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1 4nd AIAA Aerospace Scences Meetng and Exhbt 5-8 January 004, Reno, Nevada AIAA nd AIAA Aerospace Scences Meetng and Exhbt Reno, Nevada 5-8 January 004 DEVELOPMENT OF A TWO-WAY COUPLED MODEL FOR TWO PHASE RAREFIED FLOWS ABSTRACT Based on a prevously publshed model for momentum and energy transfer to a sphercal sold partcle from a locally free molecular gas, a procedure s outlned for the smulaton of one-way coupled two phase flows nvolvng a nonequlbrum gas and a dlute sold partcle phase. Followng a smple analyss of nterphase collson dynamcs, the procedure s extended for use wth a range of nonsphercal partcles. An extensve modfcaton to ths method s proposed to allow the modelng of two-way coupled flows, and a representatve test case s used to verfy that momentum and energy are conserved. The method descrbed here s thought to be the frst to allow for the smulaton of two-way coupled two phase rarefed flows, and holds promse as a tool n the analyss of a varety of hgh alttude plume flows. INTRODUCTION Over the past several decades, much research has been focused on the dstrbuton and propertes of sold partcles n rocket nozzle, spacecraft thruster, and spacecraft fuel ventng flows. A varety of partcle types can be found n such flows, ncludng soot, partcles of ce or frozen fuel condensates, molecular clusters, and alumna. Ths last type has been the subject of several recent studes -6, and s extremely mportant n the analyss of sold propellant rocket plumes. Alumna partcles usually account for a large mass fracton among the consttuents ejected through a sold rocket nozzle, and are often the domnant contrbutor to the plume radaton sgnature. Analyss and predcton of the optcal propertes of the plume are therefore hghly dependent on the accuracy of algorthms for consderaton of the partcle phase. Furthermore, alumna partcles have been shown to develop sgnfcant velocty and temperature lags wthn both the nozzle and plume, and may nfluence the overall performance and effcency of the rocket motor. Partcle mpngement on nozzle walls or other surfaces may also be mportant consderatons, and can affect nozzle effcency or system relablty. In addton, the partcles may sgnfcantly nfluence the propertes of Graduate student, AIAA student member. Professor, AIAA assocate fellow. Jonathan M. Burt and Ian D. Boyd Department of Aerospace Engneerng Unversty of Mchgan, Ann Arbor, MI 4809 the surroundng gas, so that flow characterstcs are governed by complex two-way couplng between the two phases. These same effects may also be found n other multphase plume and free expanson flows. In lqudpropellant rocket plume flows, soot partcles often contrbute sgnfcantly to the radaton sgnature and may reduce engne performance. Partcles found n spacecraft thruster or fuel ventng flows can mpnge on and damage exposed surfaces. For these reasons, there s a desre for accurate methods to model the sold partcle phase n sold propellant rocket plumes, and n other rocket exhaust, spacecraft thruster, and fuel ventng flows. Exstng procedures for smulatng such flows use Computatonal Flud Dynamcs (CFD) technques to model the gas phase. The most ambtous studes currently n the lterature consder the gas usng three dmensonal or axsymmetrc fnte volume shockcapturng methods 7,8, whch allow for the accurate smulaton of certan nonequlbrum phenomena expected n rocket exhaust plumes at low alttudes. However, none of these smulaton methods are vald for hgh alttude two phase plumes, where the gas may exhbt hghly nonequlbrum behavor through much of the flowfeld, and where vrtually all CFD-based methods wll develop sgnfcant naccuracy. For the specal case of two phase free expanson flow nto a vacuum, these methods are characterzed by numercal dvergence, and the determnaton of any soluton may be mpossble. An alternate startng pont for hgh alttude plume smulatons s the Drect Smulaton Monte Carlo (DSMC) method 9, whch models the gas phase as a large collecton of computatonal partcles and makes no assumptons of contnuum or quas-equlbrum gas flow. Ths method has n the past been used extensvely to smulate plumes from hgh alttude rockets or spacecraft thrusters 0,, and has been shown to allow for a hgh degree of accuracy n the characterzaton of gas propertes n such flows. In a recent paper by Galls et al., an extenson of the DSMC method s proposed to enable the smulaton of rarefed flows nvolvng a dlute and chemcally nert sold partcle phase. Ths method has been fully mplemented wthn the exstng Copyrght 004 by Jonathan M. Burt and Ian D. Boyd. Publshed by the, Inc., wth permsson.

