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.     

Method of calculation of interaction of wall flows

A flow of viscous compressible fluid in the neighborhood of the line of interaction of wall flows is considered. A method of calculating the line of interaction and the direction of the self-induced secondary flow is developed. Papers [1–3] are devoted to the simulation of a separation flow with singularities in the neighborhood of singular lines and points, where boundary-layer equations are invalid. However, the theories of local separation used at present have mainly been developed only for two-dimensional problems, while the models of viscous-inviscid interaction have restrictions in application for turbulent flows with developed separation. The interaction of three-dimensional wall turbulent flows is considered below. It is assumed that the thickness of the boundary layers and the scales of the interaction zones are small in comparison with the characteristic dimension of the system, while the line of discontinuity of the solutions of the three-dimensional boundary layer equations is the same as the line of interaction of the wall flows.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 2, pp. 53–59, March–April, 1987.The author is grateful to G. Yu. Stepanov and V. N. Ershov for their interest in my work and their valuable remarks.     

Boundary layer in the vicinity of the stagnation point of a body of axial symmetry in a turbulent stream

The flow in the boundary layer in the vicinity of the stagnation point of a flat plate is examined. The outer stream consists of turbulent flow of the jet type, directed normally to the plate. Assumptions concerning the connection between the pulsations in velocity and temperature in the boundary layer and the average parameters chosen on the basis of experimental data made it possible to obtain an isomorphic solution of the boundary layer equations. Equations are obtained for the friction and heat transfer at the wall in the region of gradient flow taking into account the effect of the turbulence of the impinging stream. It is shown that the friction at the wall is insensitive to the turbulence of the impinging stream, while the heat transfer is significantly increased with an increase in the pulsations of the outer flow. These properties are confirmed by the results of experimental studies [1–4].Translated from Zhurnal Prikladnoi Mekhaniki i Tekhnicheskoi Fiziki, No. 5, pp. 83–87, September–October, 1973.     

Stability of unsteady viscous flows

A study is made of the linear stability of plane-parallel unsteady flows of a viscous incompressible fluid: in the mixing layer of two flows, in a jet with constant flow rate, and near a wall suddenly set in motion [1]. The slow variation of these flows in time compared with the rate of change of the perturbations makes it possible to use the method of two-scale expansions [2]. The stability of nonparallel flows with allowance for their slow variation with respect to the longitudinal coordinate was investigated, for example, in [3–6]. The unsteady flows considered in the present paper have a number of characteristic properties of non-parallel flows [1], but in contrast to them are described by exact solutions of the Navier-Stokes equations. In addition, for unsteady planeparallel flows a criterion of neutral stability can be uniquely established by means of the energy balance equation.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 4, 138–142, July–August, 1981.I thank G. I. Petrov for suggesting the problem, and also S. Ya. Gertsenshtein and A. V. Latyshev for assisting in the work.     

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.     

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.     

Governing equation of polymer solutions on the basis of the dynamics of noninteracting relaxation oscillators

Special attention is currently being given to the study of systems formed by relatively large molecules (molecular liquids, liquid crystals, polymer solutions, etc.) in connection with different applications. A phenomenological approach proves inadequate in these cases: the nonlinear governing equations of the system are ambiguous and, no less important, the relationship between macroscopic effects and the internal characteristics of the system remains unclear. These questions are also studied by another approach, in which the structural units of the system are replaced by a suitable model. As is known, the simplest model of a macromolecule being deformed is a dumbbell — a relaxation oscillator with two centers of friction coupled by an elastic force. Such a model makes it possible to describe the basic features of the nonlinear behavior of polymer solutions [1, 2]. The goal of the present work is to derive governing equations with allowance for the hydrodynamic interaction of the friction centers of the relaxation oscillators. This approach leads to the most general form of governing equation of a dilute polymer solution, while allowance for hydrodynamic interaction leads to the discovery of new effects in the study of simple shear flow. For example, the second difference of the normal stresses is nontrivial.Translated from Zhurnal Prikladnoi Mekhaniki i Tekhnicheskoi Fiziki, No. 3, pp. 101–105, May–June, 1986.     

