Calculation of the stability of the flow in a circular pipe with injection

An experimental investigation of the transition of a laminar flow regime into a turbulent one has been carried out in [1] for a flow in a circular pipe which is organized due to injection through the porous lateral surface with a jammed leading end of the pipe. It was established as a result that injection leads to an increase in stability of the laminar flow regime and increases the Reynolds number of the transition to 10,000 instead of the value 2300 which is characteristic of flow in a circular pipe with impenetrable walls. A similar effect was discovered in [2], in which it was also obtained that the Reynolds number of stability loss under the action of injection can take values significantly larger than in pipes with impenetrable walls. The phenomenon of relaminarization of a turbulent flow in the initial section of a circular pipe under the action of injection has been experimentally detected at the entrance for relatively low Reynolds numbers in [3, 4]. Theoretical investigations of stability of flow with injection have been performed only for a plane channel [5, 6]. A calculation is made in this paper of the stability of a hydrodynamically developed flow in a circular pipe with injection through a porous lateral surface.Translated from Zhurnal Prikladnoi Mekhaniki i Tekhnicheskoi Fiziki, No. 3, pp. 82–86, May–June, 1984.     

Shear flow over a porous particle

Linear axisymmetric Stokes flow over a porous spherical particle is investigated. An exact analytic solution for the fluid velocity components and the pressure inside and outside the porous particle is obtained. The solution is generalized to include the cases of arbitrary three-dimensional linear shear flow as well as translational-shear Stokes flow. As the permeability of the particle tends to zero, the solutions obtained go over into the corresponding solutions for an impermeable particle. The problem of translational Stokes flow around a spherical drop (in the limit a gas bubble or an impermeable sphere) was considered, for example, in [1,2]. A solution of the problem of translational Stokes flow over a porous spherical particle was given in [3]. Linear shear-strain Stokes flow over a spherical drop was investigated in [2].Translated from Izvestiya Rossiiskoi Akademii Nauk, Mekhanika Zhidkosti i Gaza, No. 3, pp. 113–120, May–June, 1995.     

Existence and nonuniqueness of local separation zones in viscous jets

The plane steady problem of the flow of a viscous wall jet past a smoothed break in the contour of a body is considered. For convenience, the flow in the neighborhood of the junction between two flat plates inclined at an angle to each other is chosen for study. As a result of the small extent of the region investigated the flow field is divided into two layers: the main part of the jet, which undergoes inviscid rotation, and a thin sublayer at the wall, which ensures the satisfaction of the no-slip condition. Particular interest attaches to the flow regime in which the solution in the sublayer satisfies the Prandtl boundary layer equations with a given pressure gradient. A similar problem was studied in [1–4]. The present case is distinguished by the structure of the free interaction region in a small neighborhood of the point of zero surface friction stress. By means of the method of matched asymptotic expansions, applied to the analysis of the Navier-Stokes equations, it is established that the interaction mechanism is that described in [5–7]. As a result, an integrodifferential equation describing the behavior of the surface friction stress function is obtained. A numerical solution of this equation is presented. The range of plate angles on which solutions of the equation obtained exist and, therefore, flows of this general type are realized is determined. The essential nonuniqueness of the possible solutions is established, and in particular attention is drawn to the possible existence of six permissible friction distributions.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 1, pp. 38–45, January–February, 1986.The author wishes to thank V. V. Sychev and A. I. Ruban for their useful advice and discussion of the results.     

Three-dimensional laminar boundary layer on a permeable surface in the neighborhood of a symmetry plane

Analytical and numerical methods are used to investigate a three-dimensional laminar boundary layer near symmetry planes of blunt bodies in supersonic gas flows. In the first approximation of an integral method of successive approximation an analytic solution to the problem is obtained that is valid for an impermeable surface, for small values of the blowing parameter, and arbitrary values of the suction parameter. An asymptotic solution is obtained for large values of the blowing or suction parameters in the case when the velocity vector of the blown gas makes an acute angle with the velocity vector of the external flow on the surface of the body. Some results are given of the numerical solution of the problem for bodies of different shapes and a wide range of angles of attack and blowing and suction parameters. The analytic and numerical solutions are compared and the region of applicability of the analytic expressions is estimated. On the basis of the solutions obtained in the present work and that of other authors, a formula is proposed for calculating the heat fluxes to a perfectly catalytic surface at a symmetry plane of blunt bodies in a supersonic flow of dissociated and ionized air at different angles of attack. Flow near symmetry planes on an impermeable surface or for weak blowing was considered earlier in the framework of the theory of a laminar boundary layer in [1–4]. An asymptotic solution to the equations of a three-dimensional boundary layer in the case of strong normal blowing or suction is given in [5, 6].Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 5, pp. 37–48, September–October, 1980.     

