Semiempirical theory of turbulence of inhomogeneous flows with body forces

A closed system of equations for the second moments, determining the turbulent transfer in inhomogeneous flows with body forces, is obtained in the local equilibrium approximation. Taking diffusion and convective transfer into account does not present difficulties, since the expressions obtained for the terms containing the fluctuations of the body forces and transfer terms are preserved. The effect of the external forces is considered for two cases: turbulent motion in a stratified medium and conducting flows in a longitudinal magnetic field. In the latter case the external forces are divergenceless and they therefore do not affect the pressure fluctuations. A single-scale model, in which the transverse scale is taken as the scale in the boundary layer, is considered.Translated from Izvestiya Akademii Nauk SSSS, Mekhanika Zhidkosti i Gaza, No. 3, pp. 33–40, May–June, 1985.     

Allowance for nonstationary heat and mass transfer in the propagation of small perturbations in a liquid with bubbles

The propagation of small perturbations in a liquid with vapor-gas bubbles is studied. Heat and mass transfer between the phases is taken into account on the basis of the exact equations of heat conduction and diffusion. The aim of the present investigation is to make more precise the results of an earlier paper [1] of the author and verify the applicability of using fixed coefficients of heat and mass transfer for nonstationary heat and mass transfer between a pulsating bubble and the liquid.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 4, pp. 157–162, July–August, 1979.     

Multicomponent three-dimensional viscous shock layer on blunt bodies with a catalytic surface at angles of attack and yaw

The three-dimensional supersonic flow of nonequilibrium dissociating air past smooth blunt bodies on whose surface heterogeneous chemical reactions are taking place is investigated within the framework of thin viscous shock layer theory. An economical numerical method of solving the equations with an improved order of approximation with respect to the normal coordinate is employed. This method does not require the preliminary solution of the Stefan-Maxwell relations for the diffusion fluxes and makes it possible to calculate flows that do not possess a plane of symmetry. The effect of the angles of attack and yaw, the catalytic reaction model and a number of other parameters of the problem on the pressure, heat flux and equilibrium surface temperature distributions is analyzed with reference to the example of flow past a triaxial ellipsoid.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 1, pp. 143–150, January–February, 1990.Tha authors are grateful to G. A. Tirskii for useful discussions of their results.     

Nonreacting chemical transport in two-phase reservoirs — Factoring diffusive and wave properties

The vertical transport of mass, energy andn unreacting chemical species in a two-phase reservoir is analysed when capillarity can be ignored. This results in a singular system of equations, comprising mixed parabolic and hyperbolic equations. We derive a natural factorisation of these equations into diffusive and wave equations. If diffusive or conductive effects are important for onlyN–1 of the chemical species, then there areN parabolic equations, andn+2–N wave equations. Each wave equation allows for the corresponding variable to be discontinous, or equivalently, for shock propagation to occur. Steady flows were shown to allow for more than two (vapour and liquid dominated) saturations for a given mass, energy and chemical flux, but when thermal conduction and chemical diffusion are unimportant, only the vapour and liquid dominated cases appear likely to occur. For infinitesimal shocks there is a continuous flux vector for each diffusive variable. The existence of these continuous flux vectors results in significant simplifications of the corresponding wave equations. It remains an open question if continuous flux vectors exist for finite shocks. General expressions are obtained for the diffusion and wave matrices, which require no knowledge of continuous flux vectors.     

Lie Symmetry Analysis of Differential Equations in Finance

Lie group theory is applied to differential equations occurring as mathematical models in financial problems. We begin with the complete symmetry analysis of the one-dimensional Black–Scholes model and show that this equation is included in Sophus Lie's classification of linear second-order partial differential equations with two independent variables. Consequently, the Black–Scholes transformation of this model into the heat transfer equation follows directly from Lie's equivalence transformation formulas. Then we carry out the classification of the two-dimensional Jacobs–Jones model equations according to their symmetry groups. The classification provides a theoretical background for constructing exact (invariant) solutions, examples of which are presented.     

