On asymptotic theory of separated flows past bodies

Symmetric two-dimensional steady flow past a body in a homogeneous incompressible fluid stream at high Reynolds numbers is considered. A slow motion in the reverse flow zone is investigated and the solution for the flow in the external region is obtained in the second approximation. Additional considerations of the fact that the flow in the closure region of the separation zone and in the wake behind this zone is turbulent are presented. The laminar-turbulent transition in the mixing layer is analyzed and an analogy between this process and the propagation of perturbations upstream of the boundary layer interaction regions is revealed.     

Analysis of the transition of one-dimensional flows of a viscous heat conducting gas to asymptotic regimes

Asymptotic solutions are presented which describe the formation of one-dimensional flows of a viscous heat conducting gas at short and long times. Comparison of them with numerical calculations made in the study for the system of Navier-Stokes equations in conjunction with an estimate of the ratio of each term in the asymptotic expansion to the preceding term leads to dimensionless-number dependences that determine the limits of applicability of the obtained solutions.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 5, pp. 110–118, September–October, 1981.     

Instability and nonuniqueness of quasisteady flows of a viscoelastic liquid

Within the framework of the 4-constant Oldroyd model unidirectional steady-state and quasisteady flow of a viscoelastic liquid is considered. A new explanation, not based on the assumption of slip, is given for the well-known effect of a sharp increase in the fluid flow rates at shear velocities higher than a certain critical value. It is shown, analytically and numerically, that the phenomenon of hysteresis is closely related to this effect. The relation between the problem of stability of the flow in question and the problems of nonuniqueness of the steady-state solution is also noted. A modification of the standard Oldroyd model is proposed and the corresponding integral conditions removing the singularities related with the nonexistence and nonuniqueness of the steady-state solution are determined.Moscow. Translated from Izvestiya Rossiiskoi Akademii Nauk, Mekhanika Zhidkosti i Gaza, No. 6, pp. 29–39, November–December, 1996.     

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.     

Instability of long-wave viscous flows

Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 4, pp. 30–35, Jul–August, 1990.     

Calculation of nonstationary viscous flows

The present study will consider the problem of formation of a nonstationary flow of a viscous, thermally conductive gas around a semiinfinite plate, with the gas being set in motion instantaneously from a state of rest (problem I) and behind a shock wave incident upon the gas (problem H). A numerical method will be used, based on integration of the complete system of Navier-Stokes equations. Special attention is given in the solution to these times when the simplifying assumptions of boundary layer theory are invalid.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 6, pp. 167–171, November–December, 1978.The authors thank Yu. A. Dem'yanov for evaluating the results of the study.     

On local solutions to non-Newtonian slow viscous flows

The eigenfunction expansion method is used to obtain local solutions to some non-Newtonian slow viscous flows. The forms of viscosity variation amenable to such analysis are restricted but do include power-law fluids. Power-law flow near a sharp corner between plane boundaries is analysed and results are obtained for the critical corner angle for eddy formation. Flows near a 90° corner with either a moving boundary or a finite flow rate at the corner are also considered. The “stick-slip” behaviour of a power-law fluid at a plane solid boundary is shown to obey a simple law.     

Two-dimensional compressible viscous sink flows

A theoretical investigation of two-dimensional viscous compressible sink flow is presented. An analytic solution in parametric form is obtained for specific values of the ratio of specific heats. For the more general case, a number of asymptotic solutions were also obtained for different flow regimes including hypersonic, transonic, acoustic and low Reynolds number flow.     

Stability of periodic viscous gas flows

Translated from Izvestiya Rossiiskoi Akademii Nauk, Mekhanika Zhidkosti i Gaza, No. 1, pp. 10–16, January–February, 1992.     

Measurement of the viscous sublayer in near-separated flows

This paper investigates experimentally the development of the viscous sublayer in a two-dimensional incompressible turbulent wall boundary layer under severe pressure gradients. The wall was also moderately heated and the influence of heat transfer on the development of the viscous sublayer was included. A semi-empirical equation for the thickness of the viscous sublayer: $$\delta _s = 11.5\frac{\nu }{{U_\tau }}(0.61\frac{{T_\omega }}{{T_s }})^{1/2}$$ was derived, which holds everywhere except closely near the separation point of the boundary layer. The measurements were made on a flat plate in a test section 1.7 m long and 0.8 m wide. The height and shape of the top surface of the test section could be varied, and thus it was possible to imposed different pressure gradients on the flow. Specially designed fine probes facilitated the measurement of the velocity distribution very close to the wall.     

Three-dimensional conical viscous incompressible fluid flows

New exact solutions of the Navier-Stokes equations are obtained for steady-state three-dimensional conical flows. In this class of flows the velocity decreases in inverse proportion to the distance from the source and the input equations reduce to two-dimensional ones. It is shown that in the spherical coordinate system the equations of motion reduce to a single nonlinear equation with respect to a scalar function which depends on the polar angles. The case in which this equation reduces to the integrable Liouville equation is discussed. This makes it possible to obtain a wide class of three-dimensional solutions in analytic form. Perm’. Translated from Izvestiya Rossiiskoi Akademii Nauk, Mekhanika Zhidkosti i Gaza, No. 6, pp. 144–148, November–December, 1998. The work was carried out with financial support from the Russian Foundation for Basic Research (project No. 97-01-00063).     

Local viscous,chemically nonequilibrium flows

A study is made of a chemically nonequilibrium flow in local regions near cold walls. It is found that in this case the chemically reactions can take place only on a catalytic surface, and the gas can be regarded as a binary mixture of atoms and molecules. As an example, a study is made of the aerodynamic heating of a small step on the surface of a plate in the case when the flow past the plate is described in the first approximation when the Reynolds number tends to infinity by the Navier-Stokes equations for an incompressible gas. It is found that the presence of the step increases the drag of this section of the body, for a noncatalytic surface leads to an additional heating of it, and for a catalytic surface weakens its catalytic activity and, therefore, reduces the heating of the step.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 3, pp. 137–144, May–June, 1984.I thank V. N. Gusev for helpful discussion of the results of this paper.     

Some general properties of plane-parallel viscous flows

An expression is obtained for the integral of the equations of plane-parallel viscous incompressible flow expressing conservation of the Bernoulli function along a certain family of lines which, as the viscosity tends to zero, go over into the streamlines. These lines also determine the direction of transfer of the vorticity of the viscous flow.In conclusion the authors wish to thank V. V. Sychev and V. Ya. Neiland for discussion of the results and useful comments.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 3, pp. 176–178, May–June, 1987.     

The asymptotic behavior of solutions to the Kirchhoff equation with a viscous damping term

We study the asymptotic behavior of solutions to an equation describing non-linear vibration of a string with viscosity. In the case when the string is unstretched (the degenerate case), we determine the decay order of solutions by investigating the dynamics near an infinite-dimensional center manifold. Moreover, we classify the asymptotic behavior of all solutions from a dynamical systems point of view. We also deal with the case where the string is stretched (the nondegenerate case).