Investigation of hypersonic flow of rarefied gas past wings of infinite span

In the framework of the locally self-similar approximation of the Navier-Stokes equations an investigation is made of the flow of homogeneous gas in a hypersonic viscous shock layer, including the transition region through the shock wave, on wings of infinite span with rounded leading edge. The neighborhood of the stagnation line is considered. The boundary conditions, which take into account blowing or suction of gas, are specified on the surface of the body and in the undisturbed flow. A method of numerical solution of the problem proposed by Gershbein and Kolesnikov [1] and generalized to the case of flow past wings at different angles of slip is used. A solution to the problem is found in a wide range of variation of the Reynolds numbers, the blowing (suction) parameter, and the angle of slip. Flow past wings with rounded leading edge at different angles of slip has been investigated earlier only in the framework of the boundary layer equations (see, for example, [2], which gives a brief review of early studies) or a hypersonic viscous shock layer [3].Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 3, pp. 150–154, May–June, 1984.     

Numerical investigation of axisymmetric flows in the wake of blunt bodies in a supersonic stream of viscous gas

The axisymmetric flow in the near wake of spherically blunted cones exposed to a supersonic stream of viscous perfect heat-conducting gas is numerically investigated on the basis of the complete Navier-Stokes equations. The free-stream Mach numbers considered M = 2.3 and 4 were such that the gas can be assumed to be perfect, and the Reynolds numbers such that for these Mach numbers the flow in the wake is laminar but close to laminar-turbulent transition [1–4]. The flow structure in the near wake is described in detail and the effect of the Mach and Reynolds numbers on the base pressure, the total drag and the wake geometry is investigated. The results of calculating the flow in the wake of spherically blunted cones are compared with the experimental data [4].Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 4, pp. 42–47, July–August, 1988.     

Asymptotic solutions for the asymmetric flow in a channel with porous retractable walls under a transverse magnetic field

The self-similarity solutions of the Navier-Stokes equations are constructed for an incompressible laminar flow through a uniformly porous channel with retractable walls under a transverse magnetic field. The flow is driven by the expanding or contracting walls with different permeability. The velocities of the asymmetric flow at the upper and lower walls are different in not only the magnitude but also the direction. The asymptotic solutions are well constructed with the method of boundary layer correction in two cases with large Reynolds numbers, i.e., both walls of the channel are with suction, and one of the walls is with injection while the other one is with suction. For small Reynolds number cases, the double perturbation method is used to construct the asymptotic solution. All the asymptotic results are finally verified by numerical results.     

Laminar boundary layer on swept-back wings of infinite span at an angle of attack

A study is made of the flow of a compressible gas in a laminar boundary layer on swept-back wings of infinite span in a supersonic gas flow at different angles of attack. The surface is assumed to be either impermeable or that gas is blown or sucked through it. For this flow and an axisymmetric flow an analytic solution to the problem is obtained in the first approximation of an integral method of successive approximation. For large values of the blowing or suction parameters, asymptotic solutions are found for the boundary layer equations. Some results of numerical solution of the problem obtained by the finite-difference method are given for wings of various shapes in a wide range of angles characterizing the amount by which the wings are swept back and also the blowing or suction parameters. A numerical solution is obtained for the equations of the three-dimensional mixing layer formed in the case of strong blowing of gas from the surface of the body. The analytic and numerical solutions are compared and the regions of applicability of the analytic expressions are estimated. On the basis of the solutions obtained in the present paper and studies of other authors a formula is proposed for the calculation of the heat fluxes to a perfectly catalytic surface of swept-back wings in a supersonic flow of dissociated and ionized air at different angles of attack. Flow over swept-back wings at zero angle of attack has been considered earlier (see, for example, [1–4]) in the theory of a laminar boundary layer. In [5], a study was made of flow over swept-back wings at nonzero angle of attack at small and moderate Reynolds numbers in the framework of the theory of a hypersonic viscous shock layer.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 3, pp. 27–39, May–June, 1980.We thank G. A. Tirskii for a helpful discussion of the results.     

Asymptotic theory of flow attachment for a viscous incompressible fluid

A study is made of the two-dimensional laminar flow of a viscous incompressible fluid in the neighborhood of the point of attachment of the flow to a solid surface. The case of large Reynolds numbers is considered. It is assumed that the dimensions of the separation region are of the same order of magnitude as the characteristic dimension of the body around which the flow takes place. The asymptotic theory of such flow is constructed by applying the method of matched asymptotic expansions to the analysis of the Navier-Stokes equations. It is shown that in the neighborhood of the attachment point the flow is locally inviscid and can be described by the complete system of Euler equations. A solution to the corresponding boundary-value problem is constructed numerically.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 6, pp. 63–71, November–December, 1980.     

