Wave flow of a liquid film in a horizontal separator

The calculation of the motion of separated moisture in a linear horizontal separator is made on the basis of the analysis of the development of the waves in a flow of a thin layer of liquid along a vertical surface without allowance for the transverse flow of mass [1].Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 2, pp. 174–176, March–April, 1985.     

Influence of moisture transfer in the aeration region on the thermal regime of soil

One-dimensional heat and moisture transfer in the aeration region of reclaimed land is considered for two limiting cases. In the first, the heat transfer is mainly due to the thermal conductivity of the soil; in the second, to the motion of moisture. The influence of the water-table depth on the thermal regime is investigated. A solution to the problem of unsteady heat transfer in soil is found analytically by the method of matched asymptotic expansions.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 5, pp. 63–71, September–October, 1981.We thank V. S. Berman for discussions and valuable comments.     

Unsteady three-dimensional viscous shock layer in nonuniform gas flow in the neighborhood of a stagnation point

The combined influence of unsteady effects and free-stream nonuniformity on the variation of the flow structure near the stagnation line and the mechanical and thermal surface loads is investigated within the framework of the thin viscous shock layer model with reference to the example of the motion of blunt bodies at constant velocity through a plane temperature inhomogeneity. The dependence of the friction and heat transfer coefficients on the Reynolds number, the shape of the body and the parameters of the temperature inhomogeneity is analyzed. A number of properties of the flow are established on the basis of numerical solutions obtained over a broad range of variation of the governing parameters. By comparing the solutions obtained in the exact formulation with the calculations made in the quasisteady approximation the region of applicability of the latter is determined. In a number of cases of the motion of a body at supersonic speed in nonuniform media it is necessary to take into account the effect of the nonstationarity of the problem on the flow parameters. In particular, as the results of experiments [1] show, at Strouhal numbers of the order of unity the unsteady effects are important in the problem of the motion of a body through a temperature inhomogeneity. In a number of studies the nonstationary effect associated with supersonic motion in nonuniform media has already been investigated theoretically. In [2] the Euler equations were used, while in [3–5] the equations of a viscous shock layer were used; moreover, whereas in [3–4] the solution was limited to the neighborhood of the stagnation line, in [5] it was obtained for the entire forward surface of a sphere. The effect of free-stream nonuniformity on the structure of the viscous shock layer in steady flow past axisymmetric bodies was studied in [6, 7] and for certain particular cases of three-dimensional flow in [8–11].Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 3, pp. 175–180, May–June, 1990.     

A Numerical Study of Micro-Heterogeneity Effects on Upscaled Properties of Two-Phase Flow in Porous Media

Commonly, capillary pressure–saturation–relative permeability (P ^c–S–K _r) relationships are obtained by means of laboratory experiments carried out on soil samples that are up to 10–12 cm long. In obtaining these relationships, it is implicitly assumed that the soil sample is homogeneous. However, it is well known that even at such scales, some micro-heterogeneities may exist. These heterogeneous regions will have distinct multiphase flow properties and will affect saturation and distribution of wetting and non-wetting phases within the soil sample. This, in turn, may affect the measured two-phase flow relationships. In the present work, numerical simulations have been carried out to investigate how the variations in nature, amount, and distribution of sub-sample scale heterogeneities affect P ^c–S–K _r relationships for dense non-aqueous phase liquid (DNAPL) and water flow. Fourteen combinations of sand types and heterogeneous patterns have been defined. These include binary combinations of coarse sand imbedded in fine sand and vice versa. The domains size is chosen so that it represents typical laboratory samples used in the measurements of P ^c–S–K _r curves. Upscaled drainage and imbibition P ^c–S–K _r relationships for various heterogeneity patterns have been obtained and compared in order to determine the relative significance of the heterogeneity patterns. Our results show that for micro-heterogeneities of the type shown here, the upscaled P ^c–S curve mainly follows the corresponding curve for the background sand. Only irreducible water saturation (in drainage) and residual DNAPL saturation (in imbibition) are affected by the presence and intensity of heterogeneities.     

Investigation of the properties of a shock wave arising in a supersonic flow of plasma,passing through a transverse magnetic field

In a flow of plasma, set up by an ionizing shock wave and moving through a transverse magnetic field, under definite conditions there arises a gasdynamic shock wave. The appearance of such shock waves has been observed in experimental [1–4] and theoretical [5–7] work, where an investigation was made of the interaction between a plasma and electrical and magnetic fields. The aim of the present work was a determination of the effect of the intensity of the interaction between the plasma and the magnetic field on the velocity of the motion of this shock wave. The investigation was carried out in a magnetohydrogasdynamic unit, described in [8]. The process was recorded by the Töpler method (IAB-451 instrument) through a slit along the axis of the channel, on a film moving in a direction perpendicular to the slit. The calculation of the flow is based on the one-dimensional unsteady-state equations of magnetic gasdynamics. Using a model of the process described in [9], calculations were made for conditions close to those realized experimentally. In addition, a simplified calculation is made of the velocity of the motion of the above shock wave, under the assumption that its front moves at a constant velocity ahead of the region of interaction, while in the region of interaction itself the flow is steady-state.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 1, pp. 86–91, January–February, 1975.     

