A DIRECT SCHUR-FOURIER DECOMPOSITION FOR THE SOLUTION OF THE THREE-DIMENSIONAL POISSON EQUATION OF INCOMPRESSIBLE FLOW PROBLEMS USING LOOSELY COUPLED PARALLEL COMPUTERS

Parallel computers based on PC-class hardware (Beowulf clusters) provide a matchless computing power per cost unit. However, their network performance tends to be too low for standard parallel computational fluid dynamics (CFD) algorithms. A relevant example is the solution of the Poisson equations. The subject of this article is a direct Schur-Fourier decomposition (DSFD) algorithm that, for certain three-dimensional flows, produces an accurate solution of each Poisson equation with just one message, providing speed-ups of at least 24 in a low-cost PC cluster with a conventional network and 36 processors. Direct Numerical Simulation (DNS) of turbulent natural convection flow is used as a benchmark problem.     

GENERALIZED CONJUGATE GRADIENT METHOD FOR THE EFFICIENT SOLUTION OF THREE-DIMENSIONAL FLUID FLOW PROBLEMS

A generalized conjugate gradient (CG) method was examined for use in the numerical calculation of recirculating turbulent three-dimensional fluid flows. The proposed algorithm was found to be a very efficient solution procedure in a large number of practical applications. Furthermore, the results obtained from two other CG-like solution methods that are entirely free of recursion clearly indicate the advantages of the recommended partly recursive method even for vector computers.     

IMPROVED FINITE-DIFFERENCE SCHEME FOR THE SOLUTION OF CONVECTION-DIFFUSION PROBLEMS WITH THE SIMPLEN ALGORITHM

An improved version of the differencing scheme used in the simplen algorithm is presented. In the original scheme, source terms are upwinded in the modeled equations. This approach is extended here by application of upwinding to the terms representing cross-stream fluxes in addition to the upwinding of the source terms. The method is then applied to three case studies involving two-dimensional, incompressible, laminar flow. Excellent agreement between these numerical solutions and the corresponding analytical/benchmark solutions is achieved. The new method is shown to be superior in accuracy compared to the original method.     

COMPLETE PRESSURE CORRECTION ALGORITHM FOR SOLUTION OF INCOMPRESSIBLE NAVIER-STOKES EQUATIONS ON A NONSTAGGERED GRID

Abstract

When Navier-Stokes equations for incompressible flow are solved on a nonstaggered grid, the problem of checkerboard prediction of pressure is encountered. So far, this problem has been cured either by evaluating the cell face velocities by the momentum interpolated principle III or by evaluating an effective pressure gradient in the nodal momentum equations [2] In this article it is shown that not only are these practices unnecessary, they can lead to spurious results when the true pressure variation departs considerably from linearity. What is required instead is afresh derivation of the pressure correction equation appropriate for a nonstaggered grid. The pressure correction determined from this equation comprises two components: a mass-conserving component and a smoothing component. The former corresponds to the pressure correction predicted by a staggered grid procedure, whereas the latter simply accounts for the difference between the point value of the pressure and the cell-averaged value of the pressure. The new pressure correction equation facilitates ( in a significant way) computer coding of programs written for three-dimensional geometries employing body-fitted curvilinear coordinate grids.     

INCOMPRESSIBLE FLOW AND HEAT TRANSFER COMPUTATIONS USING A CONTINUOUS PRESSURE EQUATION AND NONSTAGGERED GRIDS

A pressure-based algorithm for incompressible flows is presented. The algorithm employs a finite-volume discretization in general curvilinear coordinates on a nonstaggered mesh. This approach is derived from a finite-element algorithm, and is here extended to the finite-volume/finite-difference context. The algorithm can be classified as a SIMPLE-like sequential method, and is validated in two classical test cases: the lid-driven cavity and the differentially heated cavity problems. Good results, with no pressure checkerboarding, are achieved up to Reynolds numbers Re = 10⁴ and Rayleigh numbers Ra = 10⁸ , respectively.     

USE OF GCG METHODS FOR THE EFFICIENT SOLUTION OF MATRIX PROBLEMS ARISING FROM THE FVM FORMULATION OF RADIATIVE TRANSFER

Preconditioned generalized conjugate gradient (GCG) iterative methods are applied to the solution of large, sparse, and unsymmetric linear algebraic equations resulting from the application of the finite-volume method to the problem of radiative heat transfer in an absorbing, emitting, and scattering gray medium, with the boundary surfaces reflecting radiation in both diffuse and specular regimes. The governing radiative transfer equation, which is a complicated integro-differential equation, has been discretized using the S N finite-volume method (FVM). Different variants of GCG methods have been tested on a problem of 2-D radiation in a cylinder, and efficiencies of the methods have been compared. Numerical results indicate that preconditioning suggested in the article dramatically improves the performance of the GCG methods. Results on test problems based on S 8 FVM agree well with exact results reported in the literature.     

