V. A. Rykov

• Russian Academy of Sciences

The problem of a rarefied plasma jet emerging from a stationary plasma engine is considered. The consideration is carried out entirely at the kinetic level; namely, the motion of all plasma components is described in terms of distribution functions. The system of kinetic equations should be solved together with Maxwell’s equations. Methods for solv...

For molecules interacting among each other with the potential having both repulsive and attractive components, a system of kinetic equations is derived using the Bogolyubov method, which takes into account the effect of forming bound states by molecules. This system implies all conservation laws and their corollaries that are invariant under the Ga...

In this article, the derivation of the Boltzmann equation from BBKIY of the chain is generalized for the case when the intermolecular interaction potential has both repulsive and attractive components. In this case, the application of the Bogolyubov method leads to situation when the term taking into account the transfer of molecules from the regio...

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...

The two-dimensional Couette flow with heat transfer was studied numerically using a non-linear nonequilibrium kinetic model of the Boltzmann equation. The effects of a maximum normal stress and a minimum streamwise energy flux were found depending on the Knudsen number.

Rarefied flows through a plane microchannel into a vacuum are numerically investigated on the basis of the model kinetic equation for a diatonic gas (nitrogen). The dependence of the gas flow rate through the channel on the Knudsen number, the wall temperature, and the channel length is determined. The energy flux transferred to the cold diatomic g...

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.

An isothermal steady rarefied gas flow in a long channel (tube) of elliptical or rectangular cross-section under the action of a given pressure gradient (Poiseuille flow) is studied on the basis of the Bhatnagar-Gross-Krook model. The solution is obtained using a conservative higher-order method. The velocity field in a channel cross-section is inv...

A system of model kinetic equations is proposed to describe flows of a diatomic rarefied gas (nitrogen). A conservative numerical method is developed for its solution. A shock wave structure in nitrogen is computed, and the results are compared with experimental data in a wide range of Mach numbers. The system of model kinetic equations is intended...

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 co...

The time-dependent one-dimensional problem of the normal reflection of a shock wave propagating at constant velocity in a gas (vapor) at rest from the plane surface of its condensed phase under steady-state condensation-evaporation conditions on the interphase plane is considered within the framework of the kinetic equation for a monatomic gas with...

Slow low-Knudsen-number monatomic-gas flow past a circular cylinder is numerically investigated on the basis of a model kinetic equation. The gas flow is described by a new kinetic equation, from which the continuum equations for slow nonisothermal gas flows containing temperature stresses follow rigorously. It is shown that a closed convective-flo...

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.

The structure of a normal (direct) shock in a gas for the parameters corresponding to nitrogen is investigated with allowance for the rotational degrees of freedom on the basis of a model kinetic equation. For various Mach numbers the structure is compared with both the known experimental results and the solutions of the Navier-Stokes approximation...

A method is proposed for averaging the Boltzmann kinetic equation with respect to transverse velocities. A system of two integro-differential equations for two desired functions depending only on the longitudinal velocity is derived. The gas particles are assumed to interact as absolutely hard spheres. The integrals in the equations are double. The...

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.

The two-dimensional supersonic rarefied gas flow past an infinite plate placed normally to the flow is analyzed. The gas possesses rotational degrees of freedom. The problem is stated for a model kinetic equation and is solved by applying a second-order accurate implicit conservative finite-difference method. The gas parameters correspond to nitrog...

Numerically, on the basis of the Krook kinetic equation, the rarefied gas flow around a circular cylinder is investigated in stationary and oscillatory regimes. The flows around a rotating cylinder and a cylinder with a nonuniformly heated surface are considered. The Knudsen numbers at which the lift force acting on the rotating cylinder changes si...

A symmetric second-order accurate splitting numerical method for the Boltzmann equation is proposed that is oriented to the examination of gas flow at low Knudsen numbers. The collision integral is linearized about a locally Maxwellian distribution function. Examples of the calculation of two-dimensional nonstationary flows around a circular cylind...

A numerical splitting method is proposed for solving the Boltzmann kinetic equation. The method is second-order accurate in time and spatial variables. It is based on a symmetric splitting of the gas motion into relaxation and free-molecular stages. This ensures a second-order accurate approximation of the Boltzmann equation in time. The free-molec...

