Throughflow Effects on Penetrative Convection in Superposed Fluid and Porous Layers

The effect of vertical throughflow on the onset of penetrative convection simulated via internal heating in a two-layer system in which a layer of fluid overlies and saturates a layer of porous medium is studied. Flow in the porous medium is governed by Forchheimer-extended Darcy equation, and Beavers?CJoseph slip condition is applied at the interface between the fluid and the porous layers. The boundaries are considered to be rigid, however permeable, and insulated to temperature perturbations. The eigenvalue problem is solved using a regular perturbation technique with wave number as a perturbation parameter. The ratio of fluid layer thickness to porous layer thickness, ??, the direction of throughflow, and the presence of volumetric internal heat source in fluid and/or porous layer play a decisive role on the stability characteristics of the system. In addition, the influence of Prandtl number arising due to throughflow is also emphasized on the stability of the system. It is observed that both stabilizing and destabilizing factors can be enhanced because of the simultaneous presence of a volumetric heat source and vertical throughflow so that a more precise control (suppress or augment) of thermal convective instability in a layer of fluid or porous medium is possible.     

Non-Darcian Effects on the Onset of Convection in a Porous Layer with Throughflow

The Brinkman extended Darcy model including Lapwood and Forchheimer inertia terms with fluid viscosity being different from effective viscosity is employed to investigate the effect of vertical throughflow on thermal convective instabilities in a porous layer. Three different types of boundary conditions (free–free, rigid–rigid and rigid–free) are considered which are either conducting or insulating to temperature perturbations. The Galerkin method is used to calculate the critical Rayleigh numbers for conducting boundaries, while closed form solutions are achieved for insulating boundaries. The relative importance of inertial resistance on convective instabilities is investigated in detail. In the case of rigid–free boundaries, it is found that throughflow is destabilizing depending on the choice of physical parameters and the model used. Further, it is noted that an increase in viscosity ratio delays the onset of convection. Standard results are also obtained as particular cases from the general model presented here.     

Effect of free‐stream turbulence on the boundary‐layer structure with diffusion combustion of ethanol

The effect of turbulization of a subsonic air flow on the boundarylayer structure was experimentally studied during evaporation and combustion of ethanol behind an obstacle 3–6 mm high. It is shown that turbulization increases the thermal boundarylayer thickness by a factor of 2, where as the dynamic boundarylayer thickness changes weakly. For 1–18% turbulence at the entrance, the change in the momentum thickness along the channel is close to the change in the momentum thickness for a laminar isothermal boundary layer without injection. Local regions of elevated turbulence with a high intensity of heat and mass transfer arise in the case of combustion behind the obstacle at a distance of 40–50 obstacle heights.     

Flow near the rear end of a slender axisymmetric body

The steady flow of a viscous incompressible fluid at high Reynolds numbers near a body of revolution of finite length whose radius coincides in order of magnitude with the thickness of the boundary layer is considered. The structure of the boundary layer in the neighborhood of the rear end of the body is investigated on the assumption that it has a power-law shape with values of the exponent n 1/2. A solution is also obtained for the near wake.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 5, pp. 10–18, September–October, 1990.     

The Laminar Boundary Layer over a Permeable Wall

An analysis is given of the laminar boundary layer over a permeable/porous wall. The porous wall is passive in the sense that no suction or blowing velocity is imposed. To describe the flow inside and above the porous wall a continuum approach is employed based on the Volume-Averaging Method (S. Whitaker The method of volume averaging). With help of an order-of-magnitude analysis the boundary-layer equations are derived. The analysis is constrained by: (a) a low wall permeability; (b) a low Reynolds number for the flow inside the porous wall; (c) a sufficiently high Reynolds number for the freestream flow above the porous wall. Two boundary layers lying on top of each other can be distinguished: the Prandtl boundary layer above the porous wall, and the Brinkman boundary layer inside the porous wall. Based on the analytical solution for the Brinkman boundary layer in combination with the momentum transfer model of Ochoa-Tapia and Whitaker (Int. J. Heat Mass Transfer 38 (1995) 2635). for the interface region, a closed set of equations is derived for the Prandtl boundary layer. For the stream function a power series expansion in the perturbation parameter is adopted, where is proportional to ratio of the Brinkman to the Prandtl boundary-layer thickness. A generalization of the Falkner–Skan equation for boundary-layer flow past a wedge is derived, in which wall permeability is incorporated. Numerical solutions of the Falkner–Skan equation for various wedge angles are presented. Up to the first order in wall permeability causes a positive streamwise velocity at the interface and inside the porous wall, but a wall-normal interface velocity is a second-order effect. Furthermore, wall permeability causes a decrease in the wall shear stress when the freestream flow accelerates, but an increase in the wall shear stress when the freestream flow decelerates. From the latter it follows that separation, as indicated by zero wall shear stress, is delayed to a larger positive pressure gradient.     

