Effect of formation of argon clusters on a supersonic outflow into a rarefied medium

. A model of the argon condensation process has been developed for the direct simulation Monte Carlo (DSMC) method. The model includes the following processes: formation of dimers, merging of clusters, sticking of monomers to clusters, evaporation of monomers from clusters, and inelastic collisions between particles. The proposed model is implemented in the DSMC code SMILE++. The model is then verified and validated. Calculations of jet flows exhausting into a rarefied medium are performed on the basis of the proposed model.


Introduction
There * are numerous scientific studies, both experimental and numerical [1][2][3][4], dealing with various issues of a steady gas outflow into a rarefied medium.Such a great research interest is associated, first, with the continuing importance of the problem and, second, with the need for a general approach to its solution.The relevance of the problem is related to the necessity of developing new engines operating directly in the open space and thrusters for satellites of various sizes.The difficulty of the problem lies in the fact that numerical simulation requires a method that would be able to take into account the multiphase nature of the flow, the transition from the continuum to the free-molecular flow regime, as well as nonequilibrium physical processes and condensation.
We study a kinetic numerical model of the process of homogeneous condensation of argon in a jet flowing into a rarefied medium for the direct simulation Monte Carlo (DSMC) method developed on the basis of the model proposed in [5].The model makes it possible to take into account the effects of rarefaction, nonequilibrium, and condensation arising in the flow; it includes the process of three-particle recombination leading to formation of dimers, attachment of monomers to clusters, evaporation of monomers from clusters, and coalescence of clusters.
The condensation model was implemented in the DSMC code SMILE++ [6,7] developed at the Laboratory of Computational Aerodynamics of ITAM SB RAS.

Nozzle Expansion
We consider a stationary process of gas outflow from a settling chamber containing a gas with a temperature T0 and pressure p0 through a nozzle into an expansion chamber, in which the gas temperature and pressure are equal to Th and ph, respectively.The pressure ph is close to vacuum.
The decrease in density during gas expansion reduces the number of intermolecular collisions, thus, hindering the instant energy exchange among the molecules and leading to thermal nonequilibrium.The decrease in temperature leads to conditions under which gas condensation begins.
The flow expansion character downstream from the nozzle throat changing from a locally equilibrium and continuum to free-molecular and nonequilibrium flow which is demonstrated in Fig. 1.The transition is determined by the Knudsen number where λ is the mean free path, d is the characteristic length.

Model description
The model of the argon condensation process for the DSMC method includes the following processes: formation of dimers, merging of clusters, sticking of monomers to clusters, evaporation of monomers from clusters, and inelastic collisions between particles.
When inelastic collisions between particles occur, the energy is redistributed between the translational and internal modes of clusters and monomers.The Larsen-Borgnakke model [8] was used to describe the energy exchange during collisions.The energy exchange between different modes occurs with a probability Z -1 , and an elastic collision without energy exchange occurs with a probability 1-Z -1 .
The evaporation process was considered as a monomolecular reaction of cluster decay 1

