二相流计算的一种差分算法

考虑两相流的力学行为,忽略相间的耗散作用,建立了Euler型的基本控制方程．状态方程采用刚性状态方程．基于Abgrall提出的准则,在流动区域内,对可压两相流提出了一个精度较高的Euler型数值方法,数值格式是Godunov型格式,对守恒型和非守恒型方程采用HLLC型和Lax-Friedrichs型近似Riemann解算器,引入了速度驰豫和压强驰豫过程来代替两相间的相互作用．在一维情形下给出数值算例,并且和Saurel的算例进行了比较,结果表明该算法不但精确而且稳定,且在间断处没有数值振荡．     

THE ASYMPTOTIC PRESERVING UNIFIED GAS KINETIC SCHEME FOR GRAY RADIATIVE TRANSFER EQUATIONS ON DISTORTED QUADRILATERAL MESHES

In this paper, we consider the multi-dimensional asymptotic preserving unified gas kinetic scheme for gray radiative transfer equations on distorted quadrilateral meshes. Different from the former scheme [J. Comput. Phys. 285(2015), 265-279] on uniform meshes, in this paper, in order to obtain the boundary fluxes based on the framework of unified gas kinetic scheme (UGKS), we use the real multi-dimensional reconstruction for the initial data and the macro-terms in the equation of the gray transfer equations. We can prove that the scheme is asymptotic preserving, and especially for the distorted quadrilateral meshes, a nine-point scheme [SIAM J. SCI. COMPUT. 30(2008), 1341-1361] for the diffusion limit equations is obtained, which is naturally reduced to standard five-point scheme for the orthogonal meshes. The numerical examples on distorted meshes are included to validate the current approach.     

Radiative transfer in a two-dimensional rectangular annulus medium

The radiative transfer equation in a two-dimensional rectangular annulus medium is solved numerically. The numerical method is based on a finite difference scheme and a product quadrature discrete-ordinate scheme. The discretized equation of transfer is solved iteratively to give the radiation intensity. The medium is assumed to absorb, emit, and anisotropically scatter radiation. It is exposed to diffusely emitting and diffusely reflecting boundaries. The results of the total intensity for various radiative parameters are presented. The method can be modified easily to solve the rectangular medium without the annulus. Our results in this case compare very well with those of Crosbie et al. [1], Thynell et al. [2], and Wu [3].     

ON LOCAL STRUCTURAL STABILITY OF ONE-DIMENSIONAL SHOCKS IN RADIATION HYDRODYNAMICS

In this paper, we are concerned with the local structural stability of one-dimensional shock waves in radiation hydrodynamics described by the isentropic Euler-Boltzmann equations. Even though in this radiation hydrodynamics model, the radiative effects can be understood as source terms to the isentropic Euler equations of hydrodynamics, in general the radiation field has singularities propagated in an angular domain issuing from the initial point across which the density is discontinuous. This is the major difficulty in the stability analysis of shocks. Under certain assumptions on the radiation parameters, we show there exists a local weak solution to the initial value problem of the one dimensional Euler-Boltzmann equations, in which the radiation intensity is continuous, while the density and velocity are piecewise Lipschitz continuous with a strong discontinuity representing the shock-front. The existence of such a solution indicates that shock waves are structurally stable, at least local in time, in radiation hydrodynamics.     

High-order accurate implicit running schemes

An approach to the construction of high-order accurate implicit predictor-corrector schemes is proposed. The accuracy is improved by choosing a special time integration step for computing numerical fluxes through cell interfaces by using an unconditionally stable implicit scheme. For smooth solutions of advection equations with constant coefficients, the scheme is second-order accurate. Implicit difference schemes for multidimensional advection equations are constructed on the basis of Godunov’s method with splitting over spatial variables as applied to the computation of “large” values at an intermediate layer. The numerical solutions obtained for advection equations and the radiative transfer equations in a vacuum are compared with their exact solutions. The comparison results confirm that the approach is efficient and that the accuracy of the implicit predictor-corrector schemes is improved.     

