Abstract
An axisymmetric supersonic flow of rarefied gas past a finite cylinder was calculated applying the direct simulation Monte Carlo method. The drag force, the coefficients of pressure, of skin friction, and of heat transfer, the fields of density, of temperature, and of velocity were calculated as function of the Reynolds number for a fixed Mach number. The variation of the Reynolds number is related to the variation of the Knudsen number, which characterizes the gas rarefaction. The present results show that all quantities in the transition regime (Knudsen number is about the unity) are significantly different from those in the hydrodynamic regime, when the Knudsen number is small.
Rarefied gas; knudsen number; distribution function; external flow
Direct Simulation Monte Carlo Method Applied to Aerothermodynamics
Felix Sharipov
Departamento de Física
Universidade Federal do ParanáCaixa Postal 19044
81531-990 Curitiba. Brazil
http://fisica.ufpr.br/sharipov
An axisymmetric supersonic flow of rarefied gas past a finite cylinder was calculated applying the direct simulation Monte Carlo method. The drag force, the coefficients of pressure, of skin friction, and of heat transfer, the fields of density, of temperature, and of velocity were calculated as function of the Reynolds number for a fixed Mach number. The variation of the Reynolds number is related to the variation of the Knudsen number, which characterizes the gas rarefaction. The present results show that all quantities in the transition regime (Knudsen number is about the unity) are significantly different from those in the hydrodynamic regime, when the Knudsen number is small.
Keywords: Rarefied gas, knudsen number, distribution function, external flow
Introduction
Nowadays, the practical necessity of numerical calculations of rarefied gas flows is very high. The fact is that, when the characteristic size of the gas flow has the order of the molecular mean free path one cannot apply the continuum mechanics equations, but one has to apply the numerical methods of rarefied gas dynamics (RGD) (Cercignani, 1988), (Bird, 1996), (Sharipov, 1998).
One of the many fields of application of the RGD is the aerothermodynamics of satellite re-entering into the atmosphere (Kogan, 1992), (Ivanov and Gimelshein, 1998). To calculate the trajectory of satellite it is necessary to know the drag force in all points of its trajectory beginning from the high atmospheric stratum. At the same time, it is necessary to know the heat flux on the satellite surface to create an adequate thermal protection. In the high atmosphere stratum the molecular mean free path has the order of several meters. That is why the drag force and the heat flux can be calculated only by the methods of the RGD. Even in the lower atmosphere stratum, some local rarefied regions can exist near sharp leading edges of space vehicle moving with a hypersonic velocity. The flowfield in these regions must be also calculated via the RGD.
Vacuum equipment is one more field where the RGD is applied. The maintenance of high vacuum in large reservoirs has a great scientific and practical interest. To optimize the equipment one needs to know exact values of the mass flow rate through various elements connecting the reservoirs to be evacuated and the pumps. At the gas pressure 10-3 Torr the mean free path has the order of few centimetres. So, under this condition the mass flow rate can be calculated via the RGD. Moreover, in this regime of the flow, some phenomena taking place only in rarefied gases can be used to pump the gas.
Other fields of application of the RGD are: microelectromechanical systems (MEMS), low pressure plasma reactor, vapour deposition reactor, separation of gases, laser industry etc.
The gas rarefaction is characterized by the Knudsen number (Kn) defined as a ratio of the mean free path to a characteristic size of the gas flow. If the Knudsen number is large (Kn® ¥) the gas rarefaction is so large that the intermolecular collisions can be neglected. It is the so-called free molecular (collisionless) regime of flow. All gasdynamic problems are easily solved in this regime and in many cases an analytical solution can be obtained. The opposite limit (Kn® 0) is called as hydrodynamic or continuous regime. In this limit both RGD methods and hydrodynamic equations are applicable. If the Knudsen number has the order of the unity we have the so-called transition regime of flow. In this case the intermolecular collisions must be taken into account but the hydrodynamic equations are not valid.
