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.     

Effect of turbulence of external flow on the turbulent boundary layer at a plate

In [1, 2] turbulence of the external flow was taken into account by specifying the turbulent energy at the external boundary of the boundary layer on integrating the energy-balance equation for the turbulence. In [3] a special correction that allowed the turbulence of the external flow to be taken into account was introduced in determining the mixture path. In [4, 5] the turbulent energy calculated from the energy-balance equation of the turbulence was added to the energy induced by turbulence of the external flow, the energy distribution of the induced turbulence being specified using an empirically selected function. In [6, 7] a method of taking into account the effect of turbulence of the external flow on a layer of mixing and a jet was proposed. In the present work, this method is applied to the boundary layer at a plate.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 3, pp. 26–31, May–June, 1977.     

Interaction of a submerged turbulent jet with a solid wall

The turbulent layer of a wall jet has been analyzed in many theoretical and experimental studies [1–10]. Most theoretical investigations are based on the simultaneous solution of the equations of the turbulent jet and the boundary layer that forms at the wall. The differences lie in the methods used to correlate the velocity and temperature distributions, as well as in the friction and heat-transfer laws employed. In this article we present a method based on the further development of the idea of conservation of the laws governing wall turbulence with respect to change in boundary conditions.     

Effect of external turbulence on the development of a turbulent jet

The effect of external turbulent agitation on jet development has been investigated in [1–3]. The difference of the method employed in the present work lies in the assumption that the turbulence scale of the external flow is substantially larger than the turbulence scales in either the jet or the mixing layer. Utilizing this assumption, it becomes possible to solve separately the energy equations for the turbulence of the external flow and of the jet. Solutions obtained on the basis of this assumption are found to be in qualitative agreement with experimental data.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 1, pp. 24–29, January–February, 1977.     

Some problems in the theory of plane jets of conducting fluid

Using the boundary-layer equations as a basis, the author considers the propagation of plane jets of conducting fluid in a transverse magnetic field (noninductive approximation).The propagation of plane jets of conducting fluid is considered in several studies [1–12]. In the first few studies jet flow in a nonuniform magnetic field is considered; here the field strength distribution along the jet axis was chosen in order to obtain self-similar solutions. The solution to such a problem given a constant conductivity of the medium is given in [1–3] for a free jet and in [4] for a semibounded jet; reference [5] contains a solution to the problem of a free jet allowing for the dependence of conductivity on temperature. References [6–8] attempt an exact solution to the problem of jet propagation in any magnetic field. An approximate solution to problems of this type can be obtained by using the integral method. References [9–10] contain the solution obtained by this method for a free jet propagating in a uniform magnetic field.The last study [10] also gives a comparison of the exact solution obtained in [3] with the solution obtained by the integral method using as an example the propagation of a jet in a nonuniform magnetic field. It is shown that for scale values of the jet velocity and thickness the integral method yields almost-exact values. In this study [10], the propagation of a free jet is considered allowing for conduction anisotropy. The solution to the problem of a free jet within the asymptotic boundary layer is obtained in [1] by applying the expansion method to the small magnetic-interaction parameter. With this method, the problem of a turbulent jet is considered in terms of the Prandtl scheme. The Boussinesq formula for the turbulent-viscosity coefficient is used in [12].This study considers the dynamic and thermal problems involved with a laminar free and semibounded jet within the asymptotic boundary layer, propagating in a magnetic field with any distribution. A system of ordinary differential equations and the integral condition are obtained from the initial partial differential equations. The solution of the derived equations is illustrated by the example of jet propagation in a uniform magnetic field. A similar solution is obtained for a turbulent free jet with the turbulent-exchange coefficient defined by the Prandtl scheme.     

Effect of surface temperature on the stability of the swept attachment line boundary layer

At fairly high Reynolds numbers instability may develop on the line of attachment of the potential flow to the leading edge of a swept wing and lead to a transition to boundary layer turbulence directly at the leading edge [1, 2]. Although the first results relating to the stability and transition of laminar flow at the leading edge of swept wings were obtained almost 30 years ago, the problem remains topical. The stability of the swept attachment line boundary layer was recently investigated numerically with allowance for compressibility effects [3, 4]. Below we examine the effect of surface temperature on the stability characteristics of the laminar viscous heat-conducting gas flow at the leading edge of a side slipping wing.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 6, pp. 78–82, November–December, 1990.     

