Euler Equations Of Compressible Flow Research Articles

An IMEX-DG solver for atmospheric dynamics simulations with adaptive mesh refinement

We present an accurate and efficient solver for atmospheric dynamics simulations that allows for non-conforming mesh refinement. The model equations are the conservative Euler equations for compressible flows. The numerical method is based on an h−adaptive Discontinuous Galerkin spatial discretization and on a second order Additive Runge Kutta IMEX method for time discretization, especially designed for low Mach regimes. The solver is implemented in the framework of the deal.II library, whose mesh refinement capabilities are employed to enhance efficiency. A number of numerical experiments based on classical benchmarks for atmosphere dynamics demonstrate the properties and advantages of the proposed method.

An efficient high-order gas-kinetic scheme (I): Euler equations

In this paper, an efficient high-order gas-kinetic scheme (EHGKS) is proposed to solve the Euler equations for compressible flows. We re-investigate the underlying mechanism of the high-order gas-kinetic scheme (HGKS) and find a new strategy to improve its efficiency. The main idea of the new scheme contains two parts. Firstly, inspired by the state-of-art simplifications on the third-order HGKS, we extend the HGKS to the case of arbitrary high-order accuracy and eliminate its unnecessary high-order dissipation terms. Secondly, instead of computing the derivatives of particle distribution function and their complex moments, we introduce a Lax-Wendroff procedure to compute the high-order derivatives of macroscopic quantities directly. The new scheme takes advantage of both HGKS and the Lax-Wendroff procedure, so that it can be easily extended to the case of arbitrary high-order accuracy with practical significance. Typical numerical tests are carried out by EHGKS, with the third, fifth and seventh order of accuracy. The presence of good resolution of the discontinuities and flow details, together with the optimal CFL numbers, validates the high accuracy and strong robustness of EHGKS. To compare the efficiency, we present the results computed by the EHGKS, the original HGKS and Runge-Kutta-WENO-GKS. This further demonstrates the advantages of EHGKS.

Interaction of a centered simple wave and a planar rarefaction wave of the two-dimensional Euler equations for pseudo-steady compressible flow

In this paper, we study the expansion problem which arises as two-dimensional (2D) pseudo-steady supersonic flow turns a sharp corner and expands into vacuum. The problem catches interaction of a centered simple wave and a backward planar rarefaction wave, which is deduced a Goursat problem for 2D self-similar Euler equations for compressible flow. By the methods of characteristic decomposition and invariant regions, we get the hyperbolicity in the wave interaction domain and prior C1 estimates of solutions to the Goursat problem. The global solution up to the interface of gas with vacuum to the expansion problem is obtained constructively.

Positivity-preserving high-order discontinuous Galerkin schemes for Ten-Moment Gaussian closure equations

Euler equations for compressible flows treats pressure as a scalar quantity. However, for several applications this description of pressure is not suitable. Many extended model based on the higher moments of Boltzmann equations are considered to overcome this issue. One such model is Ten-Moment Gaussian closure equations, which treats pressure as symmetric tensor. In this work, we develop a higher-order, positivity preserving Discontinuous Galerkin (DG) scheme for Ten-Moment Gaussian closure equations. The key challenge is to preserve positivity of density and symmetric pressure tensor. This is achieved by constructing a positivity limiter. In addition to preserve positivity, the scheme also ensures the accuracy of the approximation for smooth solutions. The theoretical results are then verified using several numerical experiments.

Expansion of a wedge of non-ideal gas into vacuum

We study the problem of expansion of a wedge of non-ideal gas into vacuum in a two-dimensional bounded domain. The non-ideal gas is characterized by a van der Waals type equation of state. The problem is modeled by standard Euler equations of compressible flow, which are simplified by a transformation to similarity variables and then to hodograph transformation to arrive at a second order quasilinear partial differential equation in phase space; this, using Riemann variants, can be expressed as a non-homogeneous linearly degenerate system provided that the flow is supersonic. For the solution of the governing system, we study the interaction of two-dimensional planar rarefaction waves, which is a two-dimensional Riemann problem with piecewise constant data in the self-similar plane. The real gas effects, which significantly influence the flow regions and boundaries and which do not show-up in the ideal gas model, are elucidated; this aspect of the problem has not been considered until now.

