Equilibrium Flux Method Research Articles

Analysis of the convergence properties for a non-linear implicit Equilibrium Flux Method using Quasi Newton–Raphson and BiCGStab techniques

In this work, an investigation is carried out into the implicit formulation of the Equilibrium Flux Method (EFM) applied to the numerical solution of the Euler Equations for an ideal inviscid gas. The discretization employs a non-linear Finite Volume Method (FVM) approach which requires the solution to a system of non-linear equations, which is solved using various modifications to the Quasi-Newton–Raphson method. The core of the analysis presented here lies in the investigation of the convergence properties of the BiConjugate Stabilized (BiCGStab) method used to solve the JΔx=R where R is our vector of residuals computed directly using the Euler Equations, Δx is the increment to our estimated solution x for our system and J is the Jacobian of the residual functions. The increment to our solution Δx is modified to ensure the solution remains bound and finite. Results are shown for multiple one dimensional and two dimensional benchmark problems, with convergence properties of both the Newton–Raphson and BiCGStab procedures shown. Several conclusions may be drawn from the results shown: (a) increasing CFL numbers result in increasing dispersion errors, (b) higher order treatment of temporal derivatives results in lower condition numbers, resulting in fewer BiCGstab iterations per Newton–Raphson iteration, (c) higher order treatment of spatial derivatives results in higher condition numbers, hence requiring additional BiCGstab iterations, and (d) preconditioning using a simple Jacobi preconditioner significantly reduces the number of BiCGstab iterations required to obtain a solution.

Analysis of the second order accurate uniform equilibrium flux method and its graphics processing unit acceleration

The extension of the Uniform Equilibrium Flux Method (UEFM) to second order accuracy in space is presented. This extension is made possible through the recasting of the original UEFM flux expressions from a volumetric flux to a surface flux, allowing for reconstruction through a Taylor series expansion of the resulting split surface fluxes at the cell interfaces. By doing so, we avoid the difficulties associated with integration of the gradient terms over velocity and physical space as required by the original UEFM fluxes. Analysis of the dissipative qualities of the renewed direction split UEFM flux expressions demonstrate that the numerical dissipation is a function of Mach number, with increasing amounts of dissipation present with increasing Mach numbers. Following this analysis, the higher order UEFM fluxes are applied to large scale parallel computation using Graphics Processing Units, or GPUs, through the use of CUDA. The vector split nature of the UEFM fluxes are well suited to GPU computation due to the high degree of locality. This parallelization is performed using a cell-based parallel paradigm through the creation of several key CUDA kernels for the calculation of split fluxes, gradient of split fluxes and state related computations. The algorithm is executed entirely on the GPU device, with the host remaining idle during the computation stage. The GPU accelerated UEFM algorithm is then applied to the solution of several two dimensional benchmark problems. Speedup of approximately 200 and 171 times for first order accuracy and second order accuracy respectively is demonstrated when using an Nvidia Tesla C2075 computing GPU compared to that of a single core of an Intel Xeon E5-2760 CPU.

On the remedy against shock anomalies in kinetic schemes

Shock-capturing schemes often exhibit anomalous behaviors, such as the carbuncle phenomenon and the post-shock oscillations, especially in the hypersonic flow regime. This paper proposes a simple and effective remedy against these shock instabilities in the case of the kinetic Lax–Wendroff scheme, where the well-known classic scheme is reinforced by means of an equilibrium distribution function of gas molecules. The pathologies are significantly improved to an acceptable level for practical purposes without any considerable side-effect by locally bringing out the robustness of the equilibrium flux method from the kinetic scheme. The remedy is applied only to the preprocessing of the data at cell-edges. The performance of the fortified kinetic scheme is demonstrated in the problem of a hypersonic inviscid or viscous flow past a blunt body. Comparisons are also made with various advanced shock-capturing schemes at present.

Extension of the Uniform Equilibrium Flux Method (UEFM) to Second Order Accuracy and its Graphics Processing Unit Acceleration

Presented is the Uniform Equilibrium Flux Method (UEFM) extended to second order spatial accuracy and applied to Graphics Processing Unit (GPU) computation. The UEFM is an approximation of the True Direction Equilibrium Method (TDEFM), in which higher-order extension through the inclusion of gradients in primitives is very challenging due to difficulty in integrating the exponential function over space when there are temperature gradients in the flow. Since UEFM does not directly employ exponential functions – instead replacing the equilibrium velocity probability distribution function with a series of uniform step (i.e. Heaviside) step functions, no such difficulties exist in its extension to higher order accuracy. Furthermore, due to the high locality of the UEFM, the method is readily applied to GPU acceleration. No communication with the host is required due to the exclusive use of the GPU for all flux and state computations. In increase in overall computational speed of approximately 9% is demonstrated with an approximate speedup of 81x when compared to a conventional single Xeon core. We also demonstrate that the UEFM solver has better dissipative properties when compared to the Quiet Direct Simulation (QDS) method.

