Analysis of critical dynamics for shock-induced adiabatic explosions by means of the Cauchy problem for the shock transformation

We present a theoretical and numerical study on the induction of adiabatic explosions by accelerated curved shocks in homogeneous explosives, and pay a special attention to critical conditions for initiation. We characterize the first stage of the decomposition process, or induction, as an initial-value problem. During induction, the reaction progress-variable remains small; the induction time is given by the runaway of the dependent variables and corresponds to a logarithmic singularity in theirs material distributions. We express these distributions as first-order expansions in the progress variable about the shock. Then, the framework of our procedure is the formal Cauchy problem for quasi-linear hyperbolic sets of first-order differential equations, such as the balance laws for adiabatic flows of inviscid fluids considered in this study. When a shock front is used as data surface, the solution to the Cauchy problem yields the flow derivatives at the shock, then the induction time, as functions of the shock normal velocity and acceleration, and , and the shock total curvature C. We next derive a necessary condition for explosion as a constraint among , and C that ensures bounded values of the induction time. This criterion is akin to Semenov's, in the sense that the critical condition for explosion is that the heat-production rate must just exceed the heat-loss rate, here given by the volumetric expansion rate at the shock. The violation of the criterion defines a critical shock dynamics as a relationship among , and C that generates infinite induction times. Depending on the rear-boundary conditions, which determine the shock dynamics, this event can be interpreted as either a non-initiation, or the decoupling of the shock and of the flame front induced by the shock. We illustrate our approach by a simple solution to the problem of the initiation by impact of a noncompressible piston. From the continuity constraint in the material speed and acceleration at the contact surface of the piston and the explosive, we first derive the initial shock dynamics, and then rewrite the induction time and the initiation condition in terms of the piston speed, acceleration and curvature. We compare these theoretical predictions to those of our direct numerical simulations, and to numerical results obtained by other authors, in the case of impacts on a gaseous explosive. Received 19 October 1998 / Accepted 1 June 1999     

Method of relaxed streamline upwinding for hyperbolic conservation laws

In this work a new finite element based Method of Relaxed Streamline Upwinding is proposed to solve hyperbolic conservation laws. Formulation of the proposed scheme is based on relaxation system which replaces hyperbolic conservation laws by semi-linear system with stiff source term also called as relaxation term. The advantage of the semi-linear system is that the nonlinearity in the convection term is pushed towards the source term on right hand side which can be handled with ease. Six symmetric discrete velocity models are introduced in two dimensions which symmetrically spread foot of the characteristics in all four quadrants thereby taking information symmetrically from all directions. Proposed scheme gives exact diffusion vectors which are very simple. Moreover, the formulation is easily extendable from scalar to vector conservation laws. Various test cases are solved for Burgers equation (with convex and non-convex flux functions), Euler equations and shallow water equations in one and two dimensions which demonstrate the robustness and accuracy of the proposed scheme. New test cases are proposed for Burgers equation, Euler and shallow water equations. Exact solution is given for two-dimensional Burgers test case which involves normal discontinuity and series of oblique discontinuities. Error analysis of the proposed scheme shows optimal convergence rate. Moreover, spectral stability analysis gives implicit expression of critical time step.     

Wave interactions and variation estimates for self-similar zero-viscosity limits in systems of conservation laws

We consider the problem of self-similar zero-viscosity limits for systems ofN conservation laws. First, we give general conditions so that the resulting boundary-value problem admits solutions. The obtained existence theory covers a large class of systems, in particular the class of symmetric hyperbolic systems. Second, we show that if the system is strictly hyperbolic and the Riemann data are sufficiently close, then the resulting family of solutions is of uniformly bounded variation and oscillation. Third, we construct solutions of the Riemann problem via self-similar zero-viscosity limits and study the structure of the emerging solution and the relation of self-similar zero-viscosity limits and shock profiles. The emerging solution consists ofN wave fans separated by constant states. Each wave fan is associated with one of the characteristic fields and consists of a rarefaction, a shock, or an alternating sequence of shocks and rarefactions so that each shock adjacent to a rarefaction on one side is a contact discontinuity on that side. At shocks, the solutions of the self-similar zero-viscosity problem have the internal structure of a traveling wave.     

