Relaxation time simulation method with internal energy exchange for perfect gas flow at near-continuum conditions

This paper presents an internal energy exchange scheme for the relaxation time simulation method (RTSM) which solves the BGK equation for the perfect gas flow at near-continuum region discrete rotational energies are introduced to model the relaxation of internal energy modes. This development improved the agreements between RTSM and DSMC with little additional computational cost. The result shows a possibility of an improved hybrid RTSM/DSMC code for the continuum/rarefied gas flow.     

A simple weighted essentially non-oscillatory limiter for the correction procedure via reconstruction (CPR) framework on unstructured meshes

In this paper, we adapt a simple weighted essentially non-oscillatory (WENO) limiter, originally designed for discontinuous Galerkin (DG) schemes on two-dimensional unstructured triangular meshes [39], to the correction procedure via reconstruction (CPR) framework for solving nonlinear hyperbolic conservation laws on two-dimensional unstructured triangular meshes with straight or curved edges. This is an extension of our earlier work [4] in which the WENO limiter was designed for the CPR framework on regular meshes. The objective of this simple WENO limiter is to simultaneously maintain uniform high order accuracy of the CPR framework in smooth regions and control spurious numerical oscillations near discontinuities. The WENO limiter we adopt in this paper uses information only from the target cell and its immediate neighbors. Hence, it is particularly simple to implement and will not harm the conservativeness and compactness of the CPR framework. Since the CPR framework with this WENO limiter does not in general satisfy the positivity preserving property, we also extend the positivity-preserving limiters [36], [33] to the CPR framework. Numerical results for both scalar equations and Euler systems of compressible gas dynamics are provided to illustrate the good behavior of this procedure.     

On the optimization of flux limiter schemes for hyperbolic conservation laws

A classic strategy to obtain high‐quality discretizations of hyperbolic partial differential equations is to use flux limiter (FL) functions for blending two types of approximations: a monotone first‐order scheme that deals with discontinuous solution features and a higher order method for approximating smooth solution parts. In this article, we study a new approach to FL methods. Relying on a classification of input data with respect to smoothness, we associate specific basis functions with the individual smoothness notions. Then, we construct a limiter as a linear combination of the members of parameter‐dependent families of basis functions, and we explore the possibility to optimize the parameters in interesting model situations to find a corresponding optimal limiter. © 2012 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq, 2013     

Generalization of Runge‐Kutta discontinuous Galerkin method to LWR traffic flow model with inhomogeneous road conditions

The discussed model is characterized by changeable lane numbers and free flow velocities that give rise to the spatially varying flux function in conservation equation. Accordingly a new numerical flux and a new limiter for the Runge‐Kutta Discontinuous Galerkin method are considered, which are compared with a natural but simple extension. It is verified that the new generalization is of high‐resolution and has wider stable and convergent ranges. © 2004 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq, 2005     

求解二维浅水方程的一种高分辨率有限体积法

本文将一种van Albada型可微的限制器函数引入到二维浅水方程的求解中，发展了一种求解二维浅水方程的有限体积法．数值实验结果表明，该方法不仅计算精度高，而且较其它求解二维浅水方程的高精度有限体积法，在数值解的收敛性能方面大有改善．     

On the numerical modelling of the transitional flow in rarefied gases

The aim of this work is to simulate rarefied gas flow in complex geometries, under flow conditions that range from the hydrodynamic, through the transitional, to the molecular regimes. Existing computational models apply to molecular or viscous flow, but the treatment of the transitional flow is still underdeveloped.To deal with the difficult transitional flow, two models with overlapping ranges of applicability are introduced. A direct simulation Monte Carlo (DSMC) type model, which can be used in the molecular and up to the lower transitional flow, has been designed. For the viscous to the upper transitional flow, a numerical model using a particle method is proposed. The objective is to obtain a smooth transition between the probabilistic simulation of particle histories and the deterministic approach of the solution of partial differential equations.The DSMC model has been successfully applied to molecular and lower transitional flow in a complex geometry with stationary and moving boundaries. The test results agree well with published data. The particle method was tested using simplified Navier-Stokes equations in a channel. Preliminary results in the low viscous range seem to indicate that the approach is viable.     

Estimating the statistical error in calculating velocity and temperature components by the direct simulation Monte Carlo method

The direct simulation Monte Carlo (DSMC) method is widely used to solve problems in rarefied gas dynamics. In solving steady state problems, a special feature of the method is in using dependent sample values of random variables to calculate the macroparameters of gas flow. In this paper, the possibilities of methods of statistical physics to estimate the statistical error of the DSMC method are theoretically analyzed. A simple approach to approximate estimation of the statistical error in calculating temperature and velocity components is proposed. The approach is tested on a number of problems.     

