Plane filtration in a laminar medium in the presence of imperfect reservoirs

Exact and approximate solutions were obtained describing filtration flow in a plane region in sectors corresponding to imperfect reservoirs and suction gaps. These equations make it possible to obtain an approximate solution to the three-dimensional problem of the filtration of ground waters by solution of the two-dimensional problem; imperfect reservoirs are permitted in the filtration region.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 3, pp. 76–83, May–June, 1973.     

Upscaling of Two-Phase Immiscible Flows in Communicating Stratified Reservoirs

A semi-analytical method for upscaling two-phase immiscible flows in heterogeneous porous media is described. This method is developed for stratified reservoirs with perfect communication between layers (the case of vertical equilibrium), in a viscous dominant regime, where the effects of capillary forces and gravity may be neglected. The method is discussed on the example of its basic application: waterflooding in petroleum reservoirs. We apply asymptotic analysis to a system of two-dimensional (2D) mass conservation equations for incompressible fluids. For high anisotropy ratios, the pressure gradient in vertical direction may be set zero, which is the only assumption of our derivation. In this way, the 2D Buckley–Leverett problem may be reduced to a one-dimensional problem for a system of quasi-linear hyperbolic equations, of a number equal to the number of layers in the reservoir. They are solved numerically, based on an upstream finite difference algorithm. Self-similarity of the solution makes it possible to compute pseudofractional flow functions depending on the average saturation. The computer partial differential equation solver COMSOL is used for comparison of the complete 2D solutions with averaged 1D simulations. Cases of both discrete and continuous (log-normal) permeability distribution are studied. Generally, saturation profiles of the 1D model are only slightly different from the 2D simulation results. Recovery curves and fractional flow curves fit well. Calculations show that at a favorable mobility ratio (displaced to displacing phase) crossflow increases the recovery, while at an unfavorable mobility ratio, the effect is the opposite. Compared with the classical Hearn method, our method is more general and more precise, since it does not assume universal relative permeabilities and piston-like displacement, and it presumes non-zero exchange between layers. The method generalizes also the study of Yortsos (Transp Porous Media 18:107–129, 1995), taking into account in a more consistent way the interactions between the layers.     

A qualitative investigation of the equations of quasi-one-dimensional magnetohydrodynamic channel flow

A qualitative investigation of the system of differential equations describing the quasi-one-dimensional flow of an electrically conducting medium at small magnetic Reynolds numbers gives an idea of the different possible flow patterns occuring when the electromagnetic field and channel shape are given in different ways. Such a treatment is essential for the calculation of one-dimensional flows, and also for the solution of variational problems [1].In the literature devoted to this question studies have been made of flow in a one-dimensional electromagnetic field and a channel of constant cross section [2], as well as of the flow when the magnetic field is described by specially given functions of the flow velocity [3]. These cases reduce to the analysis of integral curves in a plane.In the present paper the investigation is carried out for an arbitrary distribution of the electric and magnetic fields and channel shape, which leads to a consideration of the behavior of integral curves in three-dimensional space. The qualitative results are illustrated by examples.     

Effect of Boundary-Layer Thickness on the Structure of a Near-Wall Flow with a Two-Dimensional Obstacle

Results of physical and numerical experiments on investigating the effect of the depth of immersion of a two-dimensional obstacle with a square cross section into a developed turbulent boundary layer on the length of the separated flow region are presented. The numerical simulation is based on solving averaged Navier–Stokes equations with the use of the k– model of turbulence. The near-wall flow is visualized in the experiments, and the fields of mean and fluctuating velocities are measured. Flow regions where the results of numerical simulation agree with experimental data are determined. It is shown that the length of the recirculation flow region in the near wake increases with decreasing depth of immersion of the two-dimensional obstacle into the turbulent boundary layer.     

Theory of large-strain torsion of prismatic bodies with moment stresses

The problem of the torsion and tension-compression of a prismatic bar with a stress-free lateral surface is studied using three-dimensional elasticity theory for materials with moment stresses. A substitution is found that allows one to separate one variable in the nonlinear equilibrium equations for a Cosserat continuum and boundary conditions on the lateral surface. This substitution reduces the original spatial problem of the equilibrium of a micropolar body to a two-dimensional nonlinear boundary-value problem for a plane region shaped like the cross section of the prismatic bar. Variational formulations of the two-dimensional problem for the section are given that differ in the sets of varied functions and the constraints imposed on their boundary values. __________ Translated from Prikladnaya Mekhanika i Tekhnicheskaya Fizika, Vol. 47, No. 4, pp. 167–175, July–August, 2006.     