Fourier stability analysis and local Courant number of the preconditioned variational multiscale stabilization (P-VMS) for Euler compressible flow

The results of a Fourier stability analysis of the preconditioned variational multiscale stabilization (P-VMS) method introduced in Moragues et al. (2015) are presented in this paper. P-VMS combines a variational multiscale stabilized finite elements discretization together with local preconditioning. In this work, we deal with the P-VMS method using van Leer–Lee–Roe’s (vanLeer et al., 1991) and Choi–Merkle’s (Choi and Merkle, 1993) local preconditioners. We solve the Euler equations of compressible flow for steady problems. We concentrate on explicit time integration schemes. The stability analysis is performed on a two dimensional simplified problem with a structured mesh and its conclusions are applied to two and three dimensional general problems with unstructured meshes. As a result of this analysis a local Courant–Friedrichs–Lewy number is defined for the computation of the time step. The convergence rate is evaluated, and compared with the traditional constant Courant–Friedrichs–Lewy number for various test cases spanning a large range of Mach numbers.

Nesting an incompressible-flow code within a compressible-flow code: A two-dimensional study

We consider numerical algorithms appropriate for one- and two-way coupling between meso-scale and micro-scale fluid-dynamics codes for wind energy computing. At the meso-scale is a numerical weather-prediction code, which is typically based on the compressible-flow Euler equations. At the micro-scale, surrounding one or more wind turbines, is a computational fluid dynamics code, which is typically based on the incompressible-flow Navier–Stokes equations. When calculating short-duration flow around wind turbines, one-way coupling is sufficient, where the meso-scale computational model drives the micro-scale model. However, in long-duration simulations involving large wind farms, the influence of the wind farm on the meso-scale weather may no longer be insignificant and two-way coupling is warranted. In this study, we focus on a simple two-dimensional system, for which our goal is to devise one- and two-way coupling algorithms that can effectively transport a vortex propagating in laminar flow from one domain to the other. Two coupling schemes and their numerical implementation are described: partial-boundary coupling and projection coupling. In the former, the micro-scale-domain boundary is decomposed, based on the meso-scale solution, into sections corresponding to inflow and outflow. The micro-scale model has Dirichlet- and Neumann-type boundary conditions on these sections, respectively. In projection coupling, the meso-scale solution is projected onto the incompressible-flow solution space in the micro-scale domain, from which Dirichlet-type boundary conditions are derived. In these simulations, the uncoupled meso-scale solution is taken as the reference, and the best coupling method is that which produces solutions that deviate the least from the reference. In one-way coupling, under a simple two-dimensional laminar-flow test case, partial-boundary coupling was more effective than projection coupling. However, in two-way coupling, projection coupling was the best performer.

Energy and Momentum Transfer to a Clamped Elastic Plate in an Air-Blast

The momentum transfer by a planar wave impinging upon a rigid, free-standing plate in water, a largely incompressible medium, is well understood [1]. Kambouchev et al. [2] extended the results of Taylor [1] to include the nonlinear effects of compressibility whilst Hutchinson [3] has recently addressed the issues of energy and momentum transfer to a rigid, free-standing plate. In this paper, key conclusions from the aforementioned studies are critically re-examined in the context of a `fully-clamped' elastic plate. The dynamic response of an elastic plate is represented by an equivalent single-degree-of-freedom (SDOF) system. A numerical method based on a Lagrangian formulation of the Euler equations of compressible flow and conventional shock-capturing techniques, similar to that employed in [2, 3], were employed to solve numerically the interaction between the air blast wave and elastic plate. Particular emphasis is placed on elucidating the energy and momentum transfer to a `fully-clamped' elastic plate compared to its rigid, free-standing counterpart, and on whether enhancement in the beneficial effects of FSI as a result of fluid compressibility remains and to what extent.

