Singularities in the boundary layer on a cone at incidence

The singularities in the three-dimensional laminar boundary layer on a cone at incidence are studied. It is shown that these singularities are formed in the outer part of the boundary layer and described by linear equations whose solutions are obtained in analytic form. The known results for the plane of symmetry are classified on this basis. Two solutions of the non-self-similar problem are found, one of which has a singularity at zero incidence and in the sink plane. The second branch goes over continuously into the solution for axisymmetric flow. However, as the angle of attack increases, in the sink plane a singularity is formed and all the self-similar solutions existing here lose their meaning. Starting from the critical angle of attack, the flow in the vicinity of the sink plane is no longer described by the boundary layer equations, so that the results can be used to construct an adequate physical model.Translated from Izvestiya Rossiiskoi Akademii Nauk, Mekhanika Zhidkosti i Gaza, No.6, pp. 25–33, November–December, 1993.     

Boundary layer in the vicinity of the stagnation point of a body of axial symmetry in a turbulent stream

The flow in the boundary layer in the vicinity of the stagnation point of a flat plate is examined. The outer stream consists of turbulent flow of the jet type, directed normally to the plate. Assumptions concerning the connection between the pulsations in velocity and temperature in the boundary layer and the average parameters chosen on the basis of experimental data made it possible to obtain an isomorphic solution of the boundary layer equations. Equations are obtained for the friction and heat transfer at the wall in the region of gradient flow taking into account the effect of the turbulence of the impinging stream. It is shown that the friction at the wall is insensitive to the turbulence of the impinging stream, while the heat transfer is significantly increased with an increase in the pulsations of the outer flow. These properties are confirmed by the results of experimental studies [1–4].Translated from Zhurnal Prikladnoi Mekhaniki i Tekhnicheskoi Fiziki, No. 5, pp. 83–87, September–October, 1973.     

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.     

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.     

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.     

Marginal separation in three-dimensional flows

The boundary layer in the vicinity of the zero skin-friction point on the leeward symmetry line of a prolate spheroid placed at an angle of attack is considered. The existence of this flow was established by Cebeci et al. (1980) for an angle of attack =40°. The current study is based on the results of Brown (1985) who described the marginal separation in the symmetry plane for a zero skin-friction point and on the results of Zametaev (1989) who included the spatial extension of Brown's solution but without interaction between the boundary layer and the outer flow. It is found that the three-dimensional boundary-layer equations in the vicinity of the zero skin-friction point are reduced to a single nonlinear partial differential equation of hyperbolic type which governs the longitudinal skin-friction component. Smooth solutions of this equation may be found which contain separation lines as well as double-valued regions. It is likely that the latter regions are related to the tip of the separation line obtained as a result of calculations of the full boundary-layer equations. The influence of interaction is also considered, in which case the flow is governed by a partial integro-differential equation. Numerical solutions are given for each of these problems.This study was supported by the United Technologies Research Center     

Study of laminar separation in the vicinity of the point of piecewise-constant pressure distribution over the wall

A family of two-dimensional divergent channels with piecewise-constant velocity and pressure distributions over the wall is considered. The method of matched asymptotic expansions is applied to study the two-dimensional viscous incompressible flow at high but subcritical Reynolds numbers in the vicinity of a pressure jump point on the channel wall. It is shown that if the pressure difference is of the order O(Re^?1/4), then in the vicinity of this point a classical region of interaction between the viscous boundary layer on the wall and the outer inviscid flow occurs. The problem formulated for the interaction region is solved numerically. The asymptotic values of the pressure difference corresponding to separationless flow are determined and the separation flow patterns are constructed.     