Some cases of unsteady flow in porous media associated with irrigation

The results of an earlier study [1] are used to consider the motion of underground water flowing from an irrigation channel. The surface of the ground flow usually has an uneven shape, since in some sections there is stripwise irrigation, while at others evaporation may occur or rain may fall. The variations of the water level in the channel are assumed known. The subsequent variation of the ground water level is determined. If the level rises appreciably there can be salinization or swamped ground can form.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 3, pp. 130–137, May–June, 1988.     

Transonic flow of a gas in axisymmetric Laval nozzles with steep walls

A large number of articles have recently appeared [1–5] in which various numerical methods have been proposed for solution of both the direct and inverse problem of calculating the flow of an ideal gas in a Laval nozzle. An analysis of the results presented in these articles shows that, in spite of the two-dimensional character of the flow fields obtained, the distribution of the pressure at the wall and along the central line of the flow differs only slightly from the values calculated using the hydraulic theory, in which, as is well known, the transverse distribution of the parameters is neglected. In what follows, an analysis is presented of the numerical results of a calculation of the gas flow in the transitional region of round Laval nozzles with very steep walls, where the flow parameters vary considerably in a transverse direction and where their values differ strongly from those obtained in a hydraulic approximation.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 6, pp. 55–58, November–December, 1970.     

Convection of a viscous incompressible gas in rectangular regions with inlet and outlet channels

A study is made of two-dimensional problems of thermal convection of a viscous incompressible gas in rectangular regions that have gas inlet and outlet channels in the presence of a temperature difference between the bottom and the top (the bottom is heated). In contrast to the well-studied case of natural convection, when no-slip conditions are specified on all boundaries of the region and motion in the region occurs only through the temperature difference [1–4], the heat transfer in the investigated flows is complicated by the additional influence of the forced convection of the gas due to the motion of gas through the inlet and outlet channels. Flows of such type simulate well the processes that take place in many heat transfer devices and in ventilated and air-conditioned industrial premises. Two formulations of the problem are considered. In the first, the gas flow through the inlet and outlet channels is assumed given, and the solution of the problem is determined by the dimensionless Prandtl, Grashof, and Reynolds numbers. In the second case, this flow rate is not given but determined during the solution of the problem. The motion in the region arises from the difference between the temperatures of the bottom and the top of the region, and the motion, in its turn, causes a flow of gas through the inlet and outlet channels. As in the case of natural convection, the solution of the problem in this case is determined by only two dimensionless numbers — the Grashof and Prandtl numbers. By numerical solution of the boundary-value problems for the equations of heat transfer a study is made of the influence of the characteristic dimensionless numbers on the hydrodynamic and temperature fields and the heat fluxes through the boundaries of the region. The solutions of the problems in the two formulations are compared for different positions of the outlet channels.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 5, pp. 126–131, September–October, 1979.We thank G. I. Petrov for discussing the results.     

Flow in a stratum in the case of pumping from an imperfect well near a basin

A study is made of the three-dimensional stationary problem of the flow of ground water to a well of the type of a point sink in a stratum of unbounded thickness in one direction. The stratum is bounded at the top by the bottom of the basin and a stratum of impermeable ground. The problem is investigated in the framework of potential flow theory based on Darcy's law [1, 2], and the solution is obtained in the form of quadratures.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 5, pp. 161–164, September–October, 1980.     

Supersonic motion of bodies in a gas with shock waves

The authors consider the problem of supersonic unsteady flow of an inviscid stream containing shock waves round blunt shaped bodies. Various approaches are possible for solving this problem. The parameters in the shock layer on the axis of symmetry have been determined in [1, 2] by using one-dimensional theory. The authors of [3, 4] studied shock wave diffraction on a moving end plane and wedge, respectively, by the through calculation method. This method for studying flow around a wedge with attached shock was also used in [5]. But that study, unlike [4], used self-similar variables, and so was able to obtain a clearer picture of the interaction. The present study gives results of research into the diffraction of a plane shock wave on a body in supersonic motion with the separation of a bow shock. The solution to the problem was based on the grid characteristic method [6], which has been used successfully to solve steady and unsteady problems [7–10]. However a modification of the method was developed in order to improve the calculation of flows with internal discontinuities; this consisted of adopting the velocity of sound and entropy in place of enthalpy and pressure as the unknown thermodynamic parameters. Numerical calculations have shown how effective this procedure is in solving the present problem. The results are given for flow round bodies with spherical and flat (end plane) ends for various different values of the velocities of the bodies and the shock waves intersected by them. The collision and overtaking interactions are considered, and there is a comparison with the experimental data.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 5, pp. 141–147, September–October, 1984.     