Flow of viscous gas out of a cylindrical channel into vacuum

A study is made of the exhausting of a jet of viscous gas from a cylindrical channel into vacuum in the presence of a flat bounding surface outside the channel in the plane of its exit section. The problem is solved numerically using the complete system of Navier—Stokes equations. The developed flow model makes it possible to take into account the influence of an external medium into which the jet exhausts on the structure of the flow in the exit section of the channel, and also the influence of the subsonic part of the boundary layer in the channel on the flow field of the jet.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 1, pp. 122–128, January–February, 1981.     

Investigation of the microstructure a turbulent jet in a wake

The study of the characteristics of the turbulence in the boundary layer and in free jets is one of the most important problems of the aerodynamics of viscous fluids. The accumulation of information on the pulsation characteristics of jet flows and the establishment of the corresponding governing laws may serve to verify the basic hypotheses of the semiempirical theories of turbulence, and also for the development of more advanced computational methods. In many cases the measurement of the pulsation characteristics of turbulent jets is of practical interest.The studies made up till now [1–5] of the microstructure of turbulent flow in the primary region of submerged axisymmetric jets have made it possible to obtain several interesting results. In particular, in addition to the average velocity profiles, hot-wire anemometric equipment has been used to measure the normal and tangential Reynolds stresses and also the intermittency factor in cross sections of the jet, the distribution of the intensity of the longitudinal and lateral velocity pulsations along the axis, the correlation coefficients and the corresponding integral turbulence scales, etc. These measurements have made it possible to draw several important conclusions on the mechanism of turbulent exchange, on the order of the terms omitted in the equation of motion, and on the semiempirical theories of turbulence [6–9].The common deficiency of the studies mentioned above is that near the boundary of a submerged jet, where the average velocity is practically equal to zero, the intensity of the pulsations is so great that it makes the reliability of the results obtained by means of the hotwire anemometer questionable. In this connection Townsend [6] indicated the advisability of studying the microstructure of a turbulent jet issuing into a low-velocity ambient flow.The present study had as its objective the investigation of the microstructure of the primary region of an axisymmetric jet in a wake flow over quite a broad range of the flow ratio parameter m=u/u₀;here u₀ is the average velocity at the nozzle exit, u is the velocity of the ambient stream. For various values of the parameter m in the primary region of the jet measurements were made of the profiles of the three components of the pulsation velocity and the Reynolds shear stresses, and also the values of the average velocity and two components of the pulsation velocity at a large number of points on the jet axis. The measured profiles of the Reynolds shear stresses were compared with the corresponding profiles calculated on the basis of the boundary layer equations from the experimentally determined average velocity profiles. For two values of the parameter m, in one of the sections of the jet measurements were made of the correlation coefficients of the longitudinal components of the pulsation velocity and the variation across the jet of the integral turbulence scale was determined.The results obtained give an idea of the influence of the parameter m on the characteristics of the turbulent jet in an ambient stream.     

Propagation of an ion jet near a dielectric surface

Difficulties in determining experimentally the local electrical parameters of unipolar-charged jets are arousing interest in the theoretical investigation of electrogasdynamic (EGD) flows. Free EGD jets were examined, for example, in [1–3]. In order to control the charge on the dielectric parts of aircraft surfaces, which results from their static electrification and may have certain negative consequences [4], and, moreover, to influence the flow in the boundary layer use is being made of unipolar-charged jets propagating near the dielectric [5, 6]. In [6] the case of an ion jet near a dielectric surface possessing surface conductivity was investigated. In these circumstances it is possible to neglect charge diffusion, which considerably simplifies the problem. Space charge diffusion was taken into account in [7], but subject to certain very important simplifications. The author has calculated the electrical parameters of a unipolar-charged jet propagating in a viscous incompressible gas near an ideal dielectric plate, with allowance for surface and polarization charges and, moreover, the diffusion processes near the surface. An asymptotic solution is obtained for the equations of the ionic diffusion layer as the ratio of the thickness of the diffusion layer to the thickness of the hydrodynamic boundary layer tends to zero.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 5, pp. 174–180, September–October, 1984.The author is grateful to V. V. Mikhailov and A. V. Kazakov for valuable advice and comments.     