A Mach‐uniform preconditioner for incompressible and subsonic flows

SUMMARYIn this study, a novel Mach‐uniform preconditioning method is developed for the solution of Euler equations at low subsonic and incompressible flow conditions. In contrast to the methods developed earlier in which the conservation of mass equation is preconditioned, in the present method, the conservation of energy equation is preconditioned, which enforces the divergence free constraint on the velocity field even at the limiting case of incompressible, zero Mach number flows. Despite most preconditioners, the proposed Mach‐uniform preconditioning method does not have a singularity point at zero Mach number. The preconditioned system of equations preserves the strong conservation form of Euler equations for compressible flows and recovers the artificial compressibility equations in the case of zero Mach number. A two‐dimensional Euler solver is developed for validation and performance evaluation of the present formulation for a wide range of Mach number flows. The validation cases studied show the convergence acceleration, stability, and accuracy of the present Mach‐uniform preconditioner in comparison to the non‐preconditioned compressible flow solutions. The convergence acceleration obtained with the present formulation is similar to those of the well‐known preconditioned system of equations for low subsonic flows and to those of the artificial compressibility method for incompressible flows. Copyright © 2013 John Wiley & Sons, Ltd.

Energy and momentum transfer to a 'fully-clamped' elastic plate in an air-blast

The momentum transfer by a planar wave impinging upon a rigid, free-standing plate in water, a largely incompressible medium, is well understood [1]. Kambouchev et al. [2] extended the results of Taylor [1] to include the nonlinear effects of compressibility whilst Hutchinson [3] has recently addressed the issues of energy and momentum transfer to a rigid, free-standing plate. In this paper, key conclusions from the aforementioned studies are critically re-examined in the context of a 'fully-clamped' elastic plate. The dynamic response of an elastic plate is represented by an equivalent single-degree-of-freedom (SDOF) system. A numerical method based on a Lagrangian formulation of the Euler equations of compressible flow and conventional shock-capturing techn iques, similar to that employed in [2, 3], were employed to solve numerically the interaction between the air blast wave and elastic plate. Particular emphasis is placed on elucidating the energy and momentum transfer to a 'fully-clamped' elastic plate compared to its rigid, free-standing counterpart, and on whether enhancement in the beneficial effects of FSI as a result of fluid compressibility remains and to what extent.

Arbitrary Lagrangian Eulerian formulation for two-dimensional flows using dynamic meshes with edge swapping

The dynamic modification of the computational grid due to element displacement, deformation and edge swapping is described here in terms of suitably-defined continuous (in time) alterations of the geometry of the elements of the dual mesh. This new interpretation allows one to describe all mesh modifications within the arbitrary Lagrangian Eulerian framework, thus removing the need to interpolate the solution across computational meshes with different connectivity. The resulting scheme is by construction conservative and it is applied here to the solution of the Euler equations for compressible flows in two spatial dimensions. Preliminary two dimensional numerical simulations are presented to demonstrate the soundness of the approach. Numerical experiments show that this method allows for large time steps without causing element invalidation or tangling and at the same time guarantees high quality of the mesh elements without resorting to global re-meshing techniques, resulting in a very efficient solver for the analysis of e.g. fluid-structure interaction problems, even for those cases that require large mesh deformations or changes in the domain topology.

Analysis and shape optimization in incompressible flows with unstructured grids

The aim of this study is to develop and validate numerical methods that perform shape optimization in incompressible flows using unstructured meshes. The three-dimensional Euler equations for compressible flow are modified using the idea of artificial compressibility and discretized on unstructured tetrahedral grids to provide estimates of pressure distributions for aerodynamic configurations. Convergence acceleration techniques like multigrid and residual averaging are used along with parallel computing platforms to enable these simulations to be performed in a few minutes. This computational frame-work is used to analyze sail geometries. The adjoint equations corresponding to the “incompressible” field equations are derived along with the functional form of gradients. The evaluation of the gradients is reduced to an integral around the boundary to circumvent hurdles posed by adjoint-based gradient evaluations on unstructured meshes. The reduced gradient evaluations provide major computational savings for unstructured grids and its accuracy and use for canonical and industrial problems is a major contribution of this study. The design process is driven by a steepest-descent algorithm with a fixed step-size. The feasibility of the design process is demonstrated for three inverse design problems, two canonical problems and one industrial problem.

