Simulation of Rarefied Gas Flow in a Channel Applying Artificial Neuron Network

The purpose of the present paper is to confirm numerically applying our ANN algorithm for transitional gas flow between continuum and free-molecular flow. The main advantage of this approach is substantial computing time economy compared to usual DSMC procedure. The learning of ANN is based on well known rarefied gas flow examples, analytic solutions in limit cases, calculations by lattice Boltzmann method (LBM), and on quite few Monte Carlo simulations. Our previous results obtained for large Knudsen numbers have been applied near the free-molecular regime. In this region the flow has demonstrated the properties of unstable nonlinear dynamic system. The flow of nitrogen in a microchannel with aspect ratio L/h = 20 and by pressure ratio P between 1.84 and 4.14 is simulated by ANN. This example has shown a good coincidence of ANN and DSMC calculations even in the case when only a small part of original DSMC computation has been applied for learning ANN. Thus, computing time is reduced up to 10 times in this case.


Introduction
The most effective approach to save computing time calculating transitional gas flow (between continuum and free-molecular one) is using artificial neuron network (ANN).The purpose of the present paper is numerical simulation of rarefied gas flows in channels applying ANN.Preferred in rarefied gas dynamics DSMC methods need powerful computers and much computation time even for these high-efficiency computers.Furthermore, a lot of practical problems require DSMC calculation to be repeated several times, if the results are necessary for various values of the parameters.Therefore, only the methods like ANN could assure computer time economy in this case.To test ANN in rarefied gas unstable flows are considered.Unstable flow means here that a small deviation of the boundary conditions causes substantial change of the whole flow.Such unstable flows (including the flows in micro-channels) have been investigated in our previous papers [1]- [7] as nonlinear dynamic systems with possible attractors and bifurcations.Thus, it is one of the most complex types of rarefied gas flows.
Boundary conditions in rarefied gas flow are determined by the scattering function, which is supposed here to be ray-diffuse, i. e. it is the mixture of the ray reflection and of the diffuse scattering.The ray-diffuse model has better experimental confirmation in comparison with the specular-diffuse model widely applied in practical DSMC calculations.

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Journal of Physics: Conference Series 2701 (2024) 012057 IOP Publishing doi:10.1088/1742-6596/2701/1/012057 2 Rarefied gas flows in channels are specified on micro level by the laws of mutual interactions of the gas molecules and of their reflections from the surface.Hence, they could be regarded as discrete dynamic systems, whereas the continuum flows described usually by differential equations are, on the contrary, continuous dynamic systems.Therefore, these systems can expose strange attractors regardless of their dimension, while in a continuous dynamical system a strange attractor can arise due to the Poincaré-Bendixson theorem only if the system has three or more dimensions.Analytical investigations of the relevant cascade of bifurcations in rarefied gas flow in [1]- [7] are based on these assumptions.
ANN procedure is applied to locate the areas of the destabilization of the flow.Monte Carlo simulation allows analyzing the bifurcations near these regions.In our previous papers the effect of significant variation of macro parameters of gas flow in a channel by an insignificant deviation of one of the parameters of scattering function near the bifurcation points has been studied in free-molecular regime.The results of numerical calculations by ANN have demonstrated that this effect becomes comparatively less considerable in transitional regime between free-molecular and continuum flow.

