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Proceedings ArticleDOI

A collective collision operator for DSMC

19 Feb 2002-Vol. 585, Iss: 1, pp 401-407

AbstractA new scheme to simulate elastic collisions in particle simulation codes is presented. The new scheme aims at simulating the collisions in the highly collisional regime, in which particle simulation techniques typically become computationally expensive. The new scheme is based on the concept of a grid-based collision field. According to this scheme, the particles perform a single collision with the background grid during a time step. The properties of the background field are calculated from the moments of the distribution function accumulated on the grid. The collision operator is based on the Langevin equation. Based on comparisons with other methods, it is found that the Langevin method overestimates the collision frequency for dilute gases.

Topics: Coulomb collision (62%), Collision frequency (60%), Collision (56%), Langevin equation (56%), Elastic collision (53%)

Summary (2 min read)

INTRODUCTION

  • Particle simulation methods and in particular the Direct Simulation Monte Carlo method have proved very successful in simulating complicated physical phenomena in a raretied environment.
  • Many problems of practical interest involve dense flows that make DSMC simulation computationally very: demanding, if not impossible.
  • Even in the most favorable problems, particle simulation methods have the disadvantage of being computationally expensive.
  • These collisions keep reproducing the equilibrium distributions without adding any new information to the flow.
  • In a previous paper by Gallis and Torczynski (1), the BGK equation ( 2) and Cercignani's extension (3) w'ere examined as possible collision operators.

DISCLAIMER

  • Portions of this document may be illegible in electronic image products.
  • Images are produced from the best available original document.
  • For the highly collisional regime, and when used with the Chapman-Enskog expansion, it leads to the Navier-Stokes equations.
  • The term rrv~represents the collisions replenis.tig the equilibrium distribution f., and the term mfrepresents the collisions depleting the distributionT he physical interpretation of the assumption that the replenishing number of collisions is given by nvfo is that post-collision molecules are selected from a -elli~distribution at the local average velocity and temperature.
  • Several modifications to the BGK equation have been proposed at the expense of the simplicity of the model.

I LANGEVIN'S EQUATION

  • Langevin proposed a method for Brownian motion to model collisions between particles (4).
  • It can be shown that the Langevin equation forms the basis for deriving all the physically significant relations concerning the motion of Brownian particles.
  • In the DSMC method a fraction of the particles modi@ their properties signtilcantly in every time step while the rest-remain unchanged, but in the Langevin model every particle in a cell receives a small modification of its velocity in every time step.
  • The modification constitutes a small change in their properties.
  • In cases where the temperature of I'(1 -e ) collision frequency is very low, the probabilistic contribution O adds a vev small vector to the particle velocity.

APPLICATION TEST CASE

  • The flow travels from Iefl to right and the sides of the wedge are assumed to be specularly reflecting.
  • Figure 2(a-c ) in the same fashion presents the corresponding translational temperature contour lines.
  • At .x = Y = O,the location of the leading edge of the wedge, the shock layer forms, A quantitative comparison indicates that all three simulations are in agreement about the downstream conditions and the shock angle.
  • Comparing the figures in more detail, it is seen that the Langevin method predicts a very thin shock compared to the DSMC and BGKC solutions, which are seen to be in remarkable agreement throughout the domain.
  • An attempt was made to circumvent this problem by artificially decreasing the collision frequency.

CONCLUSIONS

  • From this comparison, it is clear that the Langevin approach overestimates the coilisiomlity of the flow with respect to the DSMC and BGKC methods.
  • As mentioned earlier, the paradigm of the Langevin approach is quite different from that of the Boltzmann equation.
  • The Langevin approach maybe suitable for very high densities and charged flows, in which the collisions are so frequent that they can be replaced by a Brownian force acting on the particles.
  • Its application to dilute gases does not appear to be appropriate.

