Fluent的并行计算.doc

Possibilities of Parallel Calculations in Solving Gas DynamicsProblems in the CFD Environment of FLUENT SoftwareN. Chebotarev Research Institute of Mathematics and Mechanics, Kazan State University, Kazan, RussiaReceived August 25, 2008Abstract ：The results of studying an incompressible gas flow field in a periodic element of the porous structure made up of the same radius spheres are presented; the studies were based on the solution of the NavierStokes equations using FLUENT software. The possibilities to accelerate the solution process with the use of parallel calculations are investigated and the calculation results under changes of pressure differential in the periodic element are given. DOI : 10.3103/S1068799809010103 Multiprocessor computers that make it possible to realize the parallel calculation algorithms are in recent use for scientific and engineering calculations. One of the fields in which application of parallel calculations must facilitate a considerable progress is the solution of three-dimensional problems of fluid mechanics. Many investigators use standard commercial CFD software that provide fast and convenient solutions of three-dimensional problems in complicated fields. The present-day CFD packages intended to solve the NavierStokes equations describing flows in the arbitrary regions contain possibilities of parallel processing. The objective of this paper is to test the solutions of a three-dimensional problem of gas dynamics using FLUENT software 1 by a multiprocessor computer in the mode of parallel processing. An incompressible gas flow in the porous structure made up of closely-arranged spheres is calculated. Structures of different sphere arrangement are widely used as models of porous media in the theory of filtration. Using the porous elements it is possible to realize the processes of filtration, phase separation, throttling, including those in aircraft engineering 2. The hydrodynamic flows in porous structures in the domain of small Reynolds numbers are described, as a rule, under the Stokes approximation without regard for the inertia terms in the equations of fluid motion 3. At the same time, the flow velocities in the porous media may be rather large, and the Stokes approximation will not describe a real flow pattern. In this case the solution of the complete NavierStokes equations should be invoked. The flow with regard for inertia terms in the equations of fluid motion in the structures of different sphere arrangement was theoretically studied in 6 and experimentally in 7. PROBLEM STATEMENT We consider an incompressible gas flow in the three-dimensional periodic element of the porous structure made up of closely-arranged spheres of the same diameter d , the centers of which are in the nodes of the ordered grid (Fig. 1a). The porosity of the structure under consideration determined as the ratio of the space occupied by the medium to the total volume is equal to 0.26. Taking into account symmetry and periodicity of the flow, we will separate in the space between spheres the least element of the region occupied by air (Fig. 1b). In connection with a difficulty in dividing the calculation domain, small cylindrical areas are excluded in the vicinity of points at which the element spheres are in contact. Fig. 1. Scheme of sphere arrangement (a) and a periodic element in the air space between spheres (b).The gas flow velocities inside the porous structures are so small that it is possible to neglect gas compressibility and adopt a model of incompressible fluid. The laminar flow of the incompressible gas is described by the stationary NavierStokes equations: where u are the gas velocity vector and its Cartesian components; p is the pressure; and are the dynamic viscosity coefficient and air density. At the end bounds of the periodic element we lay down the conditions of periodicity where L = d is the periodic element length along the flow (along the y axis); at the lateral faces we lay down the conditions of symmetry. The pressure at the end element bounds is described by the formula where p is the pressure differential in the element limits. The conditions of symmetry are taken not only at the lateral faces but also at the upper and lower faces. On the spherical surfaces the conditions of adhesion are specified. System of equations (1)(2) is solved with the aid of the SIMPLE algorithm in the finite volume method in FLUENT software environment (FLUENT 6.3.26 version). For the calculation domain the irregular tetrahedral grid division is used (Fig. 2). Fig. 2. Division of the periodic element into finite volumes.ANALYSIS OF CALCULATION RESULTS IN THE PARALLEL MODEThe calculations were carried out on the computational cluster of Kazan State University consisting of eight servers. Each server includes two AMD Opteron 224 processors with the clock rate 1.6 GHz and 2 GB of main memory. The servers operate under the control of the Ubuntu 7.10 version of the Linux operating system. The communication between the servers is based on the Gigabit Enternet technology. In the calculations the HP Message Passing Interface library (HP-MPI) delivered together with the FLUENT program is used. At the moment of experiments four servers were accessible for operation. To analyze the efficiency of parallel processing of the numerical solutions of the NavierStokesequations in the FLUENT environment, the calculations were performed with three variants of the grid division of the solution domain 116895, 307946, and 510889 finite volumes (variants A , B , C ). The number of iterations in all cases was taken to be equal to 760 resulting in solution convergence to10. All computation experiments were performed in the package mode. One of the basic moments in solving problem with the use of FLUENT software in the parallel processing mode is the division of the initial domain into subdomains. In this case, each computation unit, that is, a processor is responsible for its subdomain. In dividing into subdomains FLUENT software uses the method of bisection, that is, when it is necessary to divide into four subdomains, the initial domain is first divided into two and then recursively the daughter subdomains are divided into two. If it is necessary to divide into three subdomains, the initial domain is divided into two subdomains so that one subdomain is twice as large as the other and then the larger subdomain is divided into two. FLUENT software incorporates several algorithms of bisection, and the efficiency of each algorithm depends on the problem geometry. We studied the acceleration factor that is determined as the ratio of the calculation time t 1 by one processor to the calculation time tn by n processors ka .To provide the experiment purity, the acceleration factor was calculated four times for each case, and the calculation time obtained in each case was somewhat different. The minimal time for four calculations was chosen for the analysis. For the variants A , B , C without paralleling it amounted to 747, 2234, and 3600 s, respectively. The dependence of ka on the number of n processors being used is given in Fig. 3. If the number of processors is small ( n . The linear approximation obtained corresponds to the Forchheimer formula and is in a good agreement with the calculated data thus confirming the feasibility of the formula mentioned for description of flow characteristics in the porous media at the large Reynolds numbers. Using formulas (7) and (10), we will obtain from the calculated data the permeability factor kd/d2=