Inelastic collision selection procedures for direct ...€¦ · Inelastic collision selection...

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Inelastic collision selection procedures for direct simulation Monte Carlo calculations of gas mixtures Chonglin Zhang and Thomas E. Schwartzentruber Citation: Physics of Fluids (1994-present) 25, 106105 (2013); doi: 10.1063/1.4825340 View online: http://dx.doi.org/10.1063/1.4825340 View Table of Contents: http://scitation.aip.org/content/aip/journal/pof2/25/10?ver=pdfcov Published by the AIP Publishing This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to IP: 134.84.192.103 On: Sat, 25 Jan 2014 20:52:24

Transcript of Inelastic collision selection procedures for direct ...€¦ · Inelastic collision selection...

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Inelastic collision selection procedures for direct simulation Monte Carlo calculationsof gas mixturesChonglin Zhang and Thomas E. Schwartzentruber Citation: Physics of Fluids (1994-present) 25, 106105 (2013); doi: 10.1063/1.4825340 View online: http://dx.doi.org/10.1063/1.4825340 View Table of Contents: http://scitation.aip.org/content/aip/journal/pof2/25/10?ver=pdfcov Published by the AIP Publishing

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PHYSICS OF FLUIDS 25, 106105 (2013)

Inelastic collision selection procedures for directsimulation Monte Carlo calculations of gas mixtures

Chonglin Zhanga) and Thomas E. Schwartzentruberb)

Department of Aerospace Engineering and Mechanics, University of Minnesota,Minneapolis, Minnesota 55455, USA

(Received 18 February 2013; accepted 2 October 2013; published online 23 October 2013)

A modification to existing phenomenological inelastic collision selection proceduressuitable for modeling the internal energy exchange processes of gas mixtures in directsimulation Monte Carlo calculations is presented. The selection procedure does notdepend on the relative order of rotational and vibrational relaxation processes anddoes not require the solution of a quadratic equation for every collision to determinethe inelastic collision probability. The simulated relaxation process resulting fromthe selection procedure is analytically proven to be equivalent to the procedures ofHaas et al. [“Rates of thermal relaxation in direct simulation Monte Carlo methods,”Phys. Fluids 6, 2191–2201 (1994)] and the modified procedure of Gimelshein et al.[“Vibrational relaxation rates in the direct simulation Monte Carlo method,” Phys.Fluids 14, 4452–4455 (2002)]. The implementation and computational efficiency ofeach of the procedures are discussed. The proposed selection procedure is verifiedto accurately simulate rotational and vibrational processes for gas mixtures throughisothermal relaxation simulations compared with analytical solutions using the Jeansequation. C© 2013 AIP Publishing LLC. [http://dx.doi.org/10.1063/1.4825340]

I. INTRODUCTION

When implementing phenomenological models for rotational/vibrational relaxation and chem-ical reactions in direct simulation Monte Carlo (DSMC), different authors (and different DSMCcodes) generally use different inelastic collision selection procedures.1–4 For example, three widelyused inelastic collision selection procedures include pair selection,5, 6 particle selection permittingdouble relaxation,3 and particle selection prohibiting double relaxation.1 However, it has been shownthat to correctly simulate a specified relaxation rate, the inelastic collision probability expressionused within the DSMC method must depend on, or be specific to, the selection procedure.1, 2, 6

Only the consistent use of a selection procedure and its corresponding probability expression willresult in a DSMC simulation reproducing the desired relaxation rate. This subtlety can complicatethe transferability of probability expressions (collision models) between DSMC implementationsand can also lead to inconsistent comparisons of DSMC simulations with continuum simulationsinvolving internal energy relaxation in the near-equilibrium limit.

In continuum simulations, relaxation processes are usually modeled by the Jeans equation orLandau-Teller equation which have the same form,1

d E

dt= E∗(t) − E(t)

τ, (1)

where E(t) is the average energy at time t of either the rotational or vibrational mode associatedwith ζ degrees of freedom and τ is the characteristic relaxation time of the energy mode. E*(t)is the instantaneous equilibrium energy of the energy mode, which is defined according to the

a)Electronic mail: [email protected])Electronic mail: [email protected]

1070-6631/2013/25(10)/106105/13/$30.00 C©2013 AIP Publishing LLC25, 106105-1

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106105-2 C. Zhang and T. E. Schwartzentruber Phys. Fluids 25, 106105 (2013)

instantaneous translational temperature Tt(t), as

E∗(t) = ζ

2kB Tt (t). (2)

The characteristic relaxation time τ in Eq. (1) is determined as a function of the equilibrium gastemperature in previous theoretical and experimental studies.7, 8 τ is usually expressed as a functionof the mean collision time τ c, and an inelastic collision number Z in the following manner:

τ = τc Z . (3)

Thus, a rotational or vibrational inelastic collision number (Zrot, Zvib) is used to specify therelaxation rate, which in general, may be a function of temperature. To simulate these relaxationprocesses in DSMC, each particle involved in a collision is considered for internal energy exchangewith an inelastic collision probability,

Prot = f (ζrot,A, ζrot,B, ζt , Zrot ) or Pvib = f (ζvib,A, ζvib,B, ζt , Zvib). (4)

