Free surface asymptotics for thermocapillary flow at high marangoni numbers

Formal asymptotic expansions of the solution of the steady-state problem of incompressible flow in an unbounded region under the influence of a given temperature gradient along the free boundary are constructed for high Marangoni numbers. In the boundary layer near the free surface the flow satisfies a system of nonlinear equations for which in the neighborhood of the critical point self-similar solutions are found. Outside the boundary layer the slow flow approximately satisfies the equations of an inviscid fluid. A free surface equation, which when the temperature gradient vanishes determines the equilibrium free surface of the capillary fluid, is obtained. The surface of a gas bubble contiguous with a rigid wall and the shape of the capillary meniscus in the presence of nonuniform heating of the free boundary are calculated.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 3, pp. 61–67, May–June, 1989.     

Stability of the motion of a rigid body in an unsteady flow

The problem of rigid-body motion in an unsteady gas flow is considered using a flow model [1] in which the motion of the body is described by a system of integrodifferential equations. The case in which among the characteristic exponents of the fundamental system of solutions of the linearized equations there are not only negative but also one zero exponent is analyzed. The instability conditions established with respect to the second-order terms on the right sides of the equations are noted. The problem may be regarded as a generalization of the problem of the lateral instability of an airplane in the critical case solved by Chetaev [2], pp. 407–408.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 3, pp. 18–22, May–June, 1989.     

Vibrationally nonequilibrium flow of a compressible gas in an axisymmetric channel in the presence of glow discharge

A laminar subsonic flow of CO-He-N₂ mixture in an axisymmetric channel with glow discharge is investigated. The actual flow diagram is modeled on the basis of a system of narrow-channel equations. The results of the calculations are compared with the experimental data. The possibility of applying a one-dimensional system of equations is analyzed.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 6, pp. 114–119, November–December, 1985.The authors wish to thank B. S. Aleksandrov for his help in the work.     

Dynamics of a liquid with gas bubbles

The hydrodynamics and diffusion of an admixture near an isolated bubble, which simulates the rise of either a chain of identical bubbles or a system of regularly arranged bubbles of the same volume, are analyzed by solving the Navier-Stokes equations numerically. Data are presented for a specific liquid. It is shown that in both cases the maximum flow velocity on the surface of identical bubbles is practically the same, although in the former case the ascent velocity is considerably higher. The stationary admixture diffusion from a bubble also proves to be nearly the same.In relation to the bubbling of a gas through a liquid layer, it is shown that the total admixture diffusion is maximum for regularly arranged bubbles whose diameter is comparable with the liquids capillary constant. Although the flow past the bubble remains continuous, the values of the hydrodynamic parameters are no longer small.Novosibirsk. Translated from Izvestiya Rossiiskoi Akademii Nauk, Mekhanika Zhidkosti i Gaza, No. 3, pp. 75–88, May–June, 1996.     

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.     

Modeling of retrograde condensation in the process of steady radial flow through a porous medium

The problem of the steady axisymmetric two-phase flow of a multicomponent mixture through a porous medium with phase transitions is considered. It is shown that the system of equations for the two-phase multicomponent flow process, together with the equations of phase equilibrium, reduces to a system of two ordinary differential equations for the pressures in the gas and liquid phases. A family of numerical solutions is found under certain assumptions concerning the pressure dependence of the molar fraction of the liquid phase.Moscow. Translated from Izvestiya Rossiiskoi Akademii Nauk, Mekhanika Zhidkosti i Gaza, No. 6, pp. 92–97, November–December, 1994.     

The Local Analysis of Changing Force Balances in Immiscible Incompressible Two-Phase Flow

