Numerical study of unsteady rarefied diatomic gas flows in a plane microchannel

The numerical solution of a kinetic equation for a diatomic gas (nitrogen) is used to study two-dimensional unsteady gas flows in a plane microchannel caused by discontinuous in the initial distributions of macroscopic gas parameters. The plane discontinuity fronts are perpendicular to the walls of the channel. The arising flows are model ones for gas flows in a shock tube and a microchannel. The reflection of an incident shock wave from a flat end face is studied. It is found that the gas piles up at the cold wall, which slows down the shock wave detachment. The numerical results are in qualitative agreement with experimental data.     

Computation of rarefied diatomic gas flows through a plane microchannel

A numerical method based on a model kinetic equation was developed for computing diatomic rarefied gas flows in two dimensions. Nitrogen flows through a plane microchannel were computed, and the gas flow rate was constructed as a function of the Knudsen number for various channel lengths.     

Numerical study of the Couette flow of a diatomic rarefied gas

The Couette flow is numerically studied using a model kinetic equation for a diatomic rarefied gas (nitrogen). The boundary condition set on the wall takes into account that the molecular rotational energy passes into translational energy when the molecule interacts with the wall. For comparison purposes, the Couette flow is computed using the classical diffuse model of the gas-wall interaction. A comparison of the results obtained with both types of boundary conditions shows that the computed parameters of the Couette flow coincide only for sufficiently low Knudsen numbers. This suggests that transitions between rotational and translational energy in the gas-wall interaction have to be taken into account in the boundary condition.     

Nonequilibrium jet boundary layer in a polyatomic gas

The two-dimensional nonequilibrium hypersonic free jet boundary layer gas flow in the near wake of a body is studied using a closed system of macroscopic equations obtained (as a thin-layer version) from moment equations of kinetic origin for a polyatomic single-component gas with internal degrees of freedom. (This model is can be used to study flows with strong violations of equilibrium with respect to translational and internal degrees of freedom.) The solution of the problem under study (i.e., the kinetic model of a nonequilibrium homogeneous polyatomic gas flow in a free jet boundary layer) is shown to be related to the known solution of the well-studied simpler problem of a Navier-Stokes free jet boundary layer, and a method based on this relation is proposed for solving the former problem. It is established that the gas flow velocity distribution along the separating streamline in the kinetic problem of a free jet boundary layer coincides with the distribution obtained by solving the Navier-Stokes version of the problem. It is found that allowance for the nonequilibrium nature of the flow with respect to the internal and translational degrees of freedom of a single-component polyatomic gas in a hypersonic free jet boundary layer has no effect on the base pressure and the wake angle.     

Models of a linearized Boltzmann collision integral

For rarefied gas flows at moderate and low Knudsen numbers, model equations are derived that approximate the Boltzmann equation with a linearized collision integral. The new kinetic models generalize and refine the S-model kinetic equation.     

Kinetic model of the Boltzmann equation for a power-law intermolecular interaction potential

A model kinetic equation approximating the Boltzmann equation with a linearized collision integral is constructed to describe rarefied gas flows at moderate and low Knudsen numbers. The kinetic model describes gas flows with a power-law intermolecular interaction potential and involves five relaxation parameters. The structure of a shock wave is computed, and the results are compared with an experiment for argon.     

The linear stability of inviscid shear flow of a vibrationally excited diatomic gas

The problem of the linear stability of plane-parallel shear flows of a vibrationally excited compressible diatomic gas is investigated using a two-temperature gas dynamics model. The necessary and sufficient conditions for stability of the flows considered are obtained using the energy integrals of the corresponding linearized system for the perturbations. It is proved that thermal relaxation produces an additional dissipation factor, which enhances the flow stability. A region of eigenvalues of unstable perturbations is distinguished in the upper complex half-plane. Numerical calculations of the eigenvalues and eigenfunctions of the unstable inviscid modes are carried out. The dependence on the Mach number of the carrier stream, the vibrational relaxation time τ and the degree of non-equilibrium of the vibrational mode is analysed. The most unstable modes with maximum growth rate are obtained. It is shown that in the limit there is a continuous transition to well-known results for an ideal fluid as the Mach number and τ approach zero and for an ideal gas when τ → 0.     

A representation of the equations of multicomponent multiphase seepage

A mathematical model of non-isothermal multicomponent flows in a porous medium is investigated. A general case is considered when the model can be used to describe processes with an arbitrary number of components and phases. A general form of the system of mixed-type equations describing the flow, which is similar to the Godunov form for hyperbolic systems is proposed. The equations obtained are applicable to flows with gas, liquid and solid phases. The thermodynamic properties of the medium are determined solely by a single multivalued function, by changing which one can obtain models of different flows in a porous medium. A clear geometrical interpretation of the solutions of the equations is proposed. An equation for the entropy is obtained, and it is shown that in order that the model should not contradict the second law of thermodynamics, it is necessary to take into account, in the energy equation, the work of the gravity force, which is often neglected when investigating seepage.     

Nonlinear nonequilibrium kinetic model of the boltzmann equation for monatomic gases

A model kinetic equation approximating the Boltzmann equation in a wide range of nonequilibrium gas states was constructed to describe rarefied gas flows. The kinetic model was based on a distribution function depending on the absolute velocity of the gas particles. Highly efficient in numerical computations, the model kinetic equation was used to compute a shock wave structure. The numerical results were compared with experimental data for argon.     

