An efficient solver for the RANS equations and a one-equation turbulence model

A three-stage Runge-Kutta (RK) scheme with multigrid and an implicit preconditioner has been shown to be an effective solver for the fluid dynamic equations. Using the algebraic turbulence model of Baldwin and Lomax, this scheme has been used to solve the compressible Reynolds-averaged Navier–Stokes (RANS) equations for transonic and low-speed flows. In this paper we focus on the convergence of the RK/Implicit scheme when the effects of turbulence are represented by the one-equation model of Spalart and Allmaras. With the present scheme the RANS equations and the partial differential equation of the turbulence model are solved in a loosely coupled manner. This approach allows the convergence behavior of each system to be examined. Point symmetric Gauss-Seidel supplemented with local line relaxation is used to approximate the inverse of the implicit operator of the RANS solver. To solve the turbulence equation we consider three alternative methods: diagonally dominant alternating direction implicit (DDADI), symmetric line Gauss-Seidel (SLGS), and a two-stage RK scheme with implicit preconditioning. Computational results are presented for airfoil flows, and comparisons are made with experimental data. We demonstrate that the two-dimensional RANS equations and a transport-type equation for turbulence modeling can be efficiently solved with an indirectly coupled algorithm that uses RK/Implicit schemes.     

Development of a parallel implicit solver of fluid modeling equations for gas discharges

A parallel fully implicit PETSc-based fluid modeling equations solver for simulating gas discharges is developed. Fluid modeling equations include: the neutral species continuity equation, the charged species continuity equation with drift-diffusion approximation for mass fluxes, the electron energy density equation, and Poisson's equation for electrostatic potential. Except for Poisson's equation, all model equations are discretized by the fully implicit backward Euler method as a time integrator, and finite differences with the Scharfetter–Gummel scheme for mass fluxes on the spatial domain. At each time step, the resulting large sparse algebraic nonlinear system is solved by the Newton–Krylov–Schwarz algorithm. A 2D-GEC RF discharge is used as a benchmark to validate our solver by comparing the numerical results with both the published experimental data and the theoretical prediction. The parallel performance of the solver is investigated.     

Implicit treatment of molar mass equations in secondary oil migration

The hydrocarbon migration can be described by a coupled set of partial differential equations describing the dynamics of the temperature, component flow, pressure and velocity. A sequential solution procedure where the component flow is solved explicitly, gives severe restrictions on the time step given by the CFL condition. In this paper an implicit solution procedure is given and results from numerical tests are presented. The results are compared with the explicit solutions. As expected the implicit algorithm allows for substantially larger time steps. Received: 31 January 2001 / Accepted: 30 September 2001     

Stable discretization of poroelasticity problems and efficient preconditioners for arising saddle point type matrices

Poroelastic models arise in reservoir modeling and many other important applications. Under certain assumptions, they involve a time-dependent coupled system consisting of Navier–Lamé equations for the displacements, Darcy’s flow equation for the fluid velocity and a divergence constraint equation. Stability for infinite time of the continuous problem and, second and third order accurate, time discretized equations are shown. Methods to handle the lack of regularity at initial times are discussed and illustrated numerically. After discretization, at each time step this leads to a block matrix system in saddle point form. Mixed space discretization methods and a regularization method to stabilize the system and avoid locking in the pressure variable are presented. A certain block matrix preconditioner is shown to cluster the eigenvalues of the preconditioned matrix about the unit value but needs inner iterations for certain matrix blocks. The strong clustering leads to very few outer iterations. Various approaches to construct preconditioners are presented and compared. The sensitivity of the number of outer iterations to the stopping accuracy of the inner iterations is illustrated numerically.     

Mixed Volume Element-Characteristic Fractional Step Difference Method for Contamination from Nuclear Waste Disposal

A nonlinear system with boundary-initial value conditions of convection–diffusion partial differential equations is presented to describe incompressible nuclear waste disposal contamination in porous media. The flow pressure is determined by an elliptic equation, the concentrations of brine and radionuclide are formulated by convection–diffusion equations, and the transport of temperature is defined by a heat equation. The pressure appears in convection–diffusion equations and heat equation in the form of Darcy velocity and controls the physical processes. The fluid pressure and velocity are solved by the conservative mixed volume element and the computation accuracy of Darcy velocity is improved one order. A combination method of the mixed volume element and the approximation of characteristics is applied to solve the brine and heat, where the diffusion is discretized by a mixed volume element method and the convection is treated by the method of characteristics. The characteristics can confirm strong computation stability at sharp fronts and it can avoid numerical dispersion and nonphysical oscillation. Larger time-steps along the characteristics are shown to result in smaller time-truncation errors than those resulting from standard methods. The mixed volume element method has the property of conservation on each element and it can obtain numerical solutions of the brine and adjoint vectors. The radionuclide is solved by a coupled method of characteristics and fractional step difference. The computational work is reduced greatly by decomposing a three-dimensional problem into three successive one-dimensional problems and using the algorithm of speedup. Using numerical analysis of priori estimates of differential equations, we demonstrate an optimal second order estimate in \(l^2\) norm. Numerical data are appropriate with the scheme and it is shown that the method is a powerful tool to solve the well-known problems in porous media.     