Here, ζ t represents the translational degrees of freedom of the collision pair. ζ rot and ζvib are theeffective internal degrees of freedom of the rotational and vibrational energy modes (correspondingto collision partners A and B) participating in the inelastic collision.1–3, 9 As an example, for thevariable hard sphere (VHS) molecular model3 with a temperature dependent viscosity exponentof ω, the translational degrees of freedom participating in a collision is ζ t = 5 − 2ω. In general,the exact form of Eq. (4) is specific to the inelastic collision selection procedure used to model therelaxation process.1–4 A number of these inelastic collision selection procedures have been discussedin several papers, including Lumpkin et al.,6 Haas et al.,1 and Gimelshein et al.2

The effect of the collision selection procedure on the simulated relaxation process is most signif-icant for gas mixtures, since some selection procedures inherently couple the relaxation probabilitiesand internal energy redistribution processes of the different species. To remedy this, Haas et al.1

constructed a framework for rotational and vibrational relaxation suitable for mixtures. However, torelate the DSMC inelastic collision probabilities (Prot , Pvib) with the corresponding collision num-bers (Zrot , Zvib), a set of quadratic equations must be solved during each collision for the case whenthe collision numbers are temperature dependent. In an article on vibrational relaxation, Gimelsheinet al.2 used a modified version of the framework by Haas et al.1 that uses a single random number todetermine the probabilities of rotational and vibrational energy exchange between the colliding par-ticles and does not require the solution of quadratic equations. However, the simulated probabilitiesdo not explicitly appear in the algorithm by Gimelshein et al.,2 rather they result from inequalitiesevaluated using the random number. As a result, it is not clear that the techniques of Haas et al.1

and Gimelshein et al.2 produce the same simulated relaxation for gas mixtures and, furthermore, themodified algorithm by Gimelshein et al. was not tested on a gas mixture in the article.2

In this paper, we present a modification to the approach taken by Haas et al.,1 which also removesthe requirement of solving a set of quadratic equations during each collision, but where the simulatedprobabilities of rotational and vibrational energy relaxation appear explicitly in the algorithm.Ultimately, through this new algorithm, we are able to analytically prove the equivalence of all threealgorithms and demonstrate the ability to accurately simulate specified internal energy relaxationrates in gas mixtures. Section II A summarizes the most widely used inelastic collision selectionprocedures with specific discussion regarding simulated relaxation processes in gas mixtures. InSec. II B, the proposed sequential probability selection procedure is described and the simulatedrelaxation process for gas mixtures is analytically proven to be equivalent to that produced by theprocedures of Haas et al.1 and Gimelshein et al.2 A discussion of the computational efficiency ofthe three procedures is also presented. The ability of the proposed selection procedure to simulatespecified internal energy relaxation rates is demonstrated in Sec. II C through comparison withanalytical solutions for isothermal relaxation, and conclusions are contained in Sec. III.

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106105-3 C. Zhang and T. E. Schwartzentruber Phys. Fluids 25, 106105 (2013)

II. A SEQUENTIAL PROBABILITY COLLISION SELECTION PROCEDURE

A. Existing inelastic collision selection procedures

For clarity, we summarize the three widely used inelastic collision selection procedures as theyapply to rotational relaxation. For inelastic collisions involving vibrational energy exchange, theprocedures are identical, only the internal energy mode is altered.

A. Pair selection:5, 6 The collision pair is tested for rotational inelastic collision with a proba-bility, and once the collision pair is selected, the energy of both particles in the pair is re-distributed. Specifically, the total collision energy ET = Et + Er, A + Er, B is redistributedbetween translational and rotational modes, as E ′

t and E ′r , using the Borgnakke-Largsen

(BL) model,10 with ET = E ′r + E ′

t . The post collision rotational energy E ′r is then dis-

tributed between the two collision partners as E ′r,A, E ′

r,B again using the BL model withE ′

r = E ′r,A + E ′

r,B .3 Each of the three inelastic collision selection procedures has a limiton the smallest collision number that can be simulated in DSMC, which corresponds to asimulated rotational inelastic collision probability of 1. Suppose the translational degreesof freedom of the collision pair is ζt = ζtA|B , and the rotational degrees of freedom is ζ rot,with ζ rot = ζ rot, A + ζ rot, B, where ζ rot, A and ζ rot, B are the rotational degrees of freedom ofthe two participating molecules A and B. For the pair selection procedure, as developed inprevious studies,1, 6 the probability should be set as Prot = ζt +ζrot

ζt

1Zrot

. For rotational relax-

ation, this means the smallest rotational collision number is Zlimitrot = ζt +ζrot

ζt, which is approx-

imately 2 for nitrogen. Similarly, there is a smallest vibrational collision number that can besimulated.

B. Particle selection permitting double relaxation:3 Each particle in the collision pair istested with a probability for rotational inelastic collision individually. If the first particleis selected for an inelastic collision, the BL model is used to redistribute the total colli-sion energy between the translational energy of the pair and the rotational energy of onlythe selected particle. The second particle in the pair is then tested for a rotational inelas-tic collision. If selected, the total collision energy used in the BL procedure includes theredistributed translational energy of the pair from the first collision and only the rotationalenergy of the second particle. Thus, if both particles are selected for an inelastic collision,there is some degree of coupling between their relaxation processes. For this selection proce-dure, suppose the molecule (particle) under consideration has rotational degrees of freedomζ rot (ζ rot = ζ rot, A or ζ rot, B), similarly, the smallest rotational collision number that can be sim-ulated in DSMC is Zlimit

rot = ζt +ζrot

ζt. For molecule-molecule collisions, this number is slightly

smaller compared to the first selection procedure, since ζ rot only includes the degrees of free-dom of one molecule. Similarly, there is a smallest vibrational collision number that can besimulated in DSMC.