Nonlinear Disturbances and Weak Discontinuities in a Supersonic Boundary Layer

The processes of wave disturbance propagation in a supersonic boundary layer with self-induced pressure [1–4] are analyzed. The application of a new mathematical apparatus, namely, the theory of characteristics for systems of differential equations with operator coefficients [5–8], makes it possible to obtain generalized characteristics of the discrete and continuous spectra of the governing system of equations. It is shown that the discontinuities in the derivatives of the solution of the boundary layer equations are concentrated on the generalized characteristics. It is established that in the process of flow evolution the amplitude of the weak discontinuity in the derivatives may increase without bound, which indicates the possibility of breaking of nonlinear waves traveling in the boundary layer.     

Some problems of turbulent electrohydrodynamic jets

The problem of an axisymmetric turbulent electrohydrodynamic jet exhausting from a nozzle into an interelectrode gap is formulated. A numerical method of integrating the system of equations describing this flow is developed. This method is used to investigate three-dimensional effects in the jet (expansion of the jet, reverse flows). The influence on the jet characteristics (currents of the charge carried out of the nozzle, jet diameter, etc.) of the geometrical and electrical parameters and also of purely hydrodynamic factors (level of turbulence, relative velocity of parallel flow, etc.) is investigated.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 5, pp. 144–149, September–October, 1980.     

Solution to the problem of a swirling jet from an annular orifice

The problem of the propagation of a laminar immersed fan jet with swirling was considered in [1–3]. In [1], the jet source scheme was used to find a self-similar solution for a weakly swirling jet. An attempt to solve by an integral method the analogous problem for a jet emanating from a slit of finite size was made in [2]. In [3], the equations of motion for a jet with arbitrary swirling were reduced under a number of assumptions to the equations that describe the flow of a flat immersed jet. This paper gives the numerical solution to the problem of the propagation of a radial jet emanating with arbitrary swirling from a slit of finite size and an analytic solution for the main section of the jet.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 2, pp. 49–54, March–April, 1991.     

Aerodynamics of end-wall boundary layer in a vortex chamber

The need for the inclusion of end-wall boundary layers in the study of the aerodynamics of vortex chambers has been frequently mentioned in the literature. However, owing to limited experimental data [1–3] with reliable information on the wall layers, the existing computational methods for end-wall boundary layers are not well-founded. The question of which parameters determine the formation of end-wall flow remains debatable. In some studies [4, 5], the vortex chambers are conditionally divided into short and long chambers. However, there is no unique opinion on the role of end-wall flows in vortex chambers of different lengths. It has also not been established for what geometric and flow parameters the chamber could be considered long or short. In the present study, as in [1, 5–8], solution is obtained for the end-wall boundary-layer equations using integral methods, considering the boundary layer in the radial direction in the form of a submerged wall jet. Such an approach made it possible to use the laws for the development of wall jets [9], and obtain fairly simple relations for integral parameters, skin friction, mass flow in the boundary layer, and other characteristics. Results are compared with available experimental data and computations of others authors; turbulent flow is considered; results for laminar boundary layer are given in [10].Translated from Zhurnal Prikladnoi Mekhaniki i Tekhnicheskoi Fiziki, No. 5, pp. 117–126, September–October, 1986.     

Singularity in the solution of the equations of a three-dimensional boundary layer due to the collision of wall jets

The problem of the propagation of a three-dimensional jet of viscid incompressible fluid flowing from a narrow curved slot into a fluid-filled space along a rigid plane is considered within the framework of the equations of a steady laminar boundary layer. A class of initial conditions at the slot outlet which generates in the jet a velocity field without secondary flows is identified. Within this class the boundaryvalue problem for the three-dimensional boundary layer can be divided into two problems of lower dimensionality: a dynamic and a kinematic problem. As a result of the analysis of the kinematic problem the general structure of the regions of existence and uniqueness of the solution is determined. An investigation of the dynamic problem shows that as the boundaries of the region of existence are approached a singularity characterized by an infinite increase in the thickness of the jet is formed in the solution of the boundary layer equations.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 4, pp. 75–81, July–August, 1991.     

Effect of moving wall on the three-dimensional separation of an incompressible fluid in front of an obstacle

The problem of the laminar boundary layer formed on the surface of a semiinfinite plate with a perpendicular semi-infinite circular cylinder in a uniform steady incompressible flow normal to the leading edge is considered. Near its sharp edge the plate has a stationary part and, located at a finite distance further downstream, a part of the surface moving downstream at a constant velocity. The first-order boundary layer equations are solved numerically by an implicit finite-difference method. The effect of the moving wall on the variation of the dimensions of the separation zone ahead of the obstacle over a broad range of the governing parameters and flow characteristics is investigated. The flow in the laminar boundary layer on the surface of a plate ahead of such an obstacle was calculated in [1, 2] without motion of the wall. Data on the structure of the separated flow are given in [3].Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 6, pp. 49–53, November–December, 1990.     

