=-W·k(W<0)它代表激波的绝对不稳定性,是新得到的模式.另一种模式与过去工作中给出的[2,3]大体相同.本文则进一步给出了这种模式的激波不稳定性增长率,并指出j2((?V/?P)_H=1+2M为最不稳定点(即无量纲化的不稳定性增长率Г=∞).如果不假定ω是纯虚数,而是复数,其虚部为正实数Im(ω)≥0.本文也严格证明了其不稳定性判据仍有两种模式,ω仍为纯虚数.     

The vanishing viscosity method for the sensitivity analysis of an optimal control problem of conservation laws in the presence of shocks

In this article we study, by the vanishing viscosity method, the sensitivity analysis of an optimal control problem for 1-D scalar conservation laws in the presence of shocks. It is reduced to investigate the vanishing viscosity limit for the nonlinear conservation law, the corresponding linearized equation and its adjoint equation, respectively. We employ the method of matched asymptotic expansions to construct approximate solutions to those equations. It is then proved that the approximate solutions, respectively, satisfy those viscous equations in the asymptotic sense, and converge to the solutions of the corresponding inviscid problems with certain convergent rates. A new equation for the variation of shock positions is derived. It is also discussed how to identify descent directions to find the minimizer of the viscous optimal control problem in the quasi-shock case.     

Approximation of the perturbation equations of a quasi-linear hyperbolic system in the neighborhood of a bicharacteristic

In 1956 Whitham gave a nonlinear theory for computing the intensity of an acoustic pulse of an arbitrary shape. The theory has been used very successfully in computing the intensity of the sonic bang produced by a supersonic plane. [4.] derived an approximate quasi-linear equation for the propagation of a short wave in a compressible medium. These two methods are essentially nonlinear approximations of the perturbation equations of the system of gas-dynamic equations in the neighborhood of a bicharacteristic curve (or rays) for weak unsteady disturbances superimposed on a given steady solution. In this paper we have derived an approximate quasi-linear equation which is an approximation of perturbation equations in the neighborhood of a bicharacteristic curve for a weak pulse governed by a general system of first order quasi-linear partial differential equations in m + 1 independent variables (t, x₁,…, x_m) and derived Gubkin's result as a particular case when the system of equations consists of the equations of an unsteady motion of a compressible gas. We have also discussed the form of the approximate equation describing the waves propagating upsteam in an arbitrary multidimensional transonic flow.     

On discreteness of the Hopf equation

The principle aim of this essay is to illustrate how different phenomena is captured by different discretizations of the Hopf equation and general hyperbolic conservation laws. This includes dispersive schemes, shock capturing schemes as well as schemes for computing multi-valued solutions of the underlying equation. We introduce some model equations which describe the behavior of the discrete equation more accurate than the original equation. These model equations can either be conveniently discretized for producing novel numerical schemes or further analyzed to enrich the theory of nonlinear partial differential equations.     

An approximate solution for spherical and cylindrical piston problem

A new theory of shock dynamics (NTSD) has been derived in the form of a finite number of compatibility conditions along shock rays. It has been used to study the growth and decay of shock strengths for spherical and cylindrical pistons starting from a non-zero velocity. Further a weak shock theory has been derived using a simple perturbation method which admits an exact solution and also agrees with the classical decay laws for weak spherical and cylindrical shocks.     

Determination of vorticity,gradients of flow parameters and curvature of stream-lines behind unsteady curved shock waves

In this paper the gradients of flow parameters and the expression for vorticity vector behind three-dimensional unsteady curved shock waves in fluids obeying anarbitrary equation of state have been explicitly determined. A transformation of coordinates defined by Taub [1](¹), has been used and various conservation equations have been obtained in the new coordinate system, assuming the dissipative mechanisms such as viscosity and heat-conduction as absent. The flow quantities on the upstream side of the shock are assumed to be uniform and known and those on its downstream side have been determinedin principle in terms of known quantities for the flow obeying anarbitrary equation of state. For the flow of a perfect gas all the unknown quantities have been explicitly calculated in terms of known quantities. The derivatives of entropy and curvature of stream-lines behind the unsteady shock wave have also been explicitly determined.     

一种计算超声速流中由湍流剪切层脉动引起的激波振荡频率的方法

在超声速或高超声速绕流中,一种很严重的脉动压力环境是由激波边界层相互作用引起的激波振荡.这种高强度的振荡激波可能诱发结构共振.因这一现象非常复杂,已发表的文章都采用经验或半经验方法.本文首次从基本流体动力学方程出发,给出了由湍流剪切层引起的激波振荡频率的理论解,得到了振荡频率随气流Mach数M_∞和压缩折转角θ的变化规律,计算结果与实验值是相符的.本文为激波振荡导致的气动弹性问题提供了一种有价值的理论方法.     