A PROCEDURE FOR THE TREATMENT OF THE VELOCITY-PRESSURE COUPLING PROBLEM IN INCOMPRESSIBLE FLUID FLOW

A new block implicit procedure (BID) is introduced which utilizes a simple incomplete decomposition of the matrix resulting from the discretization of the momentum and mass conservation equations for incompressible fluid flow problems. In contrast to the conventional methods, the new method is not of the segregated type, and does not require an explicit equation for pressure. The complete, coupled block system is solved in its primitive form. In this way, mass and momentum conservation are satisfied simultaneously at all grid points, while pressure is calculated implicitly. Only a couple of overall iterations are required for the treatment of the nonlinearities of the problem. Tests show that the new procedure converges fast for any E value (in an E-factor formulation), and therefore virtually the E-factor formulation is not necessary.     

AN ACCURATE METHOD FOR THE SIMULATION OF THREE-DIMENSIONAL INCOMPRESSIBLE HEATED PIPE FLOW

A numerical scheme using Fourier expansions in the streamwise and azimuthal directions and Jacobi polynomials in the radial direction for the direct numerical simulation of three-dimensional incompressible pipe flow with heating is presented. The proposed basis and test functions for the thermal field in conjunction with those for the velocity field set forth by Leonard and Wary offer an accurate representation of flow variables in transitional flow. A simple test to a linear stability problem demonstrates that this method yields very accurate results with relatively few radial modes and is well suited for the simulation of nonisothermal pipe flow transition.     

简化保证出力计算的一个有效算法

梯级水电站群长序列保证出力优化计算模型是一个大规模ｍａｘｍｉｎ模型．本文针对梯级水电站群的特点给出一个行之有效的简化算法──有效计算期算法（ＭＭ—ＮＣ—Ａ）．计算结果表明该算法有效．     

QUASI-STATIC COUPLED PROBLEMS OF THERMOELASTICITY FOR CYLINDRICAL REGIONS

Abstract

The one-dimensional axisymmetric quasi-static coupled thermoelastic problem is investigated. The general solutions of its governing equations are obtained in the transform domain. The solutions in the real domain for the cases of an infinitely long solid cylinder and for an infinite medium with a cylindrical hole are also presented. The solution technique uses Laplace transform, and the inversion to the real domain is obtained by means of Cauchy's theorem of residues and the convolution theorem. Comparison with published results, when similar assumptions are made and the same boundary conditions are imposed, shows complete agreement.     

A COMPARISON OF TIME DISCRETIZATION SCHEMES FOR THE GREEN ELEMENT SOLUTION OF THE TRANSIENT HEAT CONDUCTION EQUATION

We present three time discretization schemes for the Green element solution of the linear conduction equation. Numerical results obtained from the three methods are assessed by their convergence and stability-related properties, namely, numerical amplitude and amplification factor through Fourier analysis. The main emphasis is the ability of any of the schemes to handle conduction problems typified by those properties that are known to be taxing to most numerical schemes. This was found in some cases to be related to how accurately the harmonics of the Fourier waves are propagated.     

SOLUTION OF DYNAMIC PROBLEMS FOR THERMOVISCOELASTIC PLATES BY THE FINITE-ELEMENT METHOD

A generalized variational principle for dynamic problems of thermoviscoelastic plates is formulated. The solution of the problem by the Laplace transform-finite-element method is then proposed, and properties of eigenvalues and the convergence of the method are analyzed.     

EFFICIENT AND ROBUST MATRIX SOLVER FOR THE PRESSURE CORRECTION EQUATIONS IN TWO- AND THREE-DIMENSIONAL FLUID FLOW PROBLEMS

When fluid flow problems are solved by the SIMPLE-like sequential procedures, most of the computing time is spent on handling the pressure-correction equation. As a result, the overall efficiency of those sequential procedures is strongly dependent on how good is the matrix solver for the pressure correction equation. In this article, a modified conjugate gradient solver (MCGS) is presented that is specially tuned to solve the pressure-correction equations arising from both two- and three-dimensional fluid flow problems. A new solver described here lakes the advantages of conjugate gradient solver (CGS) and strongly implicit procedure (SIP) but eliminates the disadvantages of each. In contrast to the SIP, the present MCGS adopts the partial cancellation of zeroth order and is insensitive to variation of the cancellation parameter. In general, MCGS can be two or three times faster than CGS but is an order of magnitude faster than SIP and block-corrected ADI in treating large-scale problems.     

