Mixed Convection Effect on Melting from a Vertical Plate in a Porous Medium

In the present work, the effect of mixed convection about vertical surfaces on the phenomenon of melting process in a fluid-saturated porous medium is analyzed on the basis of boundary layer approximations. Similarity solutions are obtained for aiding external flow. The final similarity equations are integrated numerically by use of the fourth-order Runge–Kutta method. Results are reported for the flow and thermal fields in the melt region. The melting phenomenon decreases the local Nusselt number at the solid–liquid interface.     

Theoretical investigation of the combined flow of a two-dimensional,high-pressure stream and a one-dimensional,low-pressure stream of ideal gas in ejector nozzles

On the basis of [1] an improved method was developed which, within the framework of the model of an ideal gas, allows one to calculate the flow in ejector nozzles without a limit on the coefficient of ejection. During the development of the method it was established, on the basis of a preliminary analysis, that the difference equations which approximate the differential equations of the flow of coaxial streams in an ejector nozzle (high-pressure and low-pressure streams, treated in two-dimensional and one-dimensional approximations, respectively) have a singular point. Owing to the finiteness of the integration step the position of this singular point differs in the general case from the position of the singular point for the differential equations describing the flow under investigation. This difference is larger the smaller the coefficient of ejection. Now allowing for this fact in the existing methods of calculation in an analogous formulation [1–4] limits the possibilities of all these methods, as a rule, to cases of relatively large coefficients of ejection.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 6, pp. 110–116, November–December, 1978.The authors thank A. N. Kraiko for useful discussions and attention to the work.     

Nonisothermal flow of a viscous incompressible fluid in a two-dimensional diffuser

The velocity and temperature distributions in a viscous incompressible fluid flow in a two-dimensional diffuser are analyzed. Fully developed flow is considered, i.e., the influence of the entrant section is disregarded. It is assumed that the diffuser walls are maintained at a temperature depending on the polar radius. The dynamic viscosity is considered to be an exponential function of the temperature. The problem is reduced to the solution of a system of ordinary differential equations, which is solved by the method of successive approximations. The convergence of the iterative scheme is proved.Translated from Zhurnal Prikladnoi Mekhaniki i Tekhnicheskoi Fiziki, No. 4, pp. 40–48, July–August, 1973.The author is indebted to L.A. Galin and N. N. Gvozdkov for assistance with the study.     

Approximation of complex dependences of quantities by selection of weights in averaging equations

A method is proposed for obtaining simple approximation equations by dimensionless averaging of prescribed values of the quantities. This method is useful for constructing approximations of the noninterpolation type. Examples of the derivation of useful three-point correlations of the pressure and projections of aerodynamical forces from various parameters are accompanied by comparisons with calculations of supersonic flow around bodies.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 1, pp. 161–165, January–February, 1986.     

On the boundary conditions for the equations of gas dynamics corresponding to the higher approximations in the Chapman-Enskog method of solution of the Boltzmann equation

When the gas-dynamic equations are derived from the solution of the Boltzmann equation by the Chapman-Enskog method, the order of the system of partial differential equations tends to increase with increasing number of the approximation. As a result, it is necessary to have more and more boundary conditions for these equations, which, however, are at present definitely known only for the first two approximations (models of an ideal gas and a viscous gas). A method is proposed for constructing additional boundary conditions; the method is illustrated in a number of examples.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 3, pp. 77–87, May–June, 1979.I should like to thank M. N. Kogan, V. S. Galkin, O. G. Fridlender, and S. A. Regirer for their interest in the work and for valuable discussions.     

Calculation of inviscid secondary flows in hydromachinery blade systems

The equations of the problem of calculating secondary flows in hydromachinery blade systems are obtained for the quasi-three-dimensional flow approximations. The results of calculating these flows for a Francis turbine and the pumping regime of a pumpturbine are presented.Translated from Izvestiya Rossiiskoi Akademii Nauk, Mekhanika Zhdkosti i Gaza, No. 5, pp. 10–15, September–October, 1993.     