Diffusion separation of chemical elements on a catalytic surface

An asymptotic expansion of the solution, for large Schmidt numbers, of the system of equations of a chemically nonequilibrium multicomponent boundary layer on the catalytic surface of a blunt body [1] is used to obtain expressions for the diffusion fluxes of the reaction products and chemical elements and the heat flux as functions of the gradients of the reaction product concentrations, chemical element concentrations and enthalpy across the boundary layer. It is shown that when the body is exposed to a supersonic air flow, the diffusion separation of the chemical element oxygen depends importantly on the atom concentration at the outer edge of the boundary layer and the nature of the homogeneous and heterogeneous catalytic reactions. If the surface promotes the rapid recombination of oxygen atoms and is chemically neutral with respect to nitrogen atoms, then an excess of the chemical element oxygen is formed on the body. Otherwise we get an enhanced concentration of the element nitrogen. As distinct from the case of an ideally catalytic wall [2–4], on a surface possessing the property of catalytic selectivity the diffusion separation of chemical elements takes place even when only atoms are present at the outer edge of the boundary layer. On a chemically neutral surface diffusion separation may be caused by homogeneous recombination reactions between oxygen and nitrogen atoms if their rate constants are essentially different.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 4, pp. 115–121, July–August, 1988.     

Anisotropic percolatory-convective diffusion in underground leaching

The two-dimensional problem of the propagation of an impurity in underground flows is investigated with allowance for chemical reactions between the solvent and the ore-bearing matrix material. The flow velocity field generated by a pair of wells (source—sink) is assumed to be steady and known. A numerical method of solving the system of equations of convective diffusion and nonequilibrium sorption is discussed. Typical isoconcentration fields are presented. The effect of the influence parameters on the principal characteristics of the leaching process is analyzed.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 1, pp. 96–101, January–February, 1987.     

Self-similar problems of convective diffusion in the presence of heterogeneous chemical reactions with allowance for thermal diffusion effects

A number of studies have been made of the problem of the effect of a temperature gradient on mass transfer in a forced viscous fluid flow. The question of allowing for thermal diffusion effects has been examined in connection with flow around bodies [1–4], duct flow [5], and jet flows [6,7]. Recently, in addition to the problem of thermal diffusion separation, the attention of investigators has been claimed by the problem of taking into account the effect of thermal diffusion on mass transfer in a convective flow in the presence of chemical reactions on the flow surfaces [4].     

Numerical calculation of the problem of cooling of a spherical volume of nonequilibrium-ionized radiating helium

The method is described and the results are presented for numerical calculations of a system of equations of nonsteady gasdynamics, radiation transfer in the continuous spectrum, and the kinetics of collisional ionization and ionization by radiation, which describe the dispersion and cooling of a spherical volume of He. A comparison is made with calculations performed on the assumption of thermodynamic equilibrium.Translated from Zhurnal Prikladnoi Mekhaniki i Tekhnicheskoi Fiziki, No. 2, pp. 36–41, March–April, 1976.     

An Approach to Transport in Heterogeneous Porous Media Using the Truncated Temporal Moment Equations: Theory and Numerical Validation

In the last decade, the characterization of transport in porous media has benefited largely from numerical advances in applied mathematics and from the increasing power of computers. However, the resolution of a transport problem often remains cumbersome, mostly because of the time-dependence of the equations and the numerical stability constraints imposed by their discretization. To avoid these difficulties, another approach is proposed based on the calculation of the temporal moments of a curve of concentration versus time. The transformation into the Laplace domain of the transport equations makes it possible to develop partial derivative equations for the calculation of complete moments or truncated moments between two finite times, and for any point of a bounded domain. The temporal moment equations are stationary equations, independent of time, and with weaker constraints on their stability and diffusion errors compared to the classical advection–dispersion equation, even with simple discrete numerical schemes. Following the complete theoretical development of these equations, they are compared firstly with analytical solutions for simple cases of transport and secondly with a well-performing transport model for advective–dispersive transport in a heterogeneous medium with rate-limited mass transfer between the free water and an immobile phase. Temporal moment equations have a common parametrization with transport equations in terms of their parameters and their spatial distribution on a grid of discretization. Therefore, they can be used to replace the transport equations and thus accelerate the achievement of studies in which a large number of simulations must be carried out, such as the inverse problem conditioned with transport data or for forecasting pollution hazards.     

