Determination of the magnitude of the separation criterion for a three-dimensional laminar boundary layer

A separation criterion, i.e., a definite relationship between the external flow and the boundary layer parameters [1], can be used to estimate the possibility of the origination of separation of a two-dimensional boundary layer. A functional form of the separation criterion has also been obtained for a three-dimensional boundary layer [2] on the basis of dimensional analysis. As in the case of the two-dimensional boundary layer, locally self-similar solutions can be used to determine the specific magnitude of the separation criterion as a function of the values of the governing parameters. Locally self-similar solutions of the two-dimensional laminar boundary-layer equations have been found at the separation point for a perfect gas with a linear dependence of the coefficient of viscosity on the temperature (Ω=1) and Prandtl number P=1 [3, 4]. The influence of blowing and suction has been studied for this case [5]. Self-similar solutions have been obtained for Ω=1, P=0.723 for the limit case of hypersonic perfect gas flow [6]. Locally self-similar solutions of the three-dimensional laminar boundary-layer equations at the separation point are presented in [7] for a perfect gas with Ω=1, P=1. There are no such computations for Ω≠1, P≠1; however, the results of computing several examples for a two-dimensional flow [8] show that the influence of the real properties of a gas can be significant and should be taken into account. Self-similar solutions of the two- and three-dimensional boundary-layer equations at the separation point are found in this paper for a perfect gas with a power-law dependence of the viscosity coefficient on the enthalpy (Ω=0.5, 0.75, 1.0) for different values of the Prandtl number (P=0.5, 0.7, 1.0) in a broad range of variation of the external stream velocity (v ₁ ² /2h₁* = 0–0.99) and the temperature of the streamlined surface. Magnitudes of the separation criterion for a laminar boundary layer have been obtained on the basis of these data.     

Influence of boundary layer on unsteady aerodynamic characteristics of blunt cones in a supersonic flow

A study is made of the influence of the boundary layer on the unsteady aerodynamic characteristics of blunt cones oscillating in a supersonic gas stream about zero angle of attack. A solution to the problem is constructed in the framework of the linear theory of bodies of finite thickness. Such an approach has been used [1–3] in the case of the equations of motion of an ideal gas to calculate the unsteady aerodynamic characteristics of sharp and blunt bodies of various configurations. The influence on these characteristics of the viscosity effects due to the presence on the surface of the body of a laminar boundary layer has been taken into account [4–6] for bodies of the simplest shapes (wedge, cone). The present paper considers the unsteady aerodynamic characteristics of cones and investigates the influence of rounding of the tips and laminar and turbulent flow regimes in the boundary layer.     

Two-phase couette flow

A study is made of plane laminar Couette flow, in which foreign particles are injected through the upper boundary. The effect of the particles on friction and heat transfer is analyzed on the basis of the equations of two-fluid theory. A two-phase boundary layer on a plate has been considered in [1, 2] with the effect of the particles on the gas flow field neglected. A solution has been obtained in [3] for a laminar boundary layer on a plate with allowance for the dynamic and thermal effects of the particles on the gas parameters. There are also solutions for the case of the impulsive motion of a plate in a two-phase medium [4–6], and local rotation of the particles is taken into account in [5, 6]. The simplest model accounting for the effect of the particles on friction and heat transfer for the general case, when the particles are not in equilibrium with the gas at the outer edge of the boundary layer, is Couette flow. This type of flow with particle injection and a fixed surface has been considered in [7] under the assumptions of constant gas viscosity and the simplest drag and heat-transfer law. A solution for an accelerated Couette flow without particle injection and with a wall has been obtained in [6]. In the present paper fairly general assumptions are used to obtain a numerical solution of the problem of two-phase Couette flow with particle injection, and simple formulas useful for estimating the effect of the particles on friction and heat transfer are also obtained.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 3, pp. 42–46, May–June, 1976.     

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.     