NUMERICAL SIMULATION OF GAS FLOW AND MIXING IN A MICROCHANNEL USING THE DIRECT SIMULATION MONTE CARLO METHOD

The direct simulation Monte Carlo (DSMC) method was employed to investigate gas flow and mixing in a microchannel at near-atmospheric pressure conditions. Simulations for pressure-driven flows were first carried out for a single-component gas flow in a microchannel. Mixing of two parallel gas streams (H 2 and O 2 ), separated by a splitter plate and then entering a microchannel, was considered. The effects of the inlet velocities, the inlet-outlet pressure difference, and the pressure ratio of the incoming streams (H 2 and O 2 ) on the mixing behavior were considered. The effect of the "accommodation coefficient" of the solid wall of the microchannel on the mixing behavior was also examined. The simulation results indicate that mixing decreases with the increase of inlet-outlet pressure difference. When the two streams enter the microchannel with different inlet pressures, mixing is found to decrease with the increase of the pressure ratio. The mixing process is found to be much slower for nearly specularly reflected walls compared to the mixing in a microchannel with completely diffuse walls.     

BODY-FORCE ANALOGY FOR ONE-DIMENSIONAL COUPLED DYNAMIC PROBLEMS OF THERMOELASTICITY

ABSTRACT

The classical body-force analogy for static problems of thermoelasticity is extended toward dynamic coupled problems. We consider two dynamic problems, namely, a thermal problem without body forces, but with a given distribution of transient sources of heat, and a force problem without sources of heat but with body forces. Both problems are treated within the coupled theory of thermoelasticity such that temperature must also be taken into account in the force problem. We restrict our considerations to the one-dimensional case, and we show that, given suitable boundary and initial conditions, a distribution of body forces can be constructed such that the dynamic displacements in both problems become equal. This analogy is checked by means of illustrative analytical examples. We also discuss the relations between the stresses and the temperature in both problems, and we mention that a similar analogy can be established, requiring the temperatures in both problems to be equal.     

APPLICATION OF THE PARTIAL ELIMINATION ALGORITHM FOR SOLVING THE COUPLED ENERGY EQUATIONS IN POROUS MEDIA

This article discusses the solution of coupled energy equations in local thermal nonequilibrium models for porous media. The decoupled solution approach, in which the interaction between the solid and the fluid temperature fields is treated in an explicit manner, converges very slowly when the interface heat transfer coefficient and/or the specific surface area of the porous medium are large (large Biot number). An attractive alternative to the decoupled approach is the partial elimination algorithm, proposed by D. B. Spalding. In this algorithm, the discretization equations are rearranged so that the resulting equations are more implicit and take directly into account the coupling between the two phases. The convergence rates of these two solution procedures are studied with reference to convective heat transfer in a two-dimensional channel filled with a porous medium. The partial elimination algorithm converges much more quickly than the decoupled procedure, with the number of iterations required for convergence becoming constant for large Biot numbers.     

梯级水电站群保证出力优化计算的一个直接算法

梯级水电站群保证出力优化计算问题是一个有非线性约束的maxmin问题max{M_f(X)|X∈G={X|g_j(x)≤0,j=1,…,N,X∈R~n}),其中M_f(X)=min{f_1(X),…,f_K(X)}。本文在文[1]的基础上,提出一个求解该问题的直接算法。所给出的算法保证了迭代序列{X~((i))_(i=1)~∞的可行性,即有{X~((i))_(i=1)~∞(?)G。本文最后计算了一个梯级水电站群(包含5个水电站)的保证出力优化的实例,计算结果表明该算法是有效的。     

A CALCULATION PROCEDURE FOR SOLUTION OF INCOMPRESSIBLE NAVIER-STOKES EQUATIONS ON CURVILINEAR NON STAGGERED GRIDS

A recently proposed pressure-correction algorithm for solution of incompressible Navier-Stokes equations on nonstaggered grids introduced the notion of smoothing pressure correction to overcome the problem of checkerboard prediction of pressure (9). The algorithm was derived for equations in Cartesian coordinates. In this article, the algorithm is extended to solution of Navier-Stokes equations in general curvilinear coordinates. By way of application, two cavity flow problems and two internal flow problems are solved. Comparisons with benchmark solutions or experimental data and (or) previous solutions employing staggered grids are made to validate the calculation procedure.