Radial Flow in a Bounded Randomly Heterogeneous Aquifer

Flow to wells in nonuniform geologic formations is of central interest to hydrogeologists and petroleum engineers. There are, however, very few mathematical analyses of such flow. We present analytical expressions for leading statistical moments of vertically averaged hydraulic head and flux under steady-state flow to a well that pumps water from a bounded, randomly heterogeneous aquifer. Like in the widely used Thiem equation, we prescribe a constant pumping rate deterministically at the well and a constant head at a circular outer boundary of radius L. We model the natural logarithm Y = lnT of aquifer transmissivity T as a statistically homogeneous random field with a Gaussian spatial correlation function. Our solution is based on exact nonlocal moment equations for multidimensional steady state flow in bounded, randomly heterogeneous porous media. Perturbation of these nonlocal equations leads to a system of local recursive moment equations that we solve analytically to second order in the standard deviation of Y. In contrast to most stochastic analyses of flow, which require that log transmissivity be multivariate Gaussian, our solution is free of any distributional assumptions. It yields expected values of head and flux, and the variance–covariance of these quantities, as functions of distance from the well. It also yields an apparent transmissivity, T _a, defined as the negative ratio between expected flux and head gradient at any radial distance. The solution is supported by numerical Monte Carlo simulations, which demonstrate that it is applicable to strongly heterogeneous aquifers, characterized by large values of log transmissivity variance. The two-dimensional nature of our solution renders it useful for relatively thin aquifers in which vertical heterogeneity tends to be of minor concern relative to that in the horizontal plane. It also applies to thicker aquifers when information about their vertical heterogeneity is lacking, as is commonly the case when measurements of head and flow rate are done in wells that penetrate much of the aquifer thickness. Potential uses include the analysis of pumping tests and tracer test conducted in such wells, the statistical delineation of their respective capture zones, and the analysis of contaminant transport toward fully penetrating wells.     

On the Errors Associated with Two-Dimensional Stochastic Solute Transport Models

Many subsurface solute transport studies employ numerical modeling techniques to estimate solute arrival times. Simplifying assumptions must be made to define the modeling domain within a mathematical framework. One common assumption is that the vertical flow is negligible such that the flow field can be simulated with a two-dimensional model. Reducing the vertical dimension reduces the number of flow paths that a solute can take. In a heterogenous medium, artificially removing the 3rd dimension may lead to erroneous results. We investigate the error in the simulated solute breakthrough associated with a two-dimensional model. We also use a stochastic solution of solute arrival time to derive a transform of a two-dimensional ln (k) field so that solute transport more closely resembles three-dimensional transport behavior. The moment equations for two- and three-dimensional domains were solved simultaneously to calculate this transform. The results indicate that the removal of the vertical variability (3D 2D) introduces a 5–10% error in the predicted solute breakthrough. The error tends to increase with increased hydraulic conductivity variance. Numerical experiments confirm that the transform developed herein decreases the relative error of particle breakthrough curves.     

Calculation of the averaged three-dimensional boundary layer at the axisymmetrical boundaries of the interblade channels of turbine machines

The article discusses the flow of a gas at the blade rim of an axial turbine, consisting of an external steady-state continuous flow of an ideal compressible liquid and a three-dimensional turbulent boundary layer of a compressible liquid at the end surfaces of the rim, averaged in a peripheral direction. It presents an example of a calculation of flow in fixed blades, with a different form of the meridional cross section. In a flow through the rim of a turbine machine between the convex and concave surfaces of adjacent blades there arises a transverse gradient of the static pressure. At the end surface in the boundary layer the lines of the flow are shifted toward the convex side of the profile, and a secondary transverse flow of the liquid arises [1–3]. The article discusses the following: an external two-dimensional steady-state adiabatic flow of an ideal compressible liquid at the surface S₂, which can be taken as the mean surface of the interblade channel, with boundary lines at the peripheral and root end surfaces of the rim; a two-dimensional steady-state adiabatic flow of an ideal compressible liquid at the end surfaces of the rim between the convex and concave sides of the profiles [3, 4]; and a three-dimensional turbulent boundary layer, averaged in a peripheral direction at the end surfaces of the blade rim. The averaged boundary layer is calculated along one coordinate line s, and a simplified model of the quasi-three-dimensional flow is used. The coefficients of friction and heat transfer, and the inclination of the bottom flow lines are averaged.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 3, pp. 22–31, May–June, 1975.The author thanks G. Yu. Stepanov for posing the problem and evaluating the results.     

