Rarefied Gas Research Articles

Kinetic model of a three-component plasma

In this work, a system of kinetic equations is constructed to study the processes in a three-component plasma. It is an analogue of the Krook model, which is widely used in the dy-namics of rarefied gases. The model is supposed to be used to study the processes in the channels of rocket electric thrusters.

Bерифікація алгоритму методу пробних частинок на задачі осьового обтікання конуса розрідженим газовим струменем

This paper is concerned with an aerodynamic calculation of supersonic gas plume flows and the determination of the forces they exert on obstacles. The paper presents a development of the test particle statistical method (TPSM) to numerically simulate supersonic gas plumes over a wide range of flow conditions. The work is based on the idea of a combined approach, i.e., the use of the gas-dynamic parameter distribution at the nozzle exit or on the conventional boundary of the dense core of the plume as input data for a TPSM algorithm adapted from homogeneous flows to plume ones. Combining methods of continual aerodynamics (inside or near the nozzle, where a continuum flow takes place) and the TPSM (where the motion is described on a molecular-kinetic level) allows one to solve supersonic plume efflux problems for arbitrarily rarefied plumes. The TPSM plume algorithm was tested to verify its reliability on the problem of axial flow past a cone. At the initial stage of the use of the combined approach, consideration was given to a rather rarefied gas flow, for which the gas-dynamic parameters at the nozzle exit can be used as TPSM input data. The force distribution over the cone surface and the static pressure upstream of the cone were calculated. The TPSM results were found to be in satisfactory agreement with the available direct simulation Monte-Carlo and experimental data. It was concluded that using the plume velocity and density distributions at the continual zone exit found from the Navier–Stokes equations as TPSM input data would significantly improve the expected results. This use of the TPSM in an aerodynamic calculation of gas plumes is the first in Ukraine. The TPM offers saving in computational resources: the TPSM running time depends on a variety of factors, but it is many times shorter than that of the direct simulation Monte Carlo method.

An evaluation of the hybrid Fokker–Planck-DSMC approach for high-speed rarefied gas flows

The Direct Simulation Monte Carlo (DSMC) method has been widely used for simulations of rarefied gas flows. It has been proven that the numerical solution of the DSMC method converges to the Boltzmann equation, providing a solid physical foundation for high Knudsen numbers. However, many aerospace engineering problems include both low and high-density regions in a single domain, making the DSMC method computationally expensive. Over the past decade, the particle Fokker–Planck (FP) method has been studied as a means to reduce computational costs near the continuum regime. While enhancing efficiency, the FP method loses physical accuracy at high Knudsen numbers. Aiming for universal accuracy and efficiency across the whole range of Knudsen numbers, the hybrid FP-DSMC method has been studied. Nevertheless, consistent comparisons among the DSMC, FP, and hybrid FP-DSMC methods have received limited attention so far. This paper presents a consistent comparative study of the DSMC, FP, and hybrid FP-DSMC methods to assess the accuracy and efficiency of the hybrid FP-DSMC method. The hybrid FP-DSMC solver is developed based on the open-source SPARTA framework. The benchmark problems include supersonic flow in a planar nozzle, hypersonic flow around a cylinder, and hypersonic flow around a THAAD-like missile. The results demonstrate that the hybrid FP-DSMC method can be more efficient than the DSMC method while being more accurate than the FP method. The speed-up achieved by the FP-DSMC method ranged from 1.5 to 14 times compared to the converged DSMC simulation.

