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.     

Three-dimensional local separation

A solution to the problem of local separation of a three-dimensional boundary layer from an arbitrary smooth surface is constructed. Separation takes place along the limiting streamline at the points of which the component of the surface friction (calculated from the boundary-layer equations) that is orthogonal to this streamline has a break. An asymptotic expansion of the solution of the Navier-Stokes equations that describes the flow field in the separation region is found. The conclusions for the two-dimensional and self-similar theory of local separation are generalized to the three-dimensional case.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 3, pp. 39–47, May–June, 1991.     

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.     

Direction of streamlines near the wall in a three-dimensional turbulent boundary layer

Expressions are derived that, for a large class of flows, enable one to determine the direction of the streamlines near the wall in a three-dimensional turbulent boundary layer without integrating the system of boundary-layer equations. Comparison with available experimental data reveals a completely satisfactory agreement between the calculations and the experimental data.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 2, pp. 19–25, March–April, 1980.     

Laminar boundary layer on a partly moving surface in the presence of blowing or suction

Planar and axisymmetric flows of a multicomponent compressible gas in a laminar boundary layer with nonzero tangential component of the velocity on a permeable surface are considered. The asymptotic solutions of the boundary-layer equations obtained earlier [1–4] for large values of the blowing and suction parameters are generalized to the case when the velocity vector of the blown or extracted gas makes an acute angle with the surface of the body, this angle depending on the longitudinal coordinate. The region of applicability of the asymptotic formulas is estimated on the basis of the results of numerical solution of the boundary-layer equations. The results are given of some calculations of the boundary layer on a partly moving surface.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 5, pp. 28–36, September–October, 1979.We thank G. A. Tirskii and G. G. Chernyi for a helpful discussion of the results.     

Asymptotic solutions of non-self-similar boundary-layer equations

In this paper asymptotic solutions near the outer boundary are found for the non-self-similar laminar boundary-layer equations in an incompressible fluid and a compressible gas. It is shown that the nature of the asymptotic solution is determined by the form of the initial velocity and enthalpy profiles.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 6, pp. 29–34, November–December, 1976.In conclusion, the author is grateful to N. M. Belyanin and F. A. Slobodkina for attention to the research and discussion of the results.     

Boundary-layer flow of Reiner–Philippoff fluids past a stretching wedge

This paper presents a numerical analysis of the steady boundary-layer flow of a Reiner–Philippoff fluid induced by a 90° stretching wedge in a variable free stream. The governing partial differential equations are converted into a set of two ordinary differential equations by the use of a similarity transformation. The flow is therefore governed by a stretching velocity parameter λ and two non-Newtonian fluid parameters γ and μ₀. The variation of the skin friction, as well as other flow characteristics, as a function of the governing parameters is presented graphically and tabulated. A stability analysis has also been performed for this self-similar flow based on linear disturbances to the steady similarity solutions. The results presented in this paper reveal that there are no multiple (dual) solutions for the present problem and the unique solution is stable.     

Theory of stability of compressible elastoplastic ground

Based on the three-dimensional linearized equations of stability, the deformation process of compressible elastoplastic ground is investigated in the case of small subcritical deformations. In the case of a homogeneous subcritical state, the general solutions of the equations of stability are constructed similarly to [1].Translated from Zhurnal Prikladnoi Mekhaniki i Tekhnicheskoi Fiziki, No. 5, pp. 148–153, September–October, 1977.     

Boundary-layer separation on a cooled body and interaction with a hypersonic flow

In previous papers, e.g., [1, 2], boundary-layer separation was investigated by analyzing solutions of the boundary-layer equations with a given external pressure distribution. In general, this kind of solution cannot be continued after the separation point. Study of the asymptotic behavior of solutions of the Navier-Stokes equations [3–5] shows that, in boundarylayer separation in supersonic flow over a smooth surface, the main effect on the flow in the immediate vicinity of the separation point is a large local pressure gradient induced by interaction with the external flow. The solution can be continued beyond the separation point and linked to the solutions in the other regions, located downstream [5]. Similar results for incompressible flow were recently obtained in [6]. We can assume that in general there is always a small region near the separation point in which separation is self-induced, and where the limiting solution of the Navier-Stokes equations does not contain unattainable singular points. However, this limiting slope picture can be more complex and can contain more regions where the behavior of the functions differed from that found in [3–6]. The present paper investigates separation on a body moving at hypersonic speed, where the ratio of the stagnation temperature to the body temperature is large. It is shown that both. for hypersonic and supersonic speeds the flow near the separation point is appreciably affected by the distribution of parameters over the entire unperturbed boundary layer, and not only in a narrow layer near the body, as was true in the flows studied earlier [3–6]. Regions may appear with appreciable transverse pressure drops within the zone occupied by layers of the unperturbed boundary layer. Similarity parameters are given, the boundary problems are formulated, and the results of computer calculation are presented. The concept of subcritical and supercritical boundary layers is refined, and the dependence of pressure coefficients responsible for separation on the temperature factor is established.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 6, pp. 99–109, November–December 1973.     