A numerical method is proposed for solving Boltzmann's equation, based on a second-order accurate splitting of the gas motion into relaxation and free-molecular stages. To speed up the relaxation stage of the solution, the collision integral is linearized about a locally Maxwellian distribution function and represented by a matrix with precalculate...

The flow of a viscous incompressible fluid between coaxial cylinders rotating with a constant rigid-body acceleration about the axis is studied numerically. A one-dimensional time-dependent solution of the Navier-Stokes equations is constructed analytically for the motion starting from the state of rest. On the initial time interval the one-dimensi...

Two-dimensional monatomic-gas channels flows between two parallel plates developing due to discontinuous initial macroparameter distributions are studied on the basis of a numerical solution of the kinetic equation. The discontinuities are plane fronts perpendicular to the walls. The flows developed simulate gas flows in a shock tube and a microcha...

Two-dimensional steady rarefied-gas channel flow between two parallel walls, from an evaporating face to a perfectly absorbing plane end face, is studied. The vapor is considered to be a monatomic gas. The corresponding problem for the kinetic equation with collision integral in BGK form is formulated and solved numerically by two different finite-...

A numerical method for calculating plane flows of a rarefied gas at low Knudsen numbers based on model kinetic equations is presented. The method is used to calculate two-dimensional gas flows in a plane channel. The conditions on surfaces which bound the flow domain are considered. In order to use the establishment method to solve the steady probl...

Vortical nonstationary viscous incompressible flows in the space between coaxial cylinders or hemispherical segments rotating with a constant angular acceleration about a stationary axis of symmetry are analyzed numerically for Reynolds numbers Re — 1–10. It is shown that laminar circulating motions are realized. Two vortices form in the flow. The...

The internal problem of the unsteady three-dimensional flow of a viscous incompressible fluid in the axisymmetric cavity of a body which is rotating round an axis of symmetry or a fixed point is considered. The velocity and acceleration of the body are given. The fluid completely fills the cavity. The Reynolds numbers are moderate. The flow of the...

A conservative numerical method for solving the nonstationary kinetic equation over a wide rarefaction range is presented. Particular attention is given to its efficiency at low Knudsen numbers. Test calculations are analysed.

The steady state problem of flow past a body of arbitrary shape is analyzed using a linear Boltzmann equation describing the perturbation of the distribution function. It is shown how the problem considered here can be stated as a problem in which the unknown functions include a part of the distribution function that is odd with respect to the velo...

Slow flows of a monatomic rarified gas which interacts with the surface of a solid according to a general law are considered. A functional is constructed, the Euler equations of which are linear kinetic equations with natural boundary conditions. The functional is the difference between the doubled power of the internal forces of the medium and the...

The paper deals with slow flows of a monoatomic rarefied gas interacting with the surface of a body in accordance with a general law. A functional is constructed whose Euler equations are linear kinetic equations with natural boundary conditions. The functional represents the difference between the doubled power of the internal medium forces and en...

The problem of the construction of a full set of divergent forms of macroscopic conservation equations for a monoatomic gas in an external force field is investigated on the basis of the Boltzmann equation. Necessary and sufficient conditions are determined for the existence of divergent conservation laws. For certain types of fields, new divergent...

The slow (relative to the medium) motion of a heat-conducting particle in a gas flow described by the Navier-Stokes equations is examined. The particle is small as compared with the macroscopic scale of the flow and the Local perturbation it creates is therefore described by a Linear Boltzmann equation. A system of thermodynamic fluxes correspondin...

A method for the analysis of flows of a rarefied gas at small Knudsen numbers, originally proposed in an earlier study (Demin and Rykov, 1986), is presented. The method is based on the use of an S-model kinetic equation. Subsonic flow past a plate positioned along the flow direction is examined as an example. Details of the analysis are given, and...

A mathematical model is proposed for solving kinetic equations in a system of polar coordinates with variable phasing along the radius. A numerical method is used to investigate the flow of a monatomic rarefied gas over a circular rotating cylinder. The typical structure of flow lines during supersonic flow over a rotating cylinder is defined. The...

The influence of boundary conditions on the distribution of macroscopic parameters of various types is studied using the problem of heat transfer for monatomic and diatomic gases as an example. The calculations are made on the basis of model kinetic equations [1, 2]. A boundary condition that takes into account mutual transitions of rotational and...