Maxwell boundary condition for discrete velocity methods: Macroscopic physical constraints and Lagrange multiplier-based implementation

In this paper, we propose an algorithm that imposes macroscopic physical constraints with Lagrange multiplier approach in implementing the Maxwell boundary condition within the framework of the discrete velocity method. For the simulation of rarefied gas flows in the presence of solid walls with complex geometry, the distribution function in the reflection region of the wall surface needs to be constructed in the discrete velocity space, to fulfill the specular reflection in the Maxwell boundary condition. The construction process should not consist of interpolation only, but include certain macroscopic physical constraints at the wall surface, so as to correctly account for gas-surface interaction on a macroscopic level. We demonstrate that for the specular reflection, keeping the symmetry of the first three moments of the distribution function between the incident and reflected region is sufficient for maintaining the conservation of mass, momentum, and energy at the wall surface. Furthermore, to strictly satisfy macroscopic physical constraints, a Lagrange multiplier method is introduced into the construction of the distribution function to correct the pure interpolation solution. In addition, the construction process requires the inversion of a large and sparse matrix (of dimension N × N, where N is the number of points in the velocity space). To improve the computational efficiency, the matrix inversion is converted into that of a much smaller matrix, i.e. (D + 2) × (D + 2) in the d-dimensional physical space. A series of numerical experiments are conducted to examine the performance of the proposed algorithm under different flow conditions. We demonstrate that the results obtained by the proposed algorithm are more accurate than the pure interpolation solution, comparing with the benchmark data. Moreover, after the validation of our results with previous studies, we find that the method significantly enhances the conservation of total mass and energy, especially for flows in an enclosed domain.

Motion and self-motion of thin bodies in rarefied gas

One of the important tasks of space flight safety is the ability of a wing to maintain stability in an oncoming turbulent flow. In this case, the spacecraft must move the greatest distance. The paper studies the motion of a thin plate near a boundary in an oncoming flow of rarefied gas. At low Reynolds numbers, the effect of the boundary on the plate is studied, and the possibility of self-propulsion is shown.

Plasma antenna with frequency adjustment

A discharge of limited length (“plasma column”) in a gas-discharge tube filled with a rarefied gas was studied. The discharge is created due to the one-sided excitation of an extended high-frequency discharge supported by a propagating azimuthally symmetric mode of the surface wave. It is shown that a “plasma column” can be an effective plasma antenna at operating frequencies below the plasma frequency (ωp), with frequency tuning due to changes in the length of the “plasma column”.

A method for numerical simulation of shock waves in rarefied gas mixtures based on direct solution of the Boltzmann kinetic equation

The paper proposes an approach to numerical simulation of shock waves in rarefied gas mixtures on the basis of direct solution of the Boltzmann kinetic equation. Software for simulating the gas flows was developed. The structure of a shock wave in a binary gas mixture was computed with an accuracy controlled by the computational parameters. The computations were performed for various molecular masses ratios and Mach numbers. The total accuracy of at least 1.4% for the local values of the molecular densities and temperatures of the mixture components was achieved. Numerical simulation of a shock wave propagation through a periodically perforated surface was performed. The distributions of the macroscopic characteristics of the mixture components at various points in time were obtained. Unsteady areas of strong separation of the gas mixture components were discovered.

Variance-reduction kinetic simulation for characterization of surface and corner effects in low-speed rarefied gas flows through long micro-ducts

Variance-reduction kinetic simulation for characterization of surface and corner effects in low-speed rarefied gas flows through long micro-ducts

Modification of temperature jump condition for modified Navier-Stokes equations included molecular diffusivity in high-speed rarefied gas flows

Modification of temperature jump condition for modified Navier-Stokes equations included molecular diffusivity in high-speed rarefied gas flows

A non-localized spatial–temporal constitutive relation in rarefied gas dynamics

Although the Boltzmann equation is instrumental in capturing the dynamics of rarefied gases, finding its solutions in engineering problems is challenging. Therefore, over the past century and a half, numerous partial differential equations based on a few macroscopic variables have been introduced. However, they not only have complicated forms but also cannot make satisfactory prediction when the Knudsen number is large. Here, we propose a non-localized spatial–temporal (NiST) constitutive relation for rarefied gas dynamics, where the stress/heat flux at time t and position x is determined by the velocity/temperature gradient in the nearby spatial–temporal coordinates, via convolution operators. Utilizing solutions of the Boltzmann equation for the Couette/Fourier/Poiseuille flow and the spontaneous Rayleigh–Brillouin scattering, we extract the universal parameters of non-locality as functions of the spatial and temporal Knudsen numbers. Subsequent validation through sound propagation and backward-facing step flow demonstrates that the NiST constitutive relation is capable of accurately forecasting rarefied gas flows across a broad spectrum of Knudsen numbers.