Asymptotic preserving HLL schemes

This work concerns the derivation of HLL schemes to approximate the solutions of systems of conservation laws supplemented by source terms. Such a system contains many models such as the Euler equations with high friction or the M1 model for radiative transfer. The main difficulty arising from these models comes from a particular asymptotic behavior. Indeed, in the limit of some suitable parameter, the system tends to a diffusion equation. This article is devoted to derive HLL methods able to approximate the associated transport regime but also to restore the suitable asymptotic diffusive regime. To access such an issue, a free parameter is introduced into the source term. This free parameter will be a useful correction to satisfy the expected diffusion equation at the discrete level. The derivation of the HLL scheme for hyperbolic systems with source terms comes from a modification of the HLL scheme for the associated homogeneous hyperbolic system. The resulting numerical procedure is robust as the source term discretization preserves the physical admissible states. The scheme is applied to several models of physical interest. The numerical asymptotic behavior is analyzed and an asymptotic preserving property is systematically exhibited. The scheme is illustrated with numerical experiments. © 2010 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 27: 1396–1422, 2011     

Eulerian Simulation of Gas‐Solid Particle Flow by BDIM

The numerical scheme based on the boundary domain integral method (BDIM) for the numerical simulation of twophase two‐component flows is presented. A program is being developed to model the hydrodynamics of fluidized bed systems by using the Eulerian approach in terms of velocity‐vorticity variables formulation. With the vorticity vector both phases motion computation scheme is partitioned into its kinematic and kinetic aspect. Influence of the drag coefficient on the two‐phase two‐component flow field is studied on the two‐phase gas‐solid particles vertical channel flow.     

Functional optimization technique for numerical solution of the radiative transfer equation

A numerical method for solving the time-independent radiative transfer problem in a flat layer with given properties and temperature distribution is proposed. This method avoids the numerical diffusion; rather, it is based on a gradient procedure for the functional minimization of the residual of the radiative transfer integral equation. Means for suppressing computational instabilities are proposed that reduce requirements for the approximation of the operators in the optimization problem but do not change the problem objective functional.     

Monte Carlo simulation of radiative transfer in a medium with varying refractive index specified at discrete points

A Monte Carlo method with discrete ray tracing is developed to simulate radiative transfer in a medium with a spatially varying refractive-index distribution available merely at a set of arbitrary discrete points. We solve the ray equation by a Runge−Kutta Dormand−Prince method to carry out the numerical ray tracing. To retrieve the refractive-index values and gradients needed in the discrete ray tracing, we apply cubic spline interpolation for one-dimensional simulation and a moving least square (MLS) method for two-dimensional simulation. The influence of the basis vectors and the numbers of sampled data used by the MLS method on ray tracing based on the retrieved refractive-index values and gradients has been examined. The results of radiative equilibrium in a planar medium and radiative transfer in two-dimensional media with different geometries and conditions obtained by the present methods are compared with those obtained by solving the integral equations of radiative transfer and the discrete ordinates method. The comparisons show that the present methods generate accurate results for radiative transfer with various geometries, parameters and refractive-index distributions specified at discrete points.     

On a simplified model for radiating flows

We consider a simplified model arising in radiation hydrodynamics which is based on the barotropic Navier–Stokes system describing the macroscopic fluid motion and a P1-approximation (see below) of the transport equation modeling the propagation of radiative intensity. We establish global-in-time existence of strong solutions for the associated Cauchy problem when initial data are close to a stable radiative equilibrium and local existence for large data with no vacuum. All our results are stated in the so-called critical Besov spaces.     

Unsteady nonisothermal seepage of steam in geothermal systems

The plane problem of unsteady nonisothermal seepage of steam in permeable subsurface layers of geothermal systems is analyzed in the framework of the theory of heterogeneous media. The equations of hydrodynamics and heat transfer are posed for a saturated porous medium and for the surrounding bedrock and the boundary conditions are formulated. A finite-difference scheme is constructed for numerical solution of the system of five nonlinear partial differential equations and their stability is analyzed. A numerical example is presented.Translated from Vychislitel'naya i Prikladnaya Matematika, No. 60, pp. 47–53, 1986.     

A simple Taylor-series expansion method for a class of second kind integral equations

A simple yet effective Taylor-series expansion method is presented for a class of Fredholm integral equations of the second kind with smooth and weakly singular kernels. The equations studied in this paper arise in a number of applications, e.g., potential theory, radiative equilibrium, radiative heat transfer, and electrostatics. The approach leads to an approximate solution of the integral equation which can be expressed explicitly in a simple, closed form. The approximate solution is of sufficient accuracy as illustrated by the numerical examples arising from radiative heat transfer and electrostatics.     

Numerical experiments on the Helmholtz equation derived from the solar radiation transfer equation in three-dimensional space

A numerical scheme using the finite-difference approach to solve the modified Helmholtz partial differential equation derived from the solar radiative transfer equation is developed and tested along with the method of evaluating the slant-path optical depth. For overhead solar incidence, we obtain good agreement between the finite-difference approach and the semianalytical solution in terms of the local intensity, local flux, average intensity, and average flux. For face-parallel oblique solar incidence in which the semianalytical method is not applicable, we compare the results with those of previous studies utilizing the Monte Carlo method and the approximate semianalytical method. We show that the present numerical scheme can be applied to any incident solar angle which the approximate semianalytical method is incapable of. Comparisons with results from Monte Carlo method reveal reasonable agreement for the averaged intensity and flux density.     

