Asymptotic behavior of the neutral curves of the linear stability problem for a laminar boundary layer

Asymptotic flow schemes corresponding to two branches of the solution for the neutral stability curve of a laminar boundary layer in an incompressible fluid are constructed. Two-term asymptotic solutions are obtained in the limit when the Reynolds number tends to infinity. The linear formulation of the problem is used and the flow is assumed to be two dimensional.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 5, pp. 39–46, September–October, 1981.I should like to thank O. V. Denisenko for making the numerical calculations.     

Influence of nonparallel flow on the development of disturbances in a supersonic boundary layer

The linear theory is used to solve the problem of the development of two-dimensional disturbances in the boundary layer of compressible fluid. In contrast to the stability theory of plane-parallel flows, the present paper takes into account the presence in the boundary layer of transverse (at right angles to the flow direction) motions, the dependence of the averaged flow parameters on the longitudinal coordinate, and also the deformation of the amplitude distribution profile of the disturbances as a function of the longitudinal coordinate. The calculations are made for Mach number M = 4.5.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 2, pp. 26–31, March–April, 1980.     

Influence of thermal inhomogeneity of the boundary on the forced motion of a viscous fluid in a plane layer

Numerical solution of the Navier-Stokes equations is used to investigate the laminar motion of an incompressible fluid in a plane layer when part of the lower boundary has a temperature different from the rest of the lower boundary. The problem is solved in the Boussinesq approximation.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 1, pp. 171–173, January–February, 1982.     

Numerical modeling of thermocapillary convection in a fluid layer with local heating of the free surface

Thermocapillary convection in a plane horizontal fluid layer with concentrated heating of the free surface is modeled numerically using the Navier-Stokes equations and the heat transport equation. This makes it possible to examine the structure of the convection throughout the fluid volume, in particular in the region where the motion is weak. The deformation of the free surface is assumed to be negligibly small. In the case of a ponderable fluid this assumption is justified given certain upper and lower constraints on the temperature difference and the thickness of the layer, respectively, [9, 10]. Under conditions of weightlessness a fluid layer of constant thickness in a rectangular channel can be realized at a contact angle of 90° [7].Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 4, pp. 108–113, July–August, 1987.     

Effect of the Coriolis force on the onset of convection in a layer with a fixed heat flux on the boundaries

The effect of the Coriolis force on the onset of convection in a plane horizontal layer of viscous fluid with a fixed heat flux on the rigid lower and free upper boundaries is investigated. Expressions for the critical Rayleigh numbers and wave number are obtained analytically in the rapid rotation limit.Perm'. Translated from Izvestiya Rossiiskoi Akademii Nauk, Mekhanika Zhidkosti i Gaza, No. 3, pp. 42–46, May–June, 1994.     

Unsteady gravity-elastic and gravity-capillary ship waves

The development of three-dimensional waves generated by a region of pressures moving uniformly and rectilinearly over the surface of a thin elastic isotropic plate covering an ideal fluid layer of finite depth is investigated. The pressures act starting at a certain instant. A qualitative similarity between the waves occurring and gravity-capillary waves is noted. The calculations are made for an ice cover. This model problem permits examining a number of properties of the oscillations of the ice cover occurring when hauling freight over ice roads, landing and takeoff of aircraft from ice fields, etc. [1]. The development of ship waves in a fluid of finite depth in the absence of a floating plate was investigated in [2, 3] and gravity-capillary waves were studied in [4–6]. Certain properties of steady three-dimensional waves occurring during movement of a load over the surface of a floating elastic plate were established in [1].Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 5, pp. 26–32, September–October, 1978.     

Localized Coherent Structures in the Boundary Layer

A Blasius laminar boundary layer and a steady turbulent boundary layer on a flat plate in an incompressible fluid are considered. The spectral characteristics of the Tollmien—Schlichting (TS) and Squire waves are numerically determined in a wide range of Reynolds numbers. Based on the spectral characteristics, relations determining the three–wave resonance of TS waves are studied. It is shown that the three–wave resonance is responsible for the appearance of a continuous low–frequency spectrum in the laminar region of the boundary layer. The spectral characteristics allow one to obtain quantities that enter the equations of dynamics of localized perturbations. By analogy with the laminar boundary layer, the three–wave resonance of TS waves in a turbulent boundary layer is considered.     

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.     

Unsteady thermocapillary convection in a nonuniformly heated fluid layer

In many technological processes, thin extended layers of nonuniformly heated fluid are used [1–3]. If they are sufficiently thin, thermocapillary forces have a decisive influence on the occurrence and development of motion of the fluid [4–6]. Investigation of convective motion in such a layer is of great interest for estimating the intensity of heat and mass transfer in technological processes. This paper is a study of unsteady thermocapillary motion in a layer of viscous incompressible fluid with free surface in which a thermal inhomogeneity is created at the initial time. Approximate expressions are obtained for the fields of the velocity, temperature, and pressure in the fluid, and also for the shape of the free surface.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 3, pp. 17–25, May–June, 1991.     