2 DSMC code MONACO 3 and modfed to allow for flows nvolvng a datomc gas. The mplementaton and valdaton of ths method are dscussed n a prevous paper 4, where comparsons are made wth results from an expermental study on the aerodynamc focusng of a partcle beam 5. One major assumpton of the Galls method s that only one-way couplng calculatons are requred, so that the partcle phase wll have a neglgble nfluence on the gas. As dscussed above, ths assumpton s often nvald for the sold propellant rocket plume flows, where nterphase momentum and energy transfer may sgnfcantly alter the gas propertes through much of the smulaton doman. These effects are ncorporated nto standard CFD codes for low-alttude analyss of sold-propellant rocket plumes, but t s thought that no approach has ever been used to model two-way coupled plumes at hgh alttudes, where the hghly nonequlbrum nature of the gas prevents accurate smulaton usng a CFD-based approach. A modfcaton to the Galls method, whch allows twoway couplng between the gas and partcle phases, can potentally overcome ths nherent lmtaton of CFD, and extend the alttudes and flow regmes for whch two phase plume flows may be accurately modeled. Ths paper presents a general set of procedures through whch a sold partcle phase may be ncluded wthn a DSMC smulaton, n order to model two phase rarefed flows. Frst, the one-way couplng method of Galls et al. s dscussed, and a summary s provded for the mplementaton of ths method as descrbed n Ref. (4). Partcle shape effects are then consdered, and the method s extended to allow for smulatons nvolvng a range of nonsphercal partcles. Followng a detaled analyss of gas molecule behavor durng an nterphase collson, a procedure s outlned to enable two-way couplng between the partcles and gas. Ths new method s then appled to model a test case, for whch condtons are smlar to those expected n a small sold propellant rocket flow. Smulaton results are dscussed, and t s shown that the method presented here s consstent wth the method of Galls et al. ONE-WAY COUPLING MODEL As descrbed n Ref. (4), the Galls model s used wthn a DSMC smulaton to calculate the rates of momentum and energy transfer from a locally free molecular gas to a sphercal sold partcle. Every computatonal gas molecule assgned to the same grd cell as the sold partcle s modeled as a large homogeneous collecton of actual gas molecules, a fracton of whch wll collde wth the partcle durng each tme step. As mplemented n Ref. (4), the gas molecules whch do collde are then ether reflected specularly off the partcle surface, or are dffusely reflected wth full thermal accommodaton to the partcle temperature. Among the basc assumptons of the model are that the partcle temperature s spatally unform (.e. the partcle Bot number s assumed to be much less than one) and that the sold partcle phase s dlute, so collsons between sold partcles are nfrequent and can be neglected. The contrbuton to the nterphase energy transfer rate of collson-nduced changes n the partcle knetc energy s assumed to be neglgble, so that the rate of change n partcle thermal energy wll equal the total rate at whch energy s transferred to the partcle from the gas. Through consderatons of momentum and energy conservaton, t can be shown that ths assumpton s vald f the partcle s much more massve than molecules n the surroundng gas, as s the case n all flows of nterest here. It s also assumed that collsons between reflected gas molecules and ncdent molecules wll have a neglgble nfluence on nterphase collson propertes, so that the surroundng flow can be modeled as locally free molecular for calculatons of momentum and energy transfer to the partcle. The assumpton of locally free molecular flow generally requres that the partcle Knudsen number, defned as the rato of the local gas mean free path to the effectve partcle radus (descrbed below), be of order one or greater. These assumptons are found to hold over a wde range of flow regmes, and are generally vald for the two phase free expanson flows of nterest. It s also assumed that the partcle s a perfect sphere. Ths assumpton can be relaxed, to nclude a range of nonsphercal partcles found under a wde varety of flow condtons. The consderaton of nonsphercal partcles s dscussed n detal below. One last assumpton of the Galls method s that the partcle phase has a neglgble effect on the surroundng gas. Ths assumpton can be relaxed as well; whle the Galls method addresses only the transfer of momentum and energy to the partcle resultng from nterphase collsons, a more general model descrbed below also consders the nfluence of these collsons on the gas. As dscussed n Ref. (4), the algorthm for ncludng sold partcles n a DSMC smulaton s based on a decouplng of nterphase momentum and energy transfer from the temporal varaton n partcle propertes. The total rates of momentum and energy transfer to a partcle are calculated durng each tme step, and afterward the temperature, velocty, and poston of the partcle are modfed. Note that vbratonal exctaton of a polyatomc gas s unlkely to have a sgnfcant nfluence on nterphase energy transfer for all flow condtons of nterest, so the vbratonal terms n the energy transfer equaton n Ref. (4) can be neglected. Wth ths modfcaton, the followng equatons are used to compute the rates of