Simplified radiative models for low-Mach number reactive flows

We introduce a class of numerical models for the simulation of radiative effects in low-Mach number reactive flows. These models are based on simplified P_N approximations for radiative heat transfer, low-Mach asymptotic in the compressible flow, and reduced four-step chemical reaction for reacting species. The models presented here remove on one hand, acoustic wave propagation while retaining the compressibilty effects resulting from combustion and on the other hand, simplify the integro-differential equation for radiative transfer to a set of differential equations independent of the angle variable and compatible to those used for modelling flow and combustion. We briefly discuss the basic discretization methodology for the combined equations and its implementation in a modified projection method. We present validation computations for a two-dimensional methane/air flame in which the computational results are compared for nongray participating media.     

Explicit-implicit difference scheme for the joint solution of the radiative transfer and energy equations by the splitting method

High-order accurate explicit and implicit conservative predictor-corrector schemes are presented for the radiative transfer and energy equations in the multigroup kinetic approximation solved together by applying the splitting method with respect to physical processes and spatial variables. The original system of integrodifferential equations is split into two subsystems: one of partial differential equations without sources and one of ordinary differential equations (ODE) with sources. The general solution of the ODE system and the energy equation is written in quadratures based on total energy conservation in a cell. A feature of the schemes is that a new approximation is used for the numerical fluxes through the cell interfaces. The fluxes are found along characteristics with the interaction between radiation and matter taken into account. For smooth solutions, the schemes approximating the transfer equations on spatially uniform grids are second-order accurate in time and space. As an example, numerical results for Fleck’s test problems are presented that confirm the increased accuracy and efficiency of the method.     

The cauchy problem for the radiative transfer equation with generalized conjugation conditions

The initial boundary problem for the nonstationary radiative transfer equation in a nonhomogeneous plane layer with generalized conjugation conditions on the material interface is studied. A generalized Monte Carlo algorithm is proposed for solving the problem, and numerical experiments are discussed.     

Numerical performance of stability enhancing and speed increasing steps in radiative transfer solution methods

Methods for solving the radiative transfer problem, which is crucial for a number of sectors of industry, involve several numerical challenges. This paper gives a systematic presentation of the effect of the steps that are needed or possible to make any discrete ordinate radiative transfer solution method numerically efficient. This is done through studies of the numerical performance of the stability enhancing and speed increasing steps used in modern tools like Disort or Dort2002.     

Analytical description of the radiative-conductive heat transfer in a gray medium contained between two diffuse parallel plates

Explicit analytical solutions for the temperature and heat flux in a gray medium contained between two diffuse parallel plates are derived for both pure thermal radiation and coupled conduction-radiation heat transfer. This is achieved by combining the integral equations for the heat flux and temperature predicted by the radiative transfer equation with the corresponding predictions of the discrete ordinates method. The algebraic formulation of this well-known method is used to derive analytical results that agree with their corresponding numerical ones with an accuracy greater than 99.9%, for a large interval of optical thicknesses and conduction-to-radiation factors. The explicit and original solutions, for both pure radiation and radiative-conductive heat transfer, therefore solve the problem of one dimensional steady-state heat transfer in gray cavities.     

On a model in radiation hydrodynamics

We consider a simplified model arising in radiation hydrodynamics based on the Navier–Stokes–Fourier system describing the macroscopic fluid motion, and a transport equation modeling the propagation of radiative intensity. We establish global-in-time existence for the associated initial–boundary value problem in the framework of weak solutions.     

A MUSCL smoothed particles hydrodynamics for compressible multi‐material flows

By incorporating the Monotone Upwind Scheme of Conservation Law (MUSCL) scheme into the smoothed particles hydrodynamics (SPH) method and making use of an interparticle contact algorithm, we present a MUSCL–SPH scheme of second order for multifluid computations, which extends the Riemann‐solved‐based SPH method. The numerical tests demonstrate high accuracy and resolution of the scheme for both shocks, contact discontinuities, and rarefaction waves in the one‐dimensional shock tube problem. For the two‐dimensional cylindrical Noh and shock‐bubble interaction problems, the MUSCL–SPH scheme can resolve shocks well. Copyright © 2015 John Wiley & Sons, Ltd.