A DSMC-CFD coupling method using surrogate modelling for low-speed rarefied gas flows

A new Micro-Macro-Surrogate (MMS) hybrid method is presented that couples the Direct Simulation Monte Carlo (DSMC) method with Computational Fluid Dynamics (CFD) to simulate low-speed rarefied gas flows. The proposed MMS method incorporates surrogate modelling instead of direct coupling of DSMC data with the CFD, addressing the limitations CFD has in accurately modelling rarefied gas flows, the computational cost of DSMC for low-speed and multiscale flows, as well as the pitfalls of noise in conventional direct coupling approaches. The surrogate models, trained on the DSMC data using Bayesian inference, provide noise-free and accurate corrections to the CFD simulation enabling it to capture the non-continuum physics. The MMS hybrid approach is validated by simulating low-speed, steady-state, force-driven rarefied gas flows in a canonical 1D parallel-plate system, where corrections to the boundary conditions and stress tensor are considered and shows excellent agreement with DSMC benchmark results. A comparison with the typical domain decomposition DSMC-CFD hybrid method is also presented, to demonstrate the advantages of noise-avoidance in the proposed approach. The method also inherently captures the uncertainty arising from micro-model fluctuations, allowing for the quantification of noise-related uncertainty in the predictions. The proposed MMS method demonstrates the potential to enable multiscale simulations where CFD is inaccurate and DSMC is prohibitively expensive.

Capturing non-equilibrium in hypersonic flows: Insights from a two-temperature model in polyatomic rarefied gases

The study utilizes a two-temperature model to analyze non-equilibrium in normal shocks within hypersonic flows in polyatomic rarefied gases. Derived from the extended second law of thermodynamics, this model separates translational and internal temperatures in polyatomic gases, providing a more accurate depiction of non-equilibrium gas flow compared to classical theories like the Navier–Stokes and Fourier (NSF) system. Notably, the analysis reveals that the two-temperature model incorporates an additional contribution to the heat flux due to the gradient of the dynamic temperature, resulting in improved accuracy, especially for high Mach numbers. Results show that the model gives satisfactory shock density and temperature profiles up to Mach 10, with very good agreement observed up to Mach 6.1 compared to the classical NSF model. We conduct an order of magnitude analysis on the dynamic temperature and heat flux gradients appearing in the new constitutive equation using the Mott-Smith method. This analysis highlights the impact of these terms on accurately modeling polyatomic gas behavior in high-speed flows. The effects of bulk viscosity and incoming temperature on shock profiles are also investigated, contributing to a better understanding of shock wave structures in polyatomic gases and their implications for hypersonic flow dynamics.

Scale dependence and non-isochoric effects on the thermohydrodynamics of rarefied Poiseuille flow