A dispersive regularization of the modulational instability of stratified gravity waves

We derive high-order corrections to a modulation theory for the propagation of internal gravity waves in a density-stratified fluid with coupling to the mean flow. The methodology we use allows for strong modulations of wavenumber and mean flow, extending previous approaches developed for the quasi-monochromatic regime. The wave mean flow modulation equations consist of a system of nonlinear conservation laws that may be hyperbolic, elliptic or of mixed type. We investigate the regularizing properties of the asymptotic correction terms in the case when the system becomes unstable and ill-posed due to a change of type (loss of hyperbolicity). A linear analysis reveals that the regularization by the added correction terms does so by introducing a short-wave cut-off of the unstable wavenumbers. We perform various numerical experiments that confirm the regularizing properties of the correction terms, and show that the growth of unstable modes is tempered by nonlinearity. We also find an excellent agreement between the solution of the corrected modulation system and the modulation variables extracted from the numerical solution of the nonlinear Boussinesq equations.     

Shock calculations for axially symmetric shear wave propagation in a hyperelastic incompressible solid

Breaking distance and time are calculated for axially symmetric shear waves in an incompressible hyperelastic solid, whose strain energy function is expressible as a truncated power series in terms of the basic strain invariants. Waves are generated by the application of tractions at the surface of a cylindrical cavity in an unbounded medium. Approximate formulas are developed for the characteristics of the governing system of hyperbolic partial differential equations. Using these formulas, the breaking time of a shock wave is calculated and the location of the shock path is estimated. The shock analysis carried out here provides insight into how the material parameters influence shock development. In order to test the validity of our shock calculations, numerical solutions are obtained using a relaxation scheme for systems of conservation laws.     

Evolution of weak shocks in one dimensional planar and non-planar gasdynamic flows

Asymptotic decay laws for planar and non-planar shock waves and the first order associated discontinuities that catch up with the shock from behind are obtained using four different approximation methods. The singular surface theory is used to derive a pair of transport equations for the shock strength and the associated first order discontinuity, which represents the effect of precursor disturbances that overtake the shock from behind. The asymptotic behaviour of both the discontinuities is completely analysed. It is noticed that the decay of a first order discontinuity is much faster than the decay of the shock; indeed, if the amplitude of the accompanying discontinuity is small then the shock decays faster as compared to the case when the amplitude of the first order discontinuity is finite (not necessarily small). It is shown that for a weak shock, the precursor disturbance evolves like an acceleration wave at the leading order. We show that the asymptotic decay laws for weak shocks and the accompanying first order discontinuity are exactly the ones obtained by using the theory of non-linear geometrical optics, the theory of simple waves using Riemann invariants, and the theory of relatively undistorted waves. It follows that the relatively undistorted wave approximation is a consequence of the simple wave formalism using Riemann invariants.     

Exact solutions of the system of the equations of thermal motion of gas in the rarefied space

We study the model describing thermal motion of gas in the rarefied space. This model can be used, in particular, in the study of the state of the medium behind the front of shock wave after very strong blast, in the study of the processes taking place inside of tornado, in the study of the motion of the gas in outer space. For any given initial distribution of the pressure a specific selection of mass Lagrange variables leads to reduction of the system of differential equations describing this motion to the system, for which the number of independent variables is less on the unit. For the obtained system we found all nontrivial conservation laws of the first order. In addition to the classical conservation laws the system has other conservation laws, which generalizes the energy conservation law. We obtained the exact solutions of this system. These solutions describe a variety of different physical processes taking place in the rarefied medium. Using the symmetry properties of the system we got the generating formulas for the receipt of the new solutions using already found earlier solutions of the system.     

Evaluation of discontinuous Galerkin and spectral volume methods for scalar and system conservation laws on unstructured grids

The discontinuous Galerkin (DG) and spectral volume (SV) methods are two recently developed high‐order methods for hyperbolic conservation laws capable of handling unstructured grids. In this paper, their overall performance in terms of efficiency, accuracy and memory requirement is evaluated using a 2D scalar conservation laws and the 2D Euler equations. To measure their accuracy, problems with analytical solutions are used. Both methods are also used to solve problems with strong discontinuities to test their ability in discontinuity capturing. Both the DG and SV methods are capable of achieving their formal order of accuracy while the DG method has a lower error magnitude and takes more memory. They are also similar in efficiency. The SV method appears to have a higher resolution for discontinuities because the data limiting can be done at the sub‐element level. Copyright © 2004 John Wiley & Sons, Ltd.     

The normal flux method at the boundary for multidimensional finite volume approximations in CFD

This paper presents a general method for imposing boundary conditions in the context of hyperbolic systems of conservation laws. This method is particularly well suited for approximations in the framework of Finite Volume Methods in the sense that it computes directly the normal flux at the boundary. We generalize our approach to nonconservative hyperbolic systems and discuss both the characteristic and the noncharacteristic cases. We present several applications to models occurring in Computational Fluid Mechanics like the Euler equations for compressible inviscid fluids with real equation of state, shallow water equations, magnetohydrodynamics equations and two fluid models.     