非结构网格上一类满足局部极值原理的三阶精度有限体积方法

对二维标量双曲型守恒律方程,发展了一类满足局部极值原理的非结构网格有限体积格式.其构造思想是,以单调数值通量为基础,通过应用基于最小二乘法的二次重构和极值限制器,使数值解满足局部极值原理.为保证数值解在光滑区域达到三阶精度,该格式可结合局部光滑探测器使用.本文从理论上分析了格式的稳定性条件,数值实验验证了格式的精度和对间断的分辨能力.     

An Exact Riemann Solver for a Fluidized Bed Model

We study a 2 x 2 hyperbolic system of conservation laws withsource term arising in a fluidized bed model. The system issolved numerically and results are presented to demonstratethe occurrence of ‘slugging’ in the full model equations.The numerical procedure is based on operator splitting and Godunov'smethod, for which we derive the exact solution of the Riemannproblem. A second-order improvement due to Davis (1988) mayproduce small oscillations near shocks and these can be reducedif the underlying flux limiter of the Davis method is replacedby the minmod limiter.     

高分辨率差分格式的数值分析与限制因子的构造

本文利用差分方法余项效应理论,分析比较了一些典型的限制因子.对不同的限制因子,格式的表现明显差异主要是由其数值耗散性、色散性强弱不同所致.在分析比较格式的数值耗散性、色散性之后,本文提出了一种新的限制因子,得到的格式在解的剧烈变化区具有更高的分辨率,在光滑区避免了由于数值色散性较强导致的失真.数值试验表明该格式具有较好的性质.     

不同马赫数Navier-Stokes方程计算方法的研究

为了在低马赫数到高马赫数范围内求解可压缩Navier-Stokes方程,给出了基于预处理算法的PLU-SGS方法.将高分辨率AUSMPW格式与三阶MUSCL格式融合,将其扩展到三阶精度,并采用特征边界条件.为了验证该方法的有效性,通过求解曲线坐标系可压缩Navier-Stokes方程,对几个典型流动问题进行了数值计算.计算结果与文献计算结果或实验数据比较表明,该方法对不同马赫数Navier-Stokes方程的计算,具有较高的计算精度和收敛速度以及良好的稳定性.     

IP-DSMC method for micro-scale flow with temperature variation

Based on the IP-DSMC method, a simple and effective temperature model is presented. By the developed IP-DSMC method with new temperature model, non-isothermal micro-flow is simulated. Comparing the developed IP results with the DSMC results, the correctness and effectiveness of the new temperature model are validated. Through simulating the orifice flow, it is found that it will cause numerical divergence if the second order central difference scheme of the mass conservation equation is used to update the information density when the local density gradient of the simulated flow is large. So the first order upwind scheme is recommended to update the information density.     

Implementing a discontinuous Galerkin method for the compressible,inviscid Euler equations in the DUNE framework

A discontinuous Galerkin scheme was implemented in the DUNE framework to solve the compressible, inviscid Euler equations in a multi-dimensional Cartesian grid. It uses a HLLC Riemann solver for the numerical fluxes in the interfaces, a total variation bounded limiter to handle discontinuities, a positivity preserving limiter for near vacuum conditions, and adaptive mesh refinement (AMR). (© 2014 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Monte Carlo Microflow Simulation for the Gas Film Lubrication in Modern Magnetic Disk Storage

A two‐dimensional unsteady problem of gas flow in an extremely narrow channel with an inclined upper wall and moving lower one is studied by the DSMC method. This is a model of gas film lubrication which occurs in modern magnetic disk storage, that is now under development. Far from the magnetic head the flow produced by the disk motion could be described by solution of the Rayleigh problem. Space and time distributions of the pressure on the upper wall as well as density and average velocity inside and outside of the channel were obtained. They show that as a result of the flow slowing‐down by the front wall of the magnetic head the region with an increased density is formed there. At the same time marked non‐homogeneity of gas velocity before the inlet of the channel is observed.     

