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.     

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.     

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.     

Method of mean-mass parameters for a three-dimensional boundary layer in a flow with vorticity

When a gas flows with hypersonic velocity over a slender blunt body, the bow shock induces large entropy gradients and vorticity near the wall in the disturbed flow region (in the high-entropy layer) [1]. The boundary layer on the body develops in an essentially inhomogeneous inviscid flow, so that it is necessary to take into account the difference between the values of the gas parameters on the outer edge of the boundary layer and their values on the wall in the inviscid flow. This vortex interaction is usually accompanied by a growth in the frictional stress and heat flux at the wall [2, 3]. In three-dimensional flows in which the spreading of the gas on the windward sections of the body causes the high-entropy layer to become narrower, the vortex interaction can be expected to be particularly important. The first investigations in this direction [4–6] studied the attachment lines of a three-dimensional boundary layer. The method proposed in the present paper for calculating the heat transfer generalizes the approach realized in [5] for the attachment lines and makes it possible to take into account this effect on the complete surface of a blunt body for three-dimensional laminar, transition, or turbulent flow regime in the boundary layer.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 1, pp. 80–87, January–February, 1981.     

The effect of Reynolds number on boundary layer turbulence

A new facility for studying high Reynolds number incompressible turbulent boundary layer flows has been constructed. It consists of a moderately sized wind tunnel, completely enclosed by a pressure vessel, which can raise the ambient air pressure in and around the wind tunnel to 8 atmospheres. This results in a Reynolds number range of about 20:1, while maintaining incompressible flow. Results are presented for the zero pressure gradient flat plate boundary layer over a momentum thickness Reynolds number range 1500–15?000. Scaling issues for high Reynolds number non-equilibrium boundary layers are discussed, with data comparing the three-dimensional turbulent boundary layer flow over a swept bump at Reynolds numbers of 3800 and 8600. It is found that successful prediction of these types of flows must include length scales which do not scale on Reynolds number, but are inherent to the geometry of the flow.     

Boundary-layer separation on conical bodies

Some characteristics of the variation in the linear dimensions of the flow separation zones on conical bodies with expanding conical skirts and of variation of the pressure within these zones as a function of variation of the Mach number, Reynolds number, and intensity of the disturbance that causes the boundary layer separation are examined. Experiments were conducted in laminar, transitional, and turbulent flows in flow separation regions. The interaction of viscous and nearly inviscid flows is quite common. This phenomenon occurs in flow past a concave corner, when a compression shock impinges on a boundary layer, and in many other cases. The characteristics of this phenomenon in flow about two-dimensional bodies have been investigated experimentally in [1, 2] and other studies. Attempts have been made to analyze the interaction of compression shocks with the boundary layer theoretically. In “free” separated flows, when the points of separation and reattachment of the boundary layer are not fixed (for example, on a flat plate with a long wedge attached to it), theoretical studies are usually made within the framework of the boundary layer theory with use of the approximate integral methods [3, 4]. In this article we examine some results from studies of free separated flows on conical bodies with conical skirts in laminar, transitional, and turbulent flows (Fig. 1).     

Turbulent boundary layer on a catalytic surface in a nonequilibrium dissociating gas

The majority of the studies which consider the flow of a dissociating gas in a turbulent boundary layer are devoted to the investigation of either frozen or equilibrium flows on a flat plate.The frozen turbulent boundary layer has been studied by Dorrance [1], Kutateladze and Leont'ev [2], and Lapin and Sergeev [3]. A study of the effect of catalytic recombination processes at the plate surface on the heat transfer in a frozen turbulent boundary layer was made by Lapin [4].Kosterin and Koshmarov [5], Ginzburg [6], Dorrance [7], and Lapin [8] have studied the turbulent boundary layer on a plate in equilibrium dissociating gas.The calculation of the heat transfer in a turbulent boundary layer on a catalytic plate surface with nonequilibrium dissociation was made by Kulgein [9]. In this study the nonequilibrium nature of the dissociation process was taken into account only in the laminar sublayer, while the flow in the turbulent core was considered frozen. The solution was found numerically using a computer by means of a laborious iteration process.The present paper reports a method for calculating the turbulent boundary layer on a flat catalytic plate with arbitrary dissociation rate. The method, constructed using the assumptions customary for turbulent boundary layer theory, is a successive approximation method. Good convergence of the method is assured by the fact that the effect of the nonequilibrium nature of the dissociation process on the parameter distribution in the boundary layer and, consequently, on the friction and heat transfer may be allowed for merely by finding corrections, usually relatively small, to the distribution of these parameters in the equilibrium or frozen flows. The basis of the study is the two-layer scheme of the turbulent boundary layer. The Prandtl and Schmidt numbers and also their turbulent analogs are taken equal to unity. As the model of the dissociating gas we use the Lighthill model of the ideal dissociating gas [10], extended by Freeman [11] to nonequilibrium flows.     