Hydrodynamic flows and heat transfer in slightly noncylindrical channels

Viscous incompressible laminar flow and heat transfer in channels with a small arbitrary deviation from a cylindrical surface are examined. A linear system of equations and boundary conditions for the disturbed dynamic and thermal fields, obtained by linearizing the complete system of Navier-Stokes equations with respect to the solution for developed flows in cylindrical tubes of arbitrary cross section, is presented. In the important practical case in which the perturbations of the channel surface are concentrated on an interval of finite length it is shown that the integral dynamic and thermal characteristics of the channel can be found without solving the three-dimensional equations by going over to effective two-dimensional boundary-value problems which are fundamentally no more difficult to solve than those for developed flows. Extensions of the theory to flows with low-efficiency power sources are given. Applications to plane channels and circular tubes with deformed surfaces are considered. Among the numerous applications requiring information about the integral characteristics of flows in channels whose initially cylindrical surface is slighty deformed, we note the problem of heat transfer intensification by slightly deforming the tube surface with careful estimation of the accompanying increase in resistance [1] and the calculation of the resistance of capillaries and biological transport systems in the form of tubes and channels when the walls are deformed [2]. Below we consider laminar flow in channels with deformed walls. Whereas for the first problem this class of flows is only one of those possible (in general it is necessary to analyze the transition, turbulence and flow separation effects), in the second case, which is characterized by low Reynolds numbers, the laminar flow model is perfectly adequate.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 2, pp. 21–30, March–April, 1990.The authors are grateful to A. Yu. Klimenko for useful discussions.     

Bifurcations of two-dimensional into three-dimensional wave regimes for a vertically flowing liquid film

The three-dimensional steady traveling wave regimes of a viscous liquid film flowing down a vertical wall which branch off from two-dimensional nonlinear waves are investigated. The numerical calculations are based on a model system of equations valid for intermediate Reynolds numbers. It is shown that there exist two fundamentally different types of three-dimensional steady traveling waves branching off from plane waves. One of these possesses checkerboard symmetry in the distribution of the maxima of the wave profile thickness and is the more interesting. An important difference in the breakdown of plane waves of the first and second families is also demonstrated. The wave characteristics of certain three-dimensional regimes are calculated as functions of the bifurcation parameter.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 5, pp. 109–114, September–October, 1990.     

Dispersion of a filtration flow

In this paper we shall consider the transport of a dynamically neutral impurity in a porous medium containing random inhomogeneities. The original versions of the equations for the mean impurity concentration [1, 2] were based on the hyphothesis that the random motions obeyed the Markov principle, use being made of the diffusion equations of A. N. Kolmogorov. Later [3, 4] the method of perturbations was used to study the complete system of equations for the impurity concentration and random filtration velocity in the case of a constant, nonrandom porosity; after an averaging process this yields a generalized equation for the average concentration. In the limiting cases of small- and large-scale inhomogeneities in the permeability of the medium, the basic integrodifferential equation may be, respectively, reduced to parabolic and hyperbolic equations of the second order. In the present analysis we shall use the perturbation method to study the transport of an impurity by a flow when the filtration velocity of the latter fluctuates around inhomogeneities in the permeability field, the porosity of the medium in which the flow is taking place also constituting a random field, correlating with the field of permeability. We shall derive equations for the average concentration and should formulate the corresponding boundary-value problems for these equations; we shall also calculate the components of the dispersion tensor and shall consider the equilibrium sorption of an impurity.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 4, pp. 65–69, July–August, 1976.The author is grateful to A. I. Shnirel'man for useful discussions.     

Three-dimensional boundary-layer equations in regions with large pressure gradient

Assuming that in regions with large pressure gradient the flow parameters in a direction perpendicular to the pressure gradient change little in comparison with changes along the pressure gradient, we show that the calculation of the three-dimensional boundary layer may be reduced to the integration of equations analogous to the two-dimensional boundary-layer equations along curves tangent to the pressure gradient.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 5, pp. 49–55, September–October, 1971.     

Exact Averaging of Stochastic Equations for Flow in Porous Media

It is well-known that at present, exact averaging of the equations for flow and transport in random porous media have been proposed for limited special fields. Moreover, approximate averaging methods—for example, the convergence behavior and the accuracy of truncated perturbation series—are not well-studied, and in addition, calculation of high-order perturbations is very complicated. These problems have for a long time stimulated attempts to find the answer to the question: Are there in existence some, exact, and sufficiently general forms of averaged equations? Here, we present an approach for finding the general exactly averaged system of basic equations for steady flow with sources in unbounded stochastically homogeneous fields. We do this by using (1) the existence and some general properties of Green’s functions for the appropriate stochastic problem, and (2) some information about the random field of conductivity. This approach enables us to find the form of the averaged equations without directly solving the stochastic equations or using the usual assumption regarding any small parameters. In the common case of a stochastically homogeneous conductivity field we present the exactly averaged new basic non-local equation with a unique kernel-vector. We show that in the case of some type of global symmetry (isotropy, transversal isotropy, or orthotropy), we can for three-dimensional and two-dimensional flow in the same way derive the exact averaged non-local equations with a unique kernel-tensor. When global symmetry does not exist, the non-local equation with a kernel-tensor involves complications and leads to an ill-posed problem.     