Interaction of rarefied gas with the walls of the channel
Widely applied specular-diffuse scattering function is generalized to the ray-diffuse scattering of gas particles from channel walls.This scattering function is defined as a mixture of diffuse and ray scattering fuctions (with the coefficients σ and 0 ≤ σ ≤ 1 correspondingly) where δ is Dirac delta function and the velocity u * of reflected from the surface gas particles is, generally speaking, different from the velocity in the specular model.Some restrictions (based on the principle of detailed balance in gas-surface interactions) are nesessary for the function u * ( u) to guarantee that scattering function V ( u, u ) satisfies the common criteria on wall interaction kernels.In our approach only the functions u * ( u) meeting these restrictions are used (the class of such functions u * ( u) is confirmed to be wide enough [4]).The advantage of the ray model in comparison to specular one consists in producing more diversified shapes of limit gas atoms trajectories.Diffuse supplement in (2) modifies the limit properties of the investigated dynamic system because of a randomization [1], [2].Experimental verification of the ray-diffuse scattering is more convincing than experimental evidence of other surface interaction models used in practical DSMC calculations, in particular of the specular-diffuse model [4].
Studying analytically and numerically the limit properties of this nonlinear dynamic system, we do not focus on the strange attractor appearing in rarefied gas flows in the form of a cascade of bifurcations, whereas usual applications of the chaos theory pay attention exactly to the strange attractor and relevant chaotic image of the system.On the contrary, in the rarefied gas dynamics specifically interesting is the transformation from regular to chaotic flow, as such transfer specifies in practice the values of the parameters matching to significant changes of the solution by a quite small variation of these parameters.In the present paper this transition is investigated not only for variated parameters, but also for different analytical approximations of momentum exchange coefficients.In terms of nonlinear dynamics it matches to the variation of iterative equations representing the path of a gas molecule.
Investigating the transformation from regular to chaotic flow requires multiple computations differing by small variation of initial conditions, i.e. we need to repeat DSMC calculation in rarefied gas many times.For example, constructing of the single bifurcation diagram requires reproducing the whole DSMC procedure for rarefied gas flow at least 10 4 -10 5 times.Using ANN to reduce computing time permits to lower multiple computation time substantially.The possibility of a rarefied gas flow in long enough channels or nozzles to become unstable on certain indicated conditions provided specified transformation values of the parameters of the scattering function V ( u, u ) is revealed in our previous analytical and numerical investigations [1]- [7].Analytical results are achieved for unstable free-molecular rarefied gas flow [3]- [7].Similar analytical study of unstable dynamic systems can be found in different applications (for example, in [9]- [10]).To take into consideration that the gas flow is not free-molecular, but it is in transition regime from continuum to free-molecular flow, i.e.Knudsen number Kn is finite, we use the local interaction approximations in rarefied gas flow [4].These approximations of momentum and energy exchange coefficients are explicit in free-molecular flow, and they have convincing experimental confirmation in discussed transition regime.The bifurcations and appropriate physical values in studied flows are also identified.But experimental verification of discovered numerically effect remains a complicated problem, as the intervals of according parameters, like the restrictions determining corresponding reflection are strictly limited.Consequently, considered reflection options are barely reproducible in the experiments.Hence, unstable flows of this kind have no empirical verification yet (since the experiments in rarefied gas are very costly).Nevertheless, considered unstable flows will be undoubtedly verified experimentally later, because numerical calculations specify all the observed physical quantities (including scattering function, Knudsen number etc.).Discussed bifurcations could influence significantly many used in practice gas flows, e. g. flows in propulsion systems and in microelectronic vacuum devices.Furthermore, the effect of reducing conductivity of the channel due to the destabilization of the flow in certain situations is important for practical use.
The characteristics of rarefied gas flow in a channel are obviously associated with the parameters of the scattering function V ( u, u ).In particular, if the geometric shape of the channel is fixed, then in a free-molecular flow (Kn = ∞) this function V sets uniquely all the parameters of a gas flow in the channel.The basic assumption for arising the attractors and the bifurcations in the flow is that the scattering function V should be close to the ray reflection.Relevant momentum exchange coefficients for the ray scattering function V are represented by the equations p = 2 cos θ(cos θ + u cos θ ), τ = 2 cos θ(sin θ − u sin θ ).Here θ and θ are the angles of incident and reflected gas atoms, and u and u are corresponding absolute values of the velocities u and u , [4].In transition regime (Kn < ∞) identical boundary conditions to the conditions in a free-molecular flow (Kn = ∞) are used in numerical computations.This approach permits to locate the areas of the destabilization of the flow and to check received results.