A Parallel Domain Decomposition Method for 3D Unsteady Incompressible Flows at High Reynolds Number

Numerical simulation of three-dimensional incompressible flows at high Reynolds number using the unsteady Navier–Stokes equations is challenging. In order to obtain accurate simulations, very fine meshes are necessary, and such simulations are increasingly important for modern engineering practices, such as understanding the flow behavior around high speed trains, which is the target application of this research. To avoid the time step size constraint imposed by the CFL number and the fine spacial mesh size, we investigate some fully implicit methods, and focus on how to solve the large nonlinear system of equations at each time step on large scale parallel computers. In most of the existing implicit Navier–Stokes solvers, segregated velocity and pressure treatment is employed. In this paper, we focus on the Newton–Krylov–Schwarz method for solving the monolithic nonlinear system arising from the fully coupled finite element discretization of the Navier–Stokes equations on unstructured meshes. In the subdomain, LU or point-block ILU is used as the local solver. We test the algorithm for some three-dimensional complex unsteady flows, including flows passing a high speed train, on a supercomputer with thousands of processors. Numerical experiments show that the algorithm has superlinear scalability with over three thousand processors for problems with tens of millions of unknowns.     

Development of a sixth-order two-dimensional convection–diffusion scheme via Cole–Hopf transformation

In this paper, we develop a two-dimensional finite-difference scheme for solving the time-dependent convection–diffusion equation. The numerical method exploits Cole–Hopf equation to transform the nonlinear scalar transport equation into the linear heat conduction equation. Within the semi-discretization context, the time derivative term in the transformed parabolic equation is approximated by a second-order accurate time-stepping scheme, resulting in an inhomogeneous Helmholtz equation. We apply the alternating direction implicit scheme of Polezhaev to solve the Helmholtz equation. As the key to success in the present simulation, we develop a Helmholtz scheme with sixth-order spatial accuracy. As is standard practice, we validated the code against test problems which were amenable to exact solutions. Results show excellent agreement for the one-dimensional test problems and good agreement with the analytical solution for the two-dimensional problem.     

Meshless local boundary integral equation method for simply supported and clamped plates resting on elastic foundation

Simply supported and clamped thin elastic plates resting on a two-parameter foundation are analyzed in the paper. The governing partial differential equation of fourth order for a plate is decomposed into two coupled partial differential equations of second order. One of them is Poisson’s equation whereas the other one is Helmholtz’s equation. The local boundary integral equation method is used with meshless approximation for both the Poisson and the Helmholtz equation. The moving least square method is employed as the meshless approximation. Independent of the boundary conditions fictitious nodal unknowns used for the approximation of bending moments and deflections are always coupled in the resulting system of algebraic equations. The Winkler foundation model follows from the Pasternak model if the second parameter is equal to zero. Numerical results for a square plate with simply and/or clamped edges are presented to prove the efficiency of the proposed formulation.     

Finite difference solution to the flow between two rotating spheres

This paper describes a numerical method for calculating incompressible viscous flows between two concentric rotating spheres. The dependent variables describing the axisymmetric flow field are the azimuthal components of the vorticity, of the velocity vector potential and of the velocity. The coupled set of governing partial differential equations is written as a system of strictly second-order equations by introducing vorticity conditions of an integral character in a meridional plane. Such conditions generalize the one-dimensional integral conditions employed by Dennis and Singh to calculate steady-state solutions of the same problem using Gegenbauer polynomials and finite differences. The basic equations are discretized in space and in time by means of the finite-difference method. A fourth-order accurate centred-difference approximation of the advection terms is employed and a nonlinearly implicit scheme for the discrete time integration is here considered. A general finite-difference algorithm for steady-state and time-dependent problems is obtained which has no relaxation parameter and makes extensive use of fast elliptic solvers. The numerical results obtained by the present method are found to be in good agreement with the literature and confirm the nonuniqueness of the steady-state solution in a narrow spherical gap at certain regimes.     