Analytical and numerical study of natural convection heat transfer in an inclined composite enclosure

Laminar natural convection heat transfer in inclined fluid layers divided by a partition with finite thickness and conductivity is studied analytically and numerically. The governing equations for the fluid layers are solved analytically in the limit of a thin layered system with constant flux boundary conditions. The study covers of the range of Ra from 10³ to 10⁷, from 0° to 180° and the thermal conductivity ratio of partition to fluid ratioK from 10^–2 to 10⁶. The Prandtl number was 0.72 (for air). Results are obtained in terms of an overall Nusselt number as a function of Rayleigh number, angle of inclination of the system, mid layer thickness, and mid layer thermal conductivity. The critical Rayleigh number for the onset of convection in a bottom-heated horizontal system is predicted. The results are compared with the numerical results obtained by solving the complete system of governing equations, using SIMPLER method, as well as with the limiting cases in the literature.     

Stability of the Horizontal Throughflow in a Power-Law Fluid Saturated Porous Layer

Transport in Porous Media - Double-diffusive convective instability of horizontal throughflow in a power-law fluid saturated porous layer is investigated. The boundaries of this horizontal porous...     

Stability of convective viscous fluid flow in porous and free vertical layers

The linear stability of convection in a system consisting of a vertical layer of fluid and an adjacent layer of porous medium saturated with that fluid is investigated. The fluid and the porous medium are bounded by isothermal surfaces heated to different temperatures.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 5, pp. 3–9, September–October, 1990.The author wishes to thank G. Z. Gershuni for supervising the work and D. V. Lyubimov for useful discussions.     

Shock Thickness in Monatomic Gases

The entropy balance of a Navier–Stokes–Fourier fluid is used to predict the thickness of a shock wave as a function of the Mach number. The results are in good agreement with experimental observations. Indeed the agreement of our thermodynamic calculations with experiments is better than that of the actual solution of the Navier–Stokes–Fourier equations by Gilbarg and Paolucci [1].     

Asymptotic behavior of solutions of the Orr-Sommerfeld equations in regions adjacent to two branches of the neutral curve

The asymptotic behavior of the upper and lower branches of the neutral stability curve of a boundary layer found by Lin [1] was determined more accurately by various authors [2–4], who, on the basis of the linearized Navien-Stokes equations, analyzed the higher approximations in the Reynolds number R. In the limit R , neutral perturbations have wavelengths that exceed in order of magnitude the boundary layer thickness. The long-wavelength asymptotic behavior of the Orr-Sommerfeld equation is, in particular, of interest because the characteristic solutions of the linearized equations of free interaction (triple-deck theory) [5–7] are a limiting form of Tollmierr-Schlichting waves in an incompressible fluid with critical layers next to the wall [8–9]. At the same time, the dispersion relation, which is identical to the secular equation of the Orr-Sommerfeld problem, contains an entire spectrum of solutions not considered in the earlier studies [2–4]. The first oscillation mode in the spectrum may be either stable or unstable. In the present paper, solutions are constructed for each of the subregions (including the critical layer) into which the perturbed velocity field in the linear stability problem is divided at large Reynolds numbers. Dispersion relations describing the neighborhood of the upper and lower branches of the neutral curve for the boundary layer are derived. These relations, which contain neutral solutions as a special case, go over asymptotically into each other in the unstable region between the two branches.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 4, pp. 3–11, July–August, 1984.     

