Development of laminar jet flows with zero excess impulse

The problem of the development of a laminar jet of viscous incompressible fluid with zero excess impulse (the wake of a hydrodynamic motor) was investigated for the first time by Birkhoff and Zarantonello [1], who found a self-similar solution to the dynamical problem for the case of a two-dimensional laminar wake. The problem of the development of turbulent wakes of hydrodynamic motors in the near and far flow regions was solved by Ginevskii [2] on the basis of an integral method. In the present paper, the method of asymptotic expansions is used on the basis of the boundary layer equations to solve nonself-similar problems of the development of laminar jet flows of a viscous incompressible fluid with zero excess impulse. The obtained solution takes into account the influence of the details of the source (finite size of the body and its geometry) and the value of the Prandtl number on the velocity and temperature distribution. In the case of a laminar axi-symmetric wake, a self-similar solution is obtained to the thermal problem, the solution being valid in a wide range of Prandtl numbers.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 3, pp. 27–33, May–June, 1984.     

Computation of the two-dimensional viscous incompressible fluid flow with separation at the nlet edge around a cascade

A complex flow consisting of an outer inviscid stream, a dead-water separation domain, and a boundary layer, which interact strongly, is formed in viscous fluid flows with separation at the streamlined profile with high Re numbers. Different jet and vortex models of separation flow are known for an inviscid fluid; numerical, asymptotic, and integral methods [1–3] are used for a viscous fluid. The plane, stationary, turbulent flow through a turbine cascade by a constant-density fluid without and with separation from the inlet edge of the profile and subsequent attachment of the stream to the profile (a short, slender separation domain) is considered in this paper.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 3, pp. 34–44, May–June, 1978.     

Use of the turbulence energy equation in the theory of jet flows

In this study some of the assumptions introduced in [1] in developing a closed system of equations for a turbulent boundary layer will be simplified. With the aid of the system of equations of [1], a theoretical solution is found for the problem of a jet in an accompanying flow, it being assumed that the structure of the jet turbulence depends solely on local conditions. Experiment has shown that the turbulence in such a jet does depend also on the prehistory of the flow. At large distances from the source, the theoretical characteristics of the jet agree well with the experimentally determined characteristics of the wake beyond a body. Also examined is the problem of the boundary layer between two homogeneous flows, flowing with different velocities.Translated from Izvestiya Akademii Nauk SSSR. Mekhanika Zhidkosti i Gaza, No. 2, pp. 75–81, March–April, 1973.     

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.     

Elliptic cylinder in inviscid shear flow

We examine the flow of a plane parallel inviscid stream about an elliptic contour. There is vorticity far ahead of the body because of nonuniformity of the velocity profile. In the case of a small vorticity parameter the velocity profile will be parabolic. In contrast with [1] and [2], we assume the existence of additional circulational flow around the contour. The magnitude of this flow circulation is determined from the condition under which the flow leaves the trailing edge of the body (the analog of the Chaplygin-Zhukovskii postulate in potential flow).The results obtained in this study can be used, in particular, to evaluate the flow past a two-dimensional body in the wake behind another body.The author wishes to thank G. A. Dombrovskii for his interest in this study.     

Development of a two-dimensional turbulent jet in an infinite space

The second and third terms in the asymptotic expansion of the stream function in the nonsimilar problem of the development of a two-dimensional turbulent jet in an unbounded space are found in final form. Results of experimental investigations of free turbulent jets are cited, and the effect of the initial velocity profile on the aerodynamic characteristics of the jet is considered. The problem of the development of a two-dimensional turbulent jet in an unbounded space has been considered in [1–3]. The existing solution is similar, and is valid only at a sufficiently large distance from the slit. Allowance for the finite dimensions of the slit leads to a nonsimilar problem. The papers [4–6] are devoted to the experimental investigation of the free two-dimensional turbulent jet.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 4, pp. 137–142, July–August, 1971.     

Propagation of a laminar jet of electrically conducting fluid in a uniform magnetic field

Several papers [1–4] have considered the propagation of a plane laminar jet of incompressible conducting fluid in a uniform magnetic field for magnetic Reynolds numbers much less than unity. These papers have investigated the flow of a free jet in a transverse magnetic field for small values of the magnetic interaction parameter. Equations for the first approximations were obtained in [1, 2] by a series expansion in the small interaction parameter close to the ordinary solution (without magnetic field) for the jet. The equations for the zero-th and first approximations were integrated in [3]. The same author also found a similar solution for a turbulent jet, the turbulent transfer coefficient being chosen according to Prandtl's method. As regards the solution found in [4], it suffers from the defect that the constant of integration which connects the real velocity profiles with those found in the paper remains undetermined. The present paper gives an approximate solution of the same dynamic problem of the propagation of a free plane jet in a uniform field, no assumption being made as to the smallness of the interaction parameter. In order to do this the integral method of solution, common in ordinary hydrodynamics [5, 6] is employed. The solution of the problem is generalized to include the case of a finite value of the Hall parameter.     