Numerical study of flows of a viscous compressible gas and of an equilibrium gas mixture in a flat nozzle in the presence of strong blowing

The problem of the interaction of a viscous supersonic stream in a flat nozzle with a transverse gas jet of the same composition blown through a slot in one wall of the nozzle is examined. The complete Navier-Stokes equations are used as the initial equations. The statement of the problem in the case of the absence of blowing coincides with [1]. The conditions at the blowing cut are obtained on the assumption that the flow of the blown jet up to the blowing cut is described by one-dimensional equations of ideal gasdynamics. The proposed model of the interaction is generalized to the case of flow of a multicomponent gas mixture in chemical equilibrium. The exact solutions found in [2] are used as the boundary conditions at the entrance to the section of the nozzle under consideration. The results of numerical calculations of the flows of a homogeneous nonreacting gas and of an equilibrium mixture of gases consisting of four components (H₂, H₂O, CO, CO₂) are given for different values of the parameters of the main stream and of the blown jet. In the latter case it is assumed that the effect of thermo- and barodiffusion can be neglected.Translated from Zhurnal Prikladnoi Mekhaniki i Tekhnicheskoi Fiziki, No. 4, pp. 55–63, July–August, 1974.     

Interaction of a submerged turbulent jet with a solid wall

The turbulent layer of a wall jet has been analyzed in many theoretical and experimental studies [1–10]. Most theoretical investigations are based on the simultaneous solution of the equations of the turbulent jet and the boundary layer that forms at the wall. The differences lie in the methods used to correlate the velocity and temperature distributions, as well as in the friction and heat-transfer laws employed. In this article we present a method based on the further development of the idea of conservation of the laws governing wall turbulence with respect to change in boundary conditions.     

Interaction between the boundary layer and a supersonic flow when part of the wall is rapidly heated

There have been many theoretical studies of aspects of the unsteady interaction of an exterior inviscid flow with a boundary layer [1–9]. The mathematical flow models obtained in these studies by the method of matched asymptotic expansions describe a wide range of phenomena observed experimentally. These include boundary layer separation near the hinge of a flap, the flow in the neighborhood of the trailing edge of an oscillating airfoil [1–2], and the development and propagation of perturbations in a boundary layer excited by an oscillating wall or some other way [3–5]. The present paper studies the interaction of an unsteady boundary layer with a supersonic flow when a small part of the surface of a body in the flow is rapidly heated.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 1, pp. 66–70, January–February, 1984.     

Asymptotic theory of short separation regions on the leading edge of a slender airfoil

The two-dimensional flow of a viscous incompressible fluid near the leading edge of a slender airfoil is considered. An asymptotic theory of this flow is constructed on the basis of an analysis of the Navier—Stokes equations at large Reynolds numbers by means of matched asymptotic expansions. A central feature of the theory is the region of interaction of the boundary layer and the exterior inviscid flow; such a region appears on the surface of the airfoil in a definite range of angles of attack. The boundary-value problem for this region is reduced to an integrodifferential equation for the distribution of the friction. This equation has been solved numerically. As a result, closed separation regions are constructed, and the angle of attack at which separation occurs is found.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 1, pp. 42–51, January–February, 1981.I thank V. V. Sychev and Vik. V, Sychev for assistance.     