Gas Near a Wall: Shortened Mean Free Path,Reduced Viscosity,and the Manifestation of the Knudsen Layer in the Navier–Stokes Solution of a Shear Flow

For the gas near a solid planar wall, we propose a scaling formula for the mean free path of a molecule as a function of the distance from the wall, under the assumption of a uniform distribution of the incident directions of the molecular free flight. We subsequently impose the same scaling onto the viscosity of the gas near the wall and compute the Navier–Stokes solution of the velocity of a shear flow parallel to the wall. Under the simplifying assumption of constant temperature of the gas, the velocity profile becomes an explicit nonlinear function of the distance from the wall and exhibits a Knudsen boundary layer near the wall. To verify the validity of the obtained formula, we perform the Direct Simulation Monte Carlo computations for the shear flow of argon and nitrogen at normal density and temperature. We find excellent agreement between our velocity approximation and the computed DSMC velocity profiles both within the Knudsen boundary layer and away from it.     

RKDG methods with WENO type limiters and conservative interfacial procedure for one-dimensional compressible multi-medium flow simulations

In this paper, we continue on studying the Runge-Kutta discontinuous Galerkin (RKDG) methods to solve compressible multi-medium flow with conservative treatment of the moving material interface. Comparing with the paper by J. Qiu, T.G. Liu and B.C. Khoo [J. Comput. Phys. 222 (2007) 353-373], we adopt the HLLC flux instead of Lax-Friedrichs numerical flux, the finite volume weighted essentially nonoscillatory (WENO) and Hermite WENO (HWENO) reconstructions as limiter instead of TVB limiter for RKDG. The HLLC flux is based on the approximate Riemann solver with little numerical viscosity and can resolve the contact discontinuity and shear wave very well. For limiter procedure, first we use the KXRCF indicator to identify the troubled cell, then apply WENO or HWENO method to reconstruct the polynomial in the troubled cell, while maintaining the cell average. This limiter procedure is more accurate and less problem dependent than the TVB limiter. Numerical results in one dimension for multi-medium flows such as gas-gas and gas-water are provided to illustrate the capability of these procedures.     

On nonlinear cross-diffusion systems: an optimal transport approach

We study a nonlinear, degenerate cross-diffusion model which involves two densities with two different drift velocities. A general framework is introduced based on its gradient flow structure in Wasserstein space to derive a notion of discrete-time solutions. Its continuum limit, due to the possible mixing of the densities, only solves a weaker version of the original system. In one space dimension, we find a stable initial configuration which allows the densities to be segregated. This leads to the evolution of a stable interface between the two densities, and to a stronger convergence result to the continuum limit. In particular derivation of a standard weak solution to the system is available. We also study the incompressible limit of the system, which addresses transport under a height constraint on the total density. In one space dimension we show that the problem leads to a two-phase Hele-Shaw type flow.     

气动计算中的紧致格式与迎风紧致格式

§1.引言 巨型计算机的发展为解决流体力学问题提供了强有力的手段.由于我们所面临的问题日趋复杂,在今天如何提高数值模拟精度和提高求解效率仍是计算流体力学工作者面临的重要课题. 自Beam和Warming提出隐式求解法以来,NS方程的求解效率大为提高.此后人们在这方面作了很多工作,如在[2]中利用线Gauss-Seidel迭代法加快了迭代的收     

A comparative analysis of approaches for investigating hypersonic flow over blunt bodies in a transitional regime

Hypersonic flows of a viscous perfect rarefied gas over blunt bodies in a transitional flow regime from continuum to free molecular, characteristic when spacecraft re-enter Earth's atmosphere at altitudes above 90-100 km, are considered. The two-dimensional problem of hypersonic flow is investigated over a wide range of free stream Knudsen numbers using both continuum and kinetic approaches: by numerical and analytical solutions of the continuum equations, by numerical solution of the Boltzmann kinetic equation with a model collision integral in the form of the S-model, and also by the direct simulation Monte Carlo method. The continuum approach is based on the use of asymptotically correct models of a thin viscous shock layer and a viscous shock layer. A refinement of the condition for a temperature jump on the body surface is proposed for the viscous shock layer model. The continuum and kinetic solutions, and also the solutions obtained by the Monte Carlo method are compared. The effectiveness, range of application, advantages and disadvantages of the different approaches are estimated.     

On the convergence of a second order approximation of conservation laws with discontinuous flux

A second order scheme is constructed for the scalar conservation laws with flux function allowed to be discontinuous in the space variable. The corresponding numerical solutionis shown to converge to the (A,B) entropysolution. Numerical results are presented to validate the convergence analysis. In addition, it is inferred that these numerical solutions are comparable with those obtained using the standard minmod limiter.