A multi-dimensional time-marching method of characteristics for viscous and heat-conducting flows

Summary A numerical scheme is presented which employs the characteristic surfaces in space-time for solving Navier-Stokes equations for compressible fluid flow. We consider the general case of a three-dimensional flow, a simplification of which yields the equations of the two-dimensional case. Emphasis is put on the method itself. We apply it to simulate a laminar hypersonic flow around a circular cylinder of a five-components gas mixture of nitrogen and oxygen with thermally perfect constituents and at chemical nonequilibrium. First, the partial differential equations are transformed into a standard form with directional derivatives, enabling to attain the compatibility conditions, including the viscosity terms. These conditions are discretized by approximating their integrals along the corresponding characteristic surfaces. The result is an explicit time-marching numerical scheme. Using a grid fitted between the shock and the cylinder, and starting from roughly estimated initial conditions, a steady solution is searched. A comparison is made with the solution obtained under the assumption of a perfect gas. Received 6 April 1999; accepted for publication 13 May 1999     

Development of magnetohydrodynamic boundary layers associated with the sudden initiation or deceleration of a supersonic flow at the boundary of a half-space

The problem of nonstationary magnetohydrodynamic flow of a viscous fluid in a half-space resulting from the motion of an infinite plate has received much attention. In [1], for example, solutions are presented for the case of isotropic conductivity, while in [2] a solution of the Rayleigh problem is offered for the case of anisotropic conductivity. In these instances the fluid was assumed incompressible and uniform, and the system of equations was found to be linear. In problems involving nonstationary flow of a gas in a transverse magnetic field resulting from the deceleration of a high-velocity gas flow at the boundary of a half-space or the motion of an infinite plate at supersonic speed relative to a stationary gas it becomes necessary to take into account the compressibility of the gas and the temperature dependence of the conductivity. It is then possible to have flows in which the gas becomes electrically conducting and begins to interact with the magnetic field solely as a result of the increase in temperature due to viscous dissipation of energy. The magnetic field, interacting with the conducting gas, exerts an effect on the drag and heat transfer to the surface of the plate. At sufficiently low gas pressures and strong magnetic fields a Hall effect may be observed. The system of equations describing the motion of a compressible gas with variable conductivity is essentially nonlinear. The present article is devoted to a study of such motions.     

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.     

Boundary layer development in a gas flow with stagnation enthalpy varying across the streamlines

We consider a laminar boundary layer for which the stagnation enthalpy specified in the initial section is variable with height. Such problems arise, for example, for bodies located in the wake behind another body, for hypersonic flow past slender blunted bodies (as a result of the large transverse entropy gradients in the highentropy layer), for stepwise variation of the temperature of a surface on which there is an already developed boundary layer, for sudden expansion of the boundary layer as a result of its flow past a corner of the surface, etc.Strictly, we should in such cases solve the boundary layer equations (if the longitudinal gradients are much smaller than the transverse) with the specified initial distribution of the quantities. However, from the physical point of view, the distributed region may be broken down into two regions, the near-wall boundary layer and an outer region which is a gas flow with constant velocity and the specified initial temperature profile, whose calculation yields the edge conditions for the boundary layer. The boundary between the regions is determined from the condition of adequately smooth matching of the solutions. This approach is much preferable to the first, since it permits avoiding (within the framework of boundary layer theory) the difficulties associated with the presence of a possible singularity at the initial point of the surface due to the discontinuity of the boundary conditions at this point, and also permits using conventional boundary layer theory if the effect of the viscosity in the outer region is not significant. However, this partition requires additional justifications of the possibility of independent determination of the solution in the outer region and the determination of the edge of the boundary layer, considered as the region of influence of the wetted surface. The boundary layer in a nonuniform flow has been considered in several works for a linear initial velocity or temperature profile [1–3].It should be noted that the linear initial enthalpy or velocity profiles for constant gas properties do not undergo changes under the influence of viscosity or thermal conductivity. Thus the fundamental characteristic features noted above which are associated with the presence of the two regions and their interaction in essence cannot be investigated using these examples.In this study we obtain and analyze the exact solutions of the equations of the compressible boundary layer for a power-law variation of the initial stagnation enthalpy profile as a function of the stream function for a constant initial velocity. Here it is shown that the influence of the boundary conditions at the wall are actually localized in the near-wall boundary layer, which is similar in dimensions to the conventional velocity or thermal boundary layers. In the region which is external with relation to this layer, in accordance with the physical picture described above, the solution coincides with the solution of the Cauchy problem for the heat conduction equation, which describes the development of the initial temperature profile in an infinite steady-state flow with constant velocity.It is shown that for the sufficiently smooth initial profiles which are of interest in practice the outer flow undergoes practically no changes until we reach the inner boundary layer, and it may be calculated using the perfect gas laws.     