Propagation of modulated nonlinear waves in a relaxing gas-liquid mixture

The propagation of nonstationary weak shock waves in a chemically active medium is essentially dispersive and dissipative. The equations for short-wavelength waves for such media were obtained and investigated in [1–4]. It is of interest to study quasimonochromatic waves with slowly varying amplitude and phase. A general method for obtaining the equations for modulated oscillations in nonlinear dispersive media without dissipation was proposed in [5–8]. In the present paper, for a dispersive, weakly nonlinear and weakly dissipative medium we derive in the three-dimensional formulation equations for waves of short wavelength and a Schrödinger equation, which describes slow modulations of the amplitude and phase of an arbitrary wave. The coefficients of the equations are particularized for the considered gas-liquid mixture. Solutions are obtained for narrow beams in a given defocusing medium as well as linear and nonlinear solutions in the neighborhood of a diffraction beam. A solution near a caustic for quasimonochromatic waves was found in [9].Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 1, pp. 133–143, January–February, 1980.     

Some self-similar solutions of Grad's equations

It is shown that the self-similar solutions of the Navier-Stokes and Burnett equations found earlier by the authors [1–9] can be extended to the case of two-dimensional flows of a weakly rarefied gas described by Grad's equations. Examples are given of numerical realization of self-similar solutions for flow in an expanding planar channel. It is found that there are appreciable differences between the behavior of the self-similar solutions of the Navier-Stokes, Burnett, and Grad equations in the neighborhood of a channel wall.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 3, pp. 88–94, May–June, 1982.     

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.     

Computation of base flow characteristics with uniform blowing

Blowing at bluff body base was considered under different conditions and for small amount of blowing this problem was solved using dividing streamline model [1]. The effect of supersonic blowing on the flow characteristics of the external supersonic stream was studied in [2–4]. The procedure and results of the solution to the problem of subsonic blowing of a homogeneous fluid at the base of a body in supersonic flow are discussed in this paper. Analysis of experimental results (see, e.g., [5]) shows that within a certain range of blowing rate the pressure distribution along the viscous region differs very little from the pressure in the free stream ahead of the base section. In this range the flow in the blown subsonic jet and in the mixing zones can be described approximately by slender channel flow. This approximation is used in the computation of nozzle flows with smooth wall inclination [6, 7]. On the other hand, boundary layer equations are used to compute separated stationary flows with developed recirculation regions [8] in order to describe the flow at the throat of the wake. The presence of blowing has significant effect on the flow structure in the base region. An increasing blowing rate reduces the size of the recirculation region [9] and increases base pressure. This leads to a widening of the flow region at the throat, usually described by boundary-layer approximations. At a certain blowing rate the recirculation region completely disappears which makes it possible to use boundary-layer equations to describe the flow in the entire viscous region in the immediate neighborhood of the base section.Translated from Zhurnal Prikladnoi Mekhaniki i Tekhnicheskoi Fiziki, No. 1, pp. 76–81, January–February, 1984.     

Influence of suspended particles on the structure of a turbulent flow in a tube

The results of an experimental investigation into the dispersed flow of a system subject to negative pressure gradients are presented. The measurements were based on an optical time-of-flight method in a water channel, using polystyrene spheres as the solid phase. The average and pulsational characteristics of the dispersed flow were obtained in the boundary (wall) region and also in the center (core) of the flow. For zero pressure gradient the influence of the solid phase expressed itself as a reduction in the level of turbulence and an increase in the extent of the viscous sublayer, leading to a fall in the coefficient of friction. For a negative pressure gradient the pressure of the solid phase generated small-scale vortices, reduced the extent of the viscous sublayer, and hence increased the coefficient of surface friction.Translated from Zhurnal Prikladnoi Mekhaniki i Tekhnicheskoi Fiziki, No. 2, pp. 111–118, March–April, 1976.The author wishes to thank Yu. A. Buevich for interest in this work and V. L. Zalukaev for participation in the experiments.     