Effects of direction decoupling in flux calculation in finite volume solvers

In a finite volume CFD method for unsteady flow fluxes of mass, momentum and energy are exchanged between cells over a series of small time steps. The conventional approach, which we will refer to as direction decoupling, is to estimate fluxes across interfaces in a regular array of cells by using a one-dimensional flux expression based on the component of flow velocity normal to the interface between cells. This means that fluxes cannot be exchanged between diagonally adjacent cells since they share no cell interface, even if the local flow conditions dictate that the fluxes should flow diagonally. The direction decoupling imposed by the numerical method requires that the fluxes reach a diagonally adjacent cell in two time-steps. To evaluate the effects of this direction decoupling, we examine two numerical methods which differ only in that one uses direction decoupling while the other does not. We examine a generalized form of Pullin’s equilibrium flux method (EFM) [D.I. Pullin, Direct simulation methods for compressible ideal gas flow, J. Comput. Phys. 34 (1980) 231–244] which we have called the true direction equilibrium flux method (TDEFM). The TDEFM fluxes, derived from kinetic theory, flow not only between cells sharing an interface, but ultimately to any cell in the grid. TDEFM is used here to simulate a blast wave and an imploding flow problem on a structured rectangular mesh and is compared with results from direction decoupled EFM. Since both EFM and TDEFM are identical in the low CFL number limit, differences between the results demonstrate the detrimental effect of direction decoupling. Differences resulting from direction decoupling are also shown in the simulation of hypersonic flow over a rectangular body. The computational cost of allowing the EFM fluxes to flow in the correct directions on the grid is minimal.

Atomically detailed simulations of surface resistances to transport of CH4, CF4, and C2H6 through silicalite membranes

Abstract We have used atomically detailed methods to examine the surface resistances associated with single-component permeation of CH 4 , CF 4 , and C 2 H 6 through single crystal silicalite membranes. Dual control volume grand canonical molecular dynamics (DCV-GCMD) simulations can be used for this purpose, but only for extremely thin membranes and for relatively rapidly diffusing species. We use calculations of this type to introduce a more computationally efficient but approximate method for describing surface resistances, the local equilibrium flux method (LEFM). Using LEFM calculations, it is possible to accurately and rapidly predict surface resistances for large ranges of membrane operating conditions. Using this approach, we identify the crystal thicknesses beyond which surface resistances are negligible for single-component permeation of CH 4 , CF 4 , and C 2 H 6 through silicalite membranes as functions of membrane feed and permeate pressure.

A Gas-Kinetic BGK Scheme for the Navier–Stokes Equations and Its Connection with Artificial Dissipation and Godunov Method

This paper presents an improved gas-kinetic scheme based on the Bhatnagar–Gross–Krook (BGK) model for the compressible Navier–Stokes equations. The current method extends the previous gas-kinetic Navier–Stokes solver developed by Xu and Prendergast (J. Comput. Phys.114, 9) by (a) implementing a general nonequilibrium state based on the Chapman–Enskog expansion of the BGK model as the initial gas distribution function at each time step, (b) using a different way in the construction of the equilibrium state at a cell interface, and (c) keeping two slopes of the macroscopic variables in the evaluation of spatial variations of the equilibrium state. As a result, the previous requirement of particle collision time τ being less than the time step Δt for the validity of the BGK Navier–Stokes solver is removed, and the scheme becomes more robust than the previous one. The current BGK method approximates the Navier–Stokes solutions accurately regardless of the ratio between the particle collision time and the numerical time step. The Kinetic Flux Vector Splitting Navier–Stokes (KFVS NS) solver developed by Chou and Baganoff comes to be the limiting case of the current method when the particle collision time is much larger than the time step. The Equilibrium Flux Method (EFM) of Pullin for the Euler equations is also a limiting case of the current method when the nonequilibrium part in the initial gas distribution function disappears and the collision time is much larger than the time step. The dissipative mechanism in the KFVS and many other FVS schemes is qualitatively analyzed from their relation with the BGK scheme. Also, in this paper, the appropriate implementation of boundary condition for the kinetic scheme, different limiting solutions, and the Prandtl number fix are presented. The connection among von Neumann and Richtmyer's artificial dissipation, Godunov method, and the gas-kinetic BGK scheme is discussed, from which the two concepts of dynamical and kinematical dissipation are introduced. The principles for constructing accurate and robust numerical schemes for the compressible flow simulation are proposed. Many numerical tests, which include highly nonequilibrium case (i.e., Mach 10 shock structure) are included to validate the BGK scheme for the viscous solutions and to support the physical and numerical analysis for different schemes.

Robustness versus accuracy in shock-wave computations

Despite constant progress in the development of upwind schemes, some failings still remain. Quirk recently reported that approximate Riemann solvers, which share the exact capture of contact discontinuities, generally suffer from such failings. One of these is the odd – even decoupling that occurs along planar shocks aligned with the mesh. First, a few results on some failings are given, namely the carbuncle phenomenon and the kinked Mach stem. Then, following Quirk’s analysis of Roe’s scheme, general criteria are derived to predict the odd – even decoupling. This analysis is applied to Roe’s scheme, the Equilibrium Flux Method, the Equilibrium Interface Method and the AUSM scheme. Strict stability is shown to be desirable to avoid most of these flaws. Finally, the link between marginal stability and accuracy on shear waves is established.