Solution of three-dimensional nonlinear problems of the theory of jets in ideal fluids

A method of successive approximations is proposed for solving three-dimensional nonlinear problems of the theory of jets in ideal fluids (see, for example, [1–3]). Each approximation includes the calculation of the flow over a known surface, i.e., the solution of the exterior Neumann problem for the Laplace equation in the velocity potential and the correction of part of that surface for the purpose of reducing the discrepancy in the constant-pressure condition at the surface of the jets. The correction takes the form of small deformations found from a system of integral equations; the shape of the cavity in plan is also refined. The results of calculating the flow past triaxial ellipsoids, obtained using the generalized Zhukovskii-Roshko method for closing the jets, are presented.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 2, pp. 175–179, March–April, 1989.The authors are grateful to V. P. Karlikov for useful comments.     

Modeling and computation of cavitation in vortical flow

The paper presents an investigation of Euler–Lagrangian methods for cavitating two-phase flows. The Euler–Euler methods, widely used for simulations of cavitating flows in ship technology, perform well in regions of moderate flow changes but fail in zones of strong, vortical flow. Reasons are the strong approximations of cavitation models in the Euler concept. Alternatively, Euler–Lagrangian concepts enable more detailed formulations for transport, dynamics and acoustic of discrete vapor bubbles. Test calculations are performed to study the influence of different parameters in the equations of motion and in the Rayleigh–Plesset equation for bubble dynamics. Results confirm that only Lagrangian models are able to describe correctly the bubble behavior in vortices, while Eulerian results deviate strongly. Lagrangian formulations enable additionally the determination of acoustic pressure of cavitation noise. Two-way coupling between the phases is required for large regions of the vapor phase. A new coupling concept between continuous fluid flow and discrete bubble phase is developed and demonstrated for flow through a nozzle. However, the iterative coupling between the phases via volume fractions is computationally expensive and should therefore be applied only in regions where Eulerian treatment fails. A corresponding local concept for combination with an Euler–Euler method is outlined and is in progress.     

Numerical study of eccentric Couette–Taylor flows and effect of eccentricity on flow patterns

In this study, the differential quadrature (DQ) method was used to simulate the eccentric Couette–Taylor vortex flow in an annulus between two eccentric cylinders with rotating inner cylinder and stationary outer cylinder. An approach combining the SIMPLE (semi-implicit method for pressure-linked equations) and DQ discretization on a non-staggered mesh was proposed to solve the time-dependent, three-dimensional incompressible Navier–Stokes equations in the primitive variable form. The eccentric steady Couette–Taylor flow patterns were obtained from the solution of three-dimensional Navier–Stokes equations. The reported numerical results for steady Couette flow were compared with those from Chou [1], and San and Szeri [2]. Very good agreement was achieved. For steady eccentric Taylor vortex flow, detailed flow patterns were obtained and analyzed. The effect of eccentricity on the eccentric Taylor vortex flow pattern was also studied.     

Numerical calculation of the coefficient of capture of aerosol particles in toroidal channels of circular section

The flow of a liquid (or gas) with aerosol particles suspended in it in channels of different configurations is of great interest in the solution of many practical problems. The aim of the present paper is to develop a method for calculating the hydrodynamics and the heat and concentration transfer of aerosol particles for steady flow of an incompressible fluid in toroidal channels of circular section. The paper uses an implicit difference scheme with different approximations of the convective terms on a nonuniform grid (directed differences, central differences, and the monotonic approximation of Samarskii), which makes it possible to reduce the solution of the system of the original nonlinear partial differential equations to the successive solution of one-dimensional systems [1]. The method proposed by Polezhaev and Gryaznov [2] is used to calculate the boundary conditions for the vorticity. The hydrodynamic equations are solved by means of the difference scheme developed by Khristov [3], and the heat and concentration transfer equations are solved by the difference scheme proposed by Val'tsiferov and Polezhaev [4]. The obtained results make possible a detailed analysis of the dependence of the basic integrated (particle capture coefficient) and local characteristics on the values of the relevant dimensionless numbers, namely, the Dini, Prandtl, and Schmidt numbers, the parameter R/R_k, which characterizes the curvature of the channel, and the dimensionless parameter W_f = fR_G(T_O–T_W)/(pM), which characterizes the rate of thermophoresis.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 2, pp. 159–164, March–April, 1984.     