3 energy and momentum addton to a sold partcle due to the presence of a sngle DSMC computatonal gas molecule wthn the same grd cell: F p = πrp Ng τ mcr+ π m kbt p u r () Vc 3 RpτNgcr Q π p = mcr+ e rot (+ Λ) kt () B p Vc Here R p s the radus of a sphercal partcle (or the effectve radus, as defned below, of a nonsphercal partcle), N g s the number of actual gas molecules represented by the computatonal molecule, τ s the thermal accommodaton coeffcent for the partcle surface, V c s the cell volume, m s the mass of a sngle gas molecule, u r s the relatve velocty of the gas molecule wth respect to the partcle, c r s the absolute value of u r, k B s Boltzmann s constant, T p s the partcle temperature, Λ s the number of rotatonal degrees of freedom of the gas, and e rot s the rotatonal energy for a sngle gas molecule. Note that, for smplcty, the word molecule s used here to descrbe ether a polyatomc gas molecule or a sngle atom n a monatomc gas. The total force and heat transfer rate on a sold partcle are found durng each tme step by evaluatng Eqs. () and () for all computatonal gas molecules n the cell, and summng the resultng values. The partcle velocty s then altered by the product of the total force vector and a factor t/m p, where t s the tme step sze and M p s the partcle mass. Smlarly, the partcle temperature s altered by the product of the total heat transfer rate and t/(c p M p ) where c p s the partcle specfc heat. Once new values of the partcle velocty u p and temperature T p have been determned, the partcle s moved through the grd by a dstance u p t. If necessary, further calculatons are then performed to reassgn the partcle to a new cell, account for a collson wth a sold wall, or remove the partcle from the smulaton. As mplemented for multphase flow smulatons, numerous partcles are tracked smultaneously through the grd, each representng a large number N p of actual sold partcles. Cell-averaged partcle propertes, such as number densty, temperature, and mean velocty, are averaged over several thousand tme steps to determne the overall characterstcs of the partcle phase. CONSIDERATION OF NONSPHERICAL PARTICLES Whle expermental studes have shown that alumna partcles n sold rocket exhaust flows tend to be nearly sphercal, nonsphercal partcles are promnent n other flows of nterest, ncludng lqud propellant rocket plume flows and spacecraft fuel ventng flows,6. Subject to the above assumptons on 3 whch the method of Galls et al. s based, these nonsphercal partcles can also be consdered through the followng analyss. Frst, a few addtonal assumptons must be made for any nonsphercal partcle: The partcle s assumed to have a convex shape, so that no outward vector orgnatng at a pont on the partcle surface wll ntersect the partcle surface at any other pont. Whle ths wll not be true for very complex partcles such as soot agglomerates, t s generally vald for a wde range of partcles, ncludng many partcles formed durng spacecraft fuel ventng. In addton, the partcle s assumed to move through the gas wth an sotropc dstrbuton of orentatons relatve to any fxed coordnate system, so that no one orentaton s more lkely than any other. Whle ths mples that a nonsphercal partcle must be rotatng, t s further assumed that any rotaton effects partcle angular momentum, the sde force due to an asymmetrc surface pressure dstrbuton, rotatonnduced tme varaton n nterphase momentum and heat transfer, etc. are relatvely small and can be neglected. These assumptons are expected to be vald over all relevant flow regmes for a varety of partcle types. Subject to the above assumptons, the convectve heat transfer rate between a sold partcle and the surroundng gas wll depend on the partcle shape only through the value of the average collson cross secton for nterphase collsons, gven here as σ. Furthermore, the average momentum transfer rate wll depend on the partcle shape only through the value of σ and through the dstrbuton functon of the collson angle θ. Here θ s defned as the angle between the relatve velocty u r = u m u p of an ncdent gas molecule and an outward normal vector at the collson pont on the partcle surface, where u m and u p are the veloctes of the gas molecule and partcle, respectvely, n a fxed reference frame. Thus, f the partcle shape dependence for σ and the dstrbuton functon f(θ) can be found, then subject to the assumptons descrbed above, the nfluence of partcle shape on the rates of nterphase momentum and energy transfer can be determned. Frst consder the dependence of f(θ) on the partcle shape. For an arbtrary convex partcle, let the partcle surface be dvded nto a large number N of flat surface elements, each of area A. Now assume that a gas molecule colldes wth the partcle on a partcular surface element. If all orentatons of the partcle for whch ths collson may occur are equally lkely, then the relatve velocty vector of the ncdent molecule has an sotropc dstrbuton over θ [0,π/]. Ths vector wll be contaned wthn a sold angle element of sze snθdθdφ, where φ s the azmuthal angle relatve to some reference drecton n the plane of the surface

4 element. Therefore f(θ) must be proportonal to the sze of the sold angle, so that f(θ) snθ. Now, f we remove the requrement that the collson occurs on the surface element, and only assume that a collson does occur somewhere on the partcle surface, then the probablty that the collson pont wll be located on element must be proportonal to the projected area of ths element n the drecton of the relatve velocty vector u r. Thus f(θ) s proportonal to ths projected area Acosθ, so that f(θ) cosθ. By the above arguments, f(θ) must then be proportonal to snθcosθ. Applyng a trgonometrc dentty and the normalzaton condton π / 0 f(θ) dθ =, we fnd that f(θ) = snθ. (3) As ths dstrbuton functon s vald for any surface element, t must also be vald for the partcle as a whole. Note that f(θ) wll have no dependence on the partcle shape, so long as the above assumptons are vald. Next consder the partcle shape dependence of the average collson cross secton σ. As above, assume that the partcle surface s made up of a large number N of flat surface elements, each wth area A. Now defne θ as the angle between an outward normal vector on a gven element and the