Development of a moving artificial compressibility solver on unified coordinates

AbstractBased on the unified Eulerian and Lagrangian coordinate transformations, the unsteady incompressible Navier–Stokes equations with artificial compressibility effects are developed. As we know, the Eulerian coordinates cause excessive numerical diffusion across flow discontinuities, slip lines in particular. The Lagrangian coordinates, on the other hand, can resolve slip lines sharply but cause severe grid deformation, resulting in large errors and even breakdown of the computation. Recently, Hui et al. (J. Comput. Phys. 1999; 153:596–637) have introduced a unified coordinate system that moves with velocity hq, q being the velocity of the fluid particle. It includes the Eulerian system as a special case when h=0, and the Lagrangian when h=1, and was shown for the two‐dimensional unsteady Euler equations of compressible flow to be superior than both Eulerian and Lagrangian systems. In the framework of unified coordinates, our work will derive the unsteady incompressible flow equations and moving geometry equations, when hq equals grid velocity in conservation form and is updated simultaneously during each time step. Thus, the accurate estimation of geometry conservation and controlling the grid velocity or the h value based on the unified approach can keep numerical stability and avoid computation breakdown caused by moving body or boundary layers (considered as slip lines in Lagrangian coordinates). Also, the existing high‐resolution Riemann solver can be extended to discretize the current unified incompressible flow equations. Our benchmark tests including the lid‐driven cavity flow and backward step flow, oscillating flat plat and pulsating stenotic tube are used to validate the computations. The results verify the accuracy and robustness of the unified artificial compressibility solver on the moving body simulation. Copyright © 2009 John Wiley & Sons, Ltd.

The efficient implementation of a finite element, multi-resolution viscosity method for hyperbolic conservation laws

It is well known that the classic Galerkin finite element method is unstable when applied to hyperbolic conservation laws such as the Euler equations for compressible flows. Adding a diffusion term to the equations stabilizes the method but sacrifices too much accuracy to be of any practical use. An elegant solution developed in the context of spectral methods by Eitan Tadmor and coworkers is to add diffusion only to the high frequency modes of the solution and can lead to stabilization without sacrificing accuracy. We incorporate this idea into the finite element framework by using hierarchical functions as a multi-frequency basis. The result is a new finite element method for solving hyperbolic conservation laws. For this method, convergence for a one-dimensional scalar conservation law has previously been proved. Here, the method is described in detail, several issues connected with its efficient implementation are considered, and numerical results for several examples involving one- and two-dimensional hyperbolic conservation laws are provided. Several advantageous features of the method are discussed, including the ease for which discontinuities can be detected and artificial diffusion can be applied anisotropically and locally in physical as well as frequency space.

Nonlinear compressibility effects in fluid-structure interaction and their implications on the air-blast loading of structures

The impulse imparted by a blast wave to a freestanding solid plate is studied analytically and numerically focusing on the case in which nonlinear compressibility effects in the fluid are important, as is the case for explosions in air. The analysis furnishes, in effect, an extension of Taylor’s pioneering contribution to the understanding of the influence of fluid-structure interaction (FSI) on the blast loading of structures [The Scientific Papers of Sir Geoffrey Ingram Taylor, edited by G. K. Batchelor (Cambridge University Press, Cambridge, 1963), Vol. III, pp. 287–303] to the nonlinear range. The limiting cases of extremely heavy and extremely light plates are explored analytically for arbitrary blast intensity, from where it is concluded that a modified nondimensional parameter representing the mass of compressed fluid relative to the mass of the plate governs the FSI. The intermediate asymptotic FSI regime is studied using a numerical method based on a Lagrangian formulation of the Euler equations of compressible flow and conventional shock-capturing techniques. Based on the analytical and numerical results, an approximate formula describing the entire range of relevant FSI conditions is proposed. The main conclusion of this work is that nonlinear fluid compressibility further enhances the beneficial effects of FSI in reducing the impulse transmitted to the structure. More specifically, it is found that transmitted impulse reductions due to FSI when compared to those obtained ignoring FSI effects are more significant than in the acoustic limit. This result can be advantageously exploited in the design and optimization of structures with increased blast resistance.