Soret and dufour effects on MHD free convective heat and mass transfer with thermophoresis and chemical reaction over a porous stretching surface: Group theory transformation

The group theoretic method is applied for solving the problem of the combined influence of the thermal diffusion and diffusion thermoeffect on magnetohydrodynamic free convective heat and mass transfer over a porous stretching surface in the presence of thermophoresis particle deposition with variable stream conditions. The application of one-parameter groups reduces the number of independent variables by one; consequently, the system of governing partial differential equations with boundary conditions reduces to a system of ordinary differential equations with appropriate boundary conditions. The equations along with the boundary conditions are solved numerically by using the Runge-Kutta-Gill integration scheme with the shooting technique. The impact of the Soret and Dufour effects in the presence of thermophoresis particle deposition with a chemical reaction plays an important role on the flow field.     

Diffusion and adsorption of a radioactive gas in a porous medium

The results of solving the one-dimensional problem of the motion of a pulse of radioactive gas, carried through a porous medium by a stream of inert carrier with constant velocity, are generalized by the case of taking diffusion processes into account. For a delta-shaped input pulse, the solution is obtained of a system of equations which describes the migration of the pulse, taking diffusion washout and nonequilibrium adsorption into account. It is shown that in the case of equilibrium adsorption the time of appearance of the concentration maximum at the adsorber outlet depends on the decay constant and the coefficient of diffusion. Approximate solutions for strong- and weak-nonequilibrium adsorption and in the case of weak diffusion are considered. An estimate is carried out of the maximum magnitude of the diffusion coefficient, when its effect can be neglected.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 5, pp. 85–90, September–October, 1976.The authors thank A. N. Gudkova for discussing the work.     

Large eddy simulation of natural convection along a vertical isothermal surface

Large eddy simulations of natural convection along a vertical isothermal surface have been carried out using a parallel CFD code SMAFS (Smoke Movement And Flame Spread) developed by the first author to study the dynamics of the natural convection flow and the associated convective heat transfer, with sub-grid scale turbulence modeled using the Smagorinsky model. In the computation, the filtered governing equations are discretized using finite volume method, with the variables at the cell faces in the finite volume discrete equations approximated by a second order bounded QUICK scheme and the diffusion term computed based on central difference scheme. The computation was time marched explicitly, with momentum equations solved based on a second order fractional-step Adams–Bashford scheme and enthalpy computed using a second order Runge–Kutta scheme. The Poisson equation for pressure from the continuity equation was solved using a multi-grid solver. The results including the temperature and velocity profiles of the boundary layer and the local heat transfer rate are analyzed. Comparison is made with experimental data and good agreement is found.     

Approximate models of salt transfer with washings of soils with water-impermeable base

On the basis of an approximate model of filtration with drainage washings of a soil layer with a water-impermeable base [1], shown schematically in Fig. 1, an investigation is made of the desalinization process of soils for several forms of the equations of salt transfer between the moving solution and the fixed phase. Neglect of the external diffusion makes it possible to reduce the investigation to the solution of one-dimensional boundary-value problems, characterizing salt transfer along fixed streamlines. Analogous problems were discussed in [2] with application to the one-dimensional filtration of wash waters in a dry soil.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 1, pp. 44–51, January–February, 1978.The author thanks V. N. Émikh and V. I. Pen'kovskii for their repeated and useful evaluations of the work.     

Formation of laminar lateral separation caused by a sustainer engine jet

The early stage of separation flow formation on a cylindrical surface immediately after the onset of jet exhaust from the base is investigated numerically for various values of the jet-to-ambient pressure ratio on the basis of the complete time-dependent Navier-Stokes equations, the energy equation and the equations for the transfer and diffusion of the exhaust gas concentration. The time dependence of the geometrical and gasdynamic parameters in the separation flow region is obtained for various flow regimes. It is found that the process is periodic with transition from the flow regime with an open separation zone to that with a closed zone, and vice versa, until the final flow regime is attained, which may be open, closed, or periodic.Moscow. Translated from Izvestiya Rossiiskoi Akademii Nauk, Mekhanika Zhidkosti i Gaza, No. 4, pp. 122–130, July–August, 1995.     