relatve velocty vector u r of an ncdent gas molecule. Under the assumpton that the gas molecule s much smaller than the partcle, the nstantaneous collson cross secton σ wll be the sum of the projected areas of all exposed faces: N σ = Amax{cosθ,0} (4) = The average collson cross secton σ can then be approxmated as the average of a large number M of σ values, each of whch corresponds to a randomly chosen partcle orentaton relatve to u r. If θ j s the value of θ on surface element for the jth realzaton of σ, then σ can be gven by M N σ = lm A max{cosθ j,0} M M j= = N M = Alm max{cosθ j,0} M = M j= N = = π ( A g(θ )max{cosθ,0} θ 0 ) d (5) Here g(θ ) s the dstrbuton functon of θ for a collson whch may occur anywhere on the partcle surface. From the sold angle argument used n the dervaton of Eq. (3), t can be shown that g(θ ) snθ f the partcle has no preferred orentaton relatve to u r. Applyng the normalzaton condton g(θ ) dθ =, we fnd that g(θ) = ½ snθ for θ [0,π], so π 0 4 π 0 g(θ )max{cosθ,0} dθ = ¼. If the total surface area of the partcle s A s = N = A then substtuton nto Eq. (5) gves the fnal result that σ = ¼ A s. Thus, for a convex partcle of arbtrary shape, the average collson cross secton wll be one fourth of the partcle surface area. As f(θ) has no partcle shape dependence and σ depends only on the partcle surface area, then subject to the assumptons descrbed above, any convex partcle can be modeled as a sphercal partcle of the same surface area for calculatons of nterphase momentum and energy transfer. Followng a standard conventon 7, let the partcle shape be characterzed by a shape factor ψ A o /A s, where A o s the surface area of a sphere wth the same volume as the partcle. Defne R o as the radus of ths same sphere, whch for most partcle shapes wll be comparable to one-half of some characterstc average partcle length. The effectve partcle radus R p, for use n momentum and energy transfer calculatons, can then be determned from known values of ψ and R o through the followng relaton: R p = R o ψ - ½ (6) As R p R o t follows that a convex nonsphercal partcle n locally free molecular flow wll behave lke a sphercal partcle of equal mass but greater volume. Thus, the larger effectve radus for a nonsphercal partcle wll be accompaned by a reducton n the effectve partcle densty. If the partcle mass M p s calculated as M p =4/3πρ p R 3 p then the effectve partcle densty ρ p can be found through the relaton ρ p = ρ o ψ 3/, where ρ o s the actual densty of the partcle materal. Through ths analyss, the sold partcle model of Galls et al., as well as a two-way couplng method dscussed below, can be extended and appled to a varety of nonsphercal partcles. Whle the analyss s not strctly vald for partcles wth hghly complcated non-convex shapes, such as soot agglomerates, t s thought that ths can provde at least a frst-order approxmaton for the propertes of such partcles when ncluded n a smulaton. TWO-WAY COUPLING MODEL As dscussed n the ntroducton, sold rocket plume flows and other two phase free expanson flows of nterest are often characterzed by a consderable transfer of momentum and energy between the gas and sold partcles, such that the propertes of each phase are sgnfcantly affected by the presence of the other. Under these condtons, the Galls model assumpton of one-way couplng s nvald, and the nfluence of partcles on the surroundng gas must be consdered. Whle the procedure outlned above may stll be used to model the tme-varaton of partcle propertes,

5 addtonal steps must be ncluded n the calculatons to account for potentally sgnfcant two-way couplng effects. The followng analyss provdes a physcal model for the effect of an nterphase collson on a gas molecule, and allows for a numercal procedure through whch two-way couplng may be consdered. Frst, note that all assumptons lsted above for the Galls method are agan used for consderaton of momentum and energy couplng from a partcle to the surroundng gas. Most mportantly, the partcle s assumed to be n a locally free molecular flow, so that any nfluence of reflected gas molecules on an ncdent molecule can be neglected durng the collson process. The characterstcs of the collson wll therefore depend only on the propertes of the partcle and the sngle gas molecule nvolved n the collson. Further, all nterphase collsons must nvolve ether specular reflecton or dffuse reflecton wth full thermal accommodaton. Whle ths s a relatvely smplstc and phenomenologcal collson model, s has been shown expermentally to allow for a hgh degree of accuracy over a wde range of condtons, and s used as well n Eqs. () and (). Now consder the collson process between an ndvdual gas molecule and a sphercal sold partcle. As shown above, the collson angle θ between the ntal relatve velocty vector u r and the local partcle surface normal at the collson pont wll have a range of [0,π/] and a dstrbuton functon gven by Eq. (3). Let δ represent the deflecton angle n the collson, defned as the angle between -u r and the post-collson relatve velocty vector u r = u m u p, where u m and u p are the absolute velocty vectors of the gas molecule and the partcle respectvely followng the collson. (The superscrpt s used here to denote any postcollson value.) Thus, a δ value of zero s equvalent to the relaton u r = -u r. For a collson nvolvng specular reflecton, any gven θ wll correspond to a δ value of θ. The dstrbuton functons for θ and δ can then be related by f(δ)dδ = f(θ)dθ, so that, from Eq. (3), the deflecton angle for a specularly reflectng collson wll have the followng dstrbuton: f(δ) = ½ snδ for δ [0,π] (7) Note that the azmuthal angle ε of the vector u r, relatve to a fxed drecton n the plane normal to u r, must have a unform dstrbuton over [0,π]. From comparson wth the dstrbuton