The plane Poiseuille flow of a rarefied gas in a finite length channel, driven by an axial pressure gradient, is analysed numerically to probe (i) the role of ‘dilatation’ ( $\varDelta ={\boldsymbol \nabla }\boldsymbol {\cdot }{\boldsymbol u}\neq 0$ ) on its thermohydrodynamics as well as to clarify (ii) the possible equivalence with its well-studied ‘dilatation-free’ or ‘isochoric’ ( ${\rm D}\rho /{\rm D}t=0$ ) counterpart driven by a constant acceleration. Focussing on the mass flow rate ${\mathcal {M}}({Kn})$ , which is an invariant quantity for both pressure-driven and acceleration-driven Poiseuille flows, it is shown that while ${\mathcal {M}}\sim \log {{Kn}}$ at ${Kn}\gg 1$ in the acceleration-driven case, the mass flow saturates to a constant value ${\mathcal {M}}\sim {{Kn}}^0$ at ${Kn}\gg 1$ in the pressure-driven case due to the finite length ( $L_x<\infty$ ) of the channel. The latter result agrees with prior theory and recent experiments, and holds irrespective of the magnitude of the axial pressure gradient ( $G_p$ ). The pressure-dilatation cooling ( $\varPhi _p=-p\varDelta <0$ ) is shown to be responsible for the absence of the bimodal shape of the temperature profile in the pressure-driven Poiseuille flow. The dilatation-driven reduction of the shear viscosity and the odd signs of two normal stress differences ( ${\mathcal {N}}_1$ and ${\mathcal {N}}_2$ ) in the pressure-driven flow in comparison with those in its acceleration-driven counterpart are explained from the Burnett-order constitutive relations for the stress tensor. While both ${\mathcal {N}}_1$ and ${\mathcal {N}}_2$ appear at the Burnett order $O({{Kn}}^2)$ in the acceleration-driven flow, the leading term in ${\mathcal N}_1$ scales as $(\mu/p)\varDelta$ due to the non-zero dilatation in the pressure-driven Poiseuille flow which confirms that the two flows are not equivalent even at the Navier–Stokes–Fourier order $O({{Kn}})$ . The heat-flow rate ( ${{\mathcal {Q}}_q}_x=\int q_x(x,y) \,{{\rm d} y}$ ) of the tangential heat flux is found to be negative (i.e. directed against the axial pressure gradient), in contrast to its positive asymptotic value (at ${Kn}\gg 1$ ) in the acceleration-driven flow. Similar to the scale-dependence of the mass flow rate, ${{\mathcal {Q}}_q}_x({{Kn}}, L_x)$ is found to saturate to a constant value at ${Kn}\gg 1$ in finite length channels. The double-well shape of the $q_x(y)$ -profile in the near-continuum limit agrees well with predictions from a generalized Fourier law. On the whole, the dilatation-driven signatures (such as the pressure-dilatation work and the ‘normal’ shear-rate differences) are shown to be the progenitor for the observed differences between the two flows with regard to (i) the hydrodynamic fields, (ii) the rheology and (iii) the flow-induced heat transfer.

Study of microfluidic heat transfer in 2D hydrophobic walls with different slip mechanisms

Numerous studies have shown that the gas at the solid-liquid interface in microflow can increase the slip length, reduce the flow resistance. However, the relationship between nanobubbles and fluid boundary slip is complicated, and the impact of nanobubble morphology must be considered. To investigate the combined effect of the bubble model on slip and heat transfer, we investigated the drag reduction and heat transfer effect of microfluidic flow on a 2D hydrophobic wall by theoretical derivation combined with numerical simulation. The comparison of the results revealed good agreement between the analytical solution and the numerical simulation results. It was shown that the gas slip generated by the rarefied gas significantly affected the flow heat transfer effect. Further analysis of the bubble meniscus shape revealed that minimizing the protruding angle of the meniscus is crucial for enhancing drag reduction. In addition, the simulation results also revealed that increasing the gas coverage fraction effectively reduces flow resistance while maintaining heat transfer performance. At the same time, the higher pressure enhanced the heat transfer effect while maintaining the stability of the drag reduction effect.

General synthetic iterative scheme for rarefied gas mixture flows

The numerical simulation of rarefied gas mixtures with disparate mass and concentration is a huge research challenge. Based on our recent kinetic modeling for monatomic gas mixture flows, this problem is tackled by the general synthetic iterative scheme (GSIS), where the mesoscopic kinetic and macroscopic synthetic equations are alternately solved by the finite-volume discrete velocity method. Three important features of GSIS are highlighted. First, the synthetic equations are precisely derived from the kinetic equation, naturally reducing to the Navier-Stokes equations in the continuum flow regime; in other flow regimes, the kinetic equation provides high-order closure of the constitutive relations to capture the rarefaction effects. Second, these synthetic equations, which can be solved quickly, help to adjust the kinetic system to relax rapidly toward the steady state. Furthermore, in such a two-way coupling, the constraint on the spatial cell size is relieved. Third, the linear Fourier stability analysis demonstrates that the error decay rate in GSIS is smaller than 0.5 for various combinations of mass, concentration and viscosity ratios, such that the error can be reduced by three orders of magnitude after 10 iterations. The efficiency and accuracy of GSIS are demonstrated through several challenging cases covering a wide range of mass ratio, species concentration, and flow speed.