Regime diagram of the development of long-wave near-wall disturbances in a hypersonic boundary layer

A regime diagram of the development of slow near-wall disturbances induced by an unsteady self-induced pressure perturbation in a hypersonic boundary layer is constructed for a disturbance wavelength greater than the boundary layer thickness. It is shown that the main factors shaping the perturbed flow are the gas enthalpy near the body surface, the intensity of the viscous-inviscid interaction, and the nature (sub- or supersonic) of the main part of the boundary layer. Nonlinear boundary-value problems are formulated for regimes in which the near-wall boundary layer region plays a decisive role. Numerical and analytical solutions are obtained in the linear approximation. It is shown that intensification of the viscous-inviscid interaction or an increase in the role of the supersonic main region of the boundary layer impart generally supersonic properties to the main part of the boundary layer, i.e. the upstream propagation of the disturbances is damped and the disturbance growth downstream becomes more intense. Damping of the viscous-inviscid interaction and an increase in the role of the subsonic main part of the boundary layer have the opposite effect. Surface cooling increases the effect of the main part of the boundary layer on the formation of pressure disturbances and surface heating leads to an increase in the effect of the near-wall boundary layer region. It is also shown that for the regimes considered disturbances propagate in a direction opposite to that of the free stream from the turbulent flow region located downstream of the local disturbance development region.Translated from Izvestiya Rossiiskoi Academii Nauk, Mekhanika Zhidkosti i Gaza, No. 6, 2004, pp. 59–71. Original Russian Text Copyright © 2004 by Bogolepov and Neiland.     

Nonuniqueness of the solution of the problem of viscous interaction on an axisymmetric body

The article discusses solutions of the equations of the hypersonic boundary layer on an axisymmetric offset slender body (with a power exponent equal to 3/4), taking account of interactions with a nonviscous flow. It is shown that, in this case, the equations of the boundary layer have solutions differing from the self-similar solution corresponding to flow around a semi-infinite body. The solutions obtained are analogous to solutions for a strong interaction on a plate with slipping and triangular vanes [1–4], but are obtained over a wide range of values of the parameter of viscous interaction. An asymptotic solution is given to the problem with the approach to zero of the interaction parameter.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 5, pp. 41–47, September–October, 1973.The authors thank V. V. Mikhailova for discussion of the work and useful advice.     

Asymptotic solutions of the Navier-Stokes equations in regions with large local perturbations

There are many problems of the dynamics of viscous flows of liquids and gases at high Reynolds numbers for the solution of which the classical theory of the boundary layer cannot be used. This applies, in particular, to all the problems with various sorts of local singularities in the stream-flows in the vicinity of corners, in regions of interaction of the boundary layer with an incident shock, flows near points of separation or attachment of the stream, etc. The purpose of the present paper is to attempt the theoretical investigation of problems of this type on the basis of the general analysis of the asymptotic behavior of the solutions of the Navier-Stokes equations. In order to do this, use is made of the familiar method of the construction and splicing of a combination of asymptotic expansions representing the solutions in the various characteristic regions of the stream with viscosity decreasing without bound [1].As an example, detailed consideration is given to the problem of viscous supersonic flow near a wall with large local curvature of the surface.     

Investigation of Flow past a Flat Plate in the Strong Interaction Regime

Viscous gas flow past a finite-length plate in the strong interaction regime is investigated. The expansions of the flow functions in the vicinity of the leading edge are derived and the boundary value problems for the viscous and inviscid flow regions are formulated and jointly solved. The effect of the adiabatic exponent and the temperature factor on the flow parameters in these regions and the eigenvalue determining the intensity of the upstream disturbance transfer is studied. It is shown that in the non-self-similar case a transitional layer arises at the outer edge of the boundary layer.     

The coefficient of regeneration of the enthalpy in a three-dimensional laminar boundary layer

Self-similar solutions of the equations of a three-dimensional laminar boundary layer are of interest from two points of view. In the first place, they can be used to construct approximate calculating methods, making it possible to analyze several variants and to consider complex flows, in which it is impossible to neglect the interaction between the boundary layer and the external flow (for example, in the region of hypersonic interaction [1–3]). In the second place, the analysis of self-similar solutions permits clarifying the effect of individual parameters on one or another characteristic of the boundary layer and representing this effect in predictable form. One of the principal characteristics of a three-dimensional boundary layer, as also of a two-dimensional, is the coefficient of regeneration of the enthalpy. The value of this coefficient is needed for determining the temperature of a thermally insulated surface, as well as for finiing the real temperature (or enthalpy) head, which determines the value of the heat flux from a heated gas to the surface of the body around which the flow takes place. The article presents the results of calculations of the coefficient of regeneration of the enthalpy for locally self-similar solutions of the equations of a three-dimensional boundary layer, forming with flow around a cylindrical thermally insulated surface at an angle. It is clarified that the dependence of the coefficient of regeneration of the enthalpy on the determining parameters is not always continuous.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 1, pp. 60–63, January–February, 1973.     