functon g(θ ) dscussed above, t can therefore be shown that Eq. (7) corresponds to a total lack of drectonal dependence n u r. Thus, followng a specularly reflectng collson, the relatve velocty of the gas molecule wll have a magntude of c r = u r and may be orented wth equal probablty n any drecton. If the collson nstead nvolves dffuse reflecton, then the collson dynamcs are far more complcated, and only a numercal approxmaton for the deflecton angle dstrbuton functon f(δ) can be determned. In order to fnd ths expresson, two coordnate systems must now be used: Frst, let a coordnate system (x,y,z) be defned so that the orgn s at the partcle center, the y-axs s parallel to the ntal relatve velocty vector u r, and the collson pont s located on the x-y plane. For the second coordnate system (x,y,z ), the orgn s at the collson pont, the y -axs s along the local surface normal, and the partcle center s on the x -y plane. Both coordnate systems are shown n Fg (), as are the relevant angles descrbed below. Fgure. Coordnate systems and angles used n the evaluaton of f(δ) for dffuse reflecton. Next, let ϕ denote the angle between the postcollson relatve velocty vector u r and the y -axs, and desgnate as χ the azmuthal angle between the x -axs and the projecton of u r onto the x -z plane. Whle specular reflecton requres that ϕ = θ and χ = 0, n the case of dffuse reflecton ϕ wll have a contnuous dstrbuton over [0,π/] and χ wll be unformly dstrbuted over [-π,π]. Through further analyss, t can be shown that the probablty that the post-collson trajectory of a dffusely reflectng molecule wll be contaned wthn the dfferental sold angle dϕ snϕ dχ centered at (ϕ, χ) must be proportonal to both cosϕ and the sze of the sold angle. By mposng the normalzaton condton and a trgonometrc dentty, we fnd the followng form for the dstrbuton functon of ϕ: f(ϕ) = sn(ϕ) for ϕ [0,π/] (8) As shown n the appendx, the angles θ, ϕ, and χ can be related to the total deflecton angle δ by: / (snϕ snχ) = ϕ cos δ (cosθ sn θ tan cos χ) + (tan ϕ cosχ) 5 (9)

6 A Monte Carlo ntegraton method may be employed to determne the shape of the dstrbuton functon for δ. Values of θ and φ are generated by applyng the acceptance-rejecton method 9 to Eqs. (3) and (8), and χ values are randomly generated wth unform probablty over the range [-π,π]. Eq. (9) s then used to calculate the correspondng values of δ. These values are sorted nto bns of fnte wdth δ, and the frequency that δ values fall wthn each bn s recorded to produce a hstogram that approxmates the shape of the dstrbuton functon for δ over [0,π]. The resultng shape s found to be closely approxmated by the followng sxth-order polynomal: f(δ) = 0.004δ δ δ 4.903δ δ +.48δ (0) As the deflecton angle δ and post-collson relatve speed c r can be shown to be statstcally ndependent n a dffusely reflectng collson, Eq. (0) s vald for any molecule-partcle collson par for whch dffuse reflecton s nvolved. Note that a sphercal partcle has been used here for smplcty, but the above analyss allows Eq. (0) to be extended to a range of nonsphercal partcles. Both the numercal soluton and the polynomal approxmaton are shown n Fg. (), along wth the equvalent dstrbuton functon for specular reflecton. Fgure. Comparson of dstrbuton functons for deflecton angle δ. The above dstrbuton functons are utlzed n the followng procedure, whch allows a sold partcle n a two phase DSMC smulaton to nfluence the surroundng gas. We frst determne whch, f any, computatonal gas molecules wll collde wth the partcle durng each tme step. A modfcaton of the No Tme Counter method of Brd 9 s used to fnd the number n s of computatonal gas molecules that are selected as potental collson partners for the partcle. The value of n s s roughly gven by 6 n s N p n g πr p (c r ) max t/v c () where N p s the number of actual sold partcles represented by the computaton partcle, n g s the number of computatonal gas molecules assgned to the same grd cell as the partcle, R p s the effectve partcle radus, t s the tme step, V c s the cell volume, and (c r ) max s the maxmum pre-collson relatve speed, over a large number of tme steps, for any molecule-partcle par n ths cell. Note that n s must be an nteger, so a probablstc samplng method s used to round the rght sde of () ether up or down such that the average values of both sdes are equal. Once n s molecules have been chosen as potental collson partners, those that do collde are selected wth probablty c r /(c r ) max. It can be shown that ths selecton scheme corresponds to a probablty P coll = πn p R p c r t/v c that the partcle wll collde wth a gven molecule n the cell. Due to tme step lmtatons nherent n DSMC, t has been found that P coll values are almost unversally several orders of magntude smaller than one, so that the number of collsons per partcle per tme step s usually zero and s rarely greater than one. If a gven computatonal gas molecule s found to collde wth the partcle, then the collson s determned to nvolve ether sothermal dffuse reflecton, wth a probablty equal to the partcle thermal accommodaton coeffcent τ, or specular reflecton, wth probablty τ. If a specularly reflectng collson takes place, then the relatve speed c r s unchanged n the collson, and the post-collson relatve velocty u r s found by multplyng c r by a unt vector n whch s sampled from an sotropc dstrbuton. (An effcent algorthm for calculatng n s descrbed n Ref. (9).) If dffuse reflecton occurs, then the acceptance-rejecton method s appled to Eq. (0) to fnd a value for δ, and the azmuthal angle ε of the post-collson relatve velocty u r around the ntal relatve velocty vector u r s randomly generated from a unform dstrbuton over [0,π]. As knetc energy s not conserved n dffusely reflectng collsons, the post-collson relatve speed c r cannot be assumed to equal the ntal relatve speed c r. Instead, a value of c r must be determned by applyng the acceptancerejecton method to the dstrbuton functon f(c r ) = β 4 c 3 r exp(-β c r ) () where β = [m/(k B T p )] ½ s the nverse of the gas thermal speed scale at the partcle temperature. For the case of a dffusely reflected polyatomc gas molecule, the post-collson value of the rotatonal energy e rot must also be altered. From Eq. (C6) n Ref. (9), the rotatonal energy of a dffusely reflected datomc molecule can be calculated as e rot = ln(r f )k B T P (3) where R f s a randomly generated number n (0,]. As noted above, vbratonal actvaton s assumed to be