Character of the boundary layer near its origin on a body rotating in a fluid at rest

The evolution of the boundary layer on bodies of revolution rotating about the symmetry axis in a fluid at rest is largely determined by the position of its origin with respect to the axis of rotation. If the origin of the boundary layer coincides with a pole of the rotating body, then under fairly general assumptions as to the shape of the body the boundary layer has a nonzero thickness in the vicinity of the pole, and the flow in it is described by a particular self-similar solution of the boundary-layer equations [1, 2]. The applicability of existing approximate methods for calculation of the boundary layer [2, 3] is restricted to this case. The results of the present article refer to the case in which the boundary originates at the leading edge at a finite distance from the rotation axis. The behavior of the solution of the boundary-layer equations near the edge is determined. A transformation of variables that reduces the system of boundary-layer equations to a form suitable for analysis and solution is derived. This transformation is used to obtain universal equations determining the local behavior of the boundary layer in the vicinity of its origin in both of the cases indicated above.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 4, pp. 137–140, July–August, 1976.     

Investigation of two-phase flow in inclined macroinhomogeneous porous media

The results of experiments carried out in order to determine the principal characteristics of the process of displacement of one (nonwetting) fluid from inclined macroinhomogeneous porous media by another (wetting) fluid are presented. Irrespective of whether flow in inclined stratified nonhomogeneous formations or in zonally nonhomogeneous media (with a corresponding well distribution) is investigated, the term oblique stratification is used for describing these processes.Translated from Izvestiya Rossiiskoi Akademii Nauk, Mekhanika Zhidkosti i Gaza, No.6, pp. 125–131, November–December, 1992.     

Laminar flow between porous disks for intense homogeneous asymmetric inflow and outflow

In [1–3], a class of self-similar solutions was considered for the flow of incompressible fluid in a plane channel with porous walls, through which there is homogeneous symmetric inflow or outflow. An analogous class of self-similar solutions for flow between porous disks with natural homogeneous conditions at the periphery was considered in [4], where the asymptotic behavior of these solutions at a small Reynolds number of the outflow R was investigated, and the limiting form of the solution for symmetric outflow with R= was noted. In the present paper, the boundary-function method is applied to the singular problem corresponding to the flow between porous disks for asymmetric inflow and outflow characterized by large R.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 6, pp. 13–19, November–December, 1976.     

Hydrodynamics in weak force fields onset of steady thermocapillary convection in a spherical fluid layer under zero-g conditions

In the study of cellular convection in an infinite plane fluid layer with a free surface, both the Archimedes and thermocapillary forces [1–3] have been cited as reasons for the onset of convection. This has also been confirmed experimentally [4], When mass forces are absent or negligibly small it is natural to pose the question of the onset of pure thermocapillary convection or convection caused only by the surface tension gradients (see [2–3]). In the present paper, this problem is examined for a spherical fluid layer under zero-g conditions.     

Use of turbulent energy balance equation in the theory of turbulent flows along walls

The turbulent flow of an incompressible fluid is considered in a plane channel, a circular tube, and the boundary layer on a flat plate. The system of equations describing the motion of the fluid consists of the Reynolds equations and the mean kinetic energy balance equation for turbulent fluctuations. On the basis of an analysis of experimental data, hypotheses are formulated with respect to the eddy kinematic viscosity and lengthl entering into the expression for specific dissipation of turbulent energy into heat. It is assumed that in the central (outer) region of the flow in a channel, andl are constants, and expressions are taken for them which are used for a free boundary layer; near the walll varies linearly and almost linearly. Results of calculations of the turbulent energy distribution, the mean velocity, and the drag coefficient are in good agreement with the existing experimental data. The values of two empirical coefficients, which enter into the system of equations as the result of the hypotheses, are close to those obtained for a free boundary layer.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 3, pp. 25–33, May–June, 1973.     