Finite Oscillations of a Cylinder in a Coaxial Duct Subjected to Annular Compressible Flow

As a generalization considering small fluid-structural vibrations, the present paper examines the finite magnitude oscillatory motion of an elastically supported rigid cylinder in a cylindrical rigid duct conveying a compressible flow. The fluid is assumed to be inviscid and irrotational and free purely transverse vibrations of the body are dealt with. The governing equations of motion are the fully nonlinear Euler equations together with the continuity equation and a state equation (here for an ideal gas), the ordinary differential equation for the vibrating cylinder, and the kinematical transition and boundary conditions at the moving contact interface between fluid and body and the outside fluid border, respectively. A pertubation analysis is performed to calculate not only the dynamic characteristics for small coupled oscillations but also the corrections due to the inherent nonlinearities of the vibroacoustic problem. To make the calculation steps more transparent, the simpler problem of a two-dimensional channel flow between a rigid wall and an elastically supported rigid plate is also included in the present study. As an outlook, the influence of flexibility of the cylinder (or the plate) is addressed and the problem of forced vibrations is touched. This revised version was published online in July 2006 with corrections to the Cover Date.     

Effect of supersonic stream parameters on the characteristic time of transient step flow

There have been many studies of viscous compressible gas flow in wakes and behind steps [1–6] in which attention has been focused on the steady-state flow regime. The problem of the supersonic flow of a viscous compressible heat-conducting gas past a plain backward-facing step is considered. The problem is solved numerically within the framework of the complete system of Navier-Stokes equations. The passage of the solution from the initial data to the steady-state regime and the effect of the gas dynamic parameters of the external flow on the characteristic flow stabilization time are investigated.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 4, pp. 137–140, July–August, 1989.     

Nonisothermal Couette flow of a non-Newtonian fluid under the action of a pressure gradient

Nonisothermal Couette flow has been studied in a number of papers [1–11] for various laws of the temperature dependence of viscosity. In [1] the viscosity of the medium was assumed constant; in [2–5] a hyperbolic law of variation of viscosity with temperature was used; in [6–8] the Reynolds relation was assumed; in [9] the investigation was performed for an arbitrary temperature dependence of viscosity. Flows of media with an exponential temperature dependence of viscosity are characterized by large temperature gradients in the flow. This permits the treatment of the temperature variation in the flow of the fluid as a hydrodynamic thermal explosion [8, 10, 11]. The conditions of the formulation of the problem of the articles mentioned were limited by the possibility of obtaining an analytic solution. In the present article we consider nonisothermal Couette flows of a non-Newtonian fluid under the action of a pressure gradient along the plates. The equations for this case do not have an analytic solution. Methods developed in [12–14] for the qualitative study of differential equations in three-dimensional phase spaces were used in the analysis. The calculations were performed by computer.Translated from Zhurnal Prikladnoi Mekhaniki i Tekhnicheskoi Fiziki, No. 3, pp. 26–30, May–June, 1981.     