Calculation of aerodynamic characteristics of a wing profile with discharge of a controllable jet into the external flow

A wing profile of infinite span, whose lower surface is replaced by a system of guide vanes, is placed in a flow of an ideal incompressible fluid. Fluid flows out through the system of guide vanes from the internal cavity of the wing into the external stream, forming a jet in the wake (Fig. 1). The total pressure in the wing cavity and in the jet differs from the total pressure in the outer free stream. The jet boundaries are streamlines extending to infinity, along which there is a discontinuity of the velocity value. The flow of fluid in the internal wing cavity is simulated by a flow caused by a system of suitably located sources, and the system of guide vanes is replaced by discrete vortices.The form of the profile arc is selected so that the fluid flow from the sources in the direction which is nearly opposite the direction of the freestream velocity is restrained by the segment of the contour with high curvature in the vicinity of the leading edge. We consider the flow regime about the profile with an exhausting jet for which the two ends of the arc the points of detachment of the stream and the velocity discontinuity line (profile arc, jet boundary) is a smooth curve, which imposes an additional condition on the magnitude of the circulation. As the model for the study of the flow about a profile with jet blowing we take the arc of a logarithmic spiral.Formulas are obtained for determining the over-all characteristics of the stream forces acting on the profile in the presence of the jet and the total pressure discontinuity. On the basis of the calculations made for a thin wing a qualitative analysis is made for the stream force acting on the profile.The authors wish to thank S. A. Khristianovich for formulating the problem and for his advice.     

Cavity detachment on a hydrofoil with the inclusion of surface tension effects

The non-linear problem of cavity flow past a hydrofoil is considered with taking into account fluid viscosity in the cavity closure region and surface tension, which affect the cavity detachment. The theoretical model is based on the concept of viscous–inviscid interaction between the outer inviscid cavity flow and the inner turbulent separated flow downstream of the cavity. The outer inviscid flow is solved by constructing the complex flow potential, and the wake model is based on the method of integral relationships for separated turbulent flows. The obtained numerical results are compared with experimental data.     

The problem of a gas bubble in plane ideal fluid flow

We study the problem of two-dimensional fluid flow past a gas bubble adjacent to an infinite rectilinear solid wall.Two-dimensional ideal fluid flow past a gas bubble on whose boundary surface-tension forces act (or a gas bubble bounded by an elastic film) has been studied by several authors. Zhukovskii, who first studied jet flows with consideration of the capillary forces, constructed an exact solution of the problem of symmetric flow past a gas bubble in a rectilinear channel [1]. However, Zhukovskii's solution is not the general solution of the problem; in particular, we cannot obtain the flow past an isolated bubble from his solution. Slezkin [2] reduced the problem of symmetric flow of an infinite fluid stream past a bubble to the study of a nonlinear integral equation. The numerical solution of this problem has recently been found by Petrova [3]. McLeod [4] obtained an exact solution under the assumption that the gas pressure p₁ in the bubble equals the flow stagnation pressure p₀. Beyer [5] proved the existence of a solution to the problem of flow of a stream having a given velocity circulation provided p₁p₀.We examine the problem of two-dimensional ideal fluid flow past a gas bubble adjacent to an infinite rectilinear solid wall. The solution depends on the value of the contact angle . The existence of a solution is proved in some range of variation of the parameters, and a technique for finding this solution is given. The situation in which =1/2 is studied in detail.     

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.     