耦合压力与温度修正算法对流体流场数值仿真

深入研究了目前流体流场的数值仿真问题,由于流体流场中可能存在着马赫数变化很大的流动情形,通常的方法不能较好地计算出准确的结果,因此找出一种能计算任意马赫数流动的算法是非常必要的.使用了一种耦合压力与温度修正算法求解Navier-Stokes方程.是通过连续方程和能量方程推得压力修正值与温度修正值的方程,并将压力修正值方程与温度修正值方程联立求解,而其它求解变量采用分离式求解的思想,求解中对流项差分格式采用了AUSM 格式,并在低马赫数时进行了改进.通过对喷管和圆弧凸包的数值仿真,较好地反映出了流场中的激波现象,计算表明方法能适应任意马赫数范围的流体流动,并且具有一致的计算精度.     

High-order time splitting for the Stokes equations

A pseudo-spectral (or collocation) approximation of the unsteady Stokes equations is presented. Using the Uzawa algorithm the spectral system is decoupled into Helmholtz equations for the velocity components and an equation with the Pseudo-Laplacian for the pressure. In order to avoid spurious modes the pressure is approximated with lower order (two degrees lower) polynomials than the velocity. Only one grid (no staggered grids) with the standard Chebyshev Gauss-Lobatto nodes is used. Here we further compare our treatment with a Neumann boundary value problem for the pressure. The highly improved accuracy of our method becomes obvious. In the time discretization a high order backward differentiation scheme for the intermediate velocity is combined with a high order extrapolant for the pressure. It is numerically shown that a stable third order method in time can be achieved.     

Parallel computational methods for 3D simulation of a parafoil with prescribed shape changes

In this paper we describe parallel computational methods for 3D simulation of the dynamics and fluid dynamics of a parafoil with prescribed, time-dependent shape changes. The mathematical model is based on the time-dependent, 3D Navier-Stokes equations governing the incompressible flow around the parafoil and Newton's law of motion governing the dynamics of the parafoil, with the aerodynamic forces acting on the parafoil calculated from the flow field. The computational methods developed for these 3D simulations include a stabilized space-time finite element formulation to accommodate for the shape changes, special mesh generation and mesh moving strategies developed for this purpose, iterative solution techniques for the large, coupled nonlinear equation systems involved, and parallel implementation of all these methods on scalable computing systems such as the Thinking Machines CM-5. As an example, we report 3D simulation of a flare maneuver in which the parafoil velocity is reduced by pulling down the flaps. This simulation requires solution of over 3.6 million coupled, nonlinear equations at every time step of the simulation.     

A numerical flow simulation based on the rate form of the equation of state

A finite-difference numerical method of solution for the unsteady, incompressible Navier-Stokes equations using primitive variables is presented. The rate form of the equation of state is used for the calculation of pressure. This form of the equation of state is well-suited for use with the unsteady form of the conservation equations (mass, momentum and energy). An implicit algorithm is used for the time integration for greater numerical stability. This method is used to solve a known benchmark problem in steady-state natural convection as a test of steady-state accuracy. The results of the simulation are compared to the benchmark.     

Numerical appraisal of three low Mach number algorithms for radiative–convective flows in enclosures

The present study investigates three different algorithms for the numerical simulation of non-Boussinesq convection with thermal radiative heat transfer based on a low-Mach number formulation. The solution methodology employs a fractional step approach based on the finite-volume method on arbitrary polyhedral meshes. The three algorithms compute the coupled governing equations in a segregated manner using the conservative form of momentum equations in conjunction with a variable coefficient pressure Poisson equation. The first algorithm (Algorithm A) uses conservation of mass and energy equation to compute density and temperature. The other two algorithms (Algorithm B) and (Algorithm C) calculates temperature and density from the equation of state respectively and solves a conservative form of the continuity and energy equation to obtain density and temperature respectively. The energy and mass conservation errors arising due to the use of Algorithms B and C are derived concerning various non-dimensional parameters governing the flow and heat transfer. The significance of these errors is highlighted by performing investigations over a range of Rayleigh, Prandtl, and Planck numbers for various two and three-dimensional natural convection problems with radiative heat transfer. Finally, the role of balancing of the pressure and buoyancy terms is emphasized for robust calculations of large temperature difference thermo-buoyant convection with radiative heat transfer.     