A boundary condition is proposed for a diatomic gas with rotational degrees of freedom. In addition to the coefficient of accomodation of translational and rotational energies, the condition also includes the coefficients of energy exchange between translational and rotational degrees of freedom in the interaction of the gas with a solid surface. T...

Based on the Boltzmann kinetic equation, a complete set of divergent forms of macroscopic equations is obtained for arbitrary motions of a monoatomic gas. By considering a gas cloud as an example, the physical meaning of the new divergent equations is made clear, and exact expressions describing characteristics of the motion of a gas cloud in free...

A set of methods has been developed for studying the translational motion of electron-excited particles in a nonequilibrium plasma from the spectral line profiles using high-resolution spectroscopy. These methods are used here to investigate the nonequilibrium distribution functions of N2(C 3Pi u) molecules and O(3 3P) atoms excited by metastable p...

Onsager's principle is established for the linear problem of the steady three-dimensional flow of a monatomic gas around an arbitrary body. The upstream velocity and the difference between the body surface temperature and the gas temperature at infinity are the small parameters that make it possible to linearize the problem.

The symmetry of Onsager kinetic coefficients (OKCs) is demonstrated for the system of thermodynamic forces and fluxes for a flow past a body on the surface of which condensation or evaporation can occur (a case of interest for aerosol particles). Attention is given to gas flow in a channel with heat-conducting walls of arbitrary shape, connecting t...

Using the kinetic approach, an expression is obtained for the generation of entropy in the problem of external flow past a body. Using the linear Boltzmann equation and the boundary conditions satisfying the reciprocity conditions, the symmetry (the Onsager principle) of the kinetic coefficients in flows is proved.

The steady equilibrium flow of a monatomic rarefied gas past a heat-conducting body of arbitrary shape is examined in the linear approximation. The symmetry of the integral kinetic coefficients is established on the basis of the linear steady Boltzmann equation for flow past a heat-conducting body. An expression for entropy production in the gas-bo...

Symmetry of integral transport coefficients is established on the basis of a linear stationary Boltzmann equation for the problem of flow past a heat conducting body. An expression is obtained for the entropy production in the gas-body system, and this determines the thermodynamic fluxes and forces. Bakanov and Roldugin [2] have considered the prob...

The kinetic approach is used to obtain an expression for energy production in the problem of flow past a body. It is shown in the linear approximation that the thermodynamic forces are the velocity components of the incoming flow and the temperature difference between the body and the gas at infinity. The symmetry of the kinetic coefficients (i.e.,...

The flow of rarefied gas past a sphere with no-flow condition on the surface has been well studied both experimentally and numerically. In the presence of blowing on the sphere into the oncoming flow, the reflection of the main flow from the body introduces new features. This problem has been considered in the continuum regime [1–3] and, in a kinet...

The steady Krook equation describing rarefied-gas flow is reduced to a second-order differential equation for an even distribution function. The state of the gas is completely determined by the distribution function and the macroscopic-velocity vector. The boundary conditions on the surface of the body are transformed into a form that contains only...

By using a numerical solution of the Boltzmann equation in the hard-sphere approximation for a gas in the thermostat, arbitrary values of the ratio of masses of the colliding particles, and an arbitrary deviation from equilibrium, it is shown that during the relaxation process, the initial Maxwell velocity distribution of particles is not conserved...

Slow flow of a rarefied gas over a nonuniformly heated plate is investigated numerically. The interaction of the oncoming stream with the flow due to the variable temperature of the gas near the body is considered.

Laws of similitude of hypersonic flows of monatomic gases have been obtained earlier from asymptotic analysis of the equations as S ? 8 and confirmed by experimental data and numerical results [1], For diatomic gases, dimensionless numbers have not been deduced by analyzing the equations but by general arguments based on analogy with monatomic gase...

A SURVEY, compiled in the Laboratory of the Theory of Transport Processes of the Computing Centre of the Academy of Sciences of the USSR, of papers on numerical methods of solving the kinetic equations for a neural gas is presented. THe main aspects of 15 years of the scientific activity of a small research group are briefly elucidated. The papers...