Isothermal viscous incompressible flow in a hydrodynamic model of an electrophoresis chamber

A numerical solution and approximate analysis of the system of Navier-Stokes equations averaged over the transverse coordinate has made it possible to obtain the dependence of the length of the hydrodynamic flow stabilization interval in a thin cell of rectangular cross section on the Reynolds number, the relative thickness of the cell, and the relative size of the inlet opening. The principal and secondary flow regimes are calculated.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 2, pp. 14–20, March–April, 1990.     

Method of meridional sections in problems of a three-dimensional boundary layer

The difficulties and clumsiness of problems of calculating the heat transfer distribution over the surface of a body in a three-dimensional flow are well known. It is shown that this problem can be considerably simplified where the influence of the three-dimensionality of the flow, which in certain applications it is important to take into account, is only weak. In this case the three-dimensional problem can be reduced to a set of two-dimensional problems along the lines of meridional sections of the body. This has been demonstrated in detail with reference to the method of effective length or local similarity, which is widely used in engineering practice and is particularly justified in the the case of turbulent heat transfer law. However, in the three-dimensional case it is complicated by the need to calculate the distribution of the streamlines over the surface of the body [1–4]. In the presence of slight asymmetry of the flow the problem can be substantially simplified, mainly as a result of the demonstrated possibility of replacing (with quadratic accuracy) the streamlines by the lines of meridional sections. The possibility of an independent solution of the exact boundary layer equations along each meridional plane is demonstrated for the above-mentioned approximation (rule of meridional sections).Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 3, pp. 67–73, May–June, 1986.     

Meshless analysis of plasticity with application to crack growth problems

The governing equations for classical rate-independent plasticity are formulated in the framework of meshless method. The special J₂ flow theory for three-dimensional, two-dimensional plane strain and plane stress problems are presented. The numerical procedures, including return mapping algorithm, to obtain the solutions of boundary-value problems in computational plasticity are outlined. For meshless analysis the special treatment of the presence of barriers and mirror symmetries is formulated. The crack growth process in elastic–plastic solid under plane strain and plane stress conditions is analyzed. Numerical results are presented and discussed.     

Shallow-water flow solver with non-hydrostatic pressure: 2D vertical plane problems

A numerical solution for shallow-water flow is developed based on the unsteady Reynolds-averaged Navier–Stokes equations without the conventional assumption of hydrostatic pressure. Instead, the non-hydrostatic pressure component may be added in regions where its influence is significant, notably where bed slope is not small and separation in a vertical plane may occur or where the free-surface slope is not small. The equations are solved in the σ-co-ordinate system with semi-implicit time stepping and the eddy viscosity is calculated using the standard k–ϵ turbulence model. Conventionally, boundary conditions at the bed for shallow-water models only include vertical diffusion terms using wall functions, but here they are extended to include horizontal diffusion terms which can be significant when bed slope is not small. This is consistent with the inclusion of non-hydrostatic pressure. The model is applied to the 2D vertical plane flow of a current over a trench for which experimental data and other numerical results are available for comparison. Computations with and without non-hydrostatic pressure are compared for the same trench and for trenches with smaller side slopes, to test the range of validity of the conventional hydrostatic pressure assumption. The model is then applied to flow over a 2D mound and again the slope of the mound is reduced to assess the validity of the hydrostatic pressure assumption. © 1998 John Wiley & Sons, Ltd.     

Numerical investigation of flow past a system of two sweptback compression wedges

Supersonic flow past a three-dimensional configuration consisting of two neighboring wedges with sweptback leading edges mounted on a preliminary compression surface is numerically investigated. The case of sweptback wedges is considered, when their beveled surfaces deflect the compressed flows to opposite directions. The calculations are carried out on the basis of the averaged Navier-Stokes equations, together with the SST k-? turbulence model, at the freestream Mach number M = 6. The results obtained are compared with the data for inviscid flow calculated using the Euler equations. The flow pattern features, due to the interaction in the plane of symmetry between the shocks formed by the wedges and the shock-induced three-dimensional quasiconical separations of the turbulent boundary layer on the preliminary compression surface along the swept leading edges, are established. Within these separation zones the flow is directed away from the plane of symmetry of the configuration and is characterized by considerably greater values of the transverse velocity component, as compared with the flow outside of the separation zone.     

On One Approach to the Solution of Stress Problems for Noncircular Hollow Cylinders

An approach is proposed to solve three-dimensional stress problems for noncircular hollow cylinders. The end conditions are such that the problem can be reduced to a two-dimensional problem. This problem is reduced to a one-dimensional problem by introducing additional functions into the resolvable system of equations. These functions are determined using discrete Fourier series. The one-dimensional problem is solved by a stable numerical method. As an example, the stress state of cylinders with an elliptic cross section is analyzed depending on their thickness and degree of ellipticity.