Universal equations of three-dimensional boundary layer theory

We consider a parametric method for investigating three-dimensional laminar motion of an incompressible fluid in a boundary layer on a curved surface. It is found that the problem solution in the general case depends on four series of parameters, constructed from two components of the outer flow velocity and the two Lamé coefficients characterizing the shape of the immersed surface. From the general equations of the three-dimensional boundary layer we obtain a system of two universal equations which do not contain the characteristics of the outer flow. This system may be solved once and for all. As an example we consider the problem of the laminar boundary layer on the walls of an axisymmetric channel in the case of swirling outer flow. For this case we obtain numerical solutions of the system of universal equations in the local two-parameter approximation.     

Electrode layers in flows of an inviscid plasma with good conductivity

The structure of the electromagnetic electrode layers that are produced in flows across a magnetic field by a completely ionized and inviscid plasma with good conductivity and a high magnetic Reynolds number is examined in a linear approximation. Flow past a corrugated wall and flow in a plane channel of slowly varying cross section with segmented electrodes are taken as specific examples. The possibility is demonstrated of the formation of nondissipative electrode layers with thicknesses on the order of the Debye distance or electron Larmor radius and of dissipative layers with thicknesses on the order of the skin thickness, as calculated from the diffusion rate in a magnetic field [2].In plasma flow in a transverse magnetic field, near the walls, along with the gasdynamie boundary layers, which owe their formation to viscosity, thermal conductivity, etc. (because of the presence of electromagnetic fields, their structures may vary considerably from that of ordinary gasdynamic layers), proper electromagnetic boundary layers may also be produced. An example of such layers is the Debye layer in which the quasi-neutrality of the plasma is upset. No less important, in a number of cases, is the quasi-neutral electromagnetic boundary layer, in which there is an abrupt change in the frozen-in parameter k=B/p (B is the magnetic field and p is the density of the medium). This layer plays a special role when we must explicitly allow for the Hall effect and the related formation of a longitudinal electric field (in the direction of the veloeiryv of the medium). We will call this the magnetic layer. The magnetic boundary layer can be dissipative as well as noudissipative (see below). The dissipative magnetic layer has been examined in a number of papers: for an incompressible medium with a given motion law in [1], for a compressible medium with good conductivity in [2], and with poor conductivity in [3]. In the present paper, particular attention will be devoted to nondissipative magnetic boundary layers.     

Effect of Free‐Stream Turbulence on the Flow Structure near a Wedge and the Windward Side of an Airfoil

The flow structure behind wire grids is studied for flows with a low subsonic velocity, and the effect of grids on the boundarylayer flow structure is considered. It is shown that the meanvelocity inhomogeneity induced by the grid does not disappear until a distance of 925 calibers downstream of the grid is reached. Liquidcrystal thermography combined with hotwire measurements made it possible to find the source of steady largescale streamwise vortex structures in the boundary layer on a wedge and on an airfoil and to determine the parameters of these structures.     

An unsteady-state thermal boundary layer on a flat isothermal plate

The Kármán-Polhausen integral method is used to investigate the problem of an unsteady-state thermal boundary layer on an isothermal plate with a stepwise change in the conditions of flow around the plate; analytical expressions are obtained for the thickness of the thermal boundary layer. A dependence is found for the rate of movement of the boundary between the steady-state and unsteady-state regions of the solution on the Prandtl number. A similar problem was solved in [1, 2] for a dynamic layer, Goodman [3] discusses the more partial problem of an unsteady-state thermal boundary layer under steady-state flow conditions. Rozenshtok [4] considers the problem in an adequate statement but, unfortunately, he permitted errors of principle to enter into the writing of the system of characteristic equations; this led to absolutely invalid results. In an evaluation of the advantages and shortcomings of the integral method under consideration, given in [4], it must only be added that the method is applicable to problems in which the initial conditions differ from zero since, in this case, approximation of the velocity and temperature profiles by polynomials is not admissible.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 4, pp. 64–69, July–August, 1970.     