Thermodynamic properties of perfluorooctane on the liquid-gas coexistence curve: calculation method and tabulated data

A technique has been developed for constructing a phase equilibrium line in the range from the triple to the critical point for a technically important substance, perfluorooctane (C8F18), which is currently used in various fields of industry and medicine. The proposed phase equilibrium line model includes the following equations: vapour pressure; saturated liquid density; saturated vapour density; “apparent” heat of vaporization. In this case, the vapour pressure equation satisfies the requirements of the scale theory of critical phenomena near the critical point, and the Wegner model in the vicinity of the triple point. The coexistence curve model in the vicinity of the critical point satisfies the Yang-Yang model and renormalization group theory for asymmetric systems, and in the region of rarefied gas it satisfies the linear model of average diameter. In the temperature range 246.15–496.99 K, table for perfluorooctane have been developed, including pressure and density of saturated vapor, density of saturated liquid, heat of vaporization, first and second derivatives of saturated vapor pressure. Using latest experimental data, as well as experimental data on the critical pressures of perfluoroalkanes as a function of their molecular weight, the following values of critical parameters were selected: critical pressure – 1.548 MPa, critical density – 596.4 kg/m3, critical temperature – 496.99 K. A number of statistical characteristics were calculated, including absolute mean deviation and standard deviation, characterizing the accuracy of the proposed phase equilibrium line model when describing experimental data obtained in generally recognized international thermophysical laboratories. The results obtained are useful for high-tech companies engaged in the development of innovative technologies: in the field of radio-electronic and electrical industries, in which perfluorooctane is used as a liquid dielectric and coolant; in medicine, where perfluorooctane is used as a gas transport liquid and is used for ophthalmic purposes; magnetic fluid sealers for the purpose of isolating hazardous substances from the environment and seals for devices operated in vacuum conditions or in contact with aggressive substances; magnetic fluid separators for separating non-ferrous metals by density, etc. The results of this study can also be used in developing the fundamental equation of state for perfluorooctane. The results obtained are useful for high-tech companies engaged in the development of innovative technologies: in the field of radio-electronic and electrical industries, in which perfluorooctane is used as a liquid dielectric and coolant; in medicine, where perfluorooctane is used as a gas transport liquid and is used for ophthalmic purposes; magnetic fluid sealers for the purpose of isolating hazardous substances from the environment and seals for devices operated in vacuum conditions or in contact with aggressive substances; magnetic fluid separators for separating non-ferrous metals by density, etc. The results of this study may also be useful in developing the fundamental equation of state for perfluorooctane.

Thermally induced oscillatory rarefied gas flow inside a rectangular cavity

Thermally induced oscillatory rarefied gas flow inside a two-dimensional rectangular cavity is investigated based on the hybrid macro-/mesoscopic scheme. The effects of the Knudsen (Kn) numbers and the oscillation frequency of lid temperature on the flow parameters are analyzed. The Shakhov model equation is solved numerically based on the mesoscopic approach in the near-wall region, and the macroscopic approach is adopted in the bulk flow region to reduce the computational cost. To close the numerical iteration procedure, the velocity distribution functions serving as the pseudo boundary between macroscopic and mesoscopic methods are reconstructed using the high-order Hermite polynomials. Numerical simulations demonstrate that the temperature profile at the central vertical of the cavity predicted by the hybrid method is in good agreement with results from the mesoscopic method, with a maximum error of 0.23%. In addition, the computational memory cost can be saved up to about 69.91%. The hybrid approach is able to capture the nonlinear phenomenon in the thermally induced oscillatory rarefied gas flow under high Kn numbers, where the horizontal velocity no longer obeys the law of periodic oscillating cosine function, and the rise time of the horizontal velocity is much longer than the fall time. The thickness of the viscous penetration layer and the disturbed region increases as the Kn number increases and decreases as the Strouhal number increases.