A Finite Element, Multiresolution Viscosity Method for Hyperbolic Conservation Laws

It is well known that the classic Galerkin finite element method is unstable when applied to hyperbolic conservation laws such as the Euler equations for compressible flow. It is also well known that naively adding artificial diffusion to the equations stabilizes the method but sacrifices too much accuracy to be of any practical use. An elegant approach, referred to as spectral viscosity methods, has been developed for spectral methods in which one adds diffusion only to the high-frequency modes of the solution, the result being that stabilization is effected without sacrificing accuracy. We extend this idea into the finite element framework by using hierarchical finite element functions as a multifrequency basis. The result is a new finite element method for solving hyperbolic conservation laws in which artificial diffusion can be applied selectively only to the high-frequency modes of the approximation. As for spectral viscosity methods, this results in stability without compromising accuracy. In the context of a one-dimensional scalar hyperbolic conservation law, we prove the convergence of approximate solutions, obtained using the new method, to the entropy solution of the conservation law. To illustrate the method, the results of a computational experiment for a one-dimensional hyperbolic conservation law are provided.

A method for tractable dynamical studies of single and double shock compression.

A new multiscale simulation method is formulated for the study of shocked materials. The method combines molecular dynamics and the Euler equations for compressible flow. Treatment of the difficult problem of the spontaneous formation of multiple shock waves due to material instabilities is enabled with this approach. The method allows the molecular dynamics simulation of the system under dynamical shock conditions for orders of magnitude longer time periods than is possible using the popular nonequilibrium molecular dynamics approach. An example calculation is given for a model potential for silicon in which a computational speedup of 10(5) is demonstrated. Results of these simulations are consistent with the recent experimental observation of an anomalously large elastic precursor on the nanosecond time scale.

Non-existence of global smooth solutions to symmetrizable nonlinear hyperbolic systems

In this paper, we consider the Cauchy problem of general symmetrizable hyperbolic systems in multi-dimensional space. When some components of the initial data have compact support, we give a sufficient condition on the non-existence of global C1 solutions. This non-existence theorem can be applied to some physical systems, such as Euler equations for compressible flow in multi-dimensional space. The blow-up phenomena here can come from the singularity developed at the interface, such as vacuum boundary, rather than the shock formation as studied in the previous works on strictly hyperbolic systems. Therefore, the systems considered here include those which are non-strictly hyperbolic.

Computational modeling of direct-drive fusion pellets and KrF-driven foil experiments

FAST is a radiation transport hydrodynamics code that simulates laser matter interactions of relevance to direct-drive laser fusion target design. FAST solves the Euler equations of compressible flow using the Flux-Corrected Transport finite volume method. The advection algorithm provides accurate computation of flows from nearly incompressible vortical flows to those that are highly compressible and dominated by strong pressure and density gradients. In this paper we describe the numerical techniques and physics packages. FAST has also been benchmarked with Nike laser facility experiments in which linearly perturbed, low adiabat planar plastic targets are ablatively accelerated to velocities approaching 107 cm/s. Over a range of perturbation wavelengths, the code results agree with the measured Rayleigh–Taylor growth from the linear through the deeply nonlinear regimes. FAST has been applied to the two-dimensional spherical simulation design to provide surface finish and laser bandwidth tolerances for a promising new direct-drive pellet that uses a foam ablator.

Parallel structures and dynamic load balancing for adaptive finite element computation

An adaptive technique for a partial differential system automatically adjusts a computational mesh or varies the order of a numerical procedure to obtain a solution satisfying prescribed accuracy criteria in an optimal fashion. We describe data structures for distributed storage of finite element mesh data as well as software for mesh adaptation, load balancing, and solving compressible flow problems. Processor load imbalances are introduced at adaptive enrichment steps during the course of a parallel computation. To correct this, we have developed three dynamic load balancing procedures based, respectively, on load imbalance trees, moment of inertia, and octree traversal. Computational results on an IBM SP2 computer are presented for steady and transient solutions of the three-dimensional Euler equations of compressible flow.