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An open source Direct Simulation Monte Carlo (DSMC) code, called as dsmcFoam in OpenFOAM, is used to study a blunt body with the shape of a space crew capsule return vehicle. The rarefied gas has the Knudsen number with 0.03. The flow with a Mach number 4.35 over the capsule was simulated by DSMC. The distributions of velocity field and temperature around the capsule were calculated. This study may provide some useful information for the reentry of the return vehicle.

When the blunt body space capsules turn back to the Earth, the reentry process will undergo the process from rarefied to continuum regimes. The atmosphere is rarefied gas when the altitude is above 80 km from the sea level on the Earth.

Normally, Knudsen number, defined in Equation (1), is employed to describe the degree of gas rarefaction.

where is the mean free path of the gas molecules; L is the characteristic dimension of the flow.

A rarefied gas may be divided into several different flow regimes according to its level of rarefaction as quantified by Kn. When Kn is greater than 0.01, the gas is sufficiently rarefied. When Kn is greater than 10, free molecule flow occurs [1,2].

In fluid flows, when the continuum fluid approximation is contented, Navier-Stokes CFD methods [3-5] can be employed to performance the simulations. However, when the continuum fluid approximation is broken down, the constitutive equations that describe the relationships of shear stress and heat transfer with other variables are untenable. Furthermore, the linear transport terms of mass, diffusion, viscosity and thermal conductivity in the differential equations are no longer valid [

The basis of the DSMC method, first formulated by Bird, is to numerically solve the Boltzmann equation. Since Graeme Bird in 1961 [

In this paper, the DSMC, which is based on OpenFOAM, is used to do the studied of a blunt body with the shape of a capsule crew return vehicle. The simulation is under supersonic conditions with a Mach number 4.35.

Direct Simulation Monte Carlo (DSMC) is a physical simulation method to solve the Boltzmann equation. During the simulation, Monte Carlo method is used to represent molecular collisions and statistics of the random samples [3-9].

In Boltzmann equation, the number of molecules, dN, will be determined by the phase space of the system, which is the function of position r (x, y, z coordinate) and velocity along the x, y, z coordinate respectively.

Normally Boltzmann equation is a traditional mathematical option for analyzing gas flows in microscopic level. In classical mechanics the entire flow can be successfully described in terms of position, velocity and internal state of each and every molecule at a particular instant. It can be described as the following equation [

where n is number density defined as, f is velocity distribution function, c is velocity, is post collision velocity distribution function. c_{i} and f_{i} is velocity and velocity distribution function of another molecules, c_{r} is relative velocity and is the differential of collision cross section.

From Boltzmann Equation (3), it can be seen that to solve Boltzmann equation is not easy. The right hand side of Equation (3) is a form of integral-differential and seven variables needs to be solved simultaneously [

However following the advantages of computer technology, the molecular models can be employed do the direct numerical simulation (DNS) directly from the physical properties of the gases. DSMC is one kind of these methods which employs probabilistic procedures for analyzing gas flows [

Based on the statistic analysis, a physical parameter of number density, which is called number of equivalent particles defined as the number of particles in sample space per unit volume, is introduced to represent the characters of gas particles in real space through a single particle in sample space. Based on the statistic analysis, an assumption can be made that, in this sample space, initially the particles are distributed randomly. Therefore it can be deduced that Boltzmann equation will have only one dependent variable that is the particle distribution function. The remaining task is to solve the collision term. Collisions can be solved through a probability basis of collision between two particles proportioned by the product of c_{r}_{ }and σ [3,6].

Due to the average value of the probability should be calculated for each cell, the cells or grids have to be used in the sample space to perform DSMC [3-9]. It is the reason why DSMC will mesh the simulation domain.

The DSMC code in OpenFOAM version 1.7.1 [

The dsmcFoam solver can deal with the steady and transient simulations with 2D and 3D geometries. The variable hard sphere (VHS) collision model and LarsenBorgnakke internal energy redistribution model [

When the simulations are starting, dsmcInitialise is firstly called to create initial configurations of DSMC particles in the flow regions. This initialization can be run as a parallel pre-processing. It is very useful for the simulations on the arbitrary geometries. After the initialization, dsmcFoam, the DSMC solver, is employed to do the calculations. If the initialization is performed as the parallel, the solver must be carried out with the parallel running. When the calculation is finished, the flow fields of velocities and the distributions of temperature are post-processed automatically for a serial running. If the computing is parallel, a reconstructPar command should be called to collect the results among the processors. The dsmcFieldsCalc can be called to calculate the intensive fields (density, velocity and temperature) from extensive fields (mass, momentum and energy) from the collected results.

Several benchmark validation cases, including 1D relaxation to equilibrium, 2D and 3D hypersonic flows, were studies by Scanlon etc. [

The geometry model of the blunt body space capsule is used same as the Crew Exploration Vehicle (CEV) [

The hexahedral mesh is generated around the space capsule. The total mesh cells are 80,000.

The case considers the flow as a mixture of Oxygen and Nitrogen at a total number density of 1 × 1020 atoms/m^{3}. The Mole fractions are of Oxygen 0.223 and Nitrogen 0.777 respectively. The flow has a temperature of 143.3 K at a velocity of 1191.9 m/s over the spherical segment heat shield. The situation can be used to mimic the reentry of the return vehicle from out space.

The radius of the shield is 0.2 m so that the Knudsen number based on the diameter of the spherical segment heat shield is 0.03,061. The Mach number of the flow is 4.35 and the surface temperature of the space capsule is 550 K. The Reynolds number of the flow is 165.62. The time-step was fixed as 1.0 × 10^{-}^{6} s. The steady-state can be reached after 72 hours running in parallel with 4 processors (Intel 2.83 GHz).

From the results, it can be seen that when the rarefied gas

flows over the space capsule, the flow fields will affect the temperature distributions. The high temperature region locates at the area of low Mach numbers, i.e. these two regions are overlapped.

From the simulations, it can be seen dsmcFoam code is able to simulate the complex 3D rarefied gas flows. In the future, more complex situations, such as different angles of attack, the effects of different Knudsen numbers and the chemical reaction for hypersonic flows, will be considered.

Supports provided by the National Natural Science Foundation of China under Grant No. 51276130, by the fundamental Research Funds for the central Universities (2012) and by the Ph.D. Programs Foundation of Ministry of Education of China (No. 20120072110037) are gratefully acknowledged.