C. Particle selection prohibiting double relaxation:1 The two particles in the collision pairare tested with a probability for rotational inelastic collision individually. In this case, thetotal collision energy is always the sum of the relative translational energy of the collisionpair and the rotational energy of only the particle being considered. However, if one particleis selected to undergo a rotational inelastic collision, the other particle is not tested for aninelastic collision, and the relaxation process for the collision pair ends. Otherwise, the sameprocedure is then applied to the second particle in the pair. For this selection procedure,to have all types of inelastic collisions correctly simulated in DSMC, we need to satisfyProt,1 + Prot,2 + Pvib,1 + Pvib,2 < 1 as given later in Eq. (9). Compared to the two previousselection procedures this will correspond to a larger Zlimit

rot and Zlimitvib that can be simulated

in DSMC. Moreover, since the inequality needs to be satisfied, all values in the sum must bedetermined as a whole.

Selection procedure (A) couples the relaxation probabilities and energy redistribution processesfor collision pairs of different species, and thus couples the simulated relaxation process. Althoughnot as direct as procedure (A), procedure (B) also couples the energy redistribution processes of

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106105-4 C. Zhang and T. E. Schwartzentruber Phys. Fluids 25, 106105 (2013)

Test inelastic collision of particle 1 first

Perform vibrationalinelastic collision for 1

Perform rotatioinal inelastic collision for 2

Perform rotatioinal inelastic collision for 1

Perform rotatioinal inelastic collisionfor 1

Perform vibrationalinelastic collision for 2

Perform vibrationalinelastic collision for 2

Perform rotatioinal inelastic collisionfor 2

Perform vibrationalinelastic collision for 1

Particle 1 + 2 R0 > 1/2 ?

Test inelastic collisionof particle 2 first

Yes

No

R1 < Prot,1 ?

R5 < Prot,2 ? R8 < Pvib,1 ?No

End

R6 < Prot,1 ?No R7 < Pvib,2 ?No

R4 < Pvib,2 ?NoR2 < Prot,2 ?

No

R3 < Pvib,1 ?No

Yes

End

Yes

End

End

End

YesYes Yes

YesYesYes

EndEndEnd

Perform elastic collision for 1 + 2

End

No

No

FIG. 1. The process to select an inelastic collision using selection procedure (C) as constructed by Haas et al.1 In thefigure, Rj(j = 0, 1, ..., 8) are uniform random numbers between 0 and 1, and Prot,i , Pvib,i (i = 1 or 2) are the rotational andvibrational inelastic collision probabilities used in DSMC for particle i.

species when both particles are selected for rotational relaxation (i.e., double relaxation). Further-more, when both rotational and vibrational relaxation processes are considered, sequential testingfor rotational followed by vibrational inelastic collision, in selection procedures (A) and (B), willinherently couple the rotational and vibrational relaxation processes. Although such coupling mayseem physically realistic, it is stressed that the DSMC collision models discussed here are phe-nomenological and are constructed to reproduce specified internal energy relaxation rates (collisionnumbers, Z) for a given energy mode and species interaction. For this reason, in order to accu-rately simulate the relaxation process of mixtures, Haas et al.1 and Gimelshein et al.2 recommendusing selection procedure (C) to decouple the rotational and vibrational relaxation of differentspecies.

The logical steps followed by selection procedure (C) are depicted in Fig. 1 (this is constructedby Haas et al.,1 and is later adapted by Gimelshein et al. and used in a modified form2). First,one of the two particles (from the collision pair) is selected with equal chance and is tested forrotational relaxation with a specified collision probability Prot, 1 (suppose particle 1 is selected first).As discussed above, if the particle is chosen to undergo an inelastic collision, the BL model isused to redistribute the post collision energy and the relaxation of the current pair will finish. Onlywhen the first particle does not undergo a rotational inelastic collision, will the second particle inthe collision pair be tested for a rotational inelastic collision with probability Prot, 2. Again, therelaxation of the current pair will end if the second particle undergoes an inelastic collision. Onlywhen the second particle is not chosen for a rotational inelastic collision, will the two particles betested individually for a vibrational inelastic collision. In the same manner, the first particle will betested with a specified probability Pvib,1 and only if not selected will the second particle be testedwith probability Pvib,2.

As initially proposed by Haas et al.,1 the probabilities Prot, 1, Prot, 2, Pvib,1, and Pvib,2 areobtained by solving the set of quadratic equations listed in Eqs. (5a)–(5d). These equations relate theprobabilities used within DSMC (Prot and Pvib) to specified continuum rotational collision numbers(Zrot and Zvib). If the collision numbers are not constant (for example, they may be temperaturedependent), then the quadratic equation must be solved for every collision. We also note that ifthe order of testing for rotational and vibrational inelastic collisions is reversed, then the quadratic

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106105-5 C. Zhang and T. E. Schwartzentruber Phys. Fluids 25, 106105 (2013)

equations and instances of Prot, 1, Prot, 2, Pvib,1, and Pvib,2 must be changed accordingly,

1

2P2

rot,1 −(

1

2

ζt + ζrot,1

ζt

1

Zrot,1− 1

2

ζt + ζrot,2

ζt

1

Zrot,2+ 1

)Prot,1 + ζt + ζrot,1

ζt

1

Zrot,1= 0, (5a)