Friction and heat transfer in an incompressible fluid near a plane stagnation point

In the neighborhood of a plane stagnation point, the flow and heat transfer of an incompressible fluid are studied. In the inner flow region, the velocity and pressure fields are described by the complete Navier-Stokes equations, and the temperature field is described by the complete energy equation. In the outer flow region, a two-term asymptotic solution of the corresponding equations is obtained. The problem is reduced to the numerical solution of ordinary differential equations. Numerical results are discussed.Moscow. Translated from Izvestiya Rossiiskoi Akademii Nauk, Mekhanika Zhidkosti i Gaza, No. 4, pp. 52–65, July–August, 1996.     

Conjugate heat transfer inside a porous channel

Analytical and numerical analyses have been performed for fully developed forced convection in a fluid-saturated porous medium channel bounded by two parallel plates. The channel walls are assumed to be finite in thickness. Conduction heat transfer inside the channel wall is also accounted and the full problem is treated as a conjugate heat transfer problem. The flow in the porous material is described by the Darcy–Brinkman momentum equation. The outer surfaces of the solid walls are treated as isothermal. A temperature dependent volumetric heat generation is considered inside the solid wall only. Analytical expressions for velocity, temperature, and Nusselt number are obtained after simplifying and solving the governing differential equations with reasonable approximations. Subsequent results obtained by numerical calculations show an excellent agreement with the analytical results.     

Chaotic Analysis of Nonlinear Viscoelastic Panel Flutter in Supersonic Flow

In this paper chaotic behavior of nonlinear viscoelastic panels in asupersonic flow is investigated. The governing equations, based on vonKàarmàn's large deflection theory of isotropic flat plates, areconsidered with viscoelastic structural damping of Kelvin's modelincluded. Quasi-steady aerodynamic panel loadings are determined usingpiston theory. The effect of constant axial loading in the panel middlesurface and static pressure differential have also been included in thegoverning equation. The panel nonlinear partial differential equation istransformed into a set of nonlinear ordinary differential equationsthrough a Galerkin approach. The resulting system of equations is solvedthrough the fourth and fifth-order Runge–Kutta–Fehlberg (RKF-45)integration method. Static (divergence) and Hopf (flutter) bifurcationboundaries are presented for various levels of viscoelastic structuraldamping. Despite the deterministic nature of the system of equations,the dynamic panel response can become random-like. Chaotic analysis isperformed using several conventional criteria. Results are indicative ofthe important influence of structural damping on the domain of chaoticregion.     

Numerical Simulation of Equilibrium Induction Plasma Flows in a Cylindrical Plasmatron Channel

Subsonic equilibrium air and argon plasma flows in the cylindrical discharge channel of an induction plasmatron are calculated over a wide range of the working parameters on the basis of a solution of the complete Navier-Stokes equations combined with a simplified equation for the high-frequency electric field. An effective method and the results of calculating the necessary transport coefficients of ionized multicomponent air and argon (including the plasma electrical conductivity) are given on the temperature interval 300–1500 K with allowance for the higher approximations in Sonine polynomials in the Chapman-Enskog method. The results of calculating the air and argon plasma flow and temperature field in the discharge channel of a 100 kW induction plasmatron are given. Two types of complex channel plasma flow vortex structures are detected.     

Motion of a gas mixture through a porous medium with sorption

Solutions are investigated of a system of linear partial differential equations describing the motion of a gaseous (liquid) mixture through an undeformable homogeneous porous medium with sorption at interfaces between gaseous (liquid) and solid phases, the kinetics of which are described by a linear equation. If the porous medium consists of spherical granules, the problem is solved in quadratures. For the case of symmetric granules with arbitrary symmetry parameter, various approximate solutions are obtained; first and central moments are used as criteria for the accuracy of the approximations.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 5, pp. 95–100, September–October, 1970.     