Turbulent boundary layer on a permeable surface

Several theoretical [1–4] and experimental [5–7] studies have been devoted to the study of the effect of distributed injection of a gaseous substance on the characteristics of the turbulent boundary layer. The primary study has been made of flow past a flat plate with gas injection. The theoretical methods are based primarily on the semiempirical theories of Prandtl [1] and Karman [2].In contrast with the previous studies, the present paper proposes a power law for the mixing length; this makes it possible to obtain velocity profiles which degenerate to the known power profiles [8] in the case of flow without blowing and heat transfer. This approach yields analytic results for flows with moderate pressure gradient.Notation x, y coordinates - U, V velocity components - density - T temperature - h enthalpy - H total enthalpy - c mass concentration - , , D coefficients of molecular viscosity, thermal conductivity, diffusion - c_p specific heat - adiabatic exponent - r distance from axis of symmetry to surface - boundary layer thickness - U velocity in stream core - friction - c_f friction coefficient - P Prandtl number - S Schmidt number - S_t Stanton number - M Mach number - j=0 plane case - j=1 axisymmetric case The indices 1 injected gas - 2 mainstream gas - w quantities at the wall - core of boundary layer - 0 flow of incompressible gas without injection - v=0 flow of compressible gas without injection - ^* quantities at the edge of the laminar sublayer - quantities at the initial section - turbulent transport coefficients     

Flow over a body with development of a steady separation zone as re → ∞

This paper discusses formulation of the total problem of flow of an incompressible liquid over a body, with formation of a closed stationary separation zone as Re . The scheme used is based on the method of matched asymptotic expansions [1]. Following [1], it is postulated that the separated zone is developed (i.e., it is not infinitely fragmented and does not vanish as Re ), and the flow inside it has a definite degree of regularity with respect to Re. With these hypotheses we can use the Prandtl-Batchelor theorem [2], which states that, in the limit as Re , a region of circulating flow becomes vortex flow of an inviscid liquid with constant vorticity . Therefore, a basis for constructing matched asymptotic expansions is the vortex-potential problem (the problem of determining a stream function , satisfying the equation = 0 in the region of translational motion and the equation = in a certain region, unknowna priori, of circulating motion). In the general case the solution of the vortex-potential problem depends on two parameters: the total pressure p_o and the vorticity in the separated zone. These parameters appear in the condition for matching the solutions of the first and second boundary-layer approximations (at the boundary of the separated zone for the end Re values) with the corresponding solutions for the inviscid flow. It is shown in the present paper that the conditions for matching the cyclic boundary layer with the external translational flow are the same additional relations which allow us to close the total problem. Thus, in using the method of matched asymptotic expansions to solve the problem of flow over a body with closed stationary separation zones one must simultaneously consider no less than two approximations.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 2, pp. 28–37, March–April, 1978.The authors thank G. Yu. Stepanov for discussion of the paper and valuable comments.     

Nonstationary filtration in partially saturated porous media

From the mathematical formulation of a one-dimensional flow through a partially saturated porous medium, we arrive at a nonlinear free boundary problem, the boundary being between the saturated and the unsaturated regions in the medium. In particular we obtain an equation which is parabolic in the unsaturated part of the domain and elliptic in the saturated part.Existence, uniqueness, a maximum principle and regularity properties are proved for weak solutions of a Cauchy-Dirichlet problem in the cylinder {(x,t): 0x1, t0} and the nature, in particular the regularity, of the free boundary is discussed.Finally, it is shown that solutions of a large class of Cauchy-Dirichlet problems converge towards a stationary solution as t and estimates are given for the rate of convergence.     

Unsteady supersonic flow around blunt bodies with detached shock wave

The numerical method of calculating the supersonic three-dimensional flow about blunt bodies with detached shock wave presented in [1–3] is applied to the case of unsteady flow. The formulation of the unsteady problem is analogous to that of [4], which assumes smallness of the unsteady disturbances.The paper presents some results of a study of the unsteady flow about blunt bodies over a wide range of variation of the Mach number M=1.50– and dimensionless oscillation frequency l/V=0–1.0. A comparison is made with the results obtained from the Newton theory.     

Investigation of the flow past wings of infinite span with a blunt leading edge in the vorticity interaction regime

An asymptotic analysis of the Navier-Stokes equations is carried out for the case of hypersonic flow past wings of infinite span with a blunt leading edge when 0, Re , and M . Analytic solutions are obtained for an inviscid shock layer and inviscid boundary layer. The results of a numerical solution of the problems of vorticity interaction at the blunt edge and on the lateral surface of the wing are presented. These solutions are compared with the solution of the equations of a thin viscous shock layer and on the basis of this comparison the boundaries of the asymptotic regions are estimated.deceasedTranslated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 6, pp. 120–127, November–December, 1987.     