7 neglgble for all flows of nterest, so that no vbratonal terms are ncluded n Eq. () and conservaton of energy requres that the gas molecule vbratonal energy not be altered durng an nterphase collson. Now let u r, v r, and w r be the components of u r n the global coordnate system used n the smulaton. The correspondng components of u r can be computed from the values of u r, v r, w r, c r and c r, and the angles δ and ε usng modfcatons of equatons derved by Brd 9 for bnary elastc collsons. As modfed for use wth the above varables, these equatons are wrtten as: cr u r = -urcosδ+snδ snε(v r+w r) c ½ r v = c r -vrcosδ+snδ(crw rcosε-u rvrsnε) (4) r c (v +w ) ½ r r r w = c r -wrcosδ-snδ(crvrcosε+urw rsnε) r c (v +w ) ½ r r r Eqs. (4) are evaluated to fnd u r f the collson nvolves dffuse reflecton. Once the components of u r have been calculated for ether type of collson, the absolute gas molecule velocty s updated to a fnal value of u m = u r +u p, where u p s the velocty assgned to the sold partcle. Note that the collson-nduced velocty dfference for the partcle s assumed much smaller than that of the molecule. Ths follows from the prevous assumpton that the partcle s much more massve than the partcle, and allows the true postcollson partcle velocty u p to be replaced by u p for the calculaton of u m. Ths procedure s repeated for each sold partcle durng every tme step. It can be shown, by ntegraton of the dstrbuton functons gven above, that the average momentum and energy mparted on a gas molecule through a collson are equal n magntude, respectvely, to the average momentum and energy transfer rates to the sold partcle (gven by Eqs. () and ()) multpled by the rato of the tme step t to the collson probablty P coll. Ths property has been verfed for both specular and dffuse reflecton, and confrms that the method descrbed here s consstent wth the force and heat transfer equatons of Galls et al. In addton, and n contrast to the one-way coupled method descrbed above, the total momentum and energy of the flow are now both conserved n an average sense. EXAMPLE SIMULATION AND RESULTS In order to demonstrate the consstency of ths new method wth the one-way couplng method dscussed above, a sample smulaton s performed. All flow propertes are based on those expected along the axs and just beyond the nozzle ext plane n the exhaust flow of a small sold propellant rocket. The smulaton geometry s smplfed n order to solate the effects of 7 gas-partcle nteracton, and only a small doman s consdered to lmt the computatonal expense. The smulaton s performed on a rectangular twodmensonal grd, consstng of 0. mm long unform nflow and outflow boundares, separated on ether end by 0 mm long specularly reflectng walls. The grd geometry s shown n Fg. (3). As no energy or longtudnal momentum may be transferred through the walls, t can be expected that, f the new two-way couplng method s physcally consstent, the total momentum and energy flux wll be the same over any transverse plane whch passes through the grd. Fgure 3. Grd dmensons and boundary types. The gas n ths smulaton s a mxture of H, CO and N, wth nflow number denstes of 0 3 m -3, 0 3 m -3 and 0 3 m -3 respectvely. At the nflow boundary the gas s assgned a bulk speed of 000 m/s and a temperature of 000K. The sold phase conssts of sphercal alumna partcles, of dameter m and m, wth a mass flow fracton of 40% dvded equally between partcles of ether sze. All partcles have a velocty of 00 m/s and a temperature of 00 K at the nflow boundary, wth a total partcle mass flow rate of 3.33 kg/s-m. The partcle materal densty s set as 3970 kg/m 3, wth a specfc heat of 765 J/kg-K and a surface thermal accommodaton coeffcent of The grd s dvded nto 5000 square cells of length 0-5 m, or approxmately two mean free paths. Collsons wthn the gas phase are consdered usng the Varable Hard Sphere model, wth reference molecular dameters gven by Brd 9. The tme step sze s s and the relatve weghts N g and N p are set so that, at steady state, about 70,000 computatonal gas molecules and 0,000 sold partcles are contaned wthn the grd. Ths corresponds to roughly 54 computatonal gas molecules and sold partcles per cell. Once steady state has been reached, varous gas and partcle propertes are evaluated, and averagng s performed over approxmately 00,000 tme steps. Smulaton results are shown n Fgs. (4), (5) and (6), based on values extracted along a lne between the centers of the nflow and outflow boundares. Due to the relatvely small changes n the characterstcs of ether phase expected over the lmted grd doman, the