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

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

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].     

Görtler vortices in Falkner–Skan flows with suction and blowing

In this paper, we use nonlinear calculations to study curved boundary‐layer flows with pressure gradients and self‐similar suction or blowing. For an accelerated outer flow, stabilization occurs in the linear region while the saturation amplitude of vortices is larger than for flows with a decelerating outer flow. The combined effects of boundary‐layer suction and a favourable pressure gradient can give a significant stabilization of the flow. Streamwise vortices can be amplified on both concave and convex walls for decelerated Falkner–Skan flow with an overshoot in the velocity profile. The disturbance amplitude is generally lower far downstream compared with profiles without overshoot. Copyright © 2007 John Wiley & Sons, Ltd.     

Asymptotic analysis of the equations of a three-dimensional boundary layer

The asymptotic solutions of the self-similar equations of two- and three-dimensional boundary layers have been investigated by many authors (see, for example, [1–3]). In [4, 5], asymptotic solutions were found for non-self-similar equations for two-dimensional flow, and the propagation of perturbations near the external edge of the boundary layer was analyzed. In the present paper, asymptotic solutions are obtained for the non-self-similar equations of a three-dimensional laminar boundary layer of an incompressible fluid. It is shown that the conclusion drawn in [5] — that the boundary conditions can be transferred from infinity to a finite distance from the wall — is also true for three-dimensional flow. The obtained solutions explain the experimentally well-known phenomenon of the conservativeness of the secondary currents. The essence of this phenomenon is that a change in the sign of the transverse (along the normal to a streamline of the external flow) pressure gradient is accompanied by a very rapid change in the direction of the secondary flow near the wall, whereas in the upper layers of the boundary layer the direction remains unchanged for a substantial time.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 5, pp. 155–157, September–October, 1979.     

Reynolds Stress Budgets in Couette and Boundary Layer Flows

Reynolds stress budgets for both Couette and boundary layer flows are evaluated and presented. Data are taken from direct numerical simulations of rotating and non-rotating plane turbulent Couette flow and turbulent boundary layer with and without adverse pressure gradient. Comparison of the total shear stress for the two types of flows suggests that the Couette case may be regarded as the high Reynolds number limit for the boundary layer flow close to the wall. The limit values of turbulence statistics close to the wall for the boundary layer for increasing Reynolds number approach the corresponding Couette flow values. The direction of rotation is chosen so that it has a stabilizing effect, whereas the adverse pressure gradient is destabilizing. The pressure-strain rate tensor in the Couette flow case is presented for a split into slow, rapid and Stokes terms. Most of the influence from rotation is located to the region close to the wall, and both the slow and rapid parts are affected. The anisotropy for the boundary layer decreases for higher Reynolds number, reflecting the larger separation of scales, and becomes close to that for Couette flow. The adverse pressure gradient has a strong weakening effect on the anisotropy. All of the data presented here are available on the web [36]. This revised version was published online in July 2006 with corrections to the Cover Date.     