Drag and Thermal Reduction of Spiked Hypersonic Vehicle in Thermochemical Nonequilibrium

This study utilizes the in-house parallel direct simulation Monte Carlo–quantum dynamics method to model the aerodynamic flowfield of a spiked reentry vehicle within a rarefied high-altitude atmosphere. Investigating the flight of spiked aircraft under various flight conditions and spike sizes and considering both the real gas effect and the rarefied gas effect, the aerodynamic characteristic of the aircraft is evaluated. Performance evaluation for effectiveness in drag reduction and thermal protection is based on drag and the maximum stagnation heat flux coefficient at the vehicle’s shoulder. Findings indicate that, in the rarefied atmosphere, the distinction between the characteristic shock intensities of spiked and unspiked vehicles is not pronounced, which results in a drag reduction of approximately 5%, accompanied by a more than 100% increase in thermal effect on the vehicle’s shoulder, obviously poor in effectiveness. Subsequently, this study examines the spiked vehicle equipped with a lateral jet, and this augmented approach achieves a marked reduction in the intensity of the shock wave impacting the vehicle’s shoulder with a drag reduction exceeding 10% and attenuates the thermal impact by over 50% in comparison to the implementation of spiking alone, thereby demonstrating superior performance in terms of effectiveness.

Centerline shock reflection for supersonic internal flows in the non-Rankine-Hugoniot zone: Overexpanded supersonic microjets

The viscous and rarefaction effects on centerline shock reflection occurring in an overexpanded axisymmetric microjet have been investigated numerically by means of a fully coupled pressure-based shock capturing scheme. Due to the low free-stream Reynolds number, the Navier–Stokes equations were coupled with slip velocity and temperature jump boundary conditions to account for rarefied gas effects in the Knudsen layer. It has been found that pronounced viscosity levels can cause a transition from a three-shock to a two-shock configuration, which is impermissible by inviscid theory. This is the first observation of such phenomena in the case of over-expanded jets. Analysis of the von Neumann and detachment criteria indicates that the transition from Mach reflection to regular reflection is analogous to the dual-solution domain transition for planar shocks. In addition, prediction of the longitudinal curvature of the incident shock has been conducted from a mathematical standpoint.

Sedimentation and photophoretic levitation of aerosol clusters in the free molecular regime

Sedimentation of fractal aerosol clusters in rarefied gas medium in the dark and under external illumination similar to sunlight is studied in a numerical experiment taking into account orientation effects, depending on fractal dimension varying within a broad range. The calculation of forces and their moments, including friction forces and photophoretic forces, was carried out on the basis of approximation of free molecular gas kinetic regime and previously developed Monte-Carlo algorithms. Calculation involved 10000 clusters, each of them containing 160 primary spherical particles and efficiently absorbing sunlight and IR radiation, similarly to the particles of soot aggregates.The velocity of multi-particle cluster settling in the absence of illumination, when particle temperatures are equal to the temperature of the gaseous environment, is a distinct function of fractal dimension and is rather close to the velocity of settling of single spherical particles. The settling velocity approximation depending on fractal dimension has been obtained.Illumination brings dramatic changes into sedimentation pattern. The range of cluster sedimentation rate variations broadens substantially, which is a manifestation of gravito-photophoretic effect, so that some clusters even start to levitate. The proportion of levitating clusters is essentially dependent on fractal dimension. For soot-like clusters under irradiation similar to sunlight and pressure equal to atmospheric at the altitude of 30 km, the proportion of levitating aggregates is approximately 7%, which is in good agreement with the experimental data. In the dark and under irradiation, the aggregates rotate, and their trajectories are shaped as spirals.In the context of photophoretic phenomena, a result of principle has been obtained: under irradiation with sunlight, gravito-photophoretic levitation of aerosol soot-like aggregates composed of identical primary particles is possible at a pressure corresponding to the stratospheric altitudes. This result points to the possibility of substantial influence of gravito-photophoretic effect on transport and localization of soot aerosol in the Earth’s stratosphere and mesosphere.