At any Knudsen number a rarefied gas flow can be calculated on the basis of the distribution function f (t,r,v), which provides all macroscopic characteristics such as: number density
hydrodynamic (bulk) velocity
pressure
stress tensor
temperature
heat flow vector
where t is the time, r is the position vector, v is the molecular velocity, m is the mass of gaseous particle, k is the Boltzmann constant and V is the peculiar velocity
The distribution function can be obtained from the Boltzmann equation (Cercignani, 1988), (Bird, 1996), which in the absence of external forces reads as
Here, the affixes to f correspond to those of their arguments v : f' = f(t,r,v'), f* = f(t,r,v*). The quantity w(v,v*;v',v'*) is the probability density that two molecules having the velocities v' and v'* will have the velocities v and v*, respectively, after a binary collision between them.
Till now a numerical solution of the Boltzmann equation is a very difficult task because of its high complexity. That is why many problems are solved using simplified kinetic equations, i.e. the model equations (Sharipov and Seleznev, 1998). Another alternative to the Boltzmann equations is the Direct Simulation Monte Carlo (DSMC) method (Bird, 1996), which becomes a powerful tool in many engineering fields.
The aim of the present paper is to demonstrate the capacities of the DSMC method applying it to a supersonic gas flow past an axisymmetric body. The problem is solved in all regimes of the gas flow: free molecular, transition and hydrodynamic. A comparison of the results corresponding to the different regimes will show the influence of the gas rarefaction on the aerothermodynamic characteristics.
Description of the Method
The region of the gas flow is divided into a network of cells. The dimensions of the cells must be such that the change in flow properties across each cell would be small. The time is advanced in discrete steps of magnitude Dt, such that Dt is small compared with the mean time between two successive collisions.
The molecular motion and intermolecular collision are uncoupled over the small time interval Dt by the repetition of the following procedure:
(i)The molecules are moved through the distance determined by their velocities v and Dt. So the new coordinates of every particle is calculated via the old ones
If the trajectory crosses a solid surface a simulation of the gas-surface interaction is performed according to a given law. In the present work the diffuse reflection is assumed, i.e. a particle is emitted by the surface in any direction with the same probability. In this stage the difference of momentum and kinetic energy of the particle before and after the collision with the surface is calculated. Then this information is used to calculate the drag force, the pressure coefficient, the skin friction coefficient, and the heat transfer coefficient at any surface point. New molecules are generated at boundaries of the computational region in accordance with the conditions at the infinity.
(ii) A number of pairs to be selected for collision is calculated as
where Np is the number molecules in a cell during the last time interval Dt, 
is the average value of Np during all previous intervals Dt, FN is the number of real gaseous particles represented by one model particle, cmax is a maximum relative velocity between two particles, VC is the cell volume, and s is the diameter of particles. So, we assume the hard sphere model of molecules. Every selected pair is accepted for collision if c/cmax > Rf, where Rf is a random number varying from 0 to 1 and c is the relative velocity for the given pair. If the pair is accepted then the pre-collision velocities of the particles are replaced by the post-collision values in accordance with the elastic collision law.
After a sufficient number of the repetitions we may calculate any macroscopic characteristics defined by (1)-(6). The characteristics of our interest are as follows: number density
bulk velocity
and temperature
The DSMC has many advantages. Since the simulations of the particle motion, the intermolecular interaction and the gas-surface interaction are separated, it is easy to modify the form of the gas flow region, the potential of intermolecular interaction, the gas-surface interaction law. The numerical program can be easily generalized to a gaseous mixture of polyatomic gases taking into account chemical reactions, evaporation and condensation of the gas, absorption of the gas on the surface. So, this allows us to apply the DSMC method in many engineering problems. Below, one of them is considered.
Statement of the Problem
Consider a cylinder with a length L and a radius R. Let the origin of Cartesian coordinates be placed at the cylinder center and the cylinder axis be directed along the x axis as is shown in Fig. 1. In the cylinder surroundings there is a gas having a pressure P0, a temperature T0 and a bulk velocity U0 far way from the cylinder. We assume that the gas velocity U0 is directed along the x axis. We are going to calculate the following characteristics: (i) The fields of the density, temperature and bulk velocity of the gas around of the cylinder; (ii) The pressure coefficient defined as
where Pd is the dynamic pressure. (iii) The skin friction coefficient given as
where t is the tangential stress on the wall; (iv). The heat transfer coefficient defined as
where qn is the normal heat flux on the cylinder surface; (v) and the drag force coefficient defined as
where F is the drag force acting on the cylinder.