The structure of a three-dimensional boundary layer subjected to streamwise-varying spanwise-homogeneous pressure gradient

A spatially-evolving three-dimensional boundary layer, subjected to a streamwise-varying spanwise-homogeneous pressure gradient, equivalent to a body force, is investigated by way of direct numerical simulation. The pressure gradient, prescribed to change its sign half-way along the boundary layer, provokes strong skewing of the velocity vector, with a layer of nearly collateral flow forming close to the wall up to the position of maximum spanwise velocity. A wide range of flow-physical properties have been studied, with particular emphasis on the near-wall layer, including second-moments, major budget contributions and wall-normal two-point correlations of velocity fluctuations and their angles, relative to wall-shear fluctuations. The results illustrate the complexity caused by skewing, including a damping in turbulent mixing and a significant lag between strains and stresses. The study has been undertaken in the context of efforts to develop and test novel hybrid LES–RANS schemes for non-equilibrium near-wall flows, with an emphasis on three-dimensional near-wall straining. Fundamental flow-physical issues aside, the data derived should be of particular relevance to a priori studies of second-moment RANS closure and the development and validation of RANS-type near-wall approximations implemented in LES schemes for high-Reynolds-number complex flows.     

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.     

Laminar boundary layer in an equilibrium dissociated gas for arbitrary external velocity distribution

The development of machine computing technology permits calculating the boundary layer by direct numerical integration of the corresponding system of partial differential equations [1, 2]. In order to derive general conclusions concerning the boundary layer with a pressure gradient we must perform the integration for each concrete form of velocity specification at the outer edge of the boundary layer. The method of calculating the boundary layer used in the present study [3], based on the solution of a universal (independent of the specification of the velocity at the outer edge of the boundary layer) system of equations, permits the clarification of several general relationships.     

Turbulent boundary layer with positive pressure gradient on a permeable surface under nonisothermal conditions

It is known that the longitudinal pressure gradient can exert a strong influence on the friction law and the characteristics of a dynamic turbulent boundary layer. The thermal and diffusion boundary layers are more conservative to the effect of the pressure gradient, and, hence, methods of analyzing them are based, in the majority of cases, on the hypothesis of conservativity of the heat- and mass-transfer laws to the longitudinal pressure gradient [1]. This hypothesis is verified by experimental results [2, 3] on heat transfer on an impermeable surface in a turbulent stream with positive pressure gradient under almost isothermal conditions. However, such investigations under nonisothermal conditions are practically nonexistent. An approximate theoretical analysis of the heat transfer in a turbulent boundary layer of a nonisothermal stream with a positive pressure gradient is given in this paper. Experimental results are presented. The experimental investigation was conducted in a burned-out graphite diffuser both with and without injection of an inert gas through the wall.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 4, pp. 43–49, July–August, 1976.     

Development of three-dimensional disturbances in a boundary layer

In the region of transition from a two-dimensional laminar boundary layer to a turbulent one, three-dimensional flow occurs [1–3]. It has been proposed that this flow is formed as the result of nonlinear interaction of two-dimensional and three-dimensional disturbances predicted by linear hydrodynamic stability theory. Using many simplifications, [4, 5] performed a calculation of this interaction for a free boundary layer and a boundary layer on a wall with a very coarse approximation of the velocity profile. The results showed some argreement with experiment. On the other hand, it is known that disturbances of the Tollmin—Schlichting wave type can be observed at sufficiently high amplitude. This present study will use the method of successive linearization to calculate the primary two- and three-dimensional disturbances, and also the average secondary flow occurring because of nonlinear interaction of the primary disturbances. The method of calculation used is close to that of [4, 5], the disturbance parameters being calculated on the basis of a Blazius velocity profile. A detailed comparison of results with experimental data [1] is made. It developed that at large disturbance amplitude the amplitude growth rate differs from that of linear theory, while the spatial distribution of disturbances agree s well with the distribution given by the natural functions and their nonlinear interaction. In calculating the secondary flow an experimental correction was made to the amplitude growth rate.     

Certain effects occurring during boundary-layer control on a section of a surface

We examine the boundary layer on a plate upon which a circular cylinder is mounted. In investigations of this problem, the solution has successfully been found by numerical methods [1–3] only outside a certain domain whose location depends on the selection of the coordinate system, the spacings of the difference mesh [1], and, possibly, the computation method. Hence, a question arises as to whether it is impossible to solve the problem in a broader domain by modifying the algorithm; in other words, what is the maximum domain in which the solution of the boundary-layer equation can be found if the flow in the separation zone ahead of the obstacle is unknown? Application of the influence principle permitted the work set forth below to provide an answer to this question. The influence of cooling a section of surface on the boundary-layer characteristic was studied in [4] for parabolic external flow streamlines. The influence of cooling strips near the plane of symmetry on separation of a three-dimensional boundary layer is investigated numerically in the present paper. Experimental results that agree qualitatively with the flow singularities in the boundary layer exposed in the computation during boundary-layer control on a section of the surface are presented.     