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Content maybe subject to copyright    Report

A Collective Collision
Operator
for
Michael A. Gallis and John R. Torczynski
Engineering Sciences Center
Sandia National Laboratories”
Albuquerque New Mexico 87111-0826
USA
Abstract. A new schemeto simulateelasticcollisionsin particlesimulationcodesis presented.The newschemeaimsat
simulatingthe collisions in the highly collisional regime, in which particle simulationtechniquestypically become
computationrdlyexpensive.The new schemeis based on the conceptof a grid-basedcollision field.Accordingto this
scheme, the particles perform a single coltision with the backgroundgrid during a time step. The properties of the
backgroundfield are calculatedfrom the momentsof the distributionfimctionaccumulatedon the grid. The collision
operator is based on the Langevinequation.Based on comparisonswith other methods, it is found that the Langevin
methodoverestimatesthe collisiontkquency for dilutegases.
INTRODUCTION
Particle simulation methods and in particular the Direct Simulation Monte Carlo method (DSLMC)have proved
very successful in simulating complicated physical phenomena in a raretied environment. However. many problems
of practical interest involve dense flows that make DSMC simulation computationally very: demanding, if not
impossible.
Even in the most favorable problems, particle simulation methods have the disadvantage of being
computationally expensive. The result is that the enormous potential of these methods in physical flow modeling has
not been exploited by industry. Although massively parallel computers have contributed significantly in speeding up
particle simulation codes, three-dimensional applications remain out of the scope of most modem computer
platforms at present.
The most important limitation in modeling the highdensity regime comes from the large number of collisions
that particles undergo, which adds a very significant overhead to the calculation. In the case of non-reacting flows.
these collisions keep reproducing the equilibrium distributions without adding any new information to the flow.
In a previous paper by Gallis and Torczynski (1), the BGK equation (2) and Cercignani’s extension (3) w’ere
examined as possible collision operators. In this paper, a method based on the work of Langevin (-!) to model the
collisions in a collective manner is examined and compared with the previous methods.
MODELS FOR THE COLLISION TERM OF THE BOLTZMANN EQUATION
The complexity of the nonlinear structure of the collision integml of the Boltzmarm equation has
encouraged the development of simpltiled models that retain most of the properties of the Boltzmann equation but
are not derived directly from it. The BGK method uses a simplified collision operator that takes the form:
H
~(nf )
=
n v(fO f)
(1)
colliszon
*Sandia is a multiprogram laboratory operated by the Ssmdia Coloration, a Lockheed Martin Company, for the United States
Department of Energy under Contract DE-AC04-94AL85000.

DISCLAIMER
This report was prepared as an account of work sponsored
by an agency of the United States Government. Neither
the United States Government nor any agency thereof, nor
any of their employees, make any warranty, express or
implied, or assumes any legal liability or responsibility for
the accuracy, completeness, or usefulness of any
information, apparatus, product, or process disclosed, or
represents that its use would not infringe privately owned
rights.
Reference herein to any specific commercial
product, process, or service by trade name, trademark,
manufacturer, or otherwise does not necessarily constitute
or imply its endorsement, recommendation, or favoring by
the United States Government or any agency thereof. The
views and opinions of authors expressed herein do not
necessarily state or reflect those of the United States
Government or any agency thereof.

DISCLAIMER
Portions of this document may be illegible
in electronic image products. Images are
produced from the best available original
document.