Perturbation solutions for asymmetric laminar flow in porous channel with expanding and contracting walls

The cases of large Reynolds number and small expansion ratio for the asym- metric laminar flow through a two-dimensional porous expanding channel are considered. The Navier-Stokes equations are reduced to a nonlinear fourth-order ordinary differential equation by introducing a time and space similar transformation. A singular perturbation method is used for the large suction Reynolds case to obtain an asymptotic solution by matching outer and inner solutions. For the case of small expansion ratios, we are able to obtain asymptotic solutions by double parameter expansion in either a small Reynolds number or a small asymmetric parameter. The asymptotic solutions indicate that the Reynolds number and expansion ratio play an important role in the flow behavior. Nu- merical methods are also designed to confirm the correctness of the present asymptotic solutions.     

Boundary layer and heat transfer in a two-phase gas-evaporating droplet medium

An asymptotic model of the flow in the laminar boundary layer of a gas-evaporating droplet mixture is constructed within the framework of the two-continuum approximation. The case of evaporation of the droplets into an atmosphere of their own vapor is examined in detail with reference to the example of longitudinal flow over a hot flat plate. Numerical and asymptotic solutions of the boundary layer equations constructed are found for a number of limiting situations (low droplet concentration, no droplet deposition, significant droplet deposition). The development of the flow with respect to the longitudinal coordinate is studied and it is shown that in the absence of droplet deposition a region of pure vapor may be formed near the surface. Similarity criteria are established and the mechanism of surface heat transfer enhancement is studied for a low evaporating droplet concentration in the boundary layer. In the inertial deposition regime the results of calculating the integral heat transfer coefficient are found to correspond with the experimental data [1].Translated from Izvestiya Rossiiskoi Akademii Nauk, Mekhanika Zhidkosti i Gaza, No.3, pp. 42–50, May–June, 1992.     

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.     

The laminar boundary layer near a body corner point

A method is developed for calculating the characteristics of a laminar boundary layer near a body contour corner point, in the vicinity of which the outer supersonic stream passes through a rarefaction flow. In the study we use the asymptotic solution of the Navier-Stokes equations in the region with large longitudinal gradients of the flow functions for large values of the Reynolds number, the general form of which was used in [1].The pressure, heat flux, and friction distributions along the body surface are obtained. For small pressure differentials near the corner the solution of the corresponding equations for small disturbances is obtained in analytic form.The conventional method for studying viscous gas flow near body surfaces for large values of the Reynolds number is the use of the Prandtl boundary layer theory. Far from the body the asymptotic solution of the Navier-Stokes equations in the first approximation reduces to the solution of the Euler equations, while near the body it reduces to the solution of the Prandtl boundary layer equations. The characteristic feature of the boundary layer region is the small variation of the flow functions in the longitudinal direction in comparison with their variation in the transverse direction. However, in many cases this condition is violated.The necessity arises for constructing additional asymptotic expansions for the region in which the longitudinal and transverse variations of the flow functions are quantities of the same order. The general method for constructing asymptotic solutions for such flows with the use of the known method of outer and inner expansions is presented in [1].In the following we consider the flow in a laminar boundary layer for the case of a viscous supersonic gas stream in the vicinity of a body corner point. Behind the corner the flow separates from the body surface and flows around a stagnant zone, in which the pressure differs by a specified amount from the pressure in the undisturbed flow ahead of the point of separation. A pressure (rarefaction) disturbance propagates in the subsonic portion of the boundary layer upstream for a distance which in order of magnitude is equal to several boundary layer thicknesses. In the disturbed region of the boundary layer the longitudinal and transverse pressure and velocity disturbances are quantities of the same order. In this study we construct additional asymptotic expansions in the first approximation and calculate the distributions of the pressure, friction stress, and thermal flux along the body surface.     

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.     

Solution of Navier-Stokes equations near the trailing edge of a plate

The appearance of a local singularity in the solutions for the neighborhood of the trailing edge of a plate in a sub- or supersonic flow makes it necessary to consider the flow in the local region which is described in the first approximation by the Navier-Stokes equations for an incompressible gas. In this paper numerical solutions are obtained for such a region for both a thin plate and a plate with thickness. The streamline patterns and the distributions of the flow functions over the surface of the plate and in the wake behind it are presented. For the plate with finite thickness, a numerical solution is obtained in a wide range of variation of the local Reynolds number.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 4, pp. 173–176, July–August, 1985.     