The balance of viscous, capillary and gravity forces strongly affects two-phase flow through porous media and can therefore influence the choice of appropriate methods for numerical simulation and upscaling. A strict separation of the effects of these various forces is not possible due to the nature of the nonlinear coupling between the various terms in the transport equations. However, approximate prediction of this force balance is often made by calculation of dimensionless quantities such as capillary and gravity numbers. We present an improved method for the numerical analysis of simulations which recognises the changing balance of forces – in both space and time – in a given domain. The classical two-phase transport equations for immiscible incompressible flow are expressed in two forms: (i) the convection–diffusion-gravity (CDG) formulation where convection and diffusion represent viscous and capillary effects, respectively, (ii) the oil pressure formulation where the viscous effects are attributed to the product of mobility difference and the oil pressure gradient. Each formulation provides a different perspective on the balance of forces although the two forms are equivalent. By discretising the different formulations, the effect of each force on the rate of change of water saturation can be calculated for each cell, and this can be analysed visually using a ternary force diagram. The methods have been applied to several simple models, and the results are presented here. When model parameters are varied to determine sensitivity of the estimators for the balance of forces, the CDG formulation agrees qualitatively with what is expected from physical intuition. However, the oil pressure formulation is dominated by the steady-state solution and cannot be used accurately. In addition to providing a physical method of visualising the relative magnitudes of the viscous, gravity and capillary forces, the local force balance may be used to guide our choice of upscaling method.     

An invariant solution of the turbulent boundary-layer equations

The problem of the group stratification of the system of equations describing motion in the laminar sublayer and the turbulent core is considered. The fundamental group admissible by the initial system is constructed; invariant solutions constructed on one of the subgroups lead to a system of ordinary differential equations. Joining of the solutions and interchange of the equations occur at the boundary of the laminar sublayer. A class of power-law flows of a turbulent boundary layer is investigated. In the region of decelerated motion a double-valued solution is found corresponding to attached or separated flow. The commonly used integral characteristics are calculated and presented in the form of an interpolation polynomial.Translated from Zhurnal Prikladnoi Mekhaniki i Teknicheskoi Fiziki, No. 4, pp. 126–132, July–August, 1975.     

Effect of supersonic stream parameters on the characteristic time of transient step flow

There have been many studies of viscous compressible gas flow in wakes and behind steps [1–6] in which attention has been focused on the steady-state flow regime. The problem of the supersonic flow of a viscous compressible heat-conducting gas past a plain backward-facing step is considered. The problem is solved numerically within the framework of the complete system of Navier-Stokes equations. The passage of the solution from the initial data to the steady-state regime and the effect of the gas dynamic parameters of the external flow on the characteristic flow stabilization time are investigated.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 4, pp. 137–140, July–August, 1989.     

Flow of a viscous fluid in an annular gap with intensive blowing from the walls

Self-similar solutions are obtained in [1, 2] to the Navier-Stokes equations in gaps with completely porous boundaries and with Reynolds number tending to infinity. Approximate asymptotic solutions are also known for the Navier-Stokes equations for plane and annular gaps in the neighborhood of the line of spreading of the flow [3, 4]. A number of authors [5–8] have discovered and studied the effect of increase in the stability of a laminar flow regime in channels of the type considered and a significant increase in the Reynolds number of the transition from the laminar regime to the turbulent in comparison with the flow in a pipe with impermeable walls. In the present study a numerical solution is given to the system of Navier-Stokes equations for plane and annular gaps with a single porous boundary in the neighborhood of the line of spreading of the flow on a section in which the values of the local Reynolds number definitely do not exceed the critical values [5–8]. Generalized dependences are obtained for the coefficients of friction and heat transfer on the impermeable boundary. A comparison is made between the solutions so obtained and the exact solutions to the boundary layer equations.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 1, pp. 21–24, January–February, 1987.     

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.     

Supersonic flow of dusty gas over blunt bodies

The results are given of a numerical investigation of the flow of dusty gas over the complete front surface of a sphere. The flow conditions are varied over a wide range in which the state of the gas suspension in the shock layer changes from a frozen to an equilibrium state. The phenomenological approach [5] is used to derive the system of equations describing the behavior of the two-phase medium. The system of conservation equations for the gas—solid-particle mixture is closed by means of relations that generalize the experimental data.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 3, pp. 73–77, May–June, 1982.     

Macroscopic Two-phase Flow in Porous Media Assuming the Diffuse-interface Model at Pore Level

The paper presents a model for two-phase flow, where liquid and gas are treated as one fluid with variable density. A one-component fluid and the diffuse-interface model for two-phase flow are assumed at pore level. The wetting properties of the fluid are described by the Cahn theory. Macroscopic equations are deduced in the framework of the Marle formalism. It is shown that two-phase flow in porous media can be described by the Cahn–Hilliard equation for the mass density. The concept of relative permeability is not needed. For non-neutral wetting, it is shown that a capillary pressure exists but that it is not a function of state. Two numerical illustrations are presented, one of them showing that the model is, at least in a simple steady-state situation, compatible with the generalized two-continuum model.     