. .
where f. is the Maxwellian distribution corresponding to the local temperature and average velocity, y is the initial
distribution of the particles, n the number density, and
v is the collision frequency. Examination of the BGK model
indicates
thatit reproduces the equilibrium solution f =f. atequilibrium. The BGK equation also reproduces the
correct moments imd satisfies the H-theorem. For the highly collisional regime, and when used with the Chapman-
Enskog expansion, it leads to the Navier-Stokes equations. In this equation, the term rrv~ represents the collisions
replenis.tig the equilibrium distribution
f., and the term mfrepresents the collisions depleting the distribution~
The physical interpretation of the assumption that the replenishing number of collisions is given by
nvfo is
that post-collision molecules are selected from a -elli~ distribution at the local average velocity and
temperature. It is evident that this behavior can readily be incorporated in a particle code. By doing so, the burden of
calculating collisions is reduced.
Despite the shortcomings of the BGK equation, it is able to reproduce the main features of the Boltzmarm
equation. Several modifications to the BGK equation have been proposed at the expense of the simplicity of the
model. Cercignani (3) proposed a modification to the BGK equation that would allow it to reproduce the same
viscosity and thermal conductivity as the fill Boltzm equatio~ here referred to as the BGKC method. To do this
the Prandtl number is introduced as a parameter in the equation. The generalization of the BGK equation is obtained
by the replacement of the Maxwellian distribution with a local anisotropic threedimensional Gaussianj referred to
as the ellipsoidal statistical (ES) model:
r. 1
[
f. = m“3/2(detA)”2 e.xp - ~ Aqu,uj
1,]=1
1
I
where the matrix A is:
Av = [(2RZ’/Pr)d,j - 2(1- Pr)p,, /(P Pr)]-l
(2)
(3)
I
where R is the universal gas constant, Z7is the velocity in the center-of-mass frame of reference. p is the local density
of the flow, Pr is the Prandtl number, and
PUis the pressure tensor. Recently this method ~vasintroduced into particle
simulations with reasomble success (1).
I
LANGEVIN’S EQUATION
Langevin proposed a method for Brownian motion to model collisions between particles (4). According to
this method, the influence of the surroundings on a particle that moves with a velocity Z7can be split up into two
parts: a deterministic part representing the dynamical fiction experienced by a particle (–A-), and a probabilistic
part characteristic of the Brownian motion (~ ). The Langevin equation takes the form:
~ = ~Oe
-@t fi
(4)
in
which Al is a time interval over which motion occurs and 10 is the \elocity of the particle at the beginning of the
interval. The dynamical friction coefficient P and the probabilistic part ~ are assumed to be independent of the
velocity of the particle. The probabilistic part ~ is chosen from the modified Mamvellian distribution (4):
P(Z7,At,Zio) =
~2~’’fle-2@’~2e’Tl-~f~~~:rJ “)
in which
T is local temperature, k the Boltzrnann constant, and m the mass of the particle. Effectively this
distribution is Maxwelliam at temperature T(l e-2@t). In this study the dynamic friction coefficient is interpreted
as the equilibrium collision frequency.
It can be shown that the Langevin equation forms the basis for deriving all the physically significant
relations concerning the motion of Brownian particles. The Langevin method has been applied successfully in the
past for the modeling of charged flows with PIC by Jones et al. (5) and for DSMC by Gallis et al. (6). It should be
noted, however, that the Brownian-motion paradigm corresponds more closely to the motion of charge particles in a
plasma than to that of neutrals in rarefied
flow, More specifically, Brownian motion describes the motion of a
particle in very dense fluid (liquid density) in which multiple collisional interactions are ve~ frequent. whereas the