Analysis of turbulent boundary layer interaction with an external supersonic flow behind a step

An integral method of analyzing turbulent flow behind plane and axisymmetric steps is proposed, which will permit calculation of the pressure distribution, the displacement thickness, the momentum-loss thickness, and the friction in the zone of boundary layer interaction with an external ideal flow. The characteristics of an incompressible turbulent equilibrium boundary layer are used to analyze the flow behind the step, and the parameters of the compressible boundary layer flow are connected with the parameters of the incompressible boundary layer flow by using the Cowles-Crocco transformation.A large number of theoretical and experimental papers devoted to this topic can be mentioned. Let us consider just two [1, 2], which are similar to the method proposed herein, wherein the parameter distribution of the flow of a plane nearby turbulent wake is analyzed. The flow behind the body in these papers is separated into a zone of isobaric flow and a zone of boundary layer interaction with an external ideal flow. The jet boundary layer in the interaction zone is analyzed by the method of integral relations.The flow behind plane and axisymmetric steps is analyzed on the basis of a scheme of boundary layer interaction with an external ideal supersonic stream. The results of the analysis by the method proposed are compared with known experimental data.Notation x, y longitudinal and transverse coordinates - X, Y transformed longitudinal and transverse coordinates - , ^*, ^** boundary layer thickness, displacement thickness, momentum-loss thickness of a boundary layer - , ^*, ^** layer thickness, displacement thickness, momentum-loss thickness of an incompressible boundary layer - u, velocity and density of a compressible boundary layer - U, velocity and density of the incompressible boundary layer - , stream function of the compressible and incompressible boundary layers - , dynamic coefficient of viscosity of the compressible and incompressible boundary layers - r₁ radius of the base part of an axisymmetric body - r radius - R transformed radius - M Mach number - friction stress - p pressure - a speed of sound - s enthalpy - v Prandtl-Mayer angle - P Prandtl number - P_t turbulent Prandtl number - r₂ radius of the base sting - b step depth - =0 for plane flow - =1 for axisymmetric flow Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 3, pp. 33–40, May–June, 1971.In conclusion, the authors are grateful to M. Ya. Yudelovich and E. N. Bondarev for useful comments and discussions.     

Influence of vorticity on the heat transfer to blunt bodies in hypersonic flight

When blunt bodies are in hypersonic flight, a high-entropy layer of gas with nonzero vorticity is formed near their surface. The transverse gradients of the entropy, density, and gas velocity in the layer are high, which makes it necessary to take into account its absorption by the boundary layer of finite thickness . This vortex interaction is usually accompanied by an increase in the heat flux q and the frictional stress on the wall compared with their values as calculated in accordance with the classical scheme of a thin boundary layer, when the parameters on the outer edge of the boundary layer are set equal to the inviscid parameters on the body. This effect has been investigated on the side surface of slender (with angle 1 to the undisturbed flow) blunt bodies in a hypersonic stream [1–3]. It is shown in the present paper that the effect can have a stronger and even qualitative influence on the flow over blunt bodies with 1 if the radius of curvature R_s of the detached shock wave on the axis is small compared with the midsection radius R of the body. It is shown that the distributions of the heat fluxes with allowance for the vorticity of the inviscid shock layer are similar in the case of slightly blunt (r₀/R 0) cones with half-angles less than a critical _*.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 2, pp. 50–57, March–April, 1981.     

Stability and Transition on a Swept Cylinder in a Supersonic Flow

Results of experimental investigations of the evolution of natural disturbances and laminar–turbulent transition in a supersonic boundary layer on the attachment line of a circular cylinder with a sweep angle of 68° and a freestream Mach number M = 2 are presented. The experimental studies are supplemented by calculations of the mean flow and stability characteristics. Flow regimes in the boundary layer on the attachment line are determined by a hotwire technique as functions of the Reynolds number and height of twodimensional roughness elements. The results are compared with NASA (Ames) experiments.     

Development of magnetohydrodynamic boundary layers

Many authors have studied the problem of the development of a hydrodynamic boundary layer when a body is suddenly set in motion. The results obtained are presented most fully in the monographs of H. Schlichting [1] and L. G. Loitsyanskii [2]. In magnetohydrodynamics the development of the boundary layer over the surface of an infinite flat plate for uniform oncoming flow has been closely studied (for example [3, 4]). Below, the problem of the development of a plane magnetohydrodynamic boundary layer is considered in a different formulation. We shall suppose that the distributions of velocity U(x) and enthalpy h(x) are given along the body contour for t=0. At that moment the viscosity and thermal conductivity mechanisms are instantaneously switched on. Viscous and thermal boundary layers begin to grow in a direction normal to the body. The medium in the boundary layer interacts with the magnetic field. This formulation corresponds to the development of a magnetohydrodynamic boundary layer on a body which is set in motion with a jerk, in the case where the rate of establishment of magnetohydrodynamic flow of the inviscid, thermally nonconducting fluid around the body is much less than the rate of development of the boundary layer. Then U(x) and h(x) are found by solving the problem of stationary magnetohydrodynamic flow of an inviscid thermally nonconducting fluid around a body, or simply the hydrodynamic flow if the medium interacts with the field only in the boundary layer.     

