Euler Equations Of Gas Dynamics Research Articles

A time-accurate explicit multi-scale technique for gas dynamics

We present a new time-accurate algorithm for the explicit numerical integration of the compressible Euler equations of gas dynamics. This technique is based on the discrete-event simulation (DES) methodology for nonlinear flux-conservative PDEs [Y.A. Omelchenko, H. Karimabadi, Self-adaptive time integration of flux-conservative equations with sources, J. Comput. Phys. 216 (1) (2006) 179–194]. DES enables adaptive distribution of CPU resources in accordance with local time scales of the underlying numerical solution. It distinctly stands apart from multiple (local) time-stepping algorithms in that it requires neither selecting a global synchronization time step nor pre-determining a sequence of time-integration operations for individual parts of a heterogeneous numerical system. In this paper we extend the DES methodology in three important directions: (i) we apply DES to a system of coupled gas dynamics equations discretized via a central-upwind scheme [A. Kurganov, E. Tadmor, New high-resolution central schemes for nonlinear conservation laws and convection–diffusion equations, J. Comput. Phys. 160 (2000) 241–282; A. Kurganov, S. Noelle, G. Petrova, Semidiscrete central-upwind schemes for hyperbolic conservation laws and Hamilton–Jacobi equations, SIAM J. Sci. Comput. 23 (3) (2001) 707–740]; (ii) we introduce a new Preemptive Event Processing (PEP) technique, which automatically enforces synchronous execution of events with sufficiently close update times; (iii) we significantly improve the accuracy of the previous algorithm [Y.A. Omelchenko, H. Karimabadi, Self-adaptive time integration of flux-conservative equations with sources, J. Comput. Phys. 216 (1) (2006) 179–194] by applying locally second-order-in-time flux-conserving corrections to the solution obtained with the forward Euler scheme. The performance of the new technique is demonstrated in a series of one-dimensional gas dynamics test problems by comparing numerical solutions obtained in event-driven and equivalent time-stepping simulations.

Une introduction au schéma de Boltzmann sur réseau

We propose an elementary introduction to the lattice Boltzmann scheme. We recall the physical (Boltzmann equation) and algorithmic (cellular automata) origins of this numerical method. For a one-dimensional example, we present in detail the two characteristic steps of the algorithm: the nonlinear collision step, local in space and the linear propagation phase with the neighbouring vertices, explicit in time. We then propose a generic Taylor-type development with the so-called equivalent partial differential equation. We obtain in this way formally a so-called Chapman-Enskog development where the small parameter is the discretization step of the scheme. At order zero, the lattice Boltzmann scheme satisfies a local thermodynamical equilibrium. At first order, it satisfies the Euler equations of gas dynamics and at second order the Navier-Stokes equations. Then we detail the classical case of the nine velocities model on a square lattice.

Stabilized shock hydrodynamics: I. A Lagrangian method

A new SUPG-stabilized formulation for Lagrangian hydrodynamics of materials satisfying Mie–Grüneisen equation of state is proposed. It allows the use of simplex-type (triangular/tetrahedral) meshes as well as the more commonly used brick-type (quadrilateral/hexahedral) meshes. The proposed method yields a globally conservative formulation, in which equal-order interpolation (P1 or Q1 isoparametric finite elements) is applied to velocities, displacements, and pressure. As a direct consequence, and in contrast to traditional cell-centered multidimensional hydrocode implementations, the proposed formulation allows a natural representation of the pressure gradient on element interiors. The SUPG stabilization involves additional design requirements, specific to the Lagrangian formulation. A discontinuity capturing operator in the form of a Noh-type viscosity with artificial heat flux is used to preserve stability and smoothness of the solution in shock regions. A set of challenging shock hydrodynamics benchmark tests for the Euler equations of gas dynamics in one and two space dimensions is presented. In the two-dimensional case, computations performed on quadrilateral and triangular grids are analyzed and compared. These results indicate that the new formulation is a promising technology for hydrocode applications.

On a practical implementation of particle methods

This paper is devoted to a practical implementation of deterministic particle methods for solving transport equations with discontinuous coefficients and/or initial data, and related problems. In such methods, the solution is sought in the form of a linear combination of the delta-functions, whose positions and coefficients represent locations and weights of the particles, respectively. The locations and weights of the particles are then evolved in time according to a system of ODEs, obtained from the weak formulation of the transport PDEs. The major theoretical difficulty in solving the resulting system of ODEs is the lack of smoothness of its right-hand side. While the existence of a generalized solution is guaranteed by the theory of Filippov, the uniqueness can only be obtained via a proper regularization. Another difficulty one may encounter is related to an interpretation of the computed solution, whose point values are to be recovered from its particle distribution. We demonstrate that some of known recovering procedures, suitable for smooth functions, may fail to produce reasonable results in the nonsmooth case, and discuss several successful strategies which may be useful in practice. Different approaches are illustrated in a number of numerical examples, including one- and two-dimensional transport equations and the reactive Euler equations of gas dynamics.