Application of the Reichardt model in problems of turbulence near the wall

A method is proposed for calculating turbulent boundary layers near the wall, based on the Reichardt semiempirical model of turbulent mixing. The article considers the problem of the turbulent boundary layer of a plate, including the case of supersonic flow around a plate, as well as the problem of the turbulent boundary layer with the nonisothermal flow of turbulent jets around a surface. Here there are introduced several almost self-similar solutions of the differential transfer equations, based on the assumption of the conservative nature of the profiles of the parameters with respect to a change in the sublayer. The results of the calculation are compared with experimental data.Moscow. Translated from Izvestiya Akademii Nauk SSSR. Mekhanika Zhidkosti i Gaza, No. 2, pp. 15–24, March–April, 1972.     

Detached flow over a step with the formation of a turbulent wake

A method of calculating the plane turbulent layer behind a step interacting with a free potential flow of incompressible fluid is developed. The method includes consideration of the initial boundary layer and injection (or suction) in the isobaric bottom region. Friction on the wall behind the step is neglected, which corresponds to symmetric quasisteady flow behind the straight edge of a plate. The inviscid flow is represented by the Keldysh-Sedov integral equations; the flow in the wake with a one-parameter velocity profile is represented by three first-order differential equations—the equations of momentum for the wake and motion along its axis and the equation of interaction (through the displacement thickness) of the viscous flow with the external potential flow. The turbulent friction in the wake is given, accurate to the single empirical constant, by the Prandtl equation. The different flow regions — on the plate behind the step, the isobaric bottom region, and the wake region — are joined with the aid of the quasi-one-dimensional momentum equation for viscous flow. The momentum equation for the flow as a whole serves as the closure condition. The obtained integrodifferential system of equations is approximated by a system of nonlinear finite-difference equations, whose solution is obtained on a computer by minimization of the sum of the squares of the discrepancies. The results of the calculations agree satisfactorily with experimental data.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 3, pp. 17–25, May–June, 1977.We are grateful to V. I. Kuptsov for consultation and help in programming and to Z. A. Donskova who assisted in the calculations and preparation of the paper.     

Numerical solution of the problem of the flow of a supersonic gas stream over the upper surface of a delta wing in the expansion region

A numerical method is described for the calculation of supersonic flow over the arbitrary upper surface of a delta wing in the expansion region. The shock wave must be attached everywhere to the leading edge of this wing from the side of the lower surface. The stream flowing over the wing is assumed to be nonviscous. A problem with initial conditions at some plane and with boundary conditions at the wing surface and the characteristic surface is set up for the nonlinear system of equations of gas dynamics. The difference system of equations, which approximates the original system of differential equations on a grid, has a second order of accuracy and is solved by the iteration system proposed in [1]. The initial conditions are determined by the method of establishment of self-similar flow. A number of examples are considered. Comparison is made with the solutions of other authors and with experiment.Translated from Zhurnal Prikladnoi Mekhaniki i Tekhnicheskoi Fiziki, No. 6, pp. 76–81, November–December, 1973.The author thanks A. S. II'ina who conducted the calculations and V. S. Tatarenchik for advice.     

Thermoelastoplastic State of Flexible Laminated Shells under Axisymmetric Loading Along Various Plane Paths

The paper presents a technique for thermoelastoplastic stress–strain analysis of flexible laminated shells of revolution under complex axisymmetric loading. The constitutive deformation equations are used to describe loading along arbitrary plane paths. The problem is solved by the method of successive approximations. A numerical example is given     

A variational analytical approach to time dependent flow of oldroyd B fluid in circular tube

In the present investigation the time dependent flow of an Oldroyd fluid B in a horizontal cylindrical pipe is stuided by the variational analytical approach developed by author. The time dependent problem is mathematically reduced to a partial differential equation of third order. Using the improved variational approach due to Kantorovich the partial differential equation can be reduced to a system of ordinary differential equations for different approximations. The ordinary differential equations are solved by the method of the Laplace transform which is led to an analytical form of the solutions. Project supported by TWAS and Chinese Academy of Sciences and the National Science Foundation of China