1

2P2

rot,2 −(

1

2

ζt + ζrot,2

ζt

1

Zrot,2− 1

2

ζt + ζrot,1

ζt

1

Zrot,1+ 1

)Prot,2 + ζt + ζrot,2

ζt

1

Zrot,2= 0, (5b)

1

2(1 − Prot,1)(1 − Prot,2)P2

vib,1

−(

1

2

ζt + ζvib,1

ζt

1

Zvib,1− 1

2

ζt + ζvib,2

ζt

1

Zvib,2+ (1 − Prot,1)(1 − Prot,2)

)Pvib,1

+ζt + ζvib,1

ζt

1

Zvib,1= 0, (5c)

1

2(1 − Prot,2)(1 − Prot,1)P2

vib,2

−(

1

2

ζt + ζvib,2

ζt

1

Zvib,2− 1

2

ζt + ζvib,1

ζt

1

Zvib,1+ (1 − Prot,2)(1 − Prot,1)

)Pvib,2

+ζt + ζvib,2

ζt

1

Zvib,2= 0. (5d)

The specific notation used in Eqs. (5a)–(5d) and throughout the remainder of this article requiresa careful description. As depicted in Fig. 1, before starting the collision procedure, the two particlesin the pair must be assigned a number (either particle 1 or 2). We use a subscript, i, to denote theparticle numbering of the pair (i = 1 or i = 2). We further note that i does not denote a specificparticle type, thus particles i = 1 and i = 2 may be the same, or different, particle type (monatomic,diatomic, or polyatomic). In this manner, all parameters denoted by a subscript i (such as Prot, i, Zrot, i,ζ rot, i, etc.) are specific to the relaxation process of particle i. For example, ζ rot, i and ζvib,i are therotational and vibrational degrees of freedom of only particle i, whereas ζ t represents the availabletranslational degrees of freedom of the collision pair (and thus has no subscript). Furthermore, Zrot, i

and Zvib,i are the rotational and vibrational inelastic collision numbers specific to the relaxation ofparticle i during a collision with the other particle in the pair. For example, if both particles are of thesame type (A for example) then the collision numbers for the two particles would be equal (i.e., Zrot, 1

= Zrot, 2 = Zrot, A|A). However, if the two particles were of different types (A for i = 1 and B fori = 2, as an example), then the collision numbers would be Zrot, 1 = Zrot, A|B and Zrot, 2 = Zrot, B|Awhere in general, Zrot, A|B may be specified as not equal to Zrot, B|A.

B. Formulation of the sequential probability selection procedure

1. Proposed modification to the selection procedure of Haas et al.

In this section, we propose a modification to the particle selection prohibiting double relaxationprocedure.1 We keep the structure of selection procedure (C) as shown in Fig. 1 unchanged, whileusing different expressions to calculate Prot, i and Pvib,i appearing in the figure. With the modifiedprocedure, we no longer need to solve the quadratic equations in Eqs. (5a)–(5d) during each collisionwhen the collision numbers are not constant. In the original procedure,1 the values of Prot, i appearingin both top and bottom branches of Fig. 1 are exactly equal (the same applies for Pvib,i ), and this iswhat necessitates the solution of a quadratic equation. In our modified selection procedure (calledthe sequential probability selection procedure) the values of Prot, i and Pvib,i appearing in the upperand lower branches of Fig. 1 are not the same. Rather, we calculate the appropriate probability forthe specific particle currently under consideration for inelastic relaxation, taking into account theprevious branching steps already completed and their associated probabilities.

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106105-6 C. Zhang and T. E. Schwartzentruber Phys. Fluids 25, 106105 (2013)

Specifically, for the sequential probability selection procedure, when particle i = 1 is testedfirst (the upper branch of Fig. 1), the probabilities Prot, 1, Prot, 2, Pvib,1, Pvib,2 used in the acceptance-rejection technique for DSMC inelastic collision selection are calculated from the following expres-sions:

A = 1, Prot,1 = AFrot,1, (6a)

B = A

1 − Prot,1, Prot,2 = B Frot,2, (6b)

C = B

1 − Prot,2, Pvib,1 = C Fvib,1, (6c)

D = C

1 − Pvib,1, Pvib,2 = DFvib,2, (6d)

where

Frot,i = ζt + ζrot,i

ζt

1

Zrot,i(7)

and

Fvib,i = ζt + �i

ζt

1

Zvib,i. (8)

For vibration, if a continuous energy distribution is used,3 then �i = ζvib,i . Whereas for the simpleharmonic oscillator (SHO) discrete energy level model,2 �i = ξv(T )2 exp(θv/T )/2, where ξv(T )= 2θv/T

exp(θv/T )−1 , T is the temperature and is usually set as the cell averaged translational temperature,i.e., T = Tt and θ is the characteristic temperature of vibration (where all parameters are specific toparticle i).

Alternatively, when particle i = 2 is tested first (the lower branch of Fig. 1), the probabilitiesProt, 1, Prot, 2, Pvib,1, Pvib,2 in Fig. 1 are still calculated using Eqs. (6a)–(6d), only now with subscripts1 and 2 interchanged.