Quasi-three-dimensional calculation of flow in blade passages of turbines

The flow in turbomachines is currently calculated either on the basis of a single successive solution of an axisymmetric problem (see, for example, [1-A]) and the problem of flow past cascades of blades in a layer of variable thickness [1, 5], or by solution of a quasi-three-dimensional problem [6–8], or on the basis of three-dimensional models of the motion [9–11]. In this paper, we derive equations of a three-dimensional model of the flow of an ideal incompressible fluid for an arbitrary curvilinear system of coordinates based on averaging the equations of motion in the Gromek–Lamb form in the azimuthal direction; the pulsation terms are taken into account in the equations of the quasi-three-dimensional motion. An algorithm for numerical solution of the problem is described. The results of calculations are given and compared with experimental data for flows in the blade passages of an axial pump and a rotating-blade turbine. The obtained results are analyzed.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 2, pp. 69–76, March–April, 1991.I thank A. I. Kuzin and A. V. Gol'din for supplying the results of the experimental investigations.     

Methods of calculation and numerical analysis of conducting-gas flow in high-current electric arcs

The mechanism of conducting-gas acceleration in an electric arc by intrinsic magnetic field was first investigated in [1]. Further theoretical study of this question was associated with the numerical calculation of arcs [2–7]. A more general approach to the solution of the problem was realized in [4], where the finite-difference method was used. Integral calculational models were developed in [5–7]. The present work proposes a modified version of the difference method [4] and a series of integral methods for the calculation of the conducting-gas flow in a high-current electric arc. The development of integral methods is of interest in that they are usually associated with adequate accuracy in determining integral values and values averaged over the cross section by a relatively simple calculation, and also allow the solution of the problem to be obtained in a number of situations when the realization of a difference method is complicated. The results of different calculation methods are compared. The effect of conditions in the initial cross section of the calculation region of the arc on its characteristics is investigated and a numerical analysis of the heating and acceleration of conducting gas is carried out.Translated from Izvestiya Akademiya Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 5, pp. 103–110, September–October, 1978.     

Three-dimensional problem for entry of a disk into a compressible liquid

The unsteady problem for the oblique entry of a disk into water is solved. The water is assumed a perfect compressible liquid and the flow is assumed adiabatic. The flow and state parameters are determined during the numerical integration of the system of nonlinear equations which describe the given flow by means of a three-dimensional finite-difference scheme [1]. The variation in time of the drag coefficient as a function of the Mach number and the angles of entry and attack, the pressure distribution and the shape of the free surface formed behind the disk are investigated. The oblique entry of a disk into water and its subsequent motion have mainly been studied for velocities at which the compressibility of the water is negligible [2–4]. The influence of compressibility on the duration of the rise time and the impact load was investigated experimentally in the range of Mach numbers 0 < M0 <–0.3 [5]. Semiempirical dependences are obtained for the maximum of the drag coefficient and its rise time.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 1, pp. 17–20, January–February, 1988.     

Nonlinear problem of flow past a source with the formation of a stagnant zone given different Bernoulli constants

The problem of free flow past a point source is considered for two streams with different Bernoulli constants whose encounter creates a bounded region of constant pressure. The theory and method of solving problems of plane ideal jet flows with different Bernoulli constants in the jets were developed in [1]. Here, in conformity with [1], a nonlinear system of equations is derived, the question of the construction of a high-accuracy numerical solution is considered, and certain calculation results are presented for various values of the Bernoulli and cavitation numbers, which are dimensionless parameters of the problem.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 3, pp. 55–60, May–June, 1986.     

Draft of a nozzle with the circulating outflow of a vortical flow of gas

The presence of circulation in an outflowing gas leads to a change in the working parameters of a nozzle. The question of the mass flow rate and the draft of a nozzle without a diffusor (a point) for twisted flows has been studied theoretically and experimentally [1–6]. The use of nozzles with a supersonic part introduces a considerable degree of complication into the method for the analytical calculation of the draft characteristics and the program for their experimental investigation. In [2, 7], a theory of a nozzle is formulated for a model of a potential circulating flow of gas; in [5, 8], an electronic computer was used to solve the complete system of the equations of gasdynamics for the motion of a rotating flow along a nozzle; in [7, 9], an investigation was made of a variational problem of the shaping of a diffusor for a circulation flow. The calculation of the draft, carried out in the above-mentioned communications (with the exception of [2], in which a study was made of a partial model of an eddyless rotational motion), is bound up with labor-consuming computer calculations. In the present article, in a development of [3, 6], a quasi-one-dimensional theory of a supersonic nozzle for a vortical flow of gas is formulated and verified experimentally.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 5, pp. 145–149, September–October, 1975.     