Analysis of unsteady compressible boundary layer flow via finite elements

This paper is concerned with the discrete formulation and numerical solution of unsteady compressible boundary layer flows using the Galerkin-finite element method. Linear interpolation functions for the velocity, density, temperature and pressure are used in the momentum equation and equations of continuity, energy and state. The coupled nonlinear finite element equations are approximated by a third order Taylor series expansion as temporal operator to integrate in time with Newton-Raphson type iterations performed until convergence within each time step. As an example, a boundary layer problem of a perfect gas behind a normal shock wave is solved. A comparison of the results with those by other method indicates a favorable agreement.     

曲率驱动的基于亥姆霍兹涡量方程的图像修复模型

图像修复模型根据已知区域信息自动修复目标区域,且保证修复后的图像满足人眼视觉系统的要求,目标区域轮廓自然.理论分析证明,流体力学中无粘亥姆霍兹涡量方程可以实现图像修复.根据曲线和曲面运动方程,使用曲率驱动亥姆霍兹修复模型中的等照度线传输方向.曲率是图像几何特征作用的结果,所以新模型可以很好地保持图像中的线性特征.图像中涡量为平滑程度的度量,二维图像域中涡量具有扩散性,对涡量方程的结果做各向异性扩散,使等照度线间的图像信息交互.亥姆霍兹涡量方程修复模型中的各个参数及扩散过程是涡量扩散性和耗散性以及图像几何特征作用的结果,扩散后的修复模型稳定且不存在错误的传输方向.理论和实验证明了曲率驱动的亥姆霍兹涡量方程模型在图像修复中的有效性.     

An implicit compact scheme solver for two-dimensional multicomponent flows

A 2D implicit compact scheme solver has been implemented for the vorticity-velocity formulation in the case of nonreacting, multicomponent, axisymmetric, low Mach number flows. To stabilize the discrete boundary value problem, two sets of boundary closures are introduced to couple the velocity and vorticity fields. A Newton solver is used for solving steady-state and time-dependent equations. In this solver, the Jacobian matrix is formulated and stored in component form. To solve the system of linearized equations within each iteration of Newton’s method, preconditioned Bi-CGSTAB is used in combination with a matrix-vector product computed in component form. The almost dense Jacobian matrix is approximated by a partial Jacobian. For the preconditioner equation, the partial Jacobian is approximately factored using several methods. In a detailed study of several preconditioning techniques, a promising method based on ILUT preconditioning in combination with reordering and double scaling using the MC64 algorithm by Duff and Koster is selected. To validate the implicit compact scheme solver, several nonreacting model problems have been considered. At least third order accuracy in space is recovered on nonuniform grids. A comparison of the results of the implicit compact scheme solver with the results of a traditional implicit low order solver shows an order of magnitude reduction of computer memory and time using the compact scheme solver in the case of time-dependent mixing problems.     

The dual reciprocity boundary element formulation for nonlinear diffusion problems

This paper presents an extension of the dual reciprocity boundary element method (DRBEM) to deal with nonlinear diffusion problems in which thermal conductivity, specific heat, and density coefficients are all functions of temperature. The DRBEM, recently applied to the solution of problems governed by parabolic and hyperbolic equations, consists in the transformation of the differential equation into an integral equation involving boundary integrals only, the solution of which is achieved by employing a standard boundary element discretization coupled with a two-level finite difference time integration scheme. Contrary to previous formulations for the diffusion equation, the dual reciprocity BEM utilizes the well-known fundamental solution to Laplace's equation, which is space-dependent only. This avoids complex time integrations that normally appear in formulations employing time-dependent fundamental solutions, and permits accurate numerical solutions to be obtained in an efficient way. For nonlinear problems, the integral of conductivity is introduced as a new variable to obtain a linear diffusion equation in the Kirchhoff transform space. This equation involves a modified time variable which is itself a function of position. The problem is solved in an iterative way by using an efficient Newton-Raphson technique which is shown to be rapidly convergent.     

A two-point difference scheme for computing steady-state solutions to the conservative one-dimensional Euler equations

An implicit finite-difference method is presented for obtaining steady-state solutions to the time-dependent, conservative Euler equations for flows containing shocks. The method uses a two-point central-difference scheme for the flux derivatives with dissipation added at supersonic points via the retarded density concept. Application of the method to 1-dimensional nozzle flow equations for various combinations of subsonic and supersonic boundary conditions show the method to be very efficient. Residuals are typically reduced to machine zero in approximately 35 time steps for 50 mesh points. For 1-dimensional Euler calculations, it is shown that the scheme offers two advantages over the more widely-used three-point schemes. The first is in regard to application of boundary conditions, and the second relates to the fact that the two-point algorithm is well-conditioned for large time steps.