The problem solution is determined by the state of the gas at the infinity, i.e. by the quantities P0, T0 and U0 , and by the transport coefficients such as: the stress viscosity m and the thermal conductivity k. Here, the results will be presented in term of the dimensionless numbers: Mach number defined as
where cp and cv are the specific heats; Reynolds number defined as
Knudsen number defined as
where l0 is the mean free path at the pressure P0 and temperature T0; and Prandtl number defined as
Further, we assume that the gas is monatomic but the method presented here can be easily generalized for any polyatomic gas. Thus, for a monatomic gas we have the following relations:
Applying the Chapman-Enskog method (Ferziger and Kaper, 1972) to the Boltzmann equations (8) for hard sphere molecules, the viscosity m and the conductivity k can be obtained as
So, the Prandtl number for the gas under consideration is equal to 2/3. Using Eq.(23) the Knudsen number can be related to the Mach and Reynolds numbers as
Thus, we have only two independent numbers among Kn, Ma and Re. In papers related to the RGD the Knudsen and Mach numbers are usually used for the result presentation. Here, we choose the Mach and Reynolds numbers to facilitate a comparison of the present results with a solution based on the Navier-Stokes equation.
Free Molecular Solution
In the free molecular regime flow, i.e. Re® 0 (or Kn® ¥) the problem can be solved analytically. The distribution function of the incident particles are given as
where n0 = P0 / kT0. The distribution function of the reflected particles can be calculated if the gas-surface interaction is given. Assuming the diffuse reflection we obtain
where nw depends on the position of the surface: on the front surface (x = - L/2) we have
where u is related to the Mach number as
On the back surface (x = L/2) nwreads
and on the lateral surface nw is equal to. Thus, substituting the distribution functions (25) and (26) into the definitions (1)-(6) any macroscopic quantity can be calculated. Then we may calculate the coefficients defined by (14)-(17).
On the front surface we have
The coefficient Cfis equal to zero. On the back surface the quantities have the same expressions with the opposite sing of u.
On the lateral surface the coefficients read
The coefficient Cp is equal to zero.
Then, the drag coefficient CD is calculated as
These expressions are valid for any Mach number and any length-to-radius ratio L/R.
Numerical Results and Discussion
The numerical calculations were carried out for the value of the length-to-radius ratio equal to L/R = 1, for the value the Mach number equal to 3 and for three values of the Reynolds number: 3, 30 and 300. These values of Re correspond to the Knudsen number equal to: 1.618, 0.1618 and 0.01618, respectively. So, the first value (Re=3) corresponds to the transition regime of the flow when the mean free path has the same order as the cylinder radius. The second value (Re=30) corresponds to the near hydrodynamic regime flow, which can be calculated via the Navier-Stokes equation with the slip and jump boundary conditions (Sharipov and Seleznev, 1998). The third value corresponds to the hydrodynamic regime, which also can be calculated via the hydrodynamic equations with the usual boundary conditions, i.e. non-slip of the velocity and continuity of the temperature on solid surfaces.
The number of the model particles in the calculation was about 106. The time increment Dt was 0.01 of the mean free time. The size of computational region had the form of cylinder with the radius equal to 4R and the length equal to 8R, i.e. the computational region stretched as 4R £ x £ 4R and 0 £ r £ 4R, where r = 
. The cells was square in the plane xy. Their size was R/15 at Re=300 and R/10 at Re=3 and Re=30. These sizes of the cells and of the computational region provide the accuracy 1% for the drag coefficient. For the local characteristics such as Cp, Cf, Chthe numerical error is larger because of the numerical fluctuations. An analysis of the data for different numerical grids and different numbers of the model particles showed that the accuracy of the local characteristics is about 5%. The precision can be improved increasing the number of the iterations.
