Conservation Laws Of Gas Dynamics Research Articles

A Boltzmann scheme with physically relevant discrete velocities for Euler equations

Kinetic or Boltzmann schemes are interesting alternatives to the macroscopic numerical methods for solving the hyperbolic conservation laws of gas dynamics. They utilize the particle-based description instead of the wave propagation models. While the continuous particle velocity based upwind schemes were developed in the earlier decades, the discrete velocity Boltzmann schemes introduced in the last decade are found to be simpler and are easier to handle. In this work, we introduce a novel way of introducing discrete velocities which correspond to the physical wave speeds and formulate a discrete velocity Boltzmann scheme for solving Euler equations.

A triangular discontinuous Galerkin method for non-Newtonian implicit constitutive models of rarefied and microscale gases

The discontinuous Galerkin (DG) method has been popular as a numerical technique for solving the conservation laws of gas dynamics. In the present study, we develop an explicit modal DG scheme for multi-dimensional conservation laws on unstructured triangular meshes in conjunction with non-Newtonian implicit nonlinear coupled constitutive relations (NCCR). Special attention is given to how to treat the complex non-Newtonian type constitutive relations arising from the high degree of thermal nonequilibrium in multi-dimensional gas flows within the Galerkin framework. The Langmuir velocity slip and temperature jump conditions are also implemented into the two-dimensional DG scheme for high Knudsen number flows. As a canonical scalar case, Newtonian and non-Newtonian convection–diffusion Burgers equations are studied to develop the basic building blocks for the scheme. In order to verify and validate the scheme, we applied the scheme to a stiff problem of the shock wave structure for all Mach numbers and to the two-dimensional hypersonic rarefied and low-speed microscale gas flows past a circular cylinder. The computational results show that the NCCR model yields the solutions in better agreement with the direct simulation Monte Carlo (DSMC) data than the Newtonian linear Navier–Stokes–Fourier (NSF) results in all cases of the problem studied.

Electromagnetic Waves from TNT Explosions

Experimental observations indicate that electromagnetic (EM) radiation is emitted after the detonation of high explosives (HE) charges. The movement of ionized atoms, particles and electrons seems to be the underlying cause. Expansion of the detonation products (DP) drives a strong (~1 kb) shock in surrounding air. This forms an intense thermal wave (T ~11,000 K) with duration of ~20 microseconds. Such temperatures create significant ionization of the air. According to Ohm’s Law, movement of ionized patches generates current; and according to the Biot-Savart Law, such currents induce electric and magnetic fields. We investigate these effects through numerical simulations of TNT explosions. A high-order Godunov scheme is used to integrate the one-dimensional conservation laws of gasdynamics. An extremely fine grid (10 microns) was needed to get converged temperature and conductivity profiles. The gasdynamic solution provided a source current, which was fed into a time-domain Green’s function code to predict three-dimensional electromagnetic waves emanating from the TNT explosion. This analysis clearly demonstrates one mechanism—the Boronin current—as the source of EM emissions from TNT explosions, but other mechanisms are also possible.

Numerical assessments of high-order accurate shock capturing schemes: Kelvin–Helmholtz type vortical structures in high-resolutions

This paper investigates performance of extensions of the state-of-the-art high-resolution shock capturing schemes by solving hyperbolic conservation laws in gas dynamics. Such numerical schemes used for the integration of compressible flow simulations should provide accurate solutions for the long time integrations these flows require. To this end, several joint solvers are developed within the framework of the reconstruction and flux-splitting approaches using the underlying MUSCL and WENO frameworks. The numerical assessments include testing and evaluation of various interpolation procedures, flux-limiters, Riemann solvers, flux-splitting schemes as well as their formal order of accuracy. A three-stage optimal TVD Runge–Kutta time stepping is employed for temporal integration. The modular development of these joint solvers provides an ease in characterizing the solution procedures. The performances of these high-resolution solvers are compared for several carefully selected two-dimensional Riemann problems including shock and rarefaction waves as well as joint discontinuities. Based on solutions obtained by all forms of five-point stencil schemes, we demonstrate that the reconstruction based WENO scheme with Roe solver is more accurate than all the versions of the flux-splitting WENO solvers tested in this study. We also show that results are highly dependent on the choice of the flux limiter. Performing benchmark quality high-resolution computations, it is shown that the Euler equations discretized by the fifth-order WENO scheme produce solutions which convect vorticity and create small-scale vortical flow structures which are usually associated with the high Reynolds number viscous flows. Surprisingly, it is found that these Kelvin–Helmholtz instability like vortical structures are not captured in any form of the third-order five-point stencil schemes.