Shape Optimization for Delay of Laminar-Turbulent Transition

A method using gradient-based optimization is introduced for the design of wing profiles with the aim of natural laminar flow, as well as minimum wave drag. The Euler equations of gasdynamics, the laminar boundary-layer equations for compressible flows on infinite swept wings, and the linear parabolized stability equations (PSE) are solved to analyze the evolution of convectively unstable disturbances. Laminar‐turbulent transition is assumed to be delayed by minimizing a measure of the disturbance kinetic energy of a chosen disturbance, which is computed using the PSE. The shape gradients of the disturbance kinetic energy are computed based on the solutions of the adjoints of the state equations just named. Numerical tests are carried out to optimize the RAE 2822 airfoil with the aim to delay simultaneously the transition, reduce the pressure drag coefficient, and maintain the coefficients of lift and pitch moments. Constraints are also applied on the geometry. Results show a reduction of the total amplification of a large number of disturbances, which is assumed to represent a delay of the transition in the boundary layer. Because delay of the transition implies reduction of the viscous drag, the present method enables shape optimization to perform viscous drag reduction.

Super spectral viscosity method for nonlinear conservation laws

In this paper, the super spectral viscosity (SSV) method is developed by introducing a spectrally small amount of high order regularization which is only activated on high frequencies. The resulting SSV approximation is stable and convergent to the exact entropy solution. A Gegenbauer-Chebyshev post-processing for the SSV solution is proposed to remove the spurious oscillations at the discontinuities and recover accuracy from the spectral approximation. The SSV method is applied to the scalar periodic Burgers equation and the one-dimensional system of Euler equations of gas dynamics. The numerical results exhibit high accuracy and resolution to the exact entropy solution.

Space-marching gridless computation of steady supersonic/hypersonic flow

Most computational fluid dynamics work uses Eulerian coordinates, which require generating a grid prior to flow field computation. Despite three decades of research, grid generation is still a bottleneck of CFD, as it is time-consuming, tedious and requires specialized training. It will be shown in this paper that using the unified coordinates introduced by Hui, there is no need for grid generation prior to flow computation; the grid is automatically generated while computing the flow. This greatly saves computing time. For steady super-sonic/hypersonic flow, the Euler equations of gas dynamics are of hyperbolic type and a space-marching gridless computation along the streamlines—coordinate lines in the unified coordinates—is shown to be a complete success in that: (a) it is most robust; (b) it resolves both slip lines (also called contact lines) and shocks sharply; (c) its computing time is more than three orders of magnitude smaller than Eulerian computation; and (d) it bypasses the tedious and time-consuming grid generation stage which is needed in Eulerian computation. Three examples are given to justify these claims.

A Class of Extended Limiters Applied to Piecewise Hyperbolic Methods

Piecewise hyperbolic methods (PHMs) have been widely used for the approximation of hyperbolic conservation laws. In this paper we have studied the "power limiters" in order to analyze their suitable use in the design of PHMs. We formulate a new PHM, based on these limiters, improving the behavior of the PHM in [SIAM J. Sci. Comput., 15 (1994), pp. 892-915], at local extrema, jump discontinuities, and discontinuities in derivative. We show that this new hyperbolic reconstruction (we call it PowerPHM) is local total variation bounded and it behaves stably with a reduced numerical viscosity compared to PHM. We show numerical evidence of the above mentioned features by means of one- and two-dimensional numerical tests with the Euler equations of gas dynamics.

Second‐order entropy diminishing scheme for the Euler equations

AbstractIn several papers of Bouchut et al., Coquel and Le Floch (Math. Comput. 1996; 65(216):1439–1461; Numer. Math. 1996; 74(01):1–34), a general methodology has been developed to construct second‐order finite volume schemes for hyperbolic systems of conservation laws satisfying the entire family of entropy inequalities. This approach is mainly based on the construction of an entropy diminishing projection. Unfortunately, the explicit computation of this projection is not always easy. In the first part of this paper, we carry out this computation in the important case of the Euler equations of gas dynamics. In the second part, we present several numerical applications of the projection in the context of finite volume schemes. Copyright © 2005 John Wiley & Sons, Ltd.