Effect of the percolation heat transfer component on the temperature field and thawing dynamics of soils

The effect of the convective heat transfer component on the temperature field and thawing front dynamics of soils is investigated for high fluid percolation velocities in the thawed zone. The steady state interchange and approximate self-similarity methods are used to obtain upper and lower bounds of the solution of the Stefan problem with a convective heat transfer component in a porous medium. From the results of the calculations conclusions are drawn concerning the accuracy and limits of applicability of the solutions obtained.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 2, pp. 166–171, March–April, 1987.     

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.     

Asymptotic structure of inviscid disturbances in a thin shock layer

The WKB method, used in [4] to analyze the short-wave instability of a supersonic mixing layer, is employed to investigate various types of inviscid three-dimensional short-wave disturbances in a thin shock layer of perfect gas with arbitrary velocity and temperature distributions across the layer. Simple analytic expressions for the dispersion relations are obtained for neutral disturbances. The results of an asymptotic analysis are compared with direct numerical calculations for a simple model of the shock layer.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 6, pp. 72–79, November–December, 1988.     

Convection in a two-layer system due to the combined action of the Rayleigh and thermocapillary instability mechanisms

In a two-layer system loss of stability may be monotonic or oscillatory in character. Increasing oscillatory perturbations have been detected in the case of both Rayleigh [1, 2] and thermocapillary convection [3–5]; however, for many systems the minimum of the neutral curve corresponds to monotonic perturbations. In [5] an example was given of a system for which oscillatory instability is most dangerous when the thermogravitational and thermocapillary instability mechanisms are simultaneously operative. In this paper the occurrence of convection in a two-layer system due to the combined action of the Rayleigh (volume) and thermocapillary (surface) instability mechanisms is systematically investigated. It is shown that when the Rayleigh mechanism operates primarily in the upper layer of fluid, in the presence of a thermocapillary effect oscillatory instability may be the more dangerous. If thermogravitational convection is excited in the lower layer of fluid, the instability will be monotonic.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 1, pp. 166–170, January–February, 1987.     

Investigation of thermocapillary and thermogravitational convection in a fluid with local heating

The results of a numerical solution of the problem of the unsteady convective motion generated in a fluid layer by the formation at the initial instant of a heated zone in the form of a thin cylindrical column, extending from the surface into the interior of the fluid, are presented. The problem is formulated with allowance for both thermocapillary and thermogravitational convection. The influence of the thermocapillary and thermogravitational effects on the fluid motion for various layer thicknesses is subjected to parametric analysis.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 6, pp. 25–29, November–December, 1989.     

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.     

Stability of a laminar boundary layer of a power-law no n-newtonian liquid

The stability of a laminar boundary layer of a power-law non-Newtonian fluid is studied. The validity of the Squire theorem on the possibility of reducing the flow stability problem for a power-law fluid relative to three-dimensional disturbances to a problem with two-dimensional disturbances is demonstrated. A numerical method of integrating the generalized Orr-Sommerfeld equation is constructed on the basis of previously proposed [1] transformations. Stability characteristics of the boundary layer on a longitudinally streamlined semiinfinite plate are considered.Translated from Zhurnal Prikladnoi Mekhaniki i Tekhnicheskoi Fiziki, No. 1, pp. 101–106, January–February, 1976.     

Method of factorization in unsteady problems of gravity-elastic wave excitation in a fluid with a partially free boundary

An approach to the solution of unsteady problems with mixed boundary conditions for a layer of heavy fluid is developed. The plane problem of wave excitation by displacements given in a certain region of the lower boundary of the layer when the upper boundary is partially covered by an elastic plate is examined by way of illustration. As distinct from [1, 2], the proposed approach makes it possible to construct a solution in the form of a sum of harmonics and to carry out an analytic investigation into the nature of the propagation and stabilization of the wave fields. The space-time regions of the forming and formed wave packets are identified.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 6, pp. 100–106, November–December, 1985.     

Convective Instability in Superposed Fluid and Porous Layers with Vertical Throughflow

A closed form solution to the convective instability in a composite system of fluid and porous layers with vertical throughflow is presented. The boundaries are considered to be rigid-permeable and insulating to temperature perturbations. Flow in the porous layer is governed by Darcy–Forchheimer equation and the Beavers–Joseph condition is applied at the interface between the fluid and the porous layer. In contrast to the single-layer system, it is found that destabilization due to throughflow arises, and the ratio of fluid layer thickness to porous layer thickness, , too, plays a crucial role in deciding the stability of the system depending on the Prandtl number.     

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.     

Experimental investigation of fluid convection in a closed toroidal channel

The results of investigating convective fluid motion in a closed toroidal channel are analyzed and compared with the results of calculations based on the Lorenz model. It is shown that this model adequately describes the fluid convection only in the case of a large aspect ratio of the mean radius of the torus to its cross-sectional dimension and small heating power. The measurements and visual observations record substantial deviations from the assumptions underlying the Lorenz model in the domain of problem parameters in which, in accordance with the calculations, bifurcations and instability of steady solutions of the Lorenz model should take place.Moscow. Translated from Izvestiya Rossiiskoi Akademii Nauk, Mekhanika Zhidkosti i Gaza, No. 4, pp. 20–28, July–August, 1995.