For such particle selection procedures prohibiting double relaxation, the constraint2

Frot,1 + Frot,2 + Fvib,1 + Fvib,2 < 1 (9)

must be satisfied. While Eq. (9) is satisfied for the majority of nonequilibrium flow problems, asevident from Eqs. (5a)–(5d) and (6a)–(6d), if Frot, i were to approach 0.5 there would be a vanishingnumber of particles available to be tested for vibrational relaxation and Pvib,i may become largerthan unity. Thus for generality, it is recommended to test for vibrational relaxation first, followed byrotational relaxation, since Pvib,i is typically much smaller than Prot, i. This ensures that vibrationalrelaxation remains accurate even in extreme cases with very fast rotational relaxation. Changing theorder of rotational and vibrational relaxation only requires interchanging the subscripts (rot, vib) inEqs. (5a)–(5d) and (6a)–(6d). Similarly, when chemical reactions are considered, the collision pairshould be tested for a chemical reaction first. In this situation, to avoid the slight bias introduced tothe rotational and vibrational relaxation rate by the chemical reaction probability Preact, the value Ain Eqs. (6a)–(6d) can be modified to A = 1/(1 − Preact).

It should be noted that, we can further simplify Eqs. (6a)–(6d) depending on the valueof A. In the case A = 1, we have Prot, 1 = Frot, 1, Prot,2 = Frot,2

1−Frot,1, Pvib,1 = Fvib,1

1−Frot,1−Frot,2, and

Pvib,2 = Fvib,2

1−Frot,1−Frot,2−Fvib,2. While in the case A = 1/(1 − Preact), we have Prot,1 = Frot,1

1−Preact,

Prot,2 = Frot,2

1−Frot,1−Preact, Pvib,1 = Fvib,1

1−Frot,1−Frot,2−Preact, and Pvib,2 = Fvib,2

1−Frot,1−Frot,2−Fvib,1−Preact. These ex-

pressions can be used in the actual implementation of the sequential probability selection procedure.Finally, collision quantity dependent models5, 11–16 do not have a direct relationship between

the collision probability P and a collision number Z. For such models, Fi is now a function ofsome collision quantities (collision energies, for example) and can simply replace the corresponding

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106105-7 C. Zhang and T. E. Schwartzentruber Phys. Fluids 25, 106105 (2013)

Fi in Eqs. (6a)–(6d). In this manner, the prescribed relaxation behavior (collision quantity based)can be exactly simulated with no coupling between internal energy modes or species. Ultimately,the sequential probability particle selection procedure (detailed in Eqs. (6a)–(6d) to (8)) is aimedat phenomenological DSMC models (either collision-number or collision-quantity based) whichcombine collision probabilities with the Borgnakke-Larsen model for energy redistribution. Forstate-to-state DSMC models, such as the recent model presented by Boyd and Josyula,17 furtherconsideration or perhaps a different strategy may be required to correctly simulate the specifiedstate-to-state processes within a DSMC simulation.

2. Equivalence of sequential probability and original Haas et al. selection procedures

The sequential probability particle selection procedure (Eqs. (6a)–(6d)) achieves the samerelaxation rate as the original Haas et al. selection procedure1 (Eqs. (5a)–(5d)). To show this, westart with Eq. (B11) from Ref. 1,

FA|B = ζtA|B + ζrA

ζtA|B

1

Z A|B, (10)

which represents the fraction of collisions (on average) that are required to be inelastic for A particlesin order to achieve a relaxation rate consistent with Jeans equation using a collision number ZA|B.Given the notation described at the end of Sec. II A, Eq. (10) is identical to Eq. (7) (and Eq. (8) inthe case of vibration). The objective is to ensure that the collision probabilities (P) used within theacceptance-rejection portion of the DSMC algorithm actually result in the correct inelastic collisionfraction (F) being simulated.

In the original Haas’ selection procedure, FA|B should be the sum of the probability of two typesof inelastic collisions: the relaxation of particle A in the upper branch of Fig. 1, and the relaxation ofparticle A in the lower branch of Fig. 1. For the case of rotation, the expression takes the followingform (Eq. (B2) of Ref. 1 with the right-hand side written in current notation):

FA|B = 1

2Prot,1 + 1

2(1 − Prot,2)Prot,1 = Prot,1(1 − 1

2Prot,2). (11)

Equating Eqs. (10) and (11) (Eqs. (B11) and (B2) from Ref. 1), along with the corresponding equationfor FB|A, the quadratic equations for collision probabilities (Prot, 1 and Prot, 2) are obtained as shownin Eqs. (5a)–(5d) (equivalently Eq. (B13) of Ref. 1).

For the sequential probability selection procedure, the collision probabilities appearing in theupper and lower branches of Fig. 1 are not equal. We denote probabilities in the upper branch asPU

rot,1, PUrot,2, PU

vib,1, and PUvib,2, and probabilities in the lower branch as P L

rot,1, P Lrot,2, P L

vib,1, andP L

vib,2. Consider the rotational relaxation of particle i = 1 (of type A) through a collision withparticle i = 2 (of type B). Using Fig. 1 as a reference, Eqs. (6a)–(6d) gives the following probabilityexpressions: PU

rot,1 = Frot,1, P Lrot,1 = Frot,1

1−P Lrot,2

, P Lrot,2 = Frot,2, and PU

rot,2 = Frot,2

1−PUrot,1

. Hence, using the

sequential probability selection procedure, the DSMC simulated inelastic collision fraction F DSMCA|B

is (by reference to Fig. 1 and Eq. (11)):

F DSMCA|B = 1

2PU

rot,1 + 1

2(1 − P L

rot,2)P Lrot,1

= 1

2Frot,1 + 1

2(1 − P L

rot,2)Frot,1

1 − P Lrot,2

= Frot,1. (12)

Thus, the simulated inelastic collision fraction resulting from the original Haas et al. selectionprocedure and the modified version (the proposed sequential probability selection procedure) arethe same, and both are consistent with the Jeans equation using a specified collision number ZA|B.