Boltzmann equation describes the motion of particles when only binary collisions are important. Thus, charged
flows due to the long range interaction of the Coulomb forces fit the description of the Brownian motion more
closely than neutral particles.
Another signillcant difference exists between the conventional DSMC collision operator and the Laugevin
model. In the DSMC method a fraction of the particles modi@ their properties signtilcantly in every time step while
the rest-remain unchanged, but in the Langevin model every particle in a cell receives a small modification of its
velocity in every time step. The modification constitutes a small change in their properties. It should be noted that
the random velocities are not selected from a M=wellian with the temperature
T of the background gas, but with a
‘2@ that depends on the time interval and the collision frequency. In cases where the
temperature of I’(1 e )
collision frequency is very low, the probabilistic contribution O adds a vev small vector to the particle velocity. the
deterministic contribution similarly applies a small modification to the particle velocity for evexytime step.
The Lrmgevin method assumes that the thermal energy in a cell is always distributed according to a
Maxwellian distribution. This assumption mturally restricts the method to equilibrium or near-equilibrium
situations. The worst-case scenario for the method would be the collision of two monochromatic particle beams.
While the DSMC method predicts a gradual depletion of particles from the initial distributions to an equilibrium
distribution, the Langevin model evolves all particles gmdually towards an equilibrium distribution at the average
velocity and tempemture.
APPLICATION TEST CASE
The first test case examined is that of the flow over a 10° wedge (see Figures 1 and 2). The flow travels
from Iefl to right and the sides of the wedge are assumed to be specularly reflecting. The gas is pure argon at an
ambient temperature of 180 K and a number density of 1022m-3with an upstream velocity of 1365 mk (see Table 1).
The domain is discretized with 25,000 cells, and a time step of 10-8s is used. The downstream boundary condition is
assumed to be non-reentrant (vacuum), which is acceptable for the supersonic flow downstream of an oblique shock.
This test case is simulated with the DSMC, BGKC and Langevin methods. Steady-state conditions are observed
after 5,000 time steps while the code executes a total of 65,000 time steps.
Molet
m
Number 1
Tempera~
Speed of ~
I Upstream Conditions
I
cular Mass 39.99 mu
ific Heat Ratio 5/3
)ensitv
1022m-3
ure 180 K
Sound
249.75 m/s
Free Stream Velocity 1365
dS
Mach Number 5.465
I
Downstream Conditions
I
~
Table 1. Shock wave conditions
Figure l(a-c) presents the number-density contour lines as calculated by the DSMC. BGKC and Langevin
methods. Figure 2(a-c) in the same fashion presents the corresponding translational temperature contour lines. At .x
= Y = O,the location of the leading edge of the wedge, the shock layer forms,
A quantitative comparison indicates that all three simulations are in agreement about the downstream
conditions and the shock angle. For example, with aIl three methods, the flow downstream eventually reaches a
maximum number density that is about double the upweam nuber density, and the shock angle is 19-20°, as

Citations
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Journal ArticleDOI
Abstract: This paper presents an internal energy exchange scheme for the relaxation time simulation method (RTSM) which solves the BGK equation for the perfect gas flow at near-continuum region discrete rotational energies are introduced to model the relaxation of internal energy modes. This development improved the agreements between RTSM and DSMC with little additional computational cost. The result shows a possibility of an improved hybrid RTSM/DSMC code for the continuum/rarefied gas flow.

6 citations


Cites background from "A collective collision operator for..."

  • ...Recently, a few new models were reported to modified the BGK equation by changing the equilibrium distribution [15,16], which were expected to give a realistic Prandtl number and improve the heat transfer modeling....

    [...]


References
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Abstract: This book is designed for the junior-senior thermodynamics course given in all departments as a standard part of the curriculum. The book is devoted to a discussion of some of the basic physical concepts and methods useful in the description of situations involving systems which consist of very many particulars. It attempts, in particular, to introduce the reader to the disciplines of thermodynamics, statistical mechanics, and kinetic theory from a unified and modern point of view. The presentation emphasizes the essential unity of the subject matter and develops physical insight by stressing the microscopic content of the theory.