Plane entry flows and energy estimates for the navier-stokes equations

One of the classic problems of laminar flow theory is the development of velocity profiles in the inlet regions of channels or pipes. Such entry flow problems have been investigated extensively, usually by approximate techniques. In a recent paper [4], Horgan & Wheeler have provided an alternative approach, based on an energy method for the stationary Navier-Stokes equations. In [4], concerned with laminar flow in a cylindrical pipe of arbitrary cross-section, an analogy is drawn between the end effect issue of concern here, called the end effect, and the celebrated Saint-Venant's Principle of the theory of elasticity.In this paper, I consider the two-dimensional analog of the problem treated in [4] with a view to providing a more explicit formulation of the energy approach to entry flow problems. The flow development in a semi-infinite channel with parallel-plates is analyzed within the framework of the stationary Navier-Stokes equations. Introduction of a stream function leads to a formulation in terms of a boundary-value problem for a single fourth order nonlinear elliptic equation. In the case of Stokes flow, this problem is formally equivalent to a boundary-value problem for the biharmonic equation considered by Knowles [5] in the analysis of Saint-Venant's Principle in plane elasticity. The main result is an explicit estimate which establishes the exponential spatial flow development and leads to an upper bound for an appropriately defined entrance length. These results are obtained using differential inequality techniques analogous to those developed in investigation of Saint-Venant's Principle.     

Semi-similar solutions to the three-dimensional laminar boundary layer

The theory of semi-similar solutions is developed for and applied to the problem of three-dimensional laminar boundary layer flow. A number of specific examples are calculated. Particular attention is given to certain flows in which separation is approached and the nature of three-dimensional laminar boundary layer separation is inferred from the behavior of these solutions close to separation. Two types of separation are observed: singular separation characterized by the vanishing of the total shear along the line of separation and ordinary separation characterized by limiting streamlines which become parallel to the line of separation.     

Multicomponent turbulent boundary layer on a chemically active surface

When a gaseous mixture flows past chemically active surfaces the boundary layer formed on the wetted body may contain a large number of components with different diffusion properties. This leads to the necessity for studying the diffusion of the components in the multicomponent boundary layer.The use of thebinary boundary layer concept in the general case cannot yield satisfactory results, since replacement of the mutual diffusion coefficients D_ij of the various pairs of components by a single diffusion coefficient D in many cases is a rough approximation.In the general case the number of different diffusion coefficients is equal to N(N–1)/2 (N is the number of components). Usually it is possible to identify groups of components with similar molecular weights. Then the number of different diffusion coefficients may be reduced without large error. However, even in the comparatively simple case when it is possible to divide all the components into two groups with similar molecular weights we must take account of three different diffusion coefficients (one diffusion coefficient in each group and also the diffusion coefficient for the components of one group relative to the components of the other group). Only in particular cases when the gaseous mixture consists of only two components with arbitrary molecular weights, or if all the components of the gaseous mixture have similar molecular weights, can we with justification introduce a single diffusion coefficient (if in this case there are no limitations on the direction of the diffusion).Studies have been published covering the laminar multicomponent boundary layer. An analytic method for solving the equations of the laminar multicomponent boundary layer was developed by Tirskii [1]. There are also studies in which concrete results were obtained by numerical methods with the use of computers (for example, [2, 3]).As far as the author knows, for turbulent flow there are studies (for example, [4, 5]) covering flow with chemical reactions only in the case when all the diffusion coefficients are equal (D_ij=D).The present paper presents a method for calculating the turbulent multicomponent boundary layer with account for several different diffusion coefficients.Notation x, y coordinates - u, v velocity components - density - T temperature - h heat content - H enthalpy - c_i mass concentration of the i-th component - c ₁ ⁽¹⁾ element concentrations in solid body - J_i diffusion flux of the i-th component - m molecular weight - dynamic viscosity coefficient - kinematic viscosity coefficient - heat conduction coefficient - c_p specific heat - adiabatic index - D_ij binary diffusion coefficients - P Prandtl number - S_ij Schmidt number - S_t Stanton number - M Mach number - friction - q radiant thermal flux - boundary layer thickness - D rate of displacement of gas-solid interface - degree of gasification - r_ij weight fraction of element i in component j - _ij stoichiometric coefficients - K_i reaction equilibrium constants - l number of components for which I_i0 Indices i, j component number - w quantities for y=0 - * quantities on the edge of the laminar sublayer - ⁽¹⁾ quantities at the solid body - quantities at the outer edge of the boundary layer - molar transport coefficients     