Propagation of a flush wave in a prismatic channel

By a water flush there is generally understood an unsteady-state flow of water, arising in millraces, with the breaching (rupture) of a dam. The special characteristics of a flush wave are also possessed by a flow in a lower millrace at some distance from the dam, with the overflow of water arising in the reservoir over the crest of the dam. Usually, the necessary information on the parameters of a flush wave with its motion in natural channels is obtained by numerical solution, in a digital computer, of the equations of not fully established one-dimensional flow [1–3]. These calculations are very labor-consuming and require rather detailed information on the channel. Therefore, it is of practical importance to clarify the overall laws governing the propagation of flush waves in schematized, in particular, in prismatic channels. In some cases, on the basis of such laws, it is possible to make a preliminary diagnosis of the expected scales of the phenomenon.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 1, pp. 39–44, January–February, 1975.     

Theory of short waves in gas dynamics

An examination is made of the two-dimensional, almost stationary flow of an ideal gas with small but clear variations in its parameters. Such gas motion is described by a system of two quasilinear equations of mixed type for the radial and tangential velocity components [1, 2]. Partial solutions [3, 4], characterizing the variation in the gas parameters in the vicinity of the shock wave front (in the short-wave region), are known for this system of equations. The motion of the initial discontinuity of the short waves derived from the velocity components with respect to polar angle and their damping are studied in the report. A solution of the equations characterizing the arrangement of the initial discontinuity derived from the velocities is presented for one particular case of the class of exact solutions of the two parameter type [4]. Functions are obtained which express the nature of the variation in velocity of the front of the damped wave and its curvature.Translation from Zhurnal Prikladnoi Mekhaniki i Tekhnicheskoi Fiziki, No. 3, pp. 55–58, May–June, 1973.     

Formation of a region of interaction and of a wake with the instantaneous start of a plate in an incompressible liquid

The flow arising in an incompressible liquid if, at the initial moment of time, a plate of finite length starts to move with a constant velocity in its plane, is discussed. For the case of an infinite plate, there is a simple exact solution of the Navier—Stokes equations, obtained by Rayleigh. The case of the motion of a semiinfinite plate has also been discussed by a number of authors. Approximate solutions have been obtained in a number of statements; for the complete unsteadystate equations of the boundary layer the statement was investigated by Stewartson (for example, [1–3]); a numerical solution of the problem by an unsteady-state method is given in [4]. The main stress in the present work is laid on investigation of the region of the interaction between a nonviscous flow and the boundary layer near the end of a plate. In passing, a solution of the problem is obtained for a wake, and a new numerical solution is also given for the boundary layer at the plate.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 2, pp. 3–8, March–April, 1977.     

Qualitative mathematical analysis of the Richards equation

The Richards equation is widely used as a model for the flow of water in unsaturated soils. For modelling one-dimensional flow in a homogeneous soil, this equation can be cast in the form of a specific nonlinear partial differential equation with a time derivative and one spatial derivative. This paper is a survey of recent progress in the pure mathematical analysis of this last equation. The emphasis is on the interpretation of the results of the analysis. These are explained in terms of the qualitative behaviour of the flow of water in an unsaturated soil which is described by the Richards equation.Nomenclature a coefficient in second-order diffusion term of equation - b coefficient in first-order advection term of equation - D soil-moisture diffusivity [L²T^-1] - h pressure head [L] - H quarter-plane domain for Cauchy-Dirichlet problem [L] x [T] - K hydraulic conductivity scalar [LT^–1] - K hydraulic conductivity tensor [LT^–1] - q soil-moisture flux scalar [LT^–1] - q soil-moisture flux vector [LT^–1] - r dummy variable - R rectangle [L] x [T] - s dummy variable - s* representative value of dummy variable - S half-plane domain for Cauchy problem [L] x [T] - t time [T] - u unknown solution of partial differential equation - u₀ initial-value function - v soil-moisture velocity scalar [LT^–1] - v soil-moisture velocity vector [LT^–1]     

Calculation of nonstationary drainage in the case of a finite number of closed horizontal drains

In the hydrodynamic formulation, a study is made of nonstationary drainage in a water-bearing stratum in the presence of a finite number of closed drains and variable infiltration. The drains are simulated by linear sinks with known time-dependent flow rates per unit length of the drains. The dependences found in the paper enable one to predict, and if necessary also regulate the dynamics of the level of ground water in a drained region, which is of practical interest in the study of salinization and swamping of land.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 4, pp. 165–170, July–August, 1979.