3,171 citations


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Abstract: I. Basic Principles of The Kinetic Theory of Gases.- 1. Introduction.- 2. Probability.- 3. Phase space and Liouville's theorem.- 4. Hard spheres and rigid walls. Mean free path.- 5. Scattering of a volume element in phase space.- 6. Time averages, ergodic hypothesis and equilibrium states.- References.- II. The Boltzmann Equation.- 1. The problem of nonequilibrium states.- 2. Equations for the many particle distribution functions for a gas of rigid spheres.- 3. The Boltzmann equation for rigid spheres.- 4. Generalizations.- 5. Details of the collision term.- 6. Elementary properties of the collision operator. Collision invariants.- 7. Solution of the equation Q(f,f) = 0.- 8. Connection between the microscopic description and the macroscopic description of gas dynamics.- 9. Non-cutoff potentials and grazing collisions. Fokker-Planck equation.- 10. Model equations.- References.- III. Gas-Surface Interaction and the H-Theorem.- 1. Boundary conditions and the gas-surface interaction.- 2. Computation of scattering kernels.- 3. Reciprocity.- 4. A remarkable inequality.- 5. Maxwell's boundary conditions. Accommodation coefficients.- 6. Mathematical models for gas-surface interaction.- 7. Physical models for gas-surface interaction.- 8. Scattering of molecular beams.- 9. The H-theorem. Irreversibility.- 10. Equilibrium states and Maxwellian distributions.- References.- IV, Linear Transport.- 1. The linearized collision operator.- 2. The linearized Boltzmann equation.- 3. The linear Boltzmann equation. Neutron transport and radiative transfer.- 4. Uniqueness of the solution for initial and boundary value problems.- 5. Further investigation of the linearized collision term.- 6. The decay to equilibrium and the spectrum of the collision operator.- 7. Steady one-dimensional problems. Transport coefficients.- 8. The general case.- 9. Linearized kinetic models.- 10. The variational principle.- 11. Green's function.- 12. The integral equation approach.- References.- V. Small and Large Mean Free Paths.- 1. The Knudsen number.- 2. The Hilbert expansion.- 3. The Chapman-Enskog expansion.- 4. Criticism of the Chapman-Enskog method.- 5. Initial, boundary and shock layers.- 6. Further remarks on the Chapman-Enskog method and the computation of transport coefficients.- 7. Free molecule flow past a convex body.- 8. Free molecule flow in presence of nonconvex boundaries.- 9. Nearly free-molecule flows.- References.- VI. Analytical Solutions of Models.- 1. The method of elementary solutions.- 2. Splitting of a one-dimensional model equation.- 3. Elementary solutions of the simplest transport equation.- 4. Application of the general method to the Kramers and Milne problems.- 5. Application to the flow between parallel plates and the critical problem of a slab.- 6. Unsteady solutions of kinetic models with constant collision frequency.- 7. Analytical solutions of specific problems.- 8. More general models.- 9. Some special cases.- 10. Unsteady solutions of kinetic models with velocity dependent collision frequency.- 11. Analytic continuation.- 12. Sound propagation in monatomic gases.- 13. Two-dimensional and three-dimensional problems. Flow past solid bodies.- 14. Fluctuations and light scattering.- References.- VII. The Transition Regime.- 1. Introduction.- 2. Moment and discrete ordinate methods.- 3. The variational method.- 4. Monte Carlo methods.- 5. Problems of flow and heat transfer in regions bounded by planes or cylinders.- 6. Shock-wave structure.- 7. External flows.- 8. Expansion of a gas into a vacuum.- References.- VIII. Theorems on the Solutions of the Boltzmann Equation.- 1. Introduction.- 2. The space homogeneous case.- 3. Mollified and other modified versions of the Boltzmann equation.- 4. Nonstandard analysis approach to the Boltzmann equation.- 5. Local existence and validity of the Boltzmann equation.- 6. Global existence near equilibrium.- 7. Perturbations of vacuum.- 8. Homoenergetic solutions.- 9. Boundary value problems. The linearized and weakly nonlinear cases.- 10. Nonlinear boundary value problems.- 11. Concluding remarks.- References.- References.- Author Index.

2,797 citations


"A collective collision operator for..." refers background or methods in this paper

  • ...Ay =[(2RT/Pr)dij-2(l-Pr)Pij/(rPr)]-(1) (3)...

    [...]

  • ...In a previous paper by Gallis and Torczynski (1), the BGK equation (2) and Cercignani's extension (3) were examined as possible collision operators....

    [...]

  • ...Cercignani (3) proposed a modification to the BGK equation that would allow it to reproduce the same viscosity and thermal conductivity as the full Boltzmann equation, here referred to as the BGKC method....

    [...]


