Fluent的并行计算_fluent的并行计算
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Poibilities of Parallel Calculations in Solving Gas Dynamics Problems in the CFD Environment of FLUENT Software
N.Chebotarev Research Institute of Mathematics and Mechanics, Kazan State
University, Kazan, Ruia Received August 25, 2008 Abstract ：The results of studying an incompreible 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 Navier–Stokes equations using FLUENT software.The poibilities to accelerate the solution proce with the use of parallel calculations are investigated and the calculation results under changes of preure differential in the periodic element are given.DOI : 10.3103/S***3
Multiproceor computers that make it poible 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 progre 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 Navier–Stokes equations describing flows in the arbitrary regions contain poibilities of parallel proceing.The objective of this paper is to test the solutions of a three-dimensional problem of gas dynamics using FLUENT software [1] by a multiproceor computer in the mode of parallel proceing.An incompreible 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 poible to realize the procees 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 Navier–Stokes 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 incompreible 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 poible to neglect gas compreibility and adopt a model of incompreible fluid.The laminar flow of the incompreible gas is described by the stationary Navier–Stokes equations:
where u are the gas velocity vector and its Cartesian components;p is the preure;μ and
ρ 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 preure at the end element bounds is described by the formula
where Δp is the preure 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 MODE The calculations were carried out on the computational cluster of Kazan State University consisting of eight servers.Each server includes two AMD Opteron 224 proceors 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 Meage Paing Interface library(HP-MPI)delivered together with the FLUENT program is used.At the moment of experiments four servers were acceible for operation.To analyze the efficiency of parallel proceing of the numerical solutions of the Navier–Stokesequations 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 proceing mode is the division of the initial domain into subdomains.In this case, each computation unit, that is, a proceor is responsible for its subdomain.In dividing into subdomains FLUENT software uses the method of bisection, that is, when it is neceary 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 neceary 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 by one proceor to the calculation time tn by n proceors 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 proceors being used is given in Fig.3.If the number of proceors is small(n
Fig.3.Dependence of the acceleration factor on the number of proceors.For a more detailed analysis of the calculation time with the different number of proceors, we study the time t of data exchange between the proceors and the factor tn of unloaded state [8] that represents the share of exchange time in the total calculation time.The le is ku , the higher is the efficiency of paralleling.The time of data exchange is determined by a share of boundary finite volumes between subdomains in the total number of finite volumes.Tables 1 and 2 present a share of boundary cells between subdomains and the values of the factor of unloaded state.It is seen that as the number of proceors grows, the share of boundary volumes between subdomains increases resulting in the growth of the relative exchange time tu , that is, the factor of unloaded state.In this case, it is clear that the variants B and C are close to each other in both variants.In the variant A , the ratio of the exchange time to the calculation time increases much faster.The flow pattern in the element under study is mainly determined by the value of preure differential along the fluid flow.Denoting by v0 the average velocity in the inlet cro-section, we will introduce the Reynolds number Re:
At low preure differential(small Re)a symmetric flow that is periodic in all directions is formed.The typical vector field of velocities for this range is shown in Fig.4a.As the Reynolds number increases(preure differential grows), the inertia effects become apparent(Fig.4b).The inertia flow regime is characterized by the presence of complex three-dimensional vortex structures in the region between the spheres being streamlined.At low velocities the gas flow in the porous structure is described by the Darcy law:
where v is the filtration velocity(the volumetric air flowrate per unit of time through unit area in the porous medium);kd is the permeability factor.The Darcy law establishes proportionality of the filtration velocity to the preure gradient.In the inertia flow regime the Darcy law is violated, and to expre the dependence of the filtration velocity on the preure differential, the well-known Forhheimer formula is widely used [9]:
Fig.4.Vector field of velocities at the periodic element faces at x = 0(a, c)and x = d /2(b, d)for
Re = 0.05(a, b)and Re = 140(c, d).The permeability factor kd and the coefficient β are usually found by approximation of experimental results.At the same time the numerical solution of the equations for gas flow in the periodic element of the porous structure can be used for their definition.Let us write the equation for momentum variation when gas paes through the periodic element in the form [6]:
where n is the unit vector that is exterior with respect to the boundary element surfaceAf;τ is the vector of viscous strees.The boundary surface of the element consists of the boundary “fluid–fluid”Aff，the boundary “fluid–solid” Afs：Af=Aff+Afs
Let us rewrite Eq.(8)with regard for symmetry in the form:
where A is the area of the inlet cro-section in the periodic element.Taking into account that in connection with flow periodicity the last integral is zero, we will write Eq.(9)in the dimensionle form:
where we introduced the dimensionle parameter λ= d2Δp/ρvL.The gas velocity distribution in the porous element found from the solution of the Navier–Stokes equations makes it poible to calculate the forces acting on the spherical surfaces in the element limits, that is, the integrals in the right-hand part of Eq.(10).The dimensionle force of resistance λRe includes the resistance that is due to the normal fp and viscous f τ strees on the spheres.Figure 5 presents the dependence of Re λ on the Reynolds number obtained from the calculations of the gas flow field at different values of preure differential(calculated points), and the curve found by linear approximation of the calculated values Re λ at Re 4 >.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=6.1 × 10-4.Fig.5.Dependence of the dimensionle resistance force on the Reynolds number.Thus, using FLUENT software in the parallel calculations mode by a multi-proceor computer, we solved a problem of the incompreible gas flow in the periodic element of the porous structure made up of the closely-arranged spheres.It is shown that the calculation time is reduced as the number of proceors grows, and the efficiency of paralleling is higher with a larger number of finite volumes in dividing the calculation domain.The results of studying a flow field under variation of the preure differential in the periodic element are presented.When the preure differential increases(the large Reynolds numbers), the inertia flow regime that is characterized by a complicated pattern of vortex structures is formed in the porous element.