A fluid dynamic formulation of the isometric embedding problem in differential geometry

The isometric embedding problem is a fundamental problem in differential geometry. A longstanding problem is considered in this paper to characterize intrinsic metrics on a two-dimensional Riemannian manifold which can be realized as isometric immersions into the three-dimensional Euclidean space. A remarkable connection between gas dynamics and differential geometry is discussed. It is shown how the fluid dynamics can be used to formulate a geometry problem. The equations of gas dynamics are first reviewed. Then the formulation using the fluid dynamic variables in conservation laws of gas dynamics is presented for the isometric embedding problem in differential geometry.

Infinite set of polynomial conservation laws in gas dynamics

Infinite set of polynomial conservation laws in gas dynamics

FLUID SIMULATION USING AN ADAPTIVE SEMI-DISCRETE CENTRAL-UPWIND SCHEME

In this paper, we propose an adaptive semi-discrete central-upwind scheme on triangular meshes to approximate the solutions of hyperbolic conservation laws in gas dynamics: a local smoothness indicator is implemented to identify discontinuous solution regions; away from the discontinuities, a simple and computationally economical flux is used; near the discontinuities, an accurate but computationally uneconomical flux is used. Several numerical experiments, Euler equations and Kelvin–Helmholtz instability, are successfully simulated. The numerical results show the desired accuracy, high efficiency and robustness of the adaptive scheme.

Improved Solutions to the Small Strain Continuum Equations Using a Modified Engquist Filter

For wave propagation problems in linear and bilinear elastic solids and a class of simple nonlinear solids the benefits of high-resolution schemes in gasdynamics can be immediately extended to solid mechanics. The Engquist filter for the system of conservation laws in gasdynamics is extended to solid mechanics and implemented in a finite-difference code. The present results show that highly resolved numerical solutions, i.e., sharp shocks with very little oscillation behind them, can be achieved by applying this nonlinear filter to the finite-difference solution at every timestep. Plane wave propagation in three types of solids is calculated by the LAYER code. A two-dimensional grid is used to compute these one-dimensional plane wave problems. For linear elastic solids the nonlinear filter performed very well in removing the Gibbs oscillations. For bilinear solids the numerical solution is compared to a known analytical solution. Because this analytical solution involves a planar shock with a decaying pressure behind it, the existing form of the Engquist filter (designed for a constant pressure behind the shock) has to be modified. The relationship between this modified filter and the flux modification in the TVD scheme is discussed. This modification is expected to be necessary when the filter is applied to multi-dimensional problems. The potential for obtaining highly resolved shocks from classical second-order finite-difference solid mechanics codes using the modified nonlinear filter is demonstrated.

A Kinetic Approach to General First Order Quasilinear Equations

This paper presents a new method for constructing entropy solutions of first order quasilinear equations of conservation type, which is illustrated in terms of the kinetic theory of gases. Regarding a quasilinear equation as a model of macroscopic conservation laws in gas dynamics, we introduce as the corresponding microscopic model an auxiliary linear equation involving a real parameter $\xi$ which plays the role of the velocity argument. Approximate solutions for the quasilinear equation are then obtained by integrating solutions of the linear equation with respect to the parameter $\xi$. All of these equations are treated in the Fréchet space $L_{{\text {loc}}}^1({R^n})$, and a convergence theorem for such approximate solutions to the entropy solutions is established with the aid of nonlinear semigroup theory.

A kinetic approach to general first order quasilinear equations

This paper presents a new method for constructing entropy solutions of first order quasilinear equations of conservation type, which is illustrated in terms of the kinetic theory of gases. Regarding a quasilinear equation as a model of macroscopic conservation laws in gas dynamics, we introduce as the corresponding microscopic model an auxiliary linear equation involving a real parameter $\xi$ which plays the role of the velocity argument. Approximate solutions for the quasilinear equation are then obtained by integrating solutions of the linear equation with respect to the parameter $\xi$. All of these equations are treated in the Fréchet space $L_{{\text {loc}}}^1({R^n})$, and a convergence theorem for such approximate solutions to the entropy solutions is established with the aid of nonlinear semigroup theory.