Analysis of the convective instability of a binary mixture in a porous medium

The problem of the stability of a binary mixture in a porous medium is investigated in the complete formulation with allowance for cross kinetic and gravitational effects. Boundary conditions of the first and second kinds for a plane horizontal layer of the porous medium are considered. The boundaries of the region of instability are determined. The region of the parameters corresponding to the stability paradox effect, i.e., the instability of a mixture that becomes heavier with depth, is described. It is established that the multicomponent nature of the mixture helps to stabilize the equilibrium state.Translated from Izvestiya Rossiiskoi Akademii Nauk, Mekhanika Zhidkosti i Gaza, No. 1, pp. 110–119, January–February, 1993.     

Stability of unsteady rotational motion of a plane layer of ideal fluid with free boundaries

The exact solution of the equations of an ideal incompressible fluid describing the unsteady rotational motion of a plane layer with free boundaries is obtained. For constant vorticity the stability problem is studied in the linear approximation. The asymptotic behavior of the free boundaries of the layer as t is calculated. It is shown that the vorticity of the basic motion stabilizes the boundaries of the layer.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 2, pp. 15–21, March–April, 1986.     

Nonisothermal displacement of oil by a solution of an active additive

In earlier work [1, 2] mathematical models have been constructed for processes of displacement of oil from a porous medium by a solution of an active additive, i.e., an additive capable of changing the hydro-dynamic characteristics of the fluid and the medium. An additive of this kind that was considered was a polymer that in the dissolved state influences the properties of the displacing fluid and in the adsorbed state the permeability of the porous medium. Self-similar solutions were obtained corresponding to the problem of frontal displacement from a homogeneous porous medium, and a number of numerical calculations were made. It is natural to generalize this treatment by introducing into the problem a second active factor, which is here taken to be the temperature of the injected fluid. The analysis of the nonisothermal displacement of oil by a solution of an active additive can be transferred without significant modifications to the general problem of displacement of oil by a solution carrying two active agents. The names additive and temperature are retained here only for convenience of exposition.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 6, pp. 90–107, November–December, 1980.We thank A. A. Barmin, A. G. Kulikovskii, and L. A. Chudov for helpful discussions.     

Intermediate asymptotic behavior of the far field of internal waves in a layer of stratified fluid lying on a homogeneous layer

A study is made of the field of internal waves excited by fixed and moving sources in a layer of stratified fluid of finite depth h lying on a homogeneous layer of thickness H. It is shown that in the limit H the asymptotic behavior of the field of the moving source can be expressed in terms of a Fresnel integral. For finite H the field asymptotic behavior expressed in terms of the Fresnel integral is intermediate – it is valid at distances h r L; the asymptotic behavior expressed in terms of the Airy function is true for r L, with the intermediate scale L = const H³/h². In the case of a fixed source the intermediate asymptotic behavior can be expressed in terms of a parabolic cylinder function, and the farfield behavior can be expressed in terms of the square of the Airy function. The transition between the intermediate and far-field asymptotic behaviors is investigated numerically.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 3, pp. 158–162, May–June, 1988.     

Supersonic drag of three-dimensional bodies with a star-shaped cross section and its calculation

The effect on the aerodynamic drag of the real properties of the gas in the shock layer around pyramidal star-shaped bodies (the viscosity, the displacement thickness of the boundary layer, its separation under the influence of the inner shocks) is considered. It is shown that the models for calculating the total drag of star-shaped bodies which do not take into account the displacement thickness of the boundary layer are applicable only at low supersonic free-stream velocities (M < 3). A model of the boundary layer displacement thickness is proposed and tested over a broad range of variation of the parameters that determine the geometry of the pyramidal bodies for high supersonic or hypersonic speeds. A comparison with the experimental data shows that the calculation procedure adequately reflects the results of experiments on the aerodynamic drag of star-shaped bodies in cases in which the inner shocks in the shock layer do not lead to boundary layer separation and can be used in optimization problems.Translated from Izvestiya Rossiiskoi Akademii Nauk, Mekhanika Zhidkosti i Gaza, No.1, pp. 57–69, January–February, 1993.