Ar
Ar Ar.
n n The formula derived from the Rice-Ramsperger-Kassel (RRK) theory [9] was used to calculate the evaporation frequency of monomers.The frequency of monomer evaporation from clusters of size n was [5] ( ) where ν is the characteristic vibration frequency of monomers in the cluster, Ns is the number of surface monomers, Eint is the internal energy of the cluster, Eevap is the monomer evaporation energy, and ξvib is the number of vibrational degrees of freedom; for the dimer, the exponent in the formula is replaced by unity.
According to equipartition theorem clusters internal energy is equal where ξint is the number of internal degrees of freedom is equal The characteristic vibration frequency ν for argon clusters is 10 12 s -1 .The number of surface monomers for argon clusters is given by the relation [10] ( ) , for 5 1, for [5,6] 36 1 , for 6 The process of evaporation of a monomer from a cluster is endothermic; therefore, in order to simulate the evaporation of a monomer from a cluster of size n, it is necessary to know the energy Eevap(n) absorbed in this process.The values of the evaporation energy of the monomer from a cluster of size n were taken from [11].
Dimer formation can be modeled as a two-step process: (1) formation of collision complexes through monomer's binary collisions, (2) creation of dimers through the collision stabilization of collision complexes.This approach is more accurate, but dimers formation is modeled here as the result of a ternary collision, which can be interpreted as instantaneous stabilization of the collision complex, which was formed during the collision of two monomers, by a third particle M chosen randomly.This approach is simpler and more efficient and satisfies the principle of detailed balance.Consequently, the dimer formation process within the proposed model is governed by the equation rec 2 Ar Ar M Ar M, where Prec is the probability of the recombination reaction (formation of the Ar2 dimer).The Total Collision Energy (TCE) model for the recombination reaction [12] was used to simulate reaction (9) in the SMILE++ program.
The TCE model assumes that the dependence of the recombination rate constant under equilibrium conditions on temperature has the form Formula ( 11) was approximated by a power law using the least squares method in the temperature range from 0 K to 300 K in order to use the TCE model where Pn,m is the probability of particle association in a collision, n > 1 and m > 1.The values of the probabilities Pn,m for argon clusters were obtained in [14] The dimer formation and cluster growth processes are exothermic.The heat released when the monomer sticks to the cluster is equal to the corresponding energy of monomer evaporation for the resulting cluster.
During evaporation of monomers from the cluster and formation of dimers, pairs of new particles are formed.Each pair has internal energy and kinetic energy of relative motion.The Larsen-Bornakke model was used to set the unknown post-collision energies.

Model verification and validation
Verification of the condensation model consisted of several parts: verification of inelastic collisions, verification of the dimer formation model, and verification of the energy conservation law.
We performed calculations of gas relaxation from a nonequilibrium state (during the calculation, the condensation process was ignored) in order to test the correctness of the energy redistribution in inelastic collisions.The computational domain consisted of one collisional cell, which was limited by solid walls with the thermal accommodation coefficient equal to zero and from which particles were reflected specularly.The collision cell was populated with monomers and dimers in equal proportions.The initial translational temperature of the particles Ttr was T0 = 50 K.The temperature of the internal degrees of freedom of the dimers was Tint = 0 K.The relaxation number Z was equal to 10.
The system relaxation to thermodynamic equilibrium with an equilibrium temperature Teq due to inelastic collisions is demonstrated in Fig. 2. The equilibrium temperature can be found from the energy conservation law.A series of the recombination rate constant calculations was performed to verify the dimer formation model.The computational domain consisted of one collisional cell, which was limited by solid walls at a temperature T with the thermal accommodation coefficient equal to unity, 1.5•10 6 model particles were used in each simulation, the time step Δt was 10 -10 s, the ratio of real molecules to simulation particles Fnum = 10 12 , the cell volume V was 10 -6 m 3 , and the monomer density was 0.1 kg/m 3 .In this case, the processes of cluster formation were disabled, that is, during the calculation, only the number of dimer formation events per time step in the considered cell ΔN was collected, which is related to the recombination constant by the formula The results of the calculations are shown in Fig. 3.We calculated argon clusterization in a single cell to test the energy conservation law.The collision cell was limited by solid walls with the thermal accommodation coefficient equal to zero and from which particles were reflected specularly.The total energy of particles in the cell at the i-th step Ei was calculated as where Ni is the number of particles in the cell at the i-th step, mk is the mass of the k-th particle, vk is the velocity vector of the k-th particle, and Eint,k is the internal energy of the k-th particle.The time step was 10 -10 s, and the ratio of real molecules to simulation particles Fnum = 10 13 .There were 6•10 4 argon monomers in the cell in the initial state.
One can find the released heat by the i-th iteration Qtot(i) knowing the number of clusters of all sizes by the formula ( ) where M is the largest cluster size, Nn is the number of clusters of the n-th size, and qk is the heat of formation of the k-mer.On the other hand, the energy conservation law yields ( ) where E0 is the initial energy.The result of calculating the released heat according to formulas ( 16) and ( 17) is presented in Fig. 4. Numerical experiments on modeling the equilibrium between the processes of formation and decay of dimers were the basis for validating the model.The equilibrium state between dimers and monomers is characterized by the equilibrium constant Keq ( ) where nAr, nAr2 are the equilibrium concentrations of monomers and dimers, respectively.Validation of the numerical model was performed by comparing the calculated equilibrium constants with literature data [5,14,15]; the results of comparisons are presented in Fig. 5.