Problem of the decay of a small-amplitude discontinuity in two-layer shallow water: First approximation

A method is proposed for the approximate solution of the problem of the decay of a small-amplitude discontinuity in the case of a strictly hyperbolic system of conservation laws which has linearly degenerate characteristic fields. The method is used to analyze the qualitatively different flow regimes occurring in the solution of the problem of the decay of a small-amplitude discontinuity for two-layer shallow water model with a free upper boundary. The most detailed study is made for the special case — the dam-break problem.     

Non-Classical Shocks and Kinetic Relations: Scalar Conservation Laws

This paper analyzes the non-classical shock waves which arise as limits of certain diffusive-dispersive approximations to hyperbolic conservation laws. Such shocks occur for non-convex fluxes and connect regions of different convexity. They have negative entropy dissipation for a single convex entropy function, but not all convex entropies, and do not obey the classical Oleinik entropy criterion. We derive necessary conditions for the existence of non-classical shock waves, and construct them as limits of traveling-wave solutions for several diffusive-dispersive approximations. We introduce a “kinetic relation” to act as a selection principle for choosing a unique non-classical solution to the Riemann problem. The convergence to non-classical weak solutions for the Cauchy problem is investigated. Using numerical experiments, we demonstrate that, for the cubic flux-function, the Beam-Warming scheme produces non-classical shocks while no such shocks are observed with the Lax-Wendroff scheme. All of these results depend crucially on the sign of the dispersion coefficient. (Accepted February 8, 1996)     

Properties of shock adiabats in the neighborhood of jouguet points

Two well-known properties of shock adiabats in a gas [1] are proved for shock adiabats corresponding to discontinuous solutions of hyperbolic systems of equations expressing conservation laws. If the state on one side of a discontinuity is fixed, then at the point of extremum of the discontinuity velocity on the shock adiabat the velocity of the discontinuity is equal to one of the velocities of the characteristics on the other side of the discontinuity and vice versa. If for the systems there is defined an entropy flux or mass density of entropy, then at the points of extremum of the velocity there is an extremum of the entropy production at the discontinuity and the entropy mass density. If the system is a symmetric hyperbolic system [2, 3], then the extrema of the entropy production at the discontinuity correspond to extrema of the velocity. These properties may be helpful in the study of discontinuities in complex media, since the sections of a shock adiabat whose points can correspond to actually existing discontinuities are frequently bounded by points corresponding to discontinuities whose velocity is equal to the velocity of a characteristic on one of the sides of the discontinuity (see, for example, [1, 4, 5]).Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 2, pp. 184–186, March–April, 1979.     

Improvement of multistep WENO scheme and its extension to higher orders of accuracy

This paper presents an efficient procedure for overcoming the deficiency of weighted essentially non‐oscillatory schemes near discontinuities. Through a thorough incorporation of smoothness indicators into the weights definition, up to ninth‐order accurate multistep methods are devised, providing weighted essentially non‐oscillatory schemes with enhanced order of convergence at transition points from smooth regions to a discontinuity, while maintaining stability and the essentially non‐oscillatory behavior. We also provide a detailed analysis of the resolution power and show that the solution enhancements of the new method at smooth regions come from their ability to render smoothness indicators closer to uniformity. The new scheme exhibits similar fidelity as other multistep schemes; however, with superior characteristics in terms of robustness and efficiency, as no logical statements or mapping function is needed. Extensions to higher orders of accuracy present no extra complexity. Numerical solutions of linear advection problems and nonlinear hyperbolic conservation laws are used to demonstrate the scheme's improved behavior for shock‐capturing problems. Copyright © 2016 John Wiley & Sons, Ltd.     

非线性双曲型守恒律的两步二阶TVD差分格式

通过Mac Cormack格式和Warming-Beam的结合,构造了一种非常简单的两步二阶TVD差分格式,该差分格式更适合于使用分量形式差分计算而无须对欧拉方程组进行特征解耦。通过对流体力学方程组的大量数值试验,并与二阶ENO格式进行了比较,充分显示了该格式高精度、高分辨并且极其简单的优良特性。     

RK—AUSM^＋混合格式在跨声速Euler方程计算中的应用

首先在一维AUSM^ 格式的基础上,推导出了AUSM^ 格式在任意曲线坐标下的二维形式,并将其与Runge-Kutta格式结合,对跨声速Euler方程进行求解,最后,为了验证RK-AUSM^ 混合格式的有效性,将典型双圆弧叶栅无粘跨声速流动作为算例,本文计算结果和文献结果符合很好。     

Solution to the Riemann problem for drift-flux model with modified Chaplygin two-phase flows

In this paper, we concern about the Riemann problem for compressible no-slip drift-flux model which represents a system of quasi-linear partial differential equations derived by averaging the mass and momentum conservation laws with modified Chaplygin two-phase flows. We obtain the exact solution of Riemann problem by elaborately analyzing characteristic fields and discuss the elementary waves namely, shock wave, rarefaction wave and contact discontinuity wave. By employing the equality of pressure and velocity across the middle characteristic field, two nonlinear algebraic equations with two unknowns as gas density ahead and behind the middle wave are formed. The Newton–Raphson method of two variables is applied to find the unknowns with a series of initial data from the literature. Finally, the exact solution for the physical quantities such as gas density, liquid density, velocity, and pressure are illustrated graphically.     