Research performed at the Computational Center of the Soviet Academy of Sciences on numerical methods for solving kinetic equations for neutral gases is surveyed. Two major directions of research are summarized: (1) the approximation of Boltzmann equations and the numerical solution of the approximating kinetic equations, and (2) the direct integra...

In the present paper, a numerical solution of a model kinetic equation for a gas with rotational degrees of freedom is applied to the analysis of the spherical expansion of the gas from an evaporating droplet. The steady evaporation of the droplet of given temperature into a vacuum and into a medium of the same gas is studied on the basis of the co...

(For abstract see issue 22, p. 4057, Accession no. A80-49604)

The problem of shear motion of a gas (Couette flow) is studied. Two infinite parallel plates with temperatures T1 and T2 separated by the distance L each move in their own plane with velocities u0 and -u0, respectively. It is assumed that there is a monatomic gas between the plates. In such a formulation, the Couette problem has been considered ear...

A method is presented for studying diatomic gas flows; this method is based on constructing S-model equation with taking into account the rotational degrees of freedom. A numerical scheme for integrating this equation is developed. As an example, the method is applied to solving the problem on a sphere submerged into a stream of nitrogen. The basic...

THE PROBLME of the one-dimensional motion of a gas between infinite plane-parallel evaporating and absorbing surfaces is considered on the basis of model kinetic equation for a monatomic gas. A specially developed numerical method makes it possible to obtain a solution of the problem for the whole range of Knudsen numbers and trace the gas flow rig...

Linear integral equations are obtained for the perturbed Maxwell distribution functions. The number of eigenfunctions of the integral operators corresponding to the null eigenvalue is equal to the number of independent macroparameters defining the state of a fluid particle. Besides the eigenfunctions coinciding with the invariants of inelastic coll...

On the basis of model kinetic equations a solution is obtained by a numerical method for the flow of attenuated gas around a sphere. The effect of rotational degrees of freedom on the energy flux to the body is investigated. Values of the ratio between the energy flux Q and its free-molecular value Q* for monatomic and diatomic gases are compared;...

The Chapman-Enskog method of perturbation theory can be used to solve the system kinetic Boltzmann equations at small Knudsen numbers. Extension of this method to gas mixtures with unconstrained (not slow) bimolecular chemical reactions is hampered by the fact that the number of microparameters defining the state of a fluid particle is greater than...

An improved iterative process based on the use of the equations of conservation is proposed for solving the Boltzmann kinetic equation for small Knudsen numbers. Microparameter profiles are obtained for spherical outflow into vacuum and into an immersed space. The results are compared with solutions obtained from the Navier-Stokes equations. The fl...

A model kinetic equation for a gas with rotational degrees of freedom is obtained. By averaging of the distribution function over quantities corresponding to the rotational degrees of freedom this equation is reduced to a closed system of two kinetic equations, each of which is analogous to the kinetic equation of a monatomic gas.

An integral iterative method for solving Navier-Stokes equations in the kinetic theory of gases at Knudsen numbers less than 1 is described. This method for solving stationary problems is applied to several one-dimensional examples: heat transfer between flat walls, Couette flow, and shock wave structure. It is shown that by using conservation laws...

Recently, kinetic model equations have become widely used. Differing from the Boltzmann equation by their simplicity, they maintain its main features. It is not possible to evaluate theoretically the magnitude of the errors incurred when substituting exact equation for the modelled one. In such a situation an all-around investigation of the structu...

The problem of heat transfer between two infinite parallel plates is investigated on the basis of equations obtained by averaging the Boltzmann kinetic equation with respect to the transverse velocity. A numerical solution of the problem is accomplished for a temperature ratio between the plates of T0/T1=1/4 and for various Knudsen numbers.

By averaging the Boltzmann kinetics equation with respect to the transverse velocities we obtain a system of two integrodifferential equations for two unknown functions that depend on the longitudinal velocity u, time t, and the x coordinate. It is assumed that the particles interact with one another like perfectly elastic spheres. The integrals ap...

The slow (with respect to the ambient medium) motion of a heat-conducting particle in a gas flow described by the Navier-Stokes equations is investigated analytically. Since the paricle size is small in comparison with the macroscopic flow scale, a local gas perturbation created by the particle is described by a linear Boltzmann equation. A system...

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...