Generalized variables in boundary-layer theory

Independent variables are widely used in boundary-layer theory to construct efficient methods of solving problems. The Dorodnitsyn variables in Lees' form [1] are the most common and general. This form combines the transformations proposed by Dorodnitsyn [2], Blasius [3], and Mangler-Stepanov [4, 5]. As is well known, transformation of the boundary-layer equations to Dorodnitsyn variables in Lees' form leads to a generalized single system of equations describing plane and axisymmetric gas flows. An analogous generalization of the Mises [6] and Crocco [7] variables is carried out below.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 5, pp. 166–168, September–October, 1976.     

Asymptotic analysis of the equations of a three-dimensional boundary layer

The asymptotic solutions of the self-similar equations of two- and three-dimensional boundary layers have been investigated by many authors (see, for example, [1–3]). In [4, 5], asymptotic solutions were found for non-self-similar equations for two-dimensional flow, and the propagation of perturbations near the external edge of the boundary layer was analyzed. In the present paper, asymptotic solutions are obtained for the non-self-similar equations of a three-dimensional laminar boundary layer of an incompressible fluid. It is shown that the conclusion drawn in [5] — that the boundary conditions can be transferred from infinity to a finite distance from the wall — is also true for three-dimensional flow. The obtained solutions explain the experimentally well-known phenomenon of the conservativeness of the secondary currents. The essence of this phenomenon is that a change in the sign of the transverse (along the normal to a streamline of the external flow) pressure gradient is accompanied by a very rapid change in the direction of the secondary flow near the wall, whereas in the upper layers of the boundary layer the direction remains unchanged for a substantial time.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 5, pp. 155–157, September–October, 1979.     

On automodel numerical and asymptotic solutions of boundary-layer equations for high rates of injection

Automodel solutions of the equations of a laminar, multicomponent, isothermal boundary layer are considered for high rates of injection. The asymptotic velocity profiles and the thickness of the boundary layer are given for various negative pressure gradients (>0), A numerical solution is presented for the boundary-layer equations when injection involves the flow of a gas mixture comprising hydrogen, nitrogen, and carbon dioxide around the surface. The asymptotic solution is compared with the numerical solution, and its ranges of applicability are established.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 3, pp. 49–52, May–June, 1971.     

Determination of the frequencies of the oscillations of a low-viscosity liquid rotating in an axisymmetric vessel

The damping rate and frequencies of the normal modes of a low-viscosity liquid rotating in an axisymmetric vessel are found. The boundary-layer method enables one to express the result in terms of the solutions of the corresponding problem for an ideal liquid.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 2, pp. 128–133, March–April, 1980.     

Boundary layer stability on a wave-periodic surface

Stability under small perturbations is investigated for flows whose velocity depends periodically on the spatial coordinate in the direction of flow. Stability calculations are carried out for the case in which the velocity distribution is a solution of the boundary-layer equations.Novosibirsk. Translated from Izvestiya Akademii Nauk SSSR. Mekhanika Zhidkosti i Gaza, No. 6, pp. 11–16, November–December, 1972.     

Three-Dimensional Analog of Prandtl-Meyer Waves

This paper studies the regular partially invariant solution of the equations of gas dynamics which extends the Prandtl-Meyer solutions to the three-dimensional case. All singular manifolds of the third-order dynamic system that defines the solution are found, and its compactification is constructed.__________Translated from Prikladnaya Mekhanika i Tekhnicheskaya Fizika, Vol. 46, No. 5, pp. 38–45, September–October, 2005.     

Heating and flow of a conducting gas

A model is proposed for the laminar flow of a conducting gas in the channel of a conical plasmatron (plasma source) and in the surrounding motionless medium. The boundary-layer approximation equations are written taking the intrinsic magnetic fields into account. The flow of an argon plasma in argon and air is analyzed numerically.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 5, pp. 190–193, September–October, 1975.     

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.     

Complete model of the process of propagation of long waves and their interaction with a vertical wall

Asymptotic analysis of the nonlinear three-dimensional boundary-value problem of potential theory is carried out and a complete system of equations describing the process of propagation of long surface waves is obtained. Approximate solutions of the problems for both traveling and standing waves are constructed to the third approximation.Translated by Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 3, pp. 173–176, May–June, 1985.     

Three-dimensional steady flows of stratified fluid and internal waves

A study is made of three-dimensional steady flows of an ideal heavy incompressible fluid stratified in each layer over a flat or asymptotically flat base. Mixed Euler-Lagrange variables are chosen in which surfaces of constant density, including the layer division boundaries, become flat and parallel to the plane of the base. The original problem is reduced to a nonlinear boundary-value problem for a system of three quasilinear equations in a plane layer. This system of equations is used to construct an asymptotic theory of long waves in the three-dimensional case, which has particular solutions in the first approximation in the form of solitons and soliton systems.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 3, pp. 127–132, May–June, 1985.     

Series solution of the temperature distribution in the Falkner–Skan wedge flow by the homotopy analysis method

A new analytical method, namely the homotopy analysis method (HAM), has been applied to investigate the temperature field associated with the Falkner–Skan boundary-layer problem, and a series solution is provided in this paper. The results of the present work show agreement with those of numerical solutions in a large range of Prandtl numbers (0 < Pr ≤ 100), which demonstrates the validity of the present analysis.