Computation of base flow characteristics with uniform blowing

Blowing at bluff body base was considered under different conditions and for small amount of blowing this problem was solved using dividing streamline model [1]. The effect of supersonic blowing on the flow characteristics of the external supersonic stream was studied in [2–4]. The procedure and results of the solution to the problem of subsonic blowing of a homogeneous fluid at the base of a body in supersonic flow are discussed in this paper. Analysis of experimental results (see, e.g., [5]) shows that within a certain range of blowing rate the pressure distribution along the viscous region differs very little from the pressure in the free stream ahead of the base section. In this range the flow in the blown subsonic jet and in the mixing zones can be described approximately by slender channel flow. This approximation is used in the computation of nozzle flows with smooth wall inclination [6, 7]. On the other hand, boundary layer equations are used to compute separated stationary flows with developed recirculation regions [8] in order to describe the flow at the throat of the wake. The presence of blowing has significant effect on the flow structure in the base region. An increasing blowing rate reduces the size of the recirculation region [9] and increases base pressure. This leads to a widening of the flow region at the throat, usually described by boundary-layer approximations. At a certain blowing rate the recirculation region completely disappears which makes it possible to use boundary-layer equations to describe the flow in the entire viscous region in the immediate neighborhood of the base section.Translated from Zhurnal Prikladnoi Mekhaniki i Tekhnicheskoi Fiziki, No. 1, pp. 76–81, January–February, 1984.     

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.     

Horizontal submerged jet in a stratified fluid

The problem of jet flow excited in a viscous density-stratified fluid by a point source of momentum acting horizontally is considered. Simplified asymptotic equations are obtained in the boundary layer approximation. It is shown that the vertical velocity component is small and the motion in the jet has a layered structure. The longitudinal velocity distributions in the jet are measured experimentally. It is shown that these distributions are affine and can be satisfactorily approximated by Schlichting's well-known boundary layer solution for a round submerged jet in a fluid uniform with respect to density.Translated from Izvestiya Rossiiskoi Akademii Nauk, Mekhanika Zhidkosti i Gaza, No.6, pp. 10–16, November–December, 1993.We are grateful to I. A. Filippov for assisting with the experiments.     

A spatial laminar boundary layer on a streamline in conical external flow of a uniform gas in the absence of sources and sinks

Theoretical study of a three-dimensional laminar boundary layer is a complex problem, but it can be substantially simplified in certain particular cases and even reduced to the solution of ordinary differential equations.One such particular case is the flow of a compressible gas on a streamline in conical external flow. The case is of considerable practical importance because the local heat fluxes may take extremal values on such lines.Such flow, except for the conical case, has been examined [1–4], and an approximate method has been given [1] on the basis of integral relationships and a special form for the approximating functions. A numerical solution has been given [2, 3] for such flow around an infinite cylinder. It was assumed in [1–3] that the Prandtl number and the specific heats were constant, and that the dynamic viscosity was proportional to temperature. Heat transfer has been examined [4] near a cylinder exposed to a flow of dissociated air.Here we give results from numerical solution of a system of ordinary differential equations for the flow of a compressible gas in a laminar boundary layer on streamlines in conical external flow, with or without influx or withdrawal of a homogeneous gas. It is assumed that the gas is perfect and that the dynamic viscosity has a power-law temperature dependence.     

Interaction of a turbulent jet and a cocurrent confined incompressible flow

The effects induced in a coaxial circular channel flow by an axisymmetric turbulent jet are investigated for various values of the velocity and radius ratios 0.16m<1 and 2.5f30.9. The problem is solved by means of an e-L model of turbulence [1, 2]. The calculation scheme differs from the usual one for boundary layers, jets and wakes in that the pressure p is assumed to be unknown and is determined by assigning the boundary conditions for the radial velocity component and the transverse gradient of the longitudinal velocity component on both boundaries. On the basis of the calculations and the experimental data of [3, 4] generalized relations are obtained. These make it possible to estimate the turbulence characteristics of an axisymmetric jet in a confined cocurrent flow when the pressure is variable along the flow.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 5, pp. 14–19, September–October, 1986.In conclusion, the author wishes to thank G. S. Glushko for constructive discussion of the results and useful advice.