Calculation of viscous compressible gas flow past a corner

We consider the problem of calculating the parameters for supersonic viscous compressible gas flow past a corner (angle greater than ). The complete system of Navier-Stokes equations for the viscous compressible gas is solved in the small vicinity Q₁. (characteristic dimensionl~1/R) of the corner point. The conditions for smooth matching of the solution of the Navier-Stokes equations and the solution of the ideal gas or boundary layer equations are specified on the boundary of Q₁. All these solutions are a priori unknown, and the conditions for smooth matching reduce to certain differential equations on the boundary of Q₁. Here account is taken of the interaction of the flows near the wall surface and in the so-called outer region [1].We note that no a priori assumptions are made in Q₁ concerning the qualitative behavior of the solution, in contrast with other studies on viscous flow past a corner (for example, [2–4]).The Navier-Stokes system in Q₁ is solved numerically, using the difference scheme suggested in [5]. This scheme permits obtaining the steady-state solution by the asymptotic method for large Reynolds numbers R, and also has an approximation accuracy adequate to account for the effects of low viscosity and thermal conductivity.     

Bifurcation of developed flow in rectangular channels rotating about the transverse axis

The linear problem of the stability of Poiseuille flow between two infinite plates rotating about an axis parallel to the plates and normal to the flow direction was studied in [1, 2]. It was established that the flow is least stable with respect to disturbances in the form of standing waves known as Taylor eddies. The experimental data of [1, 3] and the results [4] of a numerical integration of the Navier-Stokes equations for channels with cross sections highly elongated in the direction of the axis of rotation are in good agreement with the conclusions of the linear theory. In the case of channels with a side ratio of the order of unity, which is of greater practical interest, the primary flow becomes essentially three-dimensional and evolves with variation of the governing criteria: the Reynolds and Rossby numbers. Obviously, this seriously complicates the use of the methods of the linear theory. The effect of the ratio of the sides of the cross section on the stability of the primary flow regime was studied experimentally in [3]. The present article describes the results of an investigation of the problem based on a numerical method of integrating the nonlinear Navier-Stokes equations. Moreover, an asymptotic estimate of the stability limit of the primary regime, based on a local condition of inviscid instability of rotating flows, is presented.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 5, pp. 27–32, September–October, 1985.     

Analysis of flow in an annular duct with large injection at the walls

Viscous heat-conducting compressible fluid flow in an annular duct formed by two coaxial cylinders with large injection at the walls is investigated. An asymptotic solution exhibiting the influence of the axial symmetry of the duct is obtained in the vicinity of the y axis and is compared with the results of exact numerical calculations. Asymptotic solutions of the Navier-Stokes equations have been obtained earlier for flows in a plane channel with various rates of wall injection in the case of an incompressible gas [1, 2] and a compressible gas [3].Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 3, pp. 135–139, May–June, 1976.     

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.     

Flow stability in a channel with porous walls

We study the stability of the flow which forms in a plane channel with influx of an incompressible viscous fluid through its porous parallel walls. Under certain assumptions the study of the stability reduces to the solution of modified Orr-Sommerfeld equation accounting for the transverse component of the main-flow velocity. As a result of numerical integration of this equation we find the dependence of the local critical Reynolds number on the blowing Reynolds number R₀, which may be defined by two factors: the variation of the longitudinal velocity profile with R₀ and the presence of the transverse velocity component. A qualitative comparison is made of the computational results with experimental data on transition from laminar to turbulent flow regimes in channels with porous walls, which confirms that it is necessary to take into account the effect of the transverse component of the main-flow velocity on the main-flow stability in the problem in question.Flows in channels with porous walls are of interest for hydrodynamic stability theory in view of the fact that they can be described by the exact solutions of the Navier-Stokes equations by analogy with the known Poiseuille and Couette flows. However, in contrast with the latter, the flows in channels with porous walls (studies in [1], for example) will be nonparallel.The theory of hydrodynamic stability of parallel flows has frequently been applied to nonparallel flows (in the boundary layer, for example). In so doing the nonparallel nature of the flow has been taken into account only by varying the longitudinal velocity component profiles. A study was made in [2, 3] of the effect of the transverse component of the main flow on its stability. In the case of the boundary layer in a compressible gas, a considerable influence of the transverse velocity component on the critical Reynolds number was found in [2] and confirmed experimentally. A strong influence of the transverse velocity component on the instability region was also found in [3] in a study of the flow stability in a boundary layer with suction for an incompressible fluid.