Evaluation of non-oscillatory schemes based on LED principle for supersonic flow computations

The objective of the present paper is to evaluate the efficiency of several high resolution, non-oscillatory schemes based on the local extremum diminishing (LED) principle [1], [2] in the case of supersonic flow computation past obstacles in two dimensions. The Euler equations of gas dynamics have been solved in a cell-vertex finite volume framework in conjunction with the non-oscillatory dissipation schemes for simulation of supersonic flow fields. Satisfactory results have been obtained with the switched as well as the flux limited dissipation schemes, namely, SLIP and USLIP schemes using scalar diffusive flux. Four different types of flux splitting techniques have also been investigated along with the switched scheme. In particular, the wave-particle splitting proposed by Balakrishnan and Deshpande [3] for upwind schemes has been formulated and applied to the present symmetric or central scheme.

Generation of Arbitrary Lagrangian–Eulerian (ALE) velocities, based on monitor functions, for the solution of compressible fluid equations

AbstractA moving mesh method is outlined based on the use of monitor functions. The method is developed from a weak conservation principle. From this principle a conservation law for the mesh position is derived. Using the Helmholtz decomposition theorem, this conservation law can be converted into an elliptic equation for a mesh velocity potential.The moving mesh method is discretized using standard finite elements. Once the mesh velocities are obtained an arbitrary Lagrangian–Eulerian (ALE) (Journal of Computational Physics 1974; 14:227) fluid solver is used to update the solution on the adaptive mesh.Results are shown for the compressible Euler equations of gas dynamics in one and two spatial dimensions. Two monitor functions are used, the fluid density (which corresponds to a Lagrangian description), and a function which includes the density gradient. A variety of test problems are considered. Copyright © 2005 John Wiley & Sons, Ltd.

High‐resolution simulation of inviscid flow in general domains

AbstractIn this paper, a high‐resolution finite volume method for calculating solutions to a hyperbolic conservation law is presented. The method works in two space dimensions on general domains and uses curvilinear meshes. A non‐trivial estimation of gradients needed for the reconstructions is presented. The paper contains examples of numerical solutions of the Euler gas dynamics equations. Copyright © 2005 John Wiley & Sons, Ltd.

Implicit finite element discretizations based on the flux‐corrected transport algorithm

AbstractThe flux‐corrected transport (FCT) methodology is generalized to implicit finite element schemes and applied to the Euler equations of gas dynamics. For scalar equations, a local extremum diminishing scheme is constructed by adding artificial diffusion so as to eliminate negative off‐diagonal entries from the high‐order transport operator. To obtain a nonoscillatory low‐order method in the case of hyperbolic systems, the artificial viscosity tensor is designed so that all off‐diagonal blocks of the discrete Jacobians are rendered positive semi‐definite. Compensating antidiffusion is applied within a fixed‐point defect correction loop so as to recover the high accuracy of the Galerkin discretization in regions of smooth solutions. All conservative matrix manipulations are performed edge‐by‐edge which leads to an efficient algorithm for the matrix assembly. Copyright © 2005 John Wiley & Sons, Ltd.

On the modeling and simulation of boundary flow through partially open pipe ends

The determination of boundary conditions for the Euler equations of gas dynamics in a pipe with partially open pipe ends is considered. The boundary problem is formulated in terms of the exact solution of the Riemann problem and of the St. Venant equation for quasi-steady flow so that a pressure-driven calculation of boundary conditions is defined. The resulting set of equations is solved by a Newton scheme. The proposed algorithm is able to solve for all inflow and outflow situations including choked and supersonic flow.

Stabilized discontinuous Galerkin method for hyperbolic equations

In this work a new stabilization technique is proposed and studied for the discontinuous Galerkin method applied to hyperbolic equations. In order to avoid the use of slope limiters, a streamline diffusion-like term is added to control oscillations for arbitrary element orders. Thus, the scheme combines ideas from both the Runge–Kutta discontinuous Galerkin method [J. Scient. Comput. 16 (2001) 173] and the streamline diffusion method [Comput. Methods Appl. Mech. Engrg. 32 (1982)]. To increase the stability range of the method, the diffusion term is treated implicitly. The result is a scheme with higher order in space with the same stability range as the finite volume method. An optimal relation between the time step and the size of the diffusion coefficient is analyzed for numerical precision. The scheme is implemented using the object oriented programming philosophy based on the environment described in [Comput. Methods Appl. Mech. Engrg. 150 (1997)]. Accuracy and shock capturing abilities of the method are analyzed in terms of two bidimensional model problems: the rotating cone and the backward facing step problem for the Euler equations of gas dynamics.