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3. Equivalence of sequential probability and Gimelshein et al. selection procedures

In the selection procedure by Gimelshein et al.,2 the calculation of the simulated collisionprobability for inelastic collisions and the subsequent use of the acceptance-rejection technique arecombined together, and it is not immediately evident that the simulated relaxation is equivalent tothat produced by the other two selection procedures. In this subsection, we present an analyticalproof that this selection procedure is equivalent to the sequential probability selection procedure, andtherefore the three selection procedures are equivalent in that they produce the same macroscopicrelaxation rate.

We start with the equations shown in Fig. 1 of Ref. 2,

A1 = Frot,1, (13a)

A2 = Frot,1 + Frot,2, (13b)

A3 = Frot,1 + Frot,2 + Fvib,1, (13c)

A4 = Frot,1 + Frot,2 + Fvib,1 + Fvib,2. (13d)

Based on our understanding, the four terms Pr,A, Pr,B, Pv,A, Pv,B in Fig. 1 of Ref. 2 shouldcorrespond to the four terms Frot,1, Frot,2, Fvib,1, Fvib,2 in the current paper. Therefore, we havechanged the notation of these four terms to the current paper notation when rewriting the aboveequations. In the Gimelshein et al. procedure, a random number Rn between [0, 1] is first selected,and then subsequently used to test each type of relaxation collision according to the acceptance-rejection technique. In these tests, the inequalities of the form Rn < A1 and Ai < Rn < Ai + 1(i =1, 2, 3) are sequentially tested. If one inequality is true, then the corresponding inelastic collisionrelaxation is performed, and the procedure ends for the current collision pair. Only if one inequalitydoes not hold, will the subsequent inequality be tested.

(1) Rn < A1

With the random number Rn ∈ [0, 1] and the acceptance-rejection technique, we should havethe simulated DSMC collision probability for rotational relaxation of particle 1 (particle A inRef. 2) as

Psimrot,1 = A1 = Frot,1. (14)

(2) A1 < Rn < A2

If A1 < Rn, then the random number is Rn ∈ [A1, 1]. If further Rn < A2, we should have

Rn − A1 < A2 − A1. (15)

Denote R1 = Rn − A1, then R1 ∈ [0, 1 − A1], and we have

A2 − A1 > R1 ∈ [0, 1 − A1]. (16)

This is equivalent toA2 − A1

1 − A1>

R1

1 − A1= R∗

1 ∈ [0, 1], (17)

where we have defined a new variable R∗1 = R1

1−A1. As a result, the DSMC simulated collision

probability for rotational relaxation of particle 2 (particle B in Ref. 2) is

Psimrot,2 = A2 − A1

1 − A1= Frot,2

1 − Frot,1. (18)

(3) A2 < Rn < A3

If A2 < Rn, then we have

R∗1 > Psim

rot,2 = A2 − A1

1 − A1(19)

and R∗1 ∈ [ A2−A1

1−A1, 1].

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106105-9 C. Zhang and T. E. Schwartzentruber Phys. Fluids 25, 106105 (2013)

If Rn < A3, then it corresponds to

R∗1 <

A3 − A1

1 − A1. (20)

Denote R2 = R∗1 − A2−A1

1−A1, then R2 ∈ [0, 1 − A2−A1

1−A1]. With Rn < A3, we then have

R2 <A3 − A1

1 − A1− A2 − A1

1 − A1= A3 − A2

1 − A1, (21)

i.e.,

A3 − A2

1 − A1> R2 ∈ [0, 1 − A2 − A1

1 − A1]. (22)

This is equivalent to

A3−A21−A1

1 − A2−A11−A1

>R2

1 − A2−A11−A1

= R∗2 ∈ [0, 1], (23)

where we have defined a new variable R∗2 = R2

1− A2−A11−A1

= R2

1− Frot,21−Frot,1

. As a result, the DSMC simulated

collision probability for vibrational relaxation of particle 1 (particle A in Ref. 2) is

Psimvib,1 =

A3−A21−A1

1 − A2−A11−A1

=Fvib,1

1−Frot,1

1 − Frot,2

1−Frot,1

. (24)

(4) A3 < Rn < A4

If A3 < Rn, then we have

R∗2 > Psim

vib,1 =A3−A21−A1

1 − A2−A11−A1

(25)

and R∗2 ∈ [Psim

vib,1, 1].If Rn < A4, then we should have

R∗2 <

A4−A21−A1

1 − A2−A11−A1

. (26)

Denote R3 = R∗2 − Psim

vib,1 = R∗2 −

A3−A21−A1

1− A2−A11−A1

, then R3 ∈ [0, 1 − Psimvib,1] = [0, 1 −

A3−A21−A1

1− A2−A11−A1

]. With Rn

< A4, we then have

R3 <

A4−A21−A1

1 − A2−A11−A1

−A3−A21−A1

1 − A2−A11−A1

=A4−A31−A1

1 − A2−A11−A1

, (27)

i.e.,

A4−A31−A1

1 − A2−A11−A1

> R3 ∈ [0, 1 − Psimvib,1] = [0, 1 −

A3−A21−A1

1 − A2−A11−A1

]. (28)