Test examples of unstable rarefied gas flows
The iterative formula describing the trajectory of a gas particle in the channel can be derived by replacing the angles in succession θ m and θ m+1 by their tangents x m = tan θ m and x m+1 = tan θ m+1 , and expressing the angle θ from the representations of momentum exchange coefficients p and τ where m is the number of collisions of a gas molecule with the walls, the momentum exchange coefficients p and τ are determined by local approximations, and the variable l m and the function η are defined by the geometry of the channel (Fig. 1).
Discussed nonlinear system has a lot of various limit solutions -attractors [1]- [7] in the case of unstable flow.System parameters are placed here in certain small intervals, and respective parameters of the function V determine the points of singularity.The computations of the paths of gas particles in the channel prove remarkable variations of the aerodynamic characteristics of the flow near corresponding values of the parameters of the dynamic system.To construct

Application of artificial neuron network to numerical simulations
To reduce computational time and costs in rarefied gas flow in a channel, a common type of ANN is used -multilayer perceptron (MLP).The basis for training of ANN is built of comparatively primitive trial flows, of discussed in previous section analytic solutions and of certain selected numerical Monte Carlo calculations.Chosen MLP has an essential benefit over another type of ANN -Radial Basis Function (RBF) network: RBF network improves only the interpolation between given solutions, while MLP uses nonlinear activation function like a hyperbolic tangent, and, consequently, MLP increases as the accuracy of the extrapolation of known solutions, as well the precision of the discontinuous functions with gaps.Furthermore, MLP provides sufficiently precise solutions for extremely difficult problems and permits solving stochastical problems [12].
Learning is executed in MLP by varying linking weights after each part of data is handled.The result of the calculation is compared to the expected one and linking weights are modified to minimize the deviation.DSMC calculations of the rarefied gas flow in the channel are repeated several times to train MLP under next restrictions.Uniform rarefied gas flow including N 0 gas particles is proposed in the initial section of the channel, N 0 is selected from 40000 and above.Long enough channel is required to assure the existence of the attractors, therefore, minimal length of the channel is considered equal to 20.Gas particles are assumed to be hard spheres, mutual collisions between the particles are simulated like in DSMC, while the trajectory of each gas particle is processed using molecular dynamics (MD) method.The reflection of gas molecules from the walls of the channel is simulated by the ray-diffuse scattering with the parameters of the gas interaction with the surface (σ, a, b and d) selected from the areas found from the condition that analytic solution obtained for free-molecular flow should be unstable.The number of MLP layers was bounded by 2 or 3 layers.

IC-MSQUARE-2023 Journal of Physics: Conference Series 2701 (2024) 012057
The distribution of the mean number of gas particles in the sections of a channel and the angles of velocity inclination along the channel are calculated by ANN (and DSMC) simulation.
Comparative (to the free-molecular flow) graphs of the numerical density of gas molecules in certain sections of the channel as a function of the variable a are shown in Fig. 3 for transitional regime (between continuum and free-molecular flow), Kn= 1.
The ray-diffuse scattering function ( 1) is proposed with the same value σ = 0.05 in all the calculations.The parameters of the function (1) are modified smoothly in the intervals of destabilization of the flow to illustrate the fluctuation of the results.For example, the parameter a varies from 1.47 to 1.60 (Fig. 3).The most apparent difference between two graphs according to closest values a = 1.47 and a = 1.48 specifies the area of the flow destabilization in the interval 1.47 ≤ a ≤= 1.48.The effect of substantial decrease of the conductivity of the channel near the value a = 1.47 means physically that the channel becomes locked.Similar effects of the flow destabilization and of locking the channel are obtained for another values of the variables a, b and σ.
The graphs showing respective effects for the variation of the mean angle θ of inclination of gas molecules velocities along the channel are presented in [1].The flow destabilization is clearly noticeable in these graphs as well as the decrease of the conductivity of the channel near the same values of the parameters of the flow.
The most effective method to investigate the limit properties of unstable nonlinear dynamic systems and their bifurcations is building the bifurcation diagrams shown in [1]- [2].But for growing coefficient σ displaying the contribution of diffuse scattering to the function V , and for finite Knudsen numbers (for example, for Kn= 1) the bifurcation diagrams similar to the diagrams shown in [1] become quite dissolved due to the randomization.Therefore, the clarity in rendering is lost.The results of numerical computations demonstrate that for comparatively minor values of σ the influence of substantial change of the characteristics of the flow by a low modification of the parameters a and b of gas-surface interaction continues to be qualitative identical to the effect for the ray reflection (σ = 0), and it appears by the same values of the coefficients a and b (Fig. 5).
Near all the points of the bifurcation an insignificant modification of one of the parameters of the ray-diffuse model (less than 1%) results also the significant distinction in the gas flow in The results of ANN computation of test example are presented in Fig. 6.Two-dimensional flow of nitrogen in a channel is considered by the length to width ratio L/h = 20 and pressure ratio Π from 1.84 to 4.14 [13].Inlet flow with uniform velocity distribution had the number density 8 • 10 19 m −3 and the temperature 300K.ANN procedure is based on DSMC and on LBM calculations [14].The parameters of the flow are not in the region of unstable behaviour, hence bifurcation effect has not been observed.ANN results are shown in Fig. 5 by dashed lines compared with the solution of Navier-Stokes equations (dotted lines, [15]) and with LBM approximations (solid lines, [13]).Three considered values of pressure ratio Π = 1.84,Π = 2.76 and Π = 4.14 correspond to Knudsen numbers in outlet from 0.127 to 0.255, i.e. it is transitional situation between continuum and free-molecular flow.The deviation of pressure distribution along the channel from linear approximation calculated applying four methods in transition regime.
Instead of normalised pressure ratio, its deviation from linear approximation is shown in Fig. 6, because the dependence of normalised pressure ratio is close to linear function, and the difference between two graphs would be insignificant.Good agreement between DSMC and ANN result verifies the advantage of ANN method, because DSMC is certainly more exact than LBM method and especially more precise than the solution of Navier-Stokes equations.Moreover, only a small part of DSMC computation (less than 5%) applied in ANN training, allows obtaining better exactness than initial DSMC [13].Therefore, proposed ANN approach permits to achieve noticeable computer time economy in rarefied gas flow calculations.