Rotational inviscid flow with circulation zones in a channel of variable cross section

The development of the theory of rotational motion of inviscid fluids for the purposes of describing channel flow encounters certain difficulties in connection with the appearance of viscosity effects near the walls. In the potential-rotational model [1], in which the vorticity is nonzero only in a closed circulation zone surrounded by potential flow, it is assumed that the separation and attachment points are known in advance. For example, for flow around a cavity these points coincide with the extreme corner points of the contour. The problem of determining the vorticity in a closed zone for the potential-rotational model has been investigated in a number of studies [2, 3], etc. In the case of an incompressible fluid the vorticity in the circulation zone is constant for two-dimensional flow and proportional to the distance from the axis for axisymmetric flow. The value of the constant is found from the steady-state condition for the adjoining viscous layers. If the channel walls have a smooth profile without corner points, then for determining the boundaries of the circulation zones additional conditions must be used. This study employs another scheme, in which the vorticity is formed outside the region of flow and in a particular problem is specified in the form of a boundary condition. An analytic solution describing the rotational flow of an inviscid fluid in a channel with a slightly varying cross section is obtained. Three types of entrance flow nonuniformity are considered: 1) uniform shear flow, 2) wake-type flow, and 3) potential flow with a narrow wall boundary layer. Streamline patterns with circulation zones are constructed for flows in diffuser channels with the above-mentioned types of entrance nonuniformity. A model of flow separation in a channel with a turbulent boundary layer on the walls is discussed.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 2, pp. 31–37, March–April, 1985.In conclusion the author wishes to thank E. Yu. Shal'man, A. N. Kraiko, and A. B. Vatazhin for useful discussions and advice.     

Steady vortex structures in inviscid flows in channels

The problem of inviscid incompressible flow through a rectangular channel is considered under the normal velocity component given at the chanel boundaries and the fluid vorticity at the entry. Nonconventional steady flows with recirculation zones and different functional dependences of the vorticity on the stream function are found by analytical and numerical methods. The stability of the steady regimes obtained is analyzed using the numerical solution of the time-dependent problem by the particle-in-cells method.     

Calculation of hydrogen-air combustion behind detached shock wave for supersonic flow past a sphere

Many of the published theoretical studies of quasi-one-dimensional flows with combustion have been devoted to combustion in a nozzle, wake, or streamtube behind a normal shock wave [1–6].Recently, considerable interest has developed in the study of two-dimensional problems, specifically, the effective combustion of fuel in a supersonic air stream.In connection with experimental studies of the motion of bodies in combustible gas mixtures using ballistic facilities [7–9], the requirement has arisen for computer calculations of two-dimensional supersonic gas flow past bodies in the presence of combustion.In preceding studies [10–12] the present author has solved the steady-state problem under very simple assumptions concerning the structure of the combustion zone in a detonation wave.In the present paper we obtain a numerical solution of the problem of supersonic hydrogen-air flow past a sphere with account for the nonequilibrium nature of eight chemical reactions. The computations encompass only the subsonic and transonic flow regions.The author thanks G. G. Chernyi for valuable comments during discussion of the article.     

Plate cascade in subsonic unsteady gas flow

Accounting for fluid compressibility creates serious difficulties in solving the problem of oscillations of a grid of thin, slightly curved profiles in a subsonic stream. The problem has been solved in [1–3] for a widely-spaced cascade without stagger whose profiles oscillate in phase opposition. The phenomenon of aerodynamic (acoustic) resonance, which may arise in a grid in the direction transverse to the stream for definite values of the stream velocity and profile oscillation frequency, was discovered in [2]. An approximate solution of the problem in which account is not taken of the effect of the vortex trails on the gas flow has been obtained in [4]. In [5, 6] Meister studied in the exact linear formulation the problem of unsteady gas motion through an unstaggered cascade of semi-infinite plates. In [7] Meister considered a grid of profiles with finite chords, but the problem solution was not completed. The problem of subsonic gas flow through a staggered lattice whose profiles oscillate following a single law with constant phase shift was solved most completely in the studies of Kurzin [8, 9] using the method of integral equations. A method of solving the problem for the case of arbitrary harmonic oscillation laws for the lattice profiles was indicated in [10]. The results of the calculation of the unsteady aerodynamic forces for the particular case of a plate cascade without stagger are presented in [9,11], and the possibility of the occurrence of aerodynamic resonance in the cascade in the directions transverse to and along the stream is indicated.Another method of solving the problem is given in [12], in which the more general problem of unsteady subsonic gas flow through a three-dimensional cascade of plates is solved. In the present study this method is applied to the solution of the problem of oscillations of staggered plate cascades in a two-dimensional subsonic gas flow. The results are presented of an electronic computer calculation of the unsteady aerodynamic characteristics of the cascade profiles, which show the essential influence of fluid compressibility on these characteristics. In particular, a sharp decrease of the aerodynamic damping in the acoustic resonance regimes is obtained.     