This is equivalent to

A4−A31−A1

1 − A2−A11−A1

/(1 − Psimvib,1) >

R3

1 − Psimvib,1

= R∗3 ∈ [0, 1], (29)

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106105-10 C. Zhang and T. E. Schwartzentruber Phys. Fluids 25, 106105 (2013)

where we have defined a new variable R∗3 = R3

1−Psimvib,1

. As a result, the DSMC simulated collision

probability for vibrational relaxation of particle 2 (particle B in Ref. 2) is

Psimvib,2 =

A4−A31−A1

1 − A2−A11−A1

/(1 − Psim

vib,1

) =Fvib,2

1−Frot,1

1 − Frot,2

1−Frot,1

/

(1 −

Fvib,1

1−Frot,1

1 − Frot,2

1−Frot,1

). (30)

By simplification, we have Psimrot,1 = Frot,1, Psim

rot,2 = Frot,2

1−Frot,1, Psim

vib,1 = Fvib,1

1−Frot,1−Frot,2, and Psim

vib,2

= Fvib,2

1−Frot,1−Frot,2−Fvib,2, which are the same as the collision probabilities given by the sequential prob-

ability selection procedure. Therefore, we have shown that the three approaches to select potentialinelastic collisions according to the particle selection prohibiting double relaxation procedure areequivalent and will give the same macroscopic relaxation rate.

4. Time cost comparison of the three approaches

In this subsection, we give a rough estimation of the time cost of the three different approachesfor particle selection prohibiting double relaxation. It should be noted, however, that the time cost ofselecting appropriate particles to relax contributes a negligible portion of the total DSMC simulationtime.

All operations required to calculate Prot, 1, etc., are floating points. For the purpose of comparison,we prescribe that the time cost of summation/subtraction is 1, the time cost of multiplication/divisionis 2, the time cost of a square root is 3, the time cost of an IF statement is 4, and the time cost forgenerating one random number is 5. As an example, assume that the rotational collision numbersare Zrot, 1 = 5, Zrot, 2 = 10, and the vibrational collision numbers are Zvib,1 = 100, Zvib,2 = 200.By counting the total number of operations involved in each selection procedure, we arrive at anapproximate total time cost for the three different particle selection prohibiting double relaxationprocedures. The results are shown in Table I, for both the constant collision number and temperaturedependent collision number cases. It should be noted, the time costs shown are rough estimations,and only account for the operations associated with the collision probability calculation, generatingrandom numbers for acceptance-rejection techniques, and IF statements used to determine potentialinelastic collision types. Again it is stressed that these time costs represent a negligible amount ofthe total DSMC simulation time cost.

C. Verification of the sequential probability selection procedure

To test the sequential probability selection procedure, we conduct an isothermal relaxationsimulation for a mixture of two species, where the translational temperature of the system is main-tained at a constant value of Tt = 10 000 K. To maintain the translational temperature of the systemat a constant value, each time step, we regenerate the velocities of all particles contained in thesimulation domain, following a Maxwell-Boltzmann distribution at Tt = 10 000 K. The rotationaland vibrational energies of the particles are not changed during this process. Since the translationaltemperature is constant during an isothermal relaxation simulation, the mean collision time is alsoa constant, and the resulting Jeans equation has an analytical solution.

As outlined in Ref. 1, for a multi-species gas mixture, the internal energy relaxation rate of aspecies j is determined through summing all inelastic contributions due to collisions with all possible

TABLE I. Time cost of the three approaches.

Approach Haas et al.1 Current approach Gimelshein et al.2

Constant Zrot, Zvib 1 1 0.47Variable Zrot, Zvib 1 0.41 0.27

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106105-11 C. Zhang and T. E. Schwartzentruber Phys. Fluids 25, 106105 (2013)

TABLE II. Simulation parameters specific to each collision pair.

j|k Zrot Zvib ωj, k d j,kre f (×10−10 m) Tre f (K)

1|1 5 40 0.74 4.17 2731|2 8 60 0.755 4.12 2732|1 10 80 0.755 4.12 2732|2 15 60 0.77 4.07 273

collision partners k in the system,

d Erot, j

dt=

∑k

E∗rot, j (t) − Erot, j (t)

τrot, j |k=

∑k

E∗rot, j (t) − Erot, j (t)

τc, j |k Zrot, j |k, (31a)

d Evib, j

dt=

∑k

E∗vib, j (t) − Evib, j (t)

τvib, j |k=

∑k

E∗vib, j (t) − Evib, j (t)

τc, j |k Zvib, j |k. (31b)

For isothermal relaxations, E∗rot, j (t) = Erot, j (∞) and E∗

vib, j (t) = Evib, j (∞). If rotational andvibrational relaxation times are assumed to depend only on translational temperature, then τ rot

= τ cZrot and τvib = τc Zvib are constant, and Eqs. (31a) and (31b) have the following analyticalsolution:

Erot, j (∞) − Erot, j (t)

Erot, j (∞) − Erot, j (0)= exp

(−

∑k

t

τc, j |k Zrot, j |k

), (32a)

Evib, j (∞) − Evib, j (t)

Evib, j (∞) − Evib, j (0)= exp

(−

∑k

t

τc, j |k Zvib, j |k

). (32b)

Using E j = ζ j

2 kB T , the above two equations can be written in terms of temperature, as

Trot, j (∞) − Trot, j (t)

Trot, j (∞) − Trot, j (0)= exp

(−

∑k

t

τc, j |k Zrot, j |k

), (33a)

t [s]

Tem

pera

ture

[K]

0 5E-09 1E-085000

6000

7000

8000

9000

10000

Tt

Tvib,1

Tvib,2

Tvib Analytical Solution

(b)

t [s]

Tem

pera

ture

[K]

0 1E-09 2E-095000

6000

7000

8000

9000

10000

Tt

Trot,1

Trot,2

Trot Analytical Solution

(a)

FIG. 2. The rotational and vibrational relaxation temperature in an isothermal reservoir simulation using the sequentialselection procedure: (a) Rotational temperature; (b) vibrational temperature.