Equations of transonic flows of relaxing mixtures

Transonic flows of chemically active gas mixtures whose composition is defined by an arbitrary number of reactions are considered. Conditions ensuring a closeness of the frozen and equilibrium velocities of sound are imposed on the equations of state. An approximate system of equations for vectors of particle velocity and completeness of reactions is obtained as a result of an asymptotic analysis of the system of Euler equations and the equations of chemical reactions associated with them. This system reduces to two equations containing only components of the particle velocity; the order of one of them equals the number of reactions increased by one, while the second equation expresses the nonvorticity condition of the stream. Various limit cases dependent on the magnitude of eigenvalues of the relaxation matrix are pointed out.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 4, pp. 139–148, July–August, 1977.     

Conditional averaging of the equations of flow in random composite porous media

The problem of conditional averaging of the transport equations is solved for a neutral impurity in a composite medium with random porosity and impurity diffusion tensor. An unclosed system of conditionally averaged equations is constructed and closed using the globally averaged equations. The average impurity concentration fields for the individual phases of the composite medium and the phase-continuum interaction characteristics are calculated.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No, 1, pp. 75–81, January–February, 1987.     

Influence of real gas properties on the integral aerodynamic coefficients

The results of mathematically modeling axisymmetric hypersonic flow past an ellipsoid are presented. The calculation data are obtained on the basis of a numerical solution of the complete Navier—Stokes equations using a finite-difference method. The investigation is carried out for the problem of laminar flow over the windward side of the body. The effect of the elongation of the ellipsoid on the total heat flux and the viscous and pressure drag is considered. Results are obtained for three different gas (air) models: the perfect gas, chemical equilibrium and chemical nonequilibrium models. For the latter model various sets of catalyticity coefficients are considered. The effect of the real properties of air on the integral aerodynamic characteristics of ellipsoids with different elongations is analyzed.Translated from Izvestiya Rossiiskoi Akademii Nauk, Mekhanika Zhidkosti i Gaza, No.4, pp. 156–164, July–August, 1992.     

Region of applicability and the main features of the generalized Chapman-Enskog method

The present paper is made necessary by the publication of the foregoing paper in this issue by Kolesnichenko [1]. It considers the basic propositions of the generalized Chapman-Enskog method and analyzes the arguments put forward by Kolesnichenko [1] and the validity of the method. The position of the results obtained by Kolesnichenko [14–17] is indicated. Nonequilibrium flows of multiatomic gases in which there occur processes of exchange of internal energy of the molecules in collisions between them and chemical reactions (such processes are called inelastic) are encountered frequently in nature and technology. It is therefore naturally of interest to derive gas-dynamic equations for such flows. The methods of the kinetic theory of gases were first used to obtain equations describing the limiting cases of very fast inelastic processes that take place in times of the order of the molecule-molecule collision times (equilibrium case) and very slow inelastic processes that take place over times of the order of the characteristic flow time (relaxation case). In [2–5], an algorithm was proposed for deriving gas-dynamic equations valid for arbitrary ratios of the rates of the elastic and inelastic processes and reducing to the well-known equations for the limiting cases already mentioned. The algorithm is called the generalized Chapman-Enskog method (abbreviated to the generalized method). The development, modification, and analysis of its properties can be found in [4, 6–13]. In [1], Kolesnichenko has questioned the validity of this algorithm.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 3, pp. 126–136, May–June, 1984.We thank V. A. Rykov for helpful and constructive discussions of the work.     

Viscous hypersonic flows for various aerophysical models

The results of mathematical simulation of hypersonic air flow past a sphere with allowance for chemical nonequilibrium and ionization are presented. The calculated data are obtained on the basis of a numerical solution of the Navier-Stokes equations and the hypersonic viscous shock layer equations. The equilibrium radiation temperature of the body surface, the pressure, and the shock wave detachment are compared for these models. The combined effect of molecular transport and the catalytic properties of the body surface on the aerodynamic heating is analyzed. The results of the effect of certain models of the thermochemical disequilibrium of the gas on the temperature of the body surface are also analyzed along a heat-stressed interval of the reentry trajectory of an aerospace vehicle.Moscow. Translated from Izvestiya Rossiiskoi Akademii Nauk, Mekhanika Zhidkosti i Gaza, No. 4, pp. 151–161, July–August, 1996.