Investigation of laminar sublayer on permeable surfaces with injection

The thin-wall region — the laminar (or viscous) sublayer, in which the molecular mechanism of transfer predominates — plays an important role in friction and heat-transfer processes in a turbulent boundary layer. In particular, the relation between the sublayer thickness and the height of the surface roughness determines the nature of the flow — hydrodynamically smooth or rough — over the surface. This is of great practical importance and, hence, the roughness criterion has become the subject of numerous and systematic investigations. There are exhaustive data for the majority of the commonest cases encountered in practice [1]. The velocity on the boundary of the laminar sublayer appears as an important parameter in two-layer calculation schemes (e.g., [2–4], etc.). Although the theoretical analysis of a turbulent boundary layer with injection started several decades ago, there are at present hardly any reliable experimental data which can be used to determine the variation of the parameters within and at the boundary of the sublayer in relation to the injection rate. In this work we used interferometric diagnostics for precision experimental investigations of the parameters of the laminar sublayer on permeable surfaces with injection.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 5, pp. 66–72, September–October, 1977.     

Aerodynamics of end-wall boundary layer in a vortex chamber

The need for the inclusion of end-wall boundary layers in the study of the aerodynamics of vortex chambers has been frequently mentioned in the literature. However, owing to limited experimental data [1–3] with reliable information on the wall layers, the existing computational methods for end-wall boundary layers are not well-founded. The question of which parameters determine the formation of end-wall flow remains debatable. In some studies [4, 5], the vortex chambers are conditionally divided into short and long chambers. However, there is no unique opinion on the role of end-wall flows in vortex chambers of different lengths. It has also not been established for what geometric and flow parameters the chamber could be considered long or short. In the present study, as in [1, 5–8], solution is obtained for the end-wall boundary-layer equations using integral methods, considering the boundary layer in the radial direction in the form of a submerged wall jet. Such an approach made it possible to use the laws for the development of wall jets [9], and obtain fairly simple relations for integral parameters, skin friction, mass flow in the boundary layer, and other characteristics. Results are compared with available experimental data and computations of others authors; turbulent flow is considered; results for laminar boundary layer are given in [10].Translated from Zhurnal Prikladnoi Mekhaniki i Tekhnicheskoi Fiziki, No. 5, pp. 117–126, September–October, 1986.     

Interaction between an external compression shock and a blunt body in a hypersonic stream

A complex shock configuration with two triple points can occur during the interaction between an external oblique compression shock and the detached shock ahead of a blunt body (for instance, ahead of a wing or stabilizer edge). This results in the formation of a high-pressure, low-entropy supersonic gas jet [1–6]. Here two flow modes are possible [1], which differ substantially in the intensity of the thermal and dynamic effects of the stream on the blunt body: mode I corresponds to the impact of a supersonic jet [2–6], while the supersonic jet in mode II does not reach the body surface in the domain of shock interaction because of curvature under the effect of a pressure drop. Conditions for the realization of the above-mentioned flow modes are investigated experimentally and theoretically, and an approximate method is proposed to determine the magnitude of the compression shock standoff in the interaction domain. Blunt bodies with plane and cylindrical leading edges are examined. The results of a computation agree satisfactorily with experimental data.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 1, pp. 97–103, January–February, 1976.The author is grateful to V. V. Lunev for discussing the research and for useful remarks.     

Three-wave resonance and averaged equations of the interaction of two waves in media described by a cubic Schrödinger equation

A numerical solution of a cubic Schrödinger equation established the phenomenon of Mach reflection of Stokes waves from a vertical wall [1]. In [2], this phenomenon was interpreted as the resonance interaction of three waves. This was based on the derivation of averaged equations of the interaction of two waves, the region of interaction of the incident and reflected waves being treated as a self-similar solution of these equations. The present paper establishes the possibility of describing these solutions by the relations of three-wave resonance; the mathematical significance of the resonance as splitting into two waves is revealed; and the properties of the averaged system are investigated.Translated from Izvestiya Rossiiskoi Akademii Nauk, Mekhanika Zhidkosti i Gaza, No. 1, pp. 107–116, January–February, 1992.I thank A. G. Kulikovskii and A. A. Barmin for helpful discussions and also colleagues at the Mechanics Section of the V. A. Steklov Mathematics Institute of the USSR Akademy of Sciences for critical comments that stimulated interest in the paper.