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106105-12 C. Zhang and T. E. Schwartzentruber Phys. Fluids 25, 106105 (2013)

t [s]

Tem

pera

ture

[K]

0 1E-09 2E-09 3E-09 4E-095000

6000

7000

8000

9000

10000

Tt

Trot,1

Trot,2

Trot Analytical Solution

(a)

t [s]

Tem

pera

ture

[K]

0 1E-09 2E-09 3E-09 4E-095000

6000

7000

8000

9000

10000

Tt

Trot,1

Trot,2

Trot Analytical Solution

(b)

FIG. 3. Comparison of two different selection procedures for rotational relaxation of two species gas mixtures in an isothermalreservoir simulation: (a) Sequential selection procedure; (b) pair selection procedure.

ζvib, j (∞)Tvib, j (∞) − ζvib, j (t)Tvib, j (t)

ζvib, j (∞)Tvib, j (∞) − ζvib, j (0)Tvib, j (0)= exp

(−

∑k

t

τc, j |k Zvib, j |k

), (33b)

where ζvib, j (0), ζvib, j (∞) and ζvib, j (t) are the effective vibrational degrees of freedom at time 0, ∞,and t, respectively.

The specified rotational and vibrational collision numbers for the two species are listed inTable II, together with the VHS parameters used in the DSMC simulations. A characteristic vibra-tional temperature of θv = 3390 K is assumed for both species. The two species have mole fractionsof 0.3 and 0.7, respectively. The VHS model parameters used for the two species (ω, dref in Table II)correspond to those of N2 and O2, however, the rotational and vibrational collision numbers Zrot andZvib do not correspond to the values for those gas species, and are set here for demonstration pur-pose only. The simulation results are shown in Fig. 2 for the relaxation history of the rotational andvibrational temperatures of each species in the mixture. It is evident that the sequential probabilityselection procedure is able to accurately simulate the specified relaxation rate.

As a further demonstration, a similar isothermal relaxation simulation is conducted using thepair selection procedure (selection procedure (A)). Specifically the form of Eq. (4) as given by bothLumpkin et al.6 and Haas et al.1 is used. This simulation considers only rotational relaxation andthe rotational collision numbers are modified from Table II to be Zrot = 10 for collisions 1|2 and 2|1,and to be Zrot = 20 for collision 2|2. The results using the sequential probability selection procedureare shown in Fig. 3(a) and the results from the pair selection procedure (selection procedure (A))are shown in Fig. 3(b). Clearly, the results using the pair selection procedure do not agree with theanalytical solution for the mixture, whereas the sequential probability selection procedure exactlyreproduces the analytical solution in the same manner as the original procedure of Haas et al.1

III. CONCLUSIONS

A modification to existing inelastic collision selection procedures is presented, which is referredto as the sequential probability selection procedure. Simple expressions for the inelastic collisionprobabilities used in DSMC simulations are detailed that do not require the solution of a set ofquadratic equations for each collision. This modified procedure is analytically proven to be equivalentto both the original procedure of Haas et al.1 and the modified framework of Gimelshein et al.2

Thus, all three procedures (for particle selection prohibiting double relaxation) simulate the same

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106105-13 C. Zhang and T. E. Schwartzentruber Phys. Fluids 25, 106105 (2013)

internal energy relaxation processes in gas mixtures and the ability to accurately simulate prescribedphenomenological relaxation rates for mixtures is demonstrated. Accuracy of the modified selectionprocedure is verified through comparison with analytical solutions for rotational and vibrationalisothermal relaxations.

The modified procedure of Gimelshein et al.,2 that achieves specified relaxation rates using asingle random number for each collision, has the simplest implementation and is more computation-ally efficient than the sequential probability selection procedure presented in this article. However,the simulated probabilities do not explicitly appear in the algorithm by Gimelshein et al.,2 ratherthey result from inequalities evaluated using the random number. Whereas the sequential probabilityselection procedure uses the simulated probabilities directly within the algorithm. This differencedoes not affect the capabilities or predictions of either selection method and is simply a matter ofpreference between implementation styles.

It is noted that the computational cost of any of the three selection procedures is negligiblecompared to the overall cost of a DSMC simulation. Thus, the purpose of this article is to conveya clear understanding of the accuracy implications of the inelastic collision selection process forDSMC simulations of gas mixtures, to prove the equivalency of various selection procedures, toverify their ability to reproduce phenomenological internal energy relaxation rates in gas mixtures,and finally to detail the implementation of these procedures so that no inconsistencies are introducedwhen probability expressions (collision models) are transferred between DSMC implementationsand when DSMC simulations are compared with continuum simulations.

ACKNOWLEDGMENTS

This research is supported by NASA under Grant No. NNX11AC19G. The first author is alsosupported by the doctoral dissertation fellowship program at the University of Minnesota.

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