Book
01 Jan 1965

2,088 citations


"A collective collision operator for..." refers background or methods in this paper

  • ...The generalization of the BGK equation is obtained by the replacement of the Maxwellian distribution with a local anisotropic three-dimensional Gaussian, referred to as the ellipsoidal statistical (ES) model: 3 I (2) /0=p- (3)/(2)(detA)exp...

    [...]

  • ...In a previous paper by Gallis and Torczynski (1), the BGK equation (2) and Cercignani's extension (3) were examined as possible collision operators....

    [...]


Journal ArticleDOI
Abstract: A new method is presented to model the intermediate regime between collisionless and Coulomb collision dominated plasmas in particle-in-cell codes. Collisional processes between particles of different species are treated through the concept of a grid-based “collision field,” which can be particularly efficient for multi-dimensional applications. In this method, particles are scattered using a force which is determined from the moments of the distribution functions accumulated on the grid. The form of the force is such to reproduce the multi-fluid transport equations through the second (energy) moment. Collisions between particles of the same species require a separate treatment. For this, a Monte Carlo-like scattering method based on the Langevin equation is used. The details of both methods are presented, and their implementation in a new hybrid (particle ion, massless fluid electron) algorithm is described. Aspects of the collision model are illustrated through several one- and two-dimensional test problems as well as examples involving laser produced colliding plasmas.

126 citations


Proceedings ArticleDOI
14 Jun 2000
Abstract: A collision model for the Direct Simulation Monte Carlo (DSMC) method is presented. The collision model is based on the BGK1 equation and makes use of the Cercignani2 ellipsoidal distribution to incorporate the effects of heat conductivity. Results obtained by the DSMC method and its BGK and BGKC modifications for a 10° wedge and a flat plate are presented and discussed. Introduction An area of interest for particle simulation codes is that of dense near-equilibrium flows in Micro Electromechanical Systems (MEMS) in which the scales of the phenomena make the application of continuum techniques questionable. However, the extension of particle simulation methods to highdensity flows is usually associated with very small time steps needed to resolve the large number of collisions that take place each time step. The computational load that high-density flows impose on particle simulations is such that it challenges even the most modern computer platforms. MEMS flows are typically subsonic the mean velocity is significantly smaller than the thermal velocity of the particles. As a result, a particle may suffer several collisions within a single time step. Simulation of all these collisions is very timeconsuming, and it does not add any information to the flow since these flows are already in near-equilibrium or equilibrium. The same issue appears when simulating charged flows, in which a charged particle may suffer many collisions within a time step. The issue of modeling collisional processes extends back nearly three decades to the early days of digital computers. One of the early approaches to reduce the computational load was to replace the collision operator of the Boltz.mann equation with simpler forms that maintain most of the fimdamental properties of the original equation. The idea behind the simplification of the collision term of the Boltzmann equation is that most of the details of the two-body interactions are not important in reproducing some of me experimentally measured quantities. In this paper extensions of some of these methods, that were initially proposed for the analytic solution of the Boltzmann equatio~ are investigated as possible alternatives to reduce the computational load associated with modeling high-density flows. The BGK’ equation (named tier its authors Bhatnagar, Gross and Krook) is used as the starting point, and related approaches are developed by adding more sophisticated distributions and increasing its physical accuracy. The Boltzmann equation The velocity distribution function~in the sixdimensional phase space (V,7 ) provides a complete description of a gas at the molecular level. The relationship between the velocity distribution fimction and the variables it depends on is given by the Boltzmann equation: [1 ~(nf)+7.~(nf)+F.~= ~(nf)

44 citations


"A collective collision operator for..." refers background or methods in this paper

  • ...= n n ( f 0 - f ) (1) J collision where f0 is the Maxwellian distribution corresponding to the local temperature and average velocity, / is the initial distribution of the particles, n the number density, and v is the collision frequency....

    [...]

  • ...In a previous paper by Gallis and Torczynski (1), the BGK equation (2) and Cercignani's extension (3) were examined as possible collision operators....

    [...]

  • ...Recently this method was introduced into particle simulations with reasonable success (1)....

    [...]