Asymptotic solution of the three-dimensional multicomponent laminar boundary-layer equations with large suction

The flow of a multicomponent compressible gas in a three-dimensional laminar boundary layer is investigated for large values of the suction parameter. Asymptotic expressions are derived for the profiles of the velocities, temperatures, and concentration of the components across the boundary layer, as well as for the friction, heat-, and mass-transfer coefficients on the surface of the body.The authors wish to thank G. A. Tirskii for a discussion.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 4, pp. 41–45, July–August, 1975.     

Asymptotic solution of incompressible boundary layer equations with negative pressure gradient and large injection rates

During entry of bodies into the Earth's atmosphere with high velocities, the mass removal from the body surface as a result of the large convective and primarily radiation fluxes may become arbitrarily large, i. e., the injection rate into the boundary layer may approach infinity. The present article presents a solution of the Prandtl equations for the incompressible boundary layer with negative pressure gradient (dp/dx<0) for large injection rates. The existence of a solution of the boundary layer equations with arbitrary injection rate under the condition dp/dx<0 was shown in Oleinik's work [1].The asymptotic solution obtained agrees with the exact numerical solution for those values of the injection rate for which the boundary layer approximation still remains valid. An analogous solution for the self-similar equations in the vicinity of the stagnation point was previously obtained in [2]. The use of the asymptotic solution makes it possible to find an expression for the friction coefficient which is convenient for concrete calculations in the case of arbitrary negative pressure gradients.In conclusion the author wishes to thank G. A. Tirskii for guidance in the work and I. Gershbein for permitting the use of the numerical solution.     

Three-dimensional bending gravitational oscillations near moving pressure regions

Translated from Zhurnal Prikladnoi Mekhaniki i Tekhnicheskoi Fiziki, No. 3, pp. 158–166, May–June, 1989.     

Asymptotic solutions of the Navier-Stokes equations in regions with large local perturbations

There are many problems of the dynamics of viscous flows of liquids and gases at high Reynolds numbers for the solution of which the classical theory of the boundary layer cannot be used. This applies, in particular, to all the problems with various sorts of local singularities in the stream-flows in the vicinity of corners, in regions of interaction of the boundary layer with an incident shock, flows near points of separation or attachment of the stream, etc. The purpose of the present paper is to attempt the theoretical investigation of problems of this type on the basis of the general analysis of the asymptotic behavior of the solutions of the Navier-Stokes equations. In order to do this, use is made of the familiar method of the construction and splicing of a combination of asymptotic expansions representing the solutions in the various characteristic regions of the stream with viscosity decreasing without bound [1].As an example, detailed consideration is given to the problem of viscous supersonic flow near a wall with large local curvature of the surface.     

Discontinuous solutions of boundary layer equations with a positive pressure gradient

Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 5, pp. 53–62, September–October, 1991.     

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.     

A method of computing skin friction and adiabatic wall temperature in a laminar boundary-layer without pressure gradient

Meccanica - A method of computing the shear stress in a laminar boundary-layer without pressure gradient is described. The non linear term in the Von Mises' equation is replaced by a suitable...     

Three-dimensional boundary-layer transition on a cone at Mach 3.5

A boundary-layer transition study on a sharp, 5° half-angle cone at various angles of attack was conducted at Mach 3.5. Transition data were obtained with and without significantly reduced freestream acoustic disturbance levels. A progressive downstream and upstream motion of the transition front on the windward and leeward rays, respectively, of the cone with angle of attack was observed for the high noise level data in agreement with data trends obtained in conventional (noisy) wind tunnels. However, the downstream movement was not observed to the same degree for the low noise level data in the present study. Transition believed to be crossflow dominated was found to be less receptive to freestream acoustic disturbances than first-mode (Tollmien-Schlichting) dominated transition. The previously-developed crossflow transition Reynolds number criterion, _tr,max 200, was found to be inadequate for the current case. An improved criterion is offered, which includes compressibility and flow-geometry effects.     

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.     

Self-similar solutions of three-dimensional laminar magnetohydrodynamic boundary-layer equations

Self-similar solutions of three-dimensional boundary-layer equations of an incompressible fluid in ordinary hydrodynamics were considered in [1–3] et al. The present work looks for self-similar solutions of three-dimensional magnetohydrodynamic boundary-layer equations.Translated from Zhurnal Prikladnoi Mekhaniki i Tekhnicheskoi Fiziki, Vol. 9, No. 4, pp. 10–17, July–August, 1968.     

Singular solution of boundary-layer equations on a moving surface

We consider a plane steady flow of incompressible fluid in the boundary layer which develops on a surface moving downstream. We obtain a singular solution of Prandtl's equations, continuously extendable through the simultaneous vanishing point of the friction and the longitudinal velocity vector component. Such a solution is realized, in particular, in a flow past a rotating circular cylinder.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 2, pp. 43–52, March–April, 1987.The author thanks A. I. Ruban for his great interest in the work and his useful remarks.     

Calculation of separated flows by means of boundary-layer equations

Translated from Zhurnal Prikladnoi Mekhaniki i Tekhnicheskoi Fiziki, No. 6, pp. 46–54, November–December, 1990.     

Discontinuous element pressure gradient stabilizations for compressible Navier-Stokes equations based on local projections

A pressure gradient discontinuous finite element formulation for the compressible Navier-Stokes equations is derived based on local projections. The resulting finite element formulation is stable and uniquely solvable without requiring a B-B stability condition. An error estimate is Obtained.     

Uniqueness of the self-similar solutions of three-dimensional laminar boundary-layer equations

The equations of the three-dimensional laminar boundary layer on lines of flow outflow and inflow are studied for conical outer flow under the assumption that the Prandtl number and the productρμ are constant. It is shown that in the case of a positive velocity gradient of the secondary flow (α₁>0) the additional conditions which result from the physical flow pattern determine a unique solution of the system of boundary-layer equations. For a negative velocity gradient of the secondary flow (α₁≤0) these conditions are satisfied by two solutions. An approximate solution is obtained for the boundary layer equations which is in rather good agreement with the numerical integration results. Compressible gas flow in a three-dimensional laminar boundary layer is described by a system of nonlinear differential equations whose solution is not unique for given boundary conditions. Therefore additional conditions resulting from the physical pattern of the gas flow are imposed on the resulting solution. In the solution of problems with a negative pressure gradient these additional conditions are sufficient for a unique selection of the solution of the boundary-layer equations. However, in the case of a positive pressure gradient the solution of the boundary-layer equations satisfying the boundary and additional conditions may not be unique. In particular, in [1] in a study of a three-dimensional laminar boundary layer in the vicinity of the stagnation point it was shown that for $$c = {{\frac{{\partial v_e }}{{\partial y}}} \mathord{\left/ {\vphantom {{\frac{{\partial v_e }}{{\partial y}}} {\frac{{\partial u_e }}{{\partial x}}}}} \right. \kern-\nulldelimiterspace} {\frac{{\partial u_e }}{{\partial x}}}} > 0$$ the solution is unique, while for c<0 there are two solutions. In the present paper we study the question of the uniqueness of the self-similar solution of the three-dimensional laminar boundary-layer equations on lines of flow outflow and inflow for a conical outer flow.     

Boundary-value problem for a degenerate system of parabolic equations in boundary-layer theory

A problem is considered for the system describing gas flows with plate boundary layer separation in Mises variables in boundary-layer theory. The existence of generalized solutions of the problem is proved. __________ Translated from Prikladnaya Mekhanika i Tekhnicheskaya Fizika, Vol. 49, No. 4, pp. 36–41, July–August, 2008.