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.     

Method of computing the chemically nonequilibrium gas flow in a heated tube in almost equilibrium or frozen states

The solution of equations describing turbulent isobaric flow of a chemically reacting gas in a heated tube is investigated analytically. Solutions of the ordinary nonlinear differential equations are obtained for almost frozen flow by the perturbation method, and for almost equilibrium flow by an asymptotic method taking account of the zero and first approximations, Linear differential equations in variations are written down to find the subsequent approximations.Translated from Izvestiya Akademiya Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 4, pp. 8–14, July–August, 1973.     

Formula for the Flow Resistance Factor in a Pipe with a Sudden Expansion at Small Reynolds Numbers

A formula for the flow resistance factors in a pipe with a sudden expansion of the cross section at Reynolds numbers of 0.2 to 10 is obtained by numerical solution of the complete Navier–Stokes equations for incompressible fluids. The flow resistance factors obtained using the derived formula are compared to those found by numerical solution of the Navier–Stokes equations.     

Permanent Fronts in Two-Phase Flows in a Porous Medium

A family of exact solutions for a model of a one-dimensional horizontal flow of two immiscible, incompressible fluids in a porous medium, including the effects of capillary pressure, is obtained analytically by solving the governing singular parabolic nonlinear diffusion equation. Each solution has the form of a permanent front propagating with a constant velocity. It is shown that, for every propagation velocity, there exists a set of permanent fronts all of which are moving with this velocity in an inflowing wetting–outflowing non-wetting flow configuration. Global bifurcations of this set, with the front velocity as a bifurcation parameter, are investigated analytically and numerically in detail in the case when the permeabilities and the capillary pressure are linear functions of the wetting phase saturation. Main results for the nonlinear Brooks–Corey model are also presented. In both models three global bifurcations occur. By using a geometric dynamical system approach, the nonlinear stability of the permanent fronts is established analytically. Based on the permanent front solutions, an interpretation of the dynamics of an arbitrary front of finite extent in the model is given as follows. The instantaneous upstream (downstream) velocity of an arbitrary non-quasistationary front is equal to the velocity of a permanent front whose shape coincides up to two leading orders with the instantaneous shape of the non-quasistationary front at the upstream (respectively, downstream) location. The upstream and downstream locations of the front undergo instantaneous translations governed by modified nonsingular hyperbolic equations. The portion of the front in between these locations undergoes a diffusive redistribution governed by a nonsingular nonlinear parabolic diffusion equation. We have proposed a numerical approach based on a parabolic–hyperbolic domain decomposition for computing non-quasistationary fronts.     

Exact solution of the system of equations of the second-order kinetic moments for a two-scale homoenergetic affine monatomic gas flow

The exact solution of the system of equations for the second-order kinetic moments (stresses) is investigated for a flow with two microscales when an unsteady shear flow is superimposed on a one-dimensional unsteady gas flow.Translated from Izvestiya Rossiiskoi Akademii Nauk, Mekhanika Zhidkosti i Gaza, No. 3, pp. 156–166, May–June, 1995.The work was carried out with financial support from the Russian Foundation for Fundamental Research (project No. 93-013-16407).     

Regression and Dimensional Analysis for Modeling Two‐Phase Flow

Numerical models describing multiphase flow phenomena are typically used to predict the displacement of water during the infiltration of nonaqueous phase liquids (NAPLs) into a groundwater system. In this paper, the applicability of regression and dimensional analysis to develop simple tools to bypass these time consuming numerical simulations is assessed. In particular, the infiltration of NAPL through a vertical, homogeneous soil column initially saturated with water is quantified. Two output variables defining the extent of infiltration were considered – the elevation of the NAPL front and the volume of NAPL which had entered the system. Dimensional analysis was initially performed to identify dimensionless terms associated with the underlying relations between these two output variables and the input variables (independent variables and system parameters). Artificial neural network techniques were then employed to develop regression equations for approximating the input–output relationships over a given domain. Application of these equations illustrated the interrelationships among capillary, buoyancy, and viscous forces driving the NAPL infiltration process.