A Runge–Kutta discontinuous Galerkin method for the Euler equations

This paper presents a Runge–Kutta discontinuous Galerkin (RKDG) method for the Euler equations of gas dynamics from the viewpoint of kinetic theory. Like the traditional gas-kinetic schemes, our proposed RKDG method does not need to use the characteristic decomposition or the Riemann solver in computing the numerical flux at the surface of the finite elements. The integral term containing the non-linear flux can be computed exactly at the microscopic level. A limiting procedure is carefully designed to suppress numerical oscillations. It is demonstrated by the numerical experiments that the proposed RKDG methods give higher resolution in solving problems with smooth solutions. Moreover, shock and contact discontinuities can be well captured by using the proposed methods.

Variants of relaxed schemes and two-dimensional gas dynamics

We present a relaxed scheme with more precise information about local speeds of propagation and a multidimensional construction of the cell averages. The physical domain of dependence is simulated correctly and high resolution is maintained. Relaxation schemes have advantages that include high resolution, simplicity and explicitly no (approximate) Riemann solvers and no characteristic decomposition is necessary. Performance of the scheme is illustrated by tests on two-dimensional Euler equations of gas dynamics.

Hydrodynamic limits with shock waves of the Boltzmann equation

AbstractWe show that piecewise smooth solutions with shocks of the Euler equations in gas dynamics can be obtained as the zero Knudsen number limit of solutions of the Boltzmann equation for hard sphere collision model. The construction of the Boltzmann solutions is done in two steps. First we introduce a generalized Hilbert expansion with shock layer correction to construct approximations to the solutions of the Boltzmann equations with small Knudsen numbers. We then apply the recently developed macro‐micro decomposition and energy method for Boltzmann shock layers to construct the exact Boltzmann solutions through the stability analysis. © 2004 Wiley Periodicals, Inc.

Dynamics of ring vortices impinging on planar shock waves

In the present paper the encounter of vortex rings with planar shock waves is studied in a three-dimensional environment, with the objective to assess the influence of shock and vortex strength and vortex orientation on the dynamics of the process. The study relies on numerical simulations performed by means of a state-of-the-art hybrid compact/WENO (weighted essentially nonoscillatory) shock-capturing algorithm to solve the Euler equations of gas dynamics. The study focuses on the characterization of the interaction in terms of the evolution of the geometrical parameters of the vortex and the mean flow properties such as kinetic energy and enstrophy. The vortex is compressed as it passes through the shock, and its characteristic dimensions decrease; at the same time its axis is deflected and the ring follows a complex dynamic evolution, even though a nearly steady state is reached for the global quantities. In particular, it has been found that the total vortex kinetic energy is generally nonincreasing after the interaction, while the enstrophy always increases. A simple theory is developed here to shed some light onto the physical phenomena involved, which is found to compare reasonably well with the results of the computations. Some final comments are made regarding the extension of the results reported in the present study to the analysis of shock-compressed turbulence.

Concentration and cavitation in the vanishing pressure limit of solutions to the Euler equations for nonisentropic fluids

Numerical simulations [2-D Riemann problem in gas dynamics and formation of spiral, in: Nonlinear Problems in Engineering and Science—Numerical and Analytical Approach (Beijing, 1991), Science Press, Beijing, 1992, pp. 167–179; Discrete Contin. Dyn. Syst. 1 (1995) 555–584; 6 (2000) 419–430] for the Euler equations for gas dynamics in the regime of small pressure showed that, for one case, the particles seem to be more sticky and tend to concentrate near some shock locations, and for the other case, in the region of rarefaction waves, the particles seem to be far apart and tend to form cavitation in the region. In this paper we identify and analyze the phenomena of concentration and cavitation by studying the vanishing pressure limit of solutions of the full Euler equations for nonisentropic compressible fluids with a scaled pressure. It is rigorously shown that any Riemann solution containing two shocks and possibly one-contact-discontinuity to the Euler equations for nonisentropic fluids tends to a δ-shock solution to the corresponding transport equations, and the intermediate densities between the two shocks tend to a weighted δ-measure that, along with the two shocks and possibly contact-discontinuity, forms the δ-shock as the pressure vanishes. By contrast, it is also shown that any Riemann solution containing two rarefaction waves and possibly one-contact-discontinuity to the Euler equations for nonisentropic fluids tends to a two-contact-discontinuity solution to the transport equations, and the nonvacuum intermediate states between the two rarefaction waves tend to a vacuum state as the pressure vanishes. Some numerical results exhibiting the processes of concentration and cavitation are presented as the pressure decreases.