Numerical investigation of convective motion in a closed cavity

The investigation of thermal convection in a closed cavity is of considerable interest in connection with the problem of heat transfer. The problem may be solved comparatively simply in the case of small characteristic temperature difference with heating from the side, when equilibrium is not possible and when slow movement is initiated for an arbitrarily small horizontal temperature gradient. In this case the motion may be studied using the small parameter method, based on expanding the velocity, temperature, and pressure in series in powers of the Grashof number—the dimensionless parameter which characterizes the intensity of the convection [1–4]. In the problems considered it has been possible to find only two or three terms of these series. The solutions obtained in this approximation describe only weak nonlinear effects and the region of their applicability is limited, naturally, to small values of the Grashof number (no larger than 10³).With increase of the temperature difference the nature of the motion gradually changes—at the boundaries of the cavity a convective boundary layer is formed, in which the primary temperature and velocity gradients are concentrated; the remaining portion of the liquid forms the flow core. On the basis of an analysis of the equations of motion for the plane case, Batchelor [4] suggested that the core is isothermal and rotates with constant and uniform vorticity. The value of the vorticity in the core must be determined as the eigenvalue of the problem of a closed boundary layer. A closed convective boundary layer in a horizontal cylinder and in a plane vertical stratum was considered in [5, 6] using the Batchelor scheme. The boundary layer parameters and the vorticity in the core were determined with the aid of an integral method. An attempt to solve the boundary layer equations analytically for a horizontal cylinder using the Oseen linearization method was made in [7].However, the results of experiments in which a study was made of the structure of the convective motion of various liquids and gases in closed cavities of different shapes [8–13] definitely contradict the Batchelor hypothesis. The measurements show that the core is not isothermal; on the contrary, there is a constant vertical temperature gradient directed upward in the core. Further, the core is practically motionless. In the core there are found retrograde motions with velocities much smaller than the velocities in the boundary layer.The use of numerical methods may be of assistance in clarifying the laws governing the convective motion in a closed cavity with large temperature differences. In [14] the two-dimensional problem of steady air convection in a square cavity was solved by expansion in orthogonal polynomials. The author was able to progress in the calculation only to a value of the Grashof numberG=10⁴. At these values of the Grashof numberG the formation of the boundary layer and the core has really only started, therefore the author's conclusion on the agreement of the numerical results with the Batchelor hypothesis is not justified. In addition, the bifurcation of the central isotherm (Fig. 3 of [14]), on the basis of which the conclusion was drawn concerning the formation of the isothermal core, is apparently the result of a misunderstanding, since an isotherm of this form obviously contradicts the symmetry of the solution.In [5] the method of finite differences is used to obtain the solution of the problem of strong convection of a gas in a horizontal cylinder whose lateral sides have different temperatures. According to the results of the calculation and in accordance with the experimental data [9], in the cavity there is a practically stationary core. However, since the authors started from the convection equations in the boundary layer approximation they did not obtain any detailed information on the core structure, in particular on the distribution of the temperature in the core.In the following we present the results of a finite difference solution of the complete nonlinear problem of plane convective motion in a square cavity. The vertical boundaries of the cavity are held at constant temperatures; the temperature varies linearly on the horizontal boundaries. The velocity and temperature distributions are obtained for values of the Grashof number in the range 0<G4·10⁵ and for a value of the Prandtl number P=1. The results of the calculation permit following the formation of the closed boundary layer and the very slowly moving core with a constant vertical temperature gradient. The heat flux through the cavity is found as a function of the Grashof number.     

On Local Perturbations of a Three-Dimensional Boundary Layer

A high-Reynolds steady-state viscous incompressible fluid flow is investigated in the neighborhood of a small three-dimensional irregularity located on the smooth surface of a body and oriented almost along the skin-friction lines. The regime in which quasi-two-dimensional flow with a given pressure gradient is realized on the irregularity scale is studied in detail. The numerical solution of the corresponding boundary value problem for the boundary layer equations is obtained. It is shown that, as distinct from the plane case, this solution is unique.