Self-similar hypervelocity shock interactions with oblique contact discontinuities

The two-dimensional inviscid refraction of a shock wave at an oblique contact discontinuity is self-similar; i.e. it depends only upon the variables \(\xi\equiv x/t\), \(\eta\equiv y/t\). We transform the compressible Euler equations into the self-similar (\(\xi,\eta\)) coordinates and solve the resulting boundary-value problem by an implicit equilibrium flux method. We present results for strong shock (\(M\ge10\) where M is the incident shock Mach number) interactions with an oblique contact discontinuity separating an inert gas from a gas which exhibits high-temperature gas chemistry effects. To model high-temperature effects we employ Lighthill's ideal dissociating gas (IDG) model. Comparison between the frozen and equilibrium limits, both of which are self-similar, indicate large changes in peak density and temperatures. Significant differences in the overall flow pattern between the frozen and equilibrium limits are observed for interfaces with low negative Atwood ratio and high positive Atwood ratio. Results from a local analysis are presented when the shock refraction is regular. The critical angle at which the transition from regular to irregular refraction occurs is slightly larger for the equilibrium chemistry case.

A Robust Low Diffusive Kinetic Scheme for the Navier–Stokes/Euler Equations

A new kinetic scheme based on the equilibrium flux method (EFM) and modified using Osher intermediate states is proposed. This new scheme called EFMO combines the robustness of the equilibrium flux method and the accuracy of flux-difference splitting schemes. The original EFM scheme is expressed in terms of simple wave decomposition in which only the linearly degenerate subpath is calculated from Osher numerical flux while nonlinear waves are still evaluated from the regular EFM splitting. Owing to its capability of withstanding intense nonlinear waves and yet exactly resolving contact discontinuities, EFMO is particularly well suited for the resolution of the Navier–Stokes equations as demonstrated by a series of severe test cases including the high-speed viscous flow around a cone, a shock-boundary layer interaction problem, a vacuum apparition problem, the hypersonic flow around a circular cylinder at Mach 100, and the forward-facing step at Mach 3.

Computational Methods for Self-similar Solutions of the Compressible Euler Equations

Computations of self-similar solutions of the compressible Euler equations as a boundary value problem in similarity coordinates (ξ =x/t, η =y/t) are presented. Two new implicit methods namely the implicit Godunov method and the implicit Equilibrium Flux Method are presented. The Jacobians for the implicit methods are analytically evaluated. In general the self-similar solutions exhibit sharper discontinuities than corresponding solutions of the initial value problem.

Numerical modeling of dynamic powder compaction using the Kawakita equation of state

Dynamic powder compaction is analyzed using the assumption that the powder behaves, while it is being compacted, like a hydrodynamic fluid in which deviatoric stress and heat conduction effects can be ignored throughout the process. This enables techniques of computational fluid dynamics such the equilibrium flux method to be used as a modeling tool. The equation of state of the powder under compression is assumed to be a modified version of the Kawakita loading curve. Computer simulations using this model are performed for conditions matching as closely as possible with those from experiments by Page and Killen [Powder Metall. 30, 233 (1987)]. The numerical and experimental results are compared and a surprising degree of qualitative agreement is observed.

Hypervelocity flow of dissociating nitrogen downstream of a blunt nose

The nature of the non-equilibrium flow of strongly dissociating nitrogen has been investigated by a series of simulation calculations using non-equilibrium (finite rate) chemical reactions. These were made with the equilibrium flux method (EFM), and the results have been found to compare favourably with experimental results obtained with a free-piston driven shock-tube wind tunnel which was used to obtain interferograms of the flow of pure nitrogen over a blunt-nosed body, 65 mm long at three angles of incidence. No simple relation between the flow with non-equilibrium chemistry and those for frozen or equilibrium chemistry has been found. The problems of relating test flows produced in the shock tunnel to flight conditions are investigated by considering the test flows that might be produced by some ‘ideal equivalent wind tunnels’. It is shown that the degree of frozen dissociation in the test flow in a shock tunnel is not a serious matter, but that the large difference in Mach number between shock tunnel flows and flight conditions may be more important.

The equilibrium flux method for the calculation of flows with non-equilibrium chemical reactions

The equilibrium flux method (D. I. Pullin, J. Comput. Phys. 34, 231 (1980)) is a kinetic theory based finite volume method for calculating the flow of a compressible ideal gas. It is shown here that, in effect, the method solves the Euler equations with added pseudo-dissipative terms and that it is a natural upwinding scheme. The method can be easily modified so that the flow of a chemically reacting gas mixture can be calculated. Results from the method for a one-dimensional non-equilibrium reacting flow are shown to agree well with a conventional continuum solution. Results are also presented for the calculation of a plane two-dimensional flow, at hypersonic speed, of a dissociating gas around a blunt nosed body.