Laminar boundary layer on an oscillating wedge

A study is made of the nonstationary laminar boundary layer on a sharp wedge over which a compressible perfect gas flows; the wedge executes slow harmonic oscillations about its front point. It is assumed that the perturbations due to the oscillations are small, and the problem is solved in the linear approximation. It is also assumed that the thickness of the boundary layer is small compared with the thickness of the complete perturbed region. Then in a first approximation the influence of the boundary layer on the exterior inviscid flow can be ignored, and the parameters on the outer boundary of the boundary layer can be taken equal to their values on the body for the case of inviscid flow over the wedge. They are determined from the solution to the inviscid problem that is exact in the framework of the linear formulation. The wall is assumed to be isothermal. The dependence of the viscosity on the temperature is linear. Under these assumptions, the problem of calculating the nonstationary perturbations in the boundary layer on the wedge is a self-similar problem.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 4, pp. 146–151, July–August, 1980.     

Laminar boundary layer in axisymmetric swirling ducted gas flow

Several studies have been published [1–3] in which the authors solve the problem of the laminar boundary layer in an incompressible fluid on the walls of an axisymmetric duct in the presence of swirl in the outer flow. In [3], Loitsyanskii's parametric method of [4, 5], generalized to the case of three-dimensional flow, is used to solve this problem.In this article the parametric method for integrating the universal equations is extended to the solution of the problem of the laminar boundary layer on the wall of an axisymmetric channel in the case of swirling gas flow.     

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.     

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.     

An approximate analytical solution of the laminar boundary layer equations

Using the pressure gradient as the new variable instead of.the ordinarylongitudinal coordinate x,Liu transformed the ordinary laminar boundary equationsinto a new form.On this base Liu obtained the frictional stress factor by using thegraphical method.In this paper the same variable replacement as in[1]is used and an approximateanalytical solution of the laminar boundary layer equations by the series method isobtained.The author also obtains a formula of frictional stress factor.For the case ofthe main function without the term of constant,the author makes a furthersimplification.The error of the frictional stress factor obtained by the author is stillless than 10％,compared with that of[1].     

Three-dimensional rotational MHD flows in bounded volumes

Three-dimensional laminar flows of a viscous conducting gas in the neighborhood of a rotating disk are considered. The simultaneous impact of an external magnetic field, suction from the disk surface, and the axial temperature gradient as well as the action of the external axial magnetic field on three-dimensional flows in the neighborhood of rigid permeable surfaces are first studied. An exact analytic solution of the system of the boundary layer equations is obtained. It is found that the direction of the radial flow initiated in the boundary layer can be varied by changing the temperature ratio in the external flow and on the disk for various Prandtl numbers Pr. An approximate solution of the problem of flow in the rotating cylinder in the presence of a retarding cover is constructed on the basis of the approach developed for extended disks.     

Viscous flow in a partially-filled horizontal cylinder rotating at constant speed

The problem under consideration is that of the stationary shape of the free surface of a viscous fluid in a steadily rotating horizontal cylinder. In the majority of investigations of this problem the thickness of the fluid layer coating the inner surface of the cylinder is assumed to be small [1–3]. The case of a near-horizontal free surface, with the bulk of the fluid at the cylinder bottom, was considered in [4], where, after considerable simplification, the governing equations were reduced to ordinary differential equations. In the present study the behavior of the free surface is investigated using a creeping flow approximation. The controlling parameters vary over a wide range. In the numerical computations a boundary element method was used. The numerical results have been confirmed experimentally.Translated from Izvestiya Rossiiskoi Akademii Nauk, Mekhanika Zhidkosti i Gaza, No. 3, pp. 25–30, May–June, 1993.     

Effect of a space-time source structure simulating a dielectric barrier discharge on the laminar boundary layer

The time-dependent pulse-periodic action of a surface electric discharge on a flat-plate laminar boundary layer is simulated theoretically. The effect of the discharge is estimated within the framework of the numerical solution of the boundary value problem for the time-dependent two-dimensional compressible boundary layer with additional terms in the momentum and energy conservation equations simulating the force and thermal action of the discharge on the gas flow with allowance for the pressure gradient across the boundary layer induced by the corresponding body force component. The effect of certain parameters of the problem formulated above on the gas velocity induced by the discharge in the boundary layer is also estimated.     

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.