The distributions of the number density, of the temperature and of the local Mach number around of the cylinder are given in Figs. 2, 3 and 4, respectively. The local Mach number was calculated as
where U and T are the local bulk velocity and temperature of the gas, respectively. One can see that there is a significant change of the distributions by increasing the Reynolds number (or decreasing the Knudsen number). In the hydrodynamic regime (Re=300) the gas forms a shock wave in front of the cylinder. Inside the interval Dx=0.1R the reduced density n/n0 jumps from 1 to 2.8, the reduced temperature T/T0 increases from 1 to 3.6 and the Mach number drops from 3 to 0.7. In the transition (Re=3) and near hydrodynamic (Re=30) regimes all characteristics vary smoothly around the cylinder.
The pressure coefficient Cp is presented in Fig. 5. One can see that on the front surface of the cylinder the coefficient is practically uniform. It decreases by the increasing Reynolds number. On the back surface the coefficient Cp is negative and its value is significantly smaller then that on the front surface. The coefficient Cp on the lateral surface is also very small. It can be both positive and negative.
The skin friction coefficient Cf is given in Fig. 6. Its value on the front surface practically does not depend on the Reynolds number, while there is a significant dependence of Cf on Re on the lateral surface. The coefficient Cf decreases by increasing Re. The value of Cf on the back surface is very small and it has the order of the numerical fluctuation. That is why these data are not given here.
The heat transfer coefficient Ch is presented in Fig. 7. It significantly depends on the Reynolds number. Its value on the lateral surface is significantly smaller then that on the front surface. On the back surface the coefficient Ch has the order of the numerical fluctuation and is not given here.
The values of the drag coefficient CD are given in Table 1. It can be seen that this coefficient decreases by increasing the Reynolds number. Note, that we consider a fixed Mach number.
Conclusions
Thus, the rarefied gas flow past a cylinder was calculated applying the direct simulation Monte Carlo method. The fields of the density, temperature and local Mach number were presented for different values of the Reynolds number, which correspond to three regimes of the gas flow: transition, near hydrodynamic and hydrodynamic. The coefficients of pressure, of skin friction and of heat transfer were calculated at all points of the cylinder. The drag coefficient was also calculated as function of the Reynolds number. The present data show that there is a significant difference between all calculated quantities in the hydrodynamic and transition regimes of the flow.
The numerical program used for the present calculations will be modified to calculate aerothermodynamic of the satellite to be launched in the frame of the program SARA (Satélite de Reentrada Atmosférica) (Moraes and Pilchowski, 1997). Since the satellite moves in the air the numerical program will be generalized for a gaseous mixture of the diatomic gases. Moreover, the phenomena occurring at high Mach numbers, such as dissociation, recombination, chemical reaction, will be taking into account. Finally, we are going to elaborate a program for an arbitrary form of the object moving in the gas. In the future, this program can be used as a numerical wind tunnel.
Acknowledgement
The results presented in this paper were obtained in the course of research sponsored by Agência Espacial Brasileira in the frame of the project "Uniespaço". The author thanks the Conselho Nacional de Desenvolvimento Científico e Tecnológico (CNPq, Brazil) for support of his work.
Article received: February, 2001. Technical Editor: Angela Ourivio Nieckele.
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- Ivanov, M. S., and Gimelshein, S. F., 1998, Computational hypersonic rarefied flows: Annu. Rev. Fluid Mech., 30, 469-505.
- Kogan, M. N., 1992, Kinetic theory in aerothermodynamics: Prog. Aerospace Sci., 29, 271-354.
- Moraes Jr., P., and Pilchowski, H-U., 1997, Plataforma orbital recuperável para experimentação em ambiente de microgravidade: XIV Brazilian Congress of Mechanical Engineering, Bauru-SP (Brasil).
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Publication Dates
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Publication in this collection
24 Sept 2002 -
Date of issue
2001
History
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Received
Feb 2001