A kinetic formulation for a model coupling free surface and pressurised flows in closed pipes

The aim of this paper is to present a kinetic formulation of a model for the coupling of transient free surface and pressurised flows. Firstly, we revisit the system of Saint-Venant equations for free surface flow: we state some properties of Saint-Venant equations, we propose a kinetic formulation and we verify that this kinetic formulation leads to a Gibbs equilibrium that minimises (in some general case) an energy and preserves the still water steady state. Secondly, we propose a model for pressurised flows in a Saint-Venant-like conservative formulation. We then propose a kinetic formulation and we verify that this kinetic formulation leads to a Gibbs equilibrium that minimises in any case an energy and preserves the still water steady state. Finally, we propose a dual model that couples these two types of flow.     

One method for solving systems of nonlinear partial differential equations

A method for reducing systems of partial differential equations to corresponding systems of ordinary differential equations is proposed. A system of equations describing two-dimensional, cylindrical, and spherical flows of a polytropic gas; a system of dimensionless Stokes equations for the dynamics of a viscous incompressible fluid; a system of Maxwell’s equations for vacuum; and a system of gas dynamics equations in cylindrical coordinates are studied. It is shown how this approach can be used for solving certain problems (shockless compression, turbulence, etc.).     

Solvability of a mathematical model of dissociative adsorption and associative desorption type

A mathematical model of dissociative adsorption and associative desorption for diatomic molecules is generalized. The model is described by a coupled system of parabolic and ordinary differential equations. The existence and uniqueness theorem of the classical solution is proved.     

On transport equations in slow flows of a polyatomic gas in an external magnetic field

The transport equations in slow flows of a polyatomic gas in an external magnetic field varying with time are obtained by quantum kinetic theory methods. It is shown that local macroscopic variables, changing slowly with time, obey a system of linked equations. In the Burnett approximation, the Wigner operator and stress tensor are obtained. Transport coefficients depending on the magnetic field frequency are determined. Translated from Teoreticheskaya i Matematicheskaya Fizika, Vol. 110, No. 3, pp. 459–469, March, 1997.     

Multiscale approach to computation of three-dimensional gas mixture flows in engineering microchannels

A multiscale approach to computing real gas flows in engineering microchannels on high-performance computer systems in a wide range of Knudsen numbers is described. The numerical implementation of the approach combines the solution of quasigasdynamic equations and the molecular dynamics method. Following the approach, the parameters of the real gas equation of state are found at the molecular level, the kinetic gas properties are calculated, and the form of boundary conditions on the microchannel walls are determined. The technique is verified by computing several test problems. The results agree well with available theoretical and experimental data.     

Kinetic model of the Boltzmann equation with limiting gas flow regimes at low Knudsen numbers

A new model of the Boltzmann kinetic equation is constructed that describes both slow nonisothermal and Navier-Stokes continuum gas flows. The model is used to compute the slow nonisothermal flow past a circular cylinder. It is shown that the force exerted by the gas on the cylinder is affected by thermal stresses.     

A comparative analysis of approaches for investigating hypersonic flow over blunt bodies in a transitional regime

Hypersonic flows of a viscous perfect rarefied gas over blunt bodies in a transitional flow regime from continuum to free molecular, characteristic when spacecraft re-enter Earth's atmosphere at altitudes above 90-100 km, are considered. The two-dimensional problem of hypersonic flow is investigated over a wide range of free stream Knudsen numbers using both continuum and kinetic approaches: by numerical and analytical solutions of the continuum equations, by numerical solution of the Boltzmann kinetic equation with a model collision integral in the form of the S-model, and also by the direct simulation Monte Carlo method. The continuum approach is based on the use of asymptotically correct models of a thin viscous shock layer and a viscous shock layer. A refinement of the condition for a temperature jump on the body surface is proposed for the viscous shock layer model. The continuum and kinetic solutions, and also the solutions obtained by the Monte Carlo method are compared. The effectiveness, range of application, advantages and disadvantages of the different approaches are estimated.     

Implicit multiblock method for solving a kinetic equation on unstructured meshes

A parallel multiblock implementation of a second-order accurate implicit numerical method based on solving a model kinetic equation is proposed for analyzing three-dimensional rarefied gas flows. The performance of the method is illustrated by computing test examples of gas flows in a circular pipe in a wide range of Knudsen numbers. The convergence rate and scalability of the method are analyzed depending on the number of blocks in the spatial grid.     

动力学的流矢量分裂法模拟浅水波方程

Based on the analogy to gas dynamics, the kinetic flux vector splitting (KFVS) method is used to stimulate the shallow water wave equations. The flux vectors of the equations are split on the basis of the local equilibrium Maxwell-Boltzmann distribution. One dimensional examples including a dam breaking wave and flows over a ridge are calculated. The solutions exhibit second-order accuracy with no spurious oscillation.     

Modified method of splitting with respect to physical processes for solving radiation gas dynamics equations

An approach based on a modified splitting method is proposed for solving the radiation gas dynamics equations in the multigroup kinetic approximation. The idea of the approach is that the original system of equations is split using the thermal radiation transfer equation rather than the energy equation. As a result, analytical methods can be used to solve integrodifferential equations and problems can be computed in the multigroup kinetic approximation without iteration with respect to the collision integral or matrix inversion. Moreover, the approach can naturally be extended to multidimensional problems. A high-order accurate difference scheme is constructed using an approximate Godunov solver for the Riemann problem in two-temperature gas dynamics.