8 nterphase transfer of momentum and energy should be nearly constant wth downstream dstance, so that most flow propertes wll vary lnearly through the grd. The expected lnear varaton s observed n Fg. (4), n whch the average gas and partcle speeds are plotted aganst longtudnal dstance from the nflow boundary. The slower partcle phase s found to accelerate at a nearly constant rate, whle the faster gas decelerates as momentum s transferred to the partcles. Fg. (5) shows the correspondng trends n the gas and partcle number denstes. Neglectng the sgnfcant statstcal scatter, the partcle number densty s found to decrease approxmately lnearly and the gas number densty s found to ncrease lnearly, as s expected from comparson wth Fg. (4) and consderatons of mass conservaton. In Fg. (6) the longtudnal varatons n the average partcle temperature and gas translatonal temperature are shown. The gas and partcle temperatures are found to ncrease and decrease respectvely wth downstream dstance, as energy s transferred to the gas from the hgher temperature partcles. average gas speed (m/s) gas partcles x (m) Fgure 4. Longtudnal varaton n average gas and partcle speeds. gas number densty (x0 3 m -3 ) gas partcles x (m) Fgure 5. Gas and partcle number denstes.. average partcle speed (m/s) partcle number densty (x0 m -3 ) gas translatonal temperature (K) gas 0 partcles x (m) Fgure 6. Varaton n gas and partcle temperatures. As noted above, the accuracy and consstency of the two-way couplng method can be determned by verfyng that momentum and energy are transferred between the two phases at equal rates, so that the total momentum or energy transfer rate wll be the same through any transverse plane whch ntersects the grd doman. In order to show ths, momentum and energy transfer rates of each phase are calculated along nne equally spaced planes. Whle approxmatons for these rates could be easly found through algebrac manpulatons of the cell-averaged veloctes, denstes, and temperatures, the small sze of the grd and the extreme senstvty of flux values to statstcal scatter requre that a more drect approach be used. The alternate procedure s as follows: Durng every tme step for whch samplng s performed, t s determned whch computatonal gas molecules or sold partcles pass through each of the nne planes. Values are recorded for the mass, momentum, and energy transferred through each plane, to whch the mass, longtudnal momentum, knetc energy, and nternal energy of each of these objects s ether added or subtracted, dependng on the drecton n whch the object passes through the plane. Resultng values are then dvded by the tme step sze t, and averagng s performed over all samplng tme steps. Each tmeaveraged momentum and energy transfer value s then dvded by the rato of the correspondng mass transfer rate to the average mass transfer rate assgned at the nflow boundary (equal to 0.00 kg/s for the gas and kg/s for the partcles). Note that ths last step s requred to account for statstcal fluctuatons n the number of objects whch pass through these planes. Ths s also necessary to correct for a slght reducton n the gas number flux wth downstream dstance, due to the fact that computatonal gas molecules may ext the grd through the nflow boundary. The correcton s only on the order of 0.%, but s found to sgnfcantly mprove the average partcle temperature (K) 8

9 accuracy of results, as the values of nterest wll vary only slghtly through the length of the grd. The varaton n longtudnal momentum transfer rates s shown n Fg. (7). Whle dfferent scales are used for the partcles and gas, the ranges of both scales are equal, so that trends n the two profles can be easly compared. As expected, momentum s observed to be removed from the gas at nearly the exact rate that momentum s added to the partcle phase. Both data sets are closely approxmated by lnear leastsquares trend lnes, wth slopes that dffer n magntude by less than %. Smlar trends can be found n Fg. (8), whch shows the varaton n energy transfer rates wth downstream dstance. Agan, the magntudes of lnear trend lnes are nearly equal, and energy s observed to be removed from the partcles at approxmately the same rate that energy s added to the gas. gas momentum transfer rate (N) gas partcles x (m) Fgure 7. Varaton n longtudnal momentum transfer rates wth downstream dstance. gas energy transfer rate (W) gas partcles x (m) Fgure 8. Energy transfer rates for gas and partcles. The small dscrepances whch are observed n Fgs. (7) and (8), and the local varaton n the total momentum and energy transfer rates whch these dscrepances mply, can be explaned by a number of partcle momentum transfer rate (N) partcle energy transfer rate (W) 9 factors. Frst, some error may be due to the collson selecton scheme gven by Eq. (), as gas molecules most lkely to collde wth a gven partcle may not be chosen as potental collson partners. Ths may slghtly reduce the nterphase collson frequency, and s a problem nherent n the No Tme Counter method on whch the collson selecton scheme s based. A more lkely error source s the fact that momentum and energy are conserved only n a tme-averaged sense, as the nstantaneous momentum and energy transfer arsng from a sngle nterphase collson s not modeled n the same manner for calculatons used to alter propertes of the two dfferent phases. Ths dfference n collson modelng arses from the wde dsparty expected n number densty between sold partcles and gas molecules, so that the nterphase collson frequency for a sngle sold partcle s lkely several orders of magntude larger than that of a gas molecule. (For the smulaton descrbed here, these two collson frequences dffer by a factor of about 4 0.) The lack of exact momentum and energy conservaton should gve rse to random walk errors, whch are magnfed n the results of Fgs. (7) and (8) due to the extreme numercal senstvty of the observed trends. However, these errors are shown to be relatvely small, and are expected to further decrease when the samplng perod s lengthened. As the DSMC method generally requres tme (or ensemble) averagng over a large number of tme steps, nstantaneous momentum and energy conservaton should not be requred to acheve levels of accuracy n the smulaton results wthn the expected statstcal scatter. It can therefore be assumed that Fgs. (7) and (8) do adequately demonstrate the conservaton of momentum and energy, so that the two-way couplng method s physcal consstent. The ntal one-way couplng method s shown n Refs. () and (4) to exhbt a hgh degree of accuracy, so t follows from the above arguments that the new method should be reasonably accurate as well. CONCLUSIONS AND FUTURE WORK An outlne has been provded for the mplementaton of the method of Galls et al. to smulate one-way coupled two phase rarefed flows, and the method has been extended for use wth a range of nonsphercal partcles. A new method has been developed n order to account for two-way couplng effects, and to broaden the range of flow regmes whch may be accurately modeled. It s thought that the method descrbed here s the frst to allow for the smulaton of two-way coupled two phase flows nvolvng a hghly nonequlbrum gas. These condtons are commonly found n hgh alttude plume flows from sold propellant rockets, and may also occur n spacecraft fuel ventng and thruster flows. The

10 method therefore holds promse as an mportant tool n the analyss of a varety of multphase hgh alttude plumes. The work dscussed here s part of an ongong project to develop and mplement accurate modelng technques for hgh alttude plume and fuel ventng flows. Future studes wll consder reductons n computatonal cost through a seres of nterphase couplng parameters, as well as the mplementaton of models for partcle formaton, surface chemstry, and phase change. A detaled radaton model wll be developed n order to accurately account for radatve heat transfer from and wthn the partcle phase, and to provde capabltes for the analyss of plume radaton sgnatures. These models wll be used n a varety of large scale smulatons, whch wll also be developed n future work. ACKNOWLEDGMENTS The authors gratefully acknowledge the fnancal support for ths work provded by the Ar Force Research Laboratory at Edwards Ar Force Base, wth Dean Wadsworth and Tom Smth as techncal montors. REFERENCES. Smmons, F. S., Rocket Exhaust Plume Phenomenology, Aerospace Press, El Segundo, CA, 000, pp Rattenn, L., Sold Motor Plume Analyses for the STAR- Spacecraft, AIAA Paper , Cucc, A., and Iaccarno, G., Numercal Analyss of the Turbulent Flow and Alumna Partcle Trajectores n Sold Rocket Motors, AIAA Paper , Betng, E., Predcted Physcal and Optcal Characterstcs of Sold Rocket Motor Exhaust n the Stratosphere, AIAA Paper , Kovalev, O., Motor and Plume Partcle Sze Predcton n Sold-Propellant Rocket Motors, Journal of Propulson and Power, Vol. 8, No. 6, 00, pp Rodonov, A. V., Plastnn, Y. A., Drakes, J. A., Smmons, M. A., and Hers, R. S., Modelng of Multphase Alumna-Loaded Jet Flow Felds, AIAA Paper , York, B. J., Lee, R. A., Snha, N., and Dash, S. M., Progress n the Smulaton of Partculate Interactons n Sold Propellant Rocket Exhausts, AIAA Paper , Dash, S. M., Wolf, D. E., Beddn, R. A., and Pergament, H. S., Analyss of Two-Phase Flow Processes n Rocket Exhaust Plumes, Journal of Spacecraft and Rockets, Vol., No. 3, 985, pp Brd, G. A., Molecular Gas Dynamcs and the Drect Smulaton of Gas Flows, Clarendon Press, Oxford, 994, pp. 36, 03-04, 8-0, , 45, Boyd, I. D., Penko, P. F., Messner, D. L., and DeWtt, K. J., Expermental and Numercal Investgatons of Low-Densty Nozzle and Plume Flows of Ntrogen, AIAA Journal, Vol. 30, No. 0, 99, pp Penko, P. F., Boyd, I. D., Messner, D. L., and DeWtt, K. J., Measurement and Analyss of a Small Nozzle Plume n Vacuum, Journal of Propulson and Power, Vol. 9, No. 4, 993, pp Galls, M. A., Torczynsk, J. R., and Rader, D. J., An Approach for Smulatng the Transport of Sphercal Partcles n a Rarefed Gas Flow va the Drect Smulaton Monte Carlo Method, Physcs of Fluds, Vol. 3, No., 00, pp Detrch, S., and Boyd, I. D., Scalar and Parallel Optmzed Implementaton of the Drect Smulaton Monte Carlo Method, Journal of Computatonal Physcs, Vol. 6, 996, pp Burt, J. M., and Boyd, I. D., Evaluaton of a Monte Carlo Model for Two Phase Rarefed Flows, AIAA Paper , Israel, G., and Fredlander, K., Hgh-Speed Beams of Small Partcles, Journal of Collod and Interface Scence, Vol. 4, 967, pp Kassal, T. T., Scatterng Propertes of Ice Partcles Formed by Release of H O n Vacuum, Journal of Spacecraft, Vol., No., 974, pp Crowe, C., Sommerfeld, M., and Tsuj, Y., Multphase Flows wth Droplets and Partcles, CRC Press, New York, 998, pp. 9. APPENDIX A proof s provded for the relaton between the angles θ, φ, χ, and δ gven as Eq. (9), based on the geometry shown n Fg. (). Frst, defne a unt vector n n the drecton of the post-collson relatve velocty vector u r, wth components (n x, n y, n z ) and (n x, n y, n z ) n the (x,y,z) and (x,y,z ) coordnate systems respectvely. Two addtonal angles must also be defned: Denote as ξ the angle between u r and the projecton of u r onto the x -y plane, and let ω be the angle between ths projecton and the y -axs. The followng expressons can be found for ω, χ, and ϕ n terms of the components of n: 0

11 tan ω=n /n tan ϕ n +n x' y' = x' z' ny' cos χ =nx' n x' +n z' Substtuton then gves tan ω = tan ϕ cos χ (5) Smlarly, the relatons sn ξ =n sn ϕ = n +n sn χ =n n +n z' x' z' z' x' z' are used to fnd an expresson for ξ n terms of ϕ and χ: sn ξ = sn ϕ sn χ (6) Next, δ can be related to θ, ω, and ξ by recognzng that cos δ =n cos(θ+ω)=n n +n y cosξ= n +n y y Then by substtuton and a trgonometry dentty: cos δ = cos ξ cos (θ+ω) = cos ξ (cos θ cos ω - sn θ sn ω) (7) Note that both ξ and ω are confned to the range [-π/, π/], hence cosξ = -sn ξ, cos ω = ( + tan ω) /, and / sn ω = tan ω( + tan ω). Usng these relatons, Eqs. (5) and (6) are substtuted nto Eq. (7) to gve an expresson for δ n terms of θ, χ, and ϕ. After some algebrac smplfcaton, ths expresson can be wrtten as (snϕ snχ) cos δ = (cos θ sn θ tanϕ cos χ) (tan cosχ) + ϕ x x / y.