Self-similar solutions of non-Newtonian fluid boundary layer in MHD

The self-similar solutions of the boundary layer for a non-Newtonian fluid in MHD were considered in [1, 2] for a power-law velocity distribution along the outer edge of the layer and constant electrical conductivity through the entire flow. However, the MHD flows of many conducting media, which are solutions or molten metals, cannot be described by the MHD equations for non-Newtonian fluids.The self-similar solutions of the boundary layer for a non-Newtonian fluid without account for interaction with the electromagnetic field were studied in [3].In the following we present the self-similar solutions for the boundary layer of pseudoplastic and dilatant fluids with account for the interaction with an electromagnetic field for the case of a power-law velocity distribution along the outer edge of the layer, when the conductivity of the fluid is constant throughout the flow and the magnetic Reynolds number is small.Izv. AN SSSR. Mekhanika Zhidkosti i Gaza, Vol. 2, No. 6, pp. 77–82, 1967The author wishes to thank S. V. Fal'kovich for his interest in this study.     

Application of unsteady mixing equations to some aerodynamic problems

Tangential discontinuities [1] are introduced in solving several transient and steady-state problems of gas dynamics. These discontinuities are unstable [2] as a result of the effects of viscosity and thermal conductivity. Therefore it is advisable to replace the tangential discontinuity by a mixing region and account for its interaction with the inviscid flows, establishing on the boundaries of this region the conditions of vanishing friction stress and equality of the velocity and temperature components to the corresponding velocity and temperature components of the inviscid flows. This formulation improves the accuracy of the solution of such problems by posing them as problems with irregular reflection and intersection of shock waves [1].The consideration of the interaction of unsteady turbulent mixing regions with the inviscid flow also permits the formulation of several problems in which the effects of viscosity lead to complete rearrangement of the flow pattern (the lambda-configuration) with the interaction of the reflected shock wave with the boundary layer in the shock tube [3,4], the formation of zones of developed separation ahead of obstacles, etc.).In this connection, §1 presents an analysis of the self-similar solutions of the unsteady turbulent mixing equations (a corresponding analysis of the laminar mixing equations which coincide with the boundary layer equations is presented in [1]). It is shown that these self-similar solutions describe, along with the several problems noted above, the problems of the formation of steady jets and mixing zones in the base wake.As an example, §2 presents, within the framework of the proposed schematization, an approximate solution of the problem of the interaction of a shock wave reflected from a semi-infinite wall with the boundary layer on a horizontal plate behind the incident shock wave. The results obtained are applied to the analysis of reflection in a shock tube. Computational results are presented which are in qualitative agreement with experiment [3, 4].     

Development of slow disturbances in a hypersonic boundary layer

The mechanisms of development of slow time-dependent disturbances in the wall region of a hypersonic boundary layer are established and a diagram of the disturbed flow patterns is plotted; the corresponding nonlinear boundary value problem is formulated for each of these regimes. It is shown that the main factors that form the disturbed flow are the gas enthalpy near the body surface, the local viscous-inviscid interaction level, and the type, either subsonic or supersonic, of the boundary layer as a whole. Numerical and analytical solutions are obtained in the linear approximation. It is established that enhancement of the local viscous-inviscid interaction or an increased role for the main supersonic region of the boundary layer makes the disturbed flow by and large “supersonic”: the upstream propagation of the disturbances becomes weaker, while their downstream growth is amplified. Contrariwise, local viscous-inviscid interaction attenuation or an increased role for the main subsonic region of the boundary layer has the opposite effect. Surface cooling favors an increased effect of the main region of the boundary layer while heating favors an increased wall region effect. It is also found that in the regimes considered disturbances travel from the turbulent flow region downstream of the disturbed region under consideration counter to the oncoming flow, which may be of considerable significance in constructing the nonlinear stability theory.     