一种简单的精确捕捉接触间断的黎曼求解器

基于Godunov型数值格式的有限体积法是求解双曲型守恒律系统的主流方法,其中用来计算界面数值通量的黎曼求解器在很大程度上决定了数值格式在计算中的表现。单波的Rusanov求解器和双波的HLL求解器具有简单、高效和鲁棒性好等优点,但是在捕捉接触间断时耗散太大。全波的HLLC格式能够精确捕捉接触间断,但是在计算中出现的激波不稳定现象限制了其在高马赫数流动问题中的应用。本文利用双曲正切函数和五阶WENO格式来重构界面两侧的密度值,并且结合边界变差下降算法来减小Rusanov格式耗散项中的密度差,从而提高格式对于接触间断的分辨率。研究表明,相比于全波的HLLC求解器,本文构造的黎曼求解器不仅具有更高的接触分辨率,而且还具有更好的激波稳定性。     

Kinetic meshless method for compressible flows

We present a grid‐free or meshless approximation called the kinetic meshless method (KMM), for the numerical solution of hyperbolic conservation laws that can be obtained by taking moments of a Boltzmann‐type transport equation. The meshless formulation requires the domain discretization to have very little topological information; a distribution of points in the domain together with local connectivity information is sufficient. For each node, the connectivity consists of a set of nearby nodes which are used to evaluate the spatial derivatives appearing in the conservation law. The derivatives are obtained using a modified form of the least‐squares approximation. The method is applied to the Euler equations for inviscid flow and results are presented for some 2‐D problems. The ability of the new scheme to accurately compute inviscid flows is clearly demonstrated, including good shock capturing ability. Comparisons with other grid‐free methods are made showing some advantages of the current approach. Copyright © 2007 John Wiley & Sons, Ltd.     

Solution of shallow water equations using fully adaptive multiscale schemes

The concept of fully adaptive multiscale finite volume methods has been developed to increase spatial resolution and to reduce computational costs of numerical simulations. Here grid adaptation is performed by means of a multiscale analysis based on biorthogonal wavelets. In order to update the solution in time we use a local time stepping strategy that has been recently developed for hyperbolic conservation laws. The adaptive multiresolution scheme is now applied to two‐dimensional shallow water equations with source terms. The efficiency of the scheme is demonstrated on several problems with a general geometry, including circular damp breaks, oblique hydraulic jump, supercritical channel flows encountering sudden change in cross‐section, and, finally, the bore wave and its interactions. Copyright © 2005 John Wiley & Sons, Ltd.     

An adaptive multilevel wavelet framework for scale‐selective WENO reconstruction schemes

We put forth a dynamic computing framework for scale‐selective adaptation of weighted essential nonoscillatory (WENO) schemes for the simulation of hyperbolic conservation laws exhibiting strong discontinuities. A multilevel wavelet‐based multiresolution procedure, embedded in a conservative finite volume formulation, is used for a twofold purpose. (i) a dynamic grid adaptation of the solution field for redistributing grid points optimally (in some sense) according to the underlying flow structures, and (ii) a dynamic minimization of the in built artificial dissipation of WENO schemes. Taking advantage of the structure detection properties of this multiresolution algorithm, the nonlinear weights of the conventional WENO implementation are selectively modified to ensure lower dissipation in smoother areas. This modification is implemented through a linear transition from the fifth‐order upwind stencil at the coarsest regions of the adaptive grid to a fully nonlinear fifth‐order WENO scheme at areas of high irregularity. Therefore, our computing algorithm consists of a dynamic grid adaptation strategy, a scale‐selective state reconstruction, a conservative flux calculation, and a total variation diminishing Runge‐Kutta scheme for time advancement. Results are presented for canonical examples drawn from the inviscid Burgers, shallow water, Euler, and magnetohydrodynamic equations. Our findings represent a novel direction for providing a scale‐selective dissipation process without a compromise on shock capturing behavior for conservation laws, which would be a strong contender for dynamic implicit large eddy simulation approaches.