Calculation of Ionization and Radiative Characteristics of Air in the Shock Layer on the Basis of Various Models of Vibration–Dissociation Interaction

Models of vibrationdissociation interaction are verified on the basis of results of numerical simulation of nonequilibrium air flow in the shock layer near vehicles flying in the atmosphere and data of inflight and windtunnel experiments on measurement of ionization and radiative characteristics of the shock layer.     

Formulation of the inverse axisymmetric problem of steady rotational flow of an ideal incompressible fluid in turbomachinery

The problem of the design of rotor blades within the framework of the hypothesis of an infinitely large number of blades reduces to the solution of an inverse axisymmetric problem. In the Bauersfeld-Voznesenskii formulation [1–3] this problem may be stated as follows: for given meridional flow and angular rotor velocity, construct the blade surface S ₂ (Fig. 1) passing through a given inlet ab (or exit cd) edge and a given line of intersection ad of the blade with one axisymmetric stream surface of the given meridional flow. Henceforth, S ₂ will be used to designate the median (or concave) surface of the blade, which, under certain conditions, coincides with the median interblade stream surface abcd. In [1–4], in solving this problem, use is made of the condition of coplanarity of the streamline elementd r=dr, rd,dz located on the blade surface S ₂ the relative velocity vector w and the absolute vorticity vector ×c In [5, 6] it is shown that this condition is valid only for irrotational flow incident on the rotor; consequently, its use in [1, 2] is completely legitimate, while in [3, 4] its use is inadmissible in principle, since the the equi-velocity meridional flow (i. e., that stream in which along each normal n to the meridional streamlines s the meridional component w_s of the velocity is constant (w _s/ _n=0)) assumed therein is essentially rotational [=(×c) _u 0] in the curved channel leading to the rotor.In [7] Gravalos presents the formulation and method of solution of the inverse axisymmetric problem for any arbitrary rotational meridional flow (and not just an equi-velocity flow), but does not take into account the constriction of the flow by the rotor blades, or take note of special cases of degeneration of the order and type of the equations at the boundaries and within the region; moreover, the method of solution employed assumes reduction of the quasilinear hyperbolic equation to the normal form to permit its solution by the Picard method of successive approximations.Below, we present the mathematical formulation of the inverse axisymmetric problem for any arbitrary rotational meridional flow in which account is taken of flow constriction. Cases of degeneration of the order and type of the equations are considered, the case with formation of a line of parabolic degeneration is examined, and important practical cases of the formulation of the boundary and initial conditions (Goursat problem and mixed problems), which determine the possible forms of the inlet and exit edges are studied. The problems formulated for the quasilinear hyperbolic equation can be solved with the aid of the method of characteristics, the method of finite differences, the method of straight lines, and other numerical methods.The discussion is directly applicable to radial and axial hydraulic turbines; however, it can be applied in essentially the same form to pump impellers, hydraulic converters, and also to stationary guide vanes (=0).In conclusion, the author wishes to thank G. Yu. Stepanov for discussing this work.     

LDA measurements in low Reynolds number turbulent boundary layer

LDA measurements of the mean velocity in a low Reynolds number turbulent boundary layer allow a direct estimate of the friction velocity U from the value of /y at the wall. The trend of the Reynolds number dependence of / is similar to the direct numerical simulations of Spalart (1988).     

Discontinuous Solutions of the “Shallow Water$rdquo; Equations for Flow Over a Bottom Step

Flows over a depth discontinuity (bottom step) are studied within the framework of a singlelayer shallow water model. Emphasis is given to substantiation of the relations for the stationary discontinuity thus formed. Admissible stable flows over this discontinuity are distinguished. As an example, the paper gives a solution of the problem of the water flow resulting from dam failure above a bottom step over which water is flowing.