New exact solutions of a generalised Boussinesq equation with damping term and a system of variant Boussinesq equations via double reduction theory

The conservation laws of a generalised Boussinesq (GB) equation with damping term are derived via the partial Noether approach. The derived conserved vectors are adjusted to satisfy the divergence condition. We use the definition of the association of symmetries of partial differential equations with conservation laws and the relationship between symmetries and conservation laws to find a double reduction of the equation. As a result, several new exact solutions are obtained. A similar analysis is performed for a system of variant Boussinesq (VB) equations.     

Nonlinear orientational dynamics of a molecular chain

We investigate the nonlinear rotational dynamics of a molecular chain with quadrupole interaction in both the discrete and the continuous cases. Based on a system of nonlinear differential-difference equations, we obtain approximate equations describing the chain excitations and preserving the initial symmetry. We introduce an effective potential and normal coordinates, using which allows decoupling the system into linear and nonlinear parts. As a result of a strong anisotropy of the potential, narrow “valleys” occur in the angle plane. Motion along a valley corresponds to a softer interaction (nonlinear equations). Linear equations describe motion across a valley (hard interaction). We consider cases where the derived nonlinear equations reduce to the sine-Gordon equation. We find integrals of motion and exact solutions of our approximate equations. We uniformly describe the energy interval encompassing the domains of order, of orientational melting, and of rotational motion of the molecules in the chain.     

Mathematical analysis of the quasilinear effects in a hyperbolic model blood flow through compliant axi‐symmetric vessels

In this paper, we present a mathematical analysis of the quasilinear effects arising in a hyperbolic system of partial differential equations modelling blood flow through large compliant vessels. The equations are derived using asymptotic reduction of the incompressible Navier–Stokes equations in narrow, long channels. To guarantee strict hyperbolicity we first derive the estimates on the initial and boundary data which imply strict hyperbolicity in the region of smooth flow. We then prove a general theorem which provides conditions under which an initial–boundary value problem for a quasilinear hyperbolic system admits a smooth solution. Using this result we show that pulsatile flow boundary data always give rise to shock formation (high gradients in the velocity and inner vessel radius). We estimate the time and the location of the first shock formation and show that in a healthy individual, shocks form well outside the physiologically interesting region (2.8m downstream from the inlet boundary). In the end we present a study of the influence of vessel tapering on shock formation. We obtain a surprising result: vessel tapering postpones shock formation. We provide an explanation for why this is the case. Copyright © 2003 John Wiley & Sons, Ltd.     

Conservation laws of turbulence models

The conservation of mass, momentum, energy, helicity, and enstrophy in fluid flow are important because these quantities organize a flow, and characterize change in the flow's structure over time. In turbulent flow, conservation laws remain important in the inertial range of wave numbers, where viscous effects are negligible. It is in the inertial range where energy, helicity (3d), and enstrophy (2d) must be accurately cascaded for a turbulence model to be qualitatively correct. A first and necessary step for an accurate cascade is conservation; however, many turbulent flow simulations are based on turbulence models whose conservation properties are little explored and might be very different from those of the Navier-Stokes equations.We explore conservation laws and approximate conservation laws satisfied by LES turbulence models. For the Leray, Leray deconvolution, Bardina, and Nth order deconvolution models, we give exact or approximate laws for a model mass, momentum, energy, enstrophy and helicity. The possibility of cascades for model quantities is also discussed.     

Explicit asymptotic modelling of transient Love waves propagated along a thin coating

An explicit asymptotic model for transient Love waves is derived from the exact equations of anti-plane elasticity. The perturbation procedure relies upon the slow decay of low-frequency Love waves to approximate the displacement field in the substrate by a power series in the depth coordinate. When appropriate decay conditions are imposed on the series, one obtains a model equation governing the displacement at the interface between the coating and the substrate. Unusually, the model equation contains a term with a pseudo-differential operator. This result is confirmed and interpreted by analysing the exact solution obtained by integral transforms. The performance of the derived model is illustrated by numerical examples     

Explicit asymptotic modelling of transient Love waves propagated along a thin coating

An explicit asymptotic model for transient Love waves is derived from the exact equations of anti-plane elasticity. The perturbation procedure relies upon the slow decay of low-frequency Love waves to approximate the displacement field in the substrate by a power series in the depth coordinate. When appropriate decay conditions are imposed on the series, one obtains a model equation governing the displacement at the interface between the coating and the substrate. Unusually, the model equation contains a term with a pseudo-differential operator. This result is confirmed and interpreted by analysing the exact solution obtained by integral transforms. The performance of the derived model is illustrated by numerical examples     

Modeling weakly nonlinear acoustic wave propagation

Three weakly nonlinear models of lossless, compressible fluidflow—a straightforward weakly nonlinear equation (WNE),the inviscid Kuznetsov equation (IKE) and the Lighthill–Westerveltequation (LWE)—are derived from first principles and theirrelationship to each other is established. Through a numericalstudy of the blow-up of acceleration waves, the weakly nonlinearequations are compared to the ‘exact’ Euler equations,and the ranges of applicability of the approximate models areassessed. By reformulating these equations as hyperbolic systemsof conservation laws, we are able to employ a Godunov-type finite-differencescheme to obtain numerical solutions of the approximate modelsfor times beyond the instant of blow-up (that is, shock formation),for both density and velocity boundary conditions. Our studyreveals that the straightforward WNE gives the best results,followed by the IKE, with the LWE's performance being the poorestoverall.     