Conclusion
1. ANN and DSMC computation of various nonlinear dynamic systems appropriate to rarefied gas flows in channels have shown certain types of the growth of the destabilization, among them the cascades of period-doubling bifurcations.2. Suggested ANN algorithm in its common form MLP permits decreasing the time of calculation in rarefied gas flow simulation.Analytic evaluations and certain Monte Carlo simulations can provide a content for MLP training.Various problems associated with rarefied gas flow could be solved by the suggested method.3. Significant distinction in the macro parameters of gas flow in a channel by extremely small modification of the variables in ray-diffuse scattering function of gas particles from a surface is verified analytically and numerically.Experimental verification for discussed type of unstable flows is expected in the future.
4. The values of the parameters respective to possible unstable rarefied gas flows can be found using proposed algorithm; the areas of possible destabilization of the flow are quite narrow, consequently, to locate them without a special search is a complicated problem.Simulated unstable situations in considered nonlinear systems are near to physical effects noticed in the experiments.5.The effect of considerable variation of a rarefied gas flow in a channel by insignificant deviation of one of the parameters of scattering function near the points of the bifurcations becomes less noticeable in transitional flow in comparison with free-molecular analytic results.

Figure 1 .
Figure 1.The scheme of consecutive reflections of a gas particle from the walls of a channel producing the iterative equation (2).

Figure 2 .
Figure 2. Three iterative expressions appropriate to consecutive collisions of a gas molecule with the walls of a channel.Here the parameters are equal to a = 2, b = 1.7 for the first model (dashed line), a = 0.2, b = 0.9 for the second approximation (dotted line), a = 2, b = 1.8, d = 0.3 for the third model (solid line).

Figure 3 .
Figure 3.The variation of the number N of gas particles along the channel by changing the parameter from a = 1.47 to a = 1.48, a = 1.50 and a = 1.60 by constant b = 1.7, σ = 0.05, N 0 = 40000: (a) Kn= ∞, free-molecular flow; (b) Kn= 1, transitional regime between continuum and free-molecular flow.

Figure 4 .
Figure 4. Two bifurcation diagrams showing the influence of the coefficient σ of ray-diffuse scattering function (the part of diffuse scattered gas atoms, σ = 0.02 on the left and σ = 0.05 on the right graph) on the destabilization of the flow for identical geometrical scattering parameters a (from 1.1 to 1.55), b =1.8 and x = tan(θ) (from −2 to 2).

Figure 5 .
Figure 5.The deviation of pressure distribution along the channel from linear approximation calculated applying four methods in transition regime.