On the use of a two-layer model for large-eddy simulations of supersonic boundary layers with separation

The two-layer modeling approach has become one of the most promising and successful methodology for simulating turbulent boundary layers in the past ten years. In the present study, a mixed wall model for large-eddy simulations (LES) of high-speed flows is proposed which combine two approaches; the thin-Boundary Layer Equations (TBLE) model of Kawai and Larsson (1994) and the analytical wall-layer model of Duprat et al. (2011) for streamwise pressure gradients. The new hybrid model has been efficiently implemented into a three-dimensional compressible LES solver and validated against DNS of a spatially-evolving supersonic boundary layer (BL) under moderate and strong pressure gradients, before being employed for the prediction of nozzle flow separations at different flow conditions, ranging from weakly to highly over-expanded regimes. A good agreement is obtained in terms of mean and fluctuating quantities compared to the DNS results. Particularly, the current wall-modeled LES results are found to perfectly match the DNS data of supersonic BL with/out pressure gradient. It is also shown that the model can account for the effect of the large-scale turbulent motions of the outer layer, indicating a good interaction between the inner and the outer part of the wall layer. In terms of simulations costs and improvements of computing power, the obtained results highlight the capability of the current wall-modeling LES strategy in saving a considerable amount of computational time compared to the wall-resolved LES counterpart, allowing to push further the simulations limits. Furthermore, the application of these computationally low-costly LES simulations to nozzle flow separation allows to clearly identify the origin of the shock unsteadiness, and the existence of broadband and energetically-significant low-frequency oscillations (LFO) in the vicinity of the separation region.     

Viscoelastic sink flow in a wedge for the UCM and Oldroyd-B models

The steady planar sink flow through wedges of angle π/α with α≥1/2 of the upper convected Maxwell (UCM) and Oldroyd-B fluids is considered. The local asymptotic structure near the wedge apex is shown to comprise an outer core flow region together with thin elastic boundary layers at the wedge walls. A class of similarity solutions is described for the outer core flow in which the streamlines are straight lines giving stress and velocity singularities of O(r⁻²) and O(r⁻¹), respectively, where r1 is the distance from the wedge apex. These solutions are matched to wall boundary layer equations which recover viscometric behaviour and are subsequently also solved using a similarity solution. The boundary layers are shown to be of thickness O(r²), their size being independent of the wedge angle. The parametric solution of this structure is determined numerically in terms of the volume flux Q and the pressure coefficient p₀, both of which are assumed furnished by the flow away from the wedge apex in the r=O(1) region. The solutions as described are sufficiently general to accommodate a wide variety of external flows from the far-field r=O(1) region. Recirculating regions are implicitly assumed to be absent.     

Similarity Solution of Unsteady Boundary Layer Equations with a Magnetic Field

The unsteady laminar incompressible boundary layer flow of an electricallyconducting fluid in the stagnation region of two-dimensional and axisymmetricbodies with an applied magnetic field has been studied. The boundary layerequations which are parabolic partial differential equations with threeindependent variables have been reduced to a system of ordinary differential equations by using suitable transformations and then solved numerically using a shooting method. Here, we have obtained new solutions which are solutions of both the boundary layer and Navier-Stokes equations.     

Newtonian and non-Newtonian liquids rotating adjacent to a stationary surface

The interaction of a rotating flow and a stationary surface is discussed for a second-order non-Newtonian liquid. Similarity solutions of the governing partial differential equations are obtained for the case of the outer flow in solid-body rotation. The results for the Newtonian case are compared with Bödewadt's series solution of this problem. The non-Newtonian solutions indicate that for certain values of the parameters characterizing the non-linear viscous response and normal stress effects a larger secondary flow is induced in the boundary layer than in the Newtonian case.Also at North Carolina State University Raleigh (N.C.), U.S.A.