The validity of the long-wave approximation for solitary waves in fluid-filled elastic tubes

Studies of solitary waves commonly apply the long-wave approximation, which unnecessarily rigidifies the behavior of the tube, but permits the problem to be solved using certain approximate numerical techniques. In this study, an approach was developed where approximating the contribution of the axial strain as a linear function of the radial strain reduced the system of exact governing differential equations to a single equation of a single dependent variable. The approximated solution was found to agree with the exact solution to within 3%. This approach would be useful for considering more complex problems where the exact solution technique may not be reasonably applied.     

The Interaction of Shocks with Dispersive Waves II. Incompressible-Integrable Limit

This is the second in a two-part series of articles in which we analyze a system similar in structure to the well-known Zakharov equations from weak plasma turbulence theory, but with a nonlinear conservation equation allowing finite time shock formation. In this article we analyze the incompressible limit in which the shock speed is large compared to the underlying group velocity of the dispersive wave (a situation typically encountered in applications). After presenting some exact solutions of the full system, a multiscale perturbation method is used to resolve several basic wave interactions. The analysis breaks down into two categories: the nonlinear limit and the linear limit, corresponding to the form of the equations when the group velocity to shock speed ratio, denoted by ε, is zero. The former case is an integrable limit in which the model reduces to the cubic nonlinear Schrödinger equation governing the dispersive wave envelope. We focus on the interaction of a “fast” shock wave and a single hump soliton. In the latter case, the ε=0 problem reduces to the linear Schrödinger equation, and the focus is on a fast shock interacting with a dispersive wave whose amplitude is cusped and exponentially decaying. To motivate the time scales and structure of the shock-dispersive wave interactions at lowest orders, we first analyze a simpler system of ordinary differential equations structurally similar to the original system. Then we return to the fully coupled partial differential equations and develop a multiscale asymptotic method to derive the effective leading-order shock equations and the leading-order modulation equations governing the phase and amplitude of the dispersive wave envelope. The leading-order interaction equations admit a fairly complete analysis based on characteristic methods. Conditions are derived in which: (a) the shock passes through the soliton, (b) the shock is completely blocked by the soliton, or (c) the shock reverses direction. In the linear limit, a phenomenon is described in which the dispersive wave induces the formation of a second, transient shock front in the rapidly moving hyperbolic wave. In all cases, we can characterize the long-time dynamics of the shock. The influence of the shock on the dispersive wave is manifested, to leading order, in the generalized frequency of the dispersive wave: the fast-time part of the frequency is the shock wave itself. Hence, the frequency undergoes a sudden jump across the shock layer.In the last section, a sequence of numerical experiments depicting some of the interesting interactions predicted by the analysis is performed on the leading-order shock equations.     

On algorithms for estimating computable error bounds for approximate periodic solutions of an autonomous delay differential equation

Machine tool chatter has been characterized as isolated periodic solutions or limit cycles of delay differential equations. Determining the amplitude and frequency of the limit cycle is sometimes crucial to understanding and controlling the stability of machining operations. In Gilsinn [Gilsinn DE. Computable error bounds for approximate periodic solutions of autonomous delay differential equations, Nonlinear Dyn 2007;50:73–92] a result was proven that says that, given an approximate periodic solution and frequency of an autonomous delay differential equation that satisfies a certain non-criticality condition, there is an exact periodic solution and frequency in a computable neighborhood of the approximate solution and frequency. The proof required the estimation of a number of parameters and the verification of three inequalities. In this paper the details of the algorithms will be given for estimating the parameters required to verify the inequalities and to compute the final approximation errors. An application will be given to a Van der Pol oscillator with delay in the non-linear terms.     

A rational generalization of Fan’s method and its application to generalized shallow water wave equations

In this paper we employ a rational expansion to generalize Fan’s method for exact travelling wave solutions for nonlinear partial differential equations (PDEs). To verify the reliability of the proposed method, the generalized shallow water wave (GSWW) equation has been investigated as an example. Kinds of new exact travelling wave solutions of a rational form have been obtained. This indicates that the proposed method provides a more general result for exact solution of nonlinear equations.     

Linearization of the Riemann problem for a triangular system of conservation laws and delta shock wave formation process

Using the weak asymptotic method, we approximate a triangular system of conservation laws arising from the so‐called generalized pressureless gas dynamics by a diagonal linear system. Then, we apply the usual method of characteristics to find approximate solution to the original system. As a consequence, we shall see how the delta shock wave naturally arises along the characteristics. Also, we propose a procedure that could be applied to more general systems of conservation laws. Copyright © 2009 John Wiley & Sons, Ltd.