Numerical simulation results
The proposed model was used for a series of calculations of the argon outflow into vacuum.
There are, however, some difficulties with DSMC computations of the flow in the subsonic flow part.In the DSMC method, macroscopic quantities are calculated by averaging the velocities of model particles.In the case of a low-speed flow, the mean gas velocity is much smaller than the mean velocity of molecules; therefore, it is calculated with low accuracy.Also, imposing boundary conditions on the subsonic inflow boundary is a nontrivial problem in the DSMC method.One possible solution to overcome these difficulties is to use a hybrid approach, where the continuum and molecular approaches are used simultaneously, each in its own spatial subdomain, and the solutions are coupled through some interface between the subdomains.
Therefore, the gas flow inside the nozzle part was modelled using the numerical solution of the Navier-Stokes equations for a compressible gas using the ANSYS Fluent program.The DSMC method was used to calculate the gas flow in the remaining part of the nozzle, as well as in the jet itself.All calculations were carried out within the framework of solving a plane (two-dimensional) problem.
The flow from a conical diverging nozzle with a length L of 17 mm, throat diameter of 0.44 mm, and outlet diameter da of 7.6 mm was considered.The stagnation pressure P0 was 10 5 Pa, and the stagnation temperature was 300 K.The walls were assumed to be isothermal.
The time step Δt was 10 -10 s, the number of real molecules in one model molecule was 10 14 , and the number of model molecules was 1.1•10 6 .Vacuum boundary conditions were used in the SMILE++ program.The computational domain is schematically shown in Fig. 6.The nozzle critical cross section is located on the coordinate along axis of symmetry x = 0, the DSMC starting surface is located at coordinate x = 10 mm.The gas temperature along the jet axis is presented in Fig. 7.It can be seen from the temperature plot that the gas temperature increases significantly due to formation of clusters and release of latent heat.Since the gas temperature decreases during expansion, but latent heat is released, in the vicinity of the point x/da = 5 there is an extremum of temperature at which the cooling process due to expansion begins to prevail over the heating process due to the release of latent heat.
The release of latent heat reduced the density of the gas along the axis of the jet, which is shown in Fig. 8. Accordingly, more energy is converted into the kinetic energy of the flow due to the increase in temperature, as is seen from the increase in the flow velocity (see Fig. 9).

Conclusions
In the present study, the model of the condensation process for the DSMC method developed on the basis of the materials from [5] was implemented in the SMILE++ software package.The model includes the following processes: sticking and evaporation of monomers, merging of clusters, and inelastic collisions between particles.
The procedures for inelastic collisions, dimer formation, and the entire condensation model were verified.A comparison of the equilibrium constant calculated for different temperatures with the literature data showed reasonable agreement in the temperature range considered.
Calculations for a conical diverging nozzle were carried out to analyse the effect of condensation on the jet.A significant difference in the temperature distribution was observed in calculations with and without condensation.The average gas density in the jet turned out to be much smaller than the density obtained with condensation being ignored in the case of a supersonic nozzle.
The research was supported by the Russian Science Foundation (project No. 22-19-00750).
where a and b are constants.To calculate the probability Prec, one needs to know the constants a and b.According to the results of[13], the recombination constant for argon has the following form:

Fig. 5 .
Fig. 5. Comparison of the calculated values of Keq with literature data.

Fig. 7 .
Fig. 7. Comparison of the gas temperature along the jet axis.

Fig. 8 .
Fig. 8.Comparison of gas velocity along the jet axis.

Fig. 9 .
Fig. 9. Comparison of the gas velocity along the jet axis.