Lattice of plates in an unsteady supersonic flow

The supersonic unsteady flow of a gas past a lattice of thin, slightly curved profiles has been investigated in several studies. The paper [1] is devoted to an evaluation of the effect of wind tunnel walls on the unsteady aerodynamic characteristics of a profile, and [2] investigates the effects of the boundaries of a free jet. These cases are equivalent respectively to the anti-phase and in-phase oscillations of the profiles of an unstaggered grid. In [3] consideration is given to a more general case of gas flow past a profile in a wind tunnel with perforated walls. Flow past a lattice of profiles with stagger is studied in [4], where the magnitude of the stagger angle is limited by the condition that the lattice leading edge is located in the undisturbed stream.In the present paper we present a method of calculation of the unsteady supersonic flow of a gas past a lattice of profiles with arbitrary stagger. As an example the results are presented of the calculation of the aerodynamic forces and moments acting on an oscillating profile in a wind tunnel with solid walls and in a free jet.     

Relaxation oscillations in the turbulent near wake

The method of [2] is used to examine the interaction of a shock or an expansion wave in the outer plane supersonic stream with a turbulent near wake. It is shown that the flow in the disturbed near wake may be two-valued, and for fixed conditions in the outer stream quasisteady transitional flows between these two steady states are possible. Under the influence of some external disturbing mechanism (believed to be turbulent pulsations of the large-vortex type) these quasi-steady flows become relaxation oscillations which lead to the experimentally observed base-pressure oscillations.The theoretical conclusions are confirmed by an experimental study of the base-pressure fluctuations which develop in a two-dimensional expansion-deflection nozzle with interaction of the off-design shock with the near wake.In conclusion the author wishes to thank G. G. Chernyi and G. N. Abramovich for helpful discussions of the study and L. V. Kapranova, T. S. Soboleva, and Z. A. Donskova for assistance in performing the calculations.     

均匀、振荡及组合流绕平板流动的尾迹研究

本文利用离散涡模型及改进的新生涡产生机制对三种不同来流绕平板的近尾迹进行数值研究。计算讨论了定常流中平板绕流流动的总体特性和近尾迹流场;对于简谐振荡来流,相应于K_c=2.0、4.0 和10.0 分别得到两种不同的尾迹形态。给出了小 K_c 数平板尾迹涡配对、运动的新模式而相应的阻力、惯性力系数计算比以前涡模拟结果更接近于 U 型管实验结果。对于流向组合来流本文模拟了涡锁定及其动力特性并于实验相符,给出了流向扰动对平板绕流流动的影响。     

Model of two-dimensional vortex motion with an entrainment mechanism

A model for describing the vertically averaged vortex motions of an incompressible viscous fluid with an arbitrary vertical structure of the bottom Ekman boundary layer is proposed. An approach similar to that adopted in [1] is used: the second moments of the deviations from the average velocities required in order to close the vorticity equation are calculated by means of the Ekman solution for gradient flows, which makes it possible to take the integral bottom boundary layer effect into account. As a result, these terms lead to a specific form of nonlinear friction with a coefficient that depends on the vorticity of the average flow. In the particular case of a constant vertical turbulent transfer coefficient the inaccuracies of the model described in [1] can be eliminated. The generalized vorticity equation obtained has solutions of the vorticity spot type with a uniform internal vorticity distribution, which can be effectively investigated by means of appropriate algorithms [2]. The mechanism of entrainment at the vorticity front is illustrated with reference to the example of the evolution of vorticity spots. An exact solution of the problem of the evolution of an elliptic vortex (generalized Kirchhoff vortex), which in the case of fairly strong anticyclonic vorticity degenerates first into a line segment (vortex sheet) and then into a point vortex, is constructed. Equations describing the dynamics of an elliptic vorticity spot in an external field with a linear dependence of the velocity on the horizontal coordinates and generalizing the classical Chaplygin-Kida model [3, 4] are constructed.Translated from Izvestiya Rossiiskoi Akademii Nauk, Mekhanika Zhidkosti i Gaza, No.6, pp. 49–56, November–December, 1992.     

Numerical solutions for impulsively started and decelerated viscous flow past a circular cylinder

A finite difference study of the unsteady two-dimensional flow past a circular cylinder has been conducted using vorticity and streamfunction as the dependent variables. The two cases considered were impulsively started and decelerated flows. The impulsively started problem was considered to validate the method and has yielded results which agree quite closely with existing results from both calculations and experiments. The decelerated flow analysis produced results which can be explained in terms of induced velocity effects from existing wake vortices for both suddenly stopped and uniformly decelerated flows.