Detached flow over a step with the formation of a turbulent wake

A method of calculating the plane turbulent layer behind a step interacting with a free potential flow of incompressible fluid is developed. The method includes consideration of the initial boundary layer and injection (or suction) in the isobaric bottom region. Friction on the wall behind the step is neglected, which corresponds to symmetric quasisteady flow behind the straight edge of a plate. The inviscid flow is represented by the Keldysh-Sedov integral equations; the flow in the wake with a one-parameter velocity profile is represented by three first-order differential equations—the equations of momentum for the wake and motion along its axis and the equation of interaction (through the displacement thickness) of the viscous flow with the external potential flow. The turbulent friction in the wake is given, accurate to the single empirical constant, by the Prandtl equation. The different flow regions — on the plate behind the step, the isobaric bottom region, and the wake region — are joined with the aid of the quasi-one-dimensional momentum equation for viscous flow. The momentum equation for the flow as a whole serves as the closure condition. The obtained integrodifferential system of equations is approximated by a system of nonlinear finite-difference equations, whose solution is obtained on a computer by minimization of the sum of the squares of the discrepancies. The results of the calculations agree satisfactorily with experimental data.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 3, pp. 17–25, May–June, 1977.We are grateful to V. I. Kuptsov for consultation and help in programming and to Z. A. Donskova who assisted in the calculations and preparation of the paper.     

Area rule for the wake behind a three-dimensional body

In three-dimensional flows, the local characteristics of the medium in the wake behind a supersonic body at Reynolds numbers Re 5·10⁴ depend in a complicated manner on the coordinates. However, in a number of cases it is important to know only the parameters of the medium averaged over the transverse section of the wake. For example, for the diagnostics of the plasma of the wake behind a body by means of microwave resonators the electron density averaged over the section is used [1]. The chemiluminescent radia tion of the wake is also obtained in an averaged form [2]. It is therefore worthwhile obtaining the average characteristics of the medium without detailed study of the local parameters. In the present paper, a rule is established that makes it possible to use the results obtained for the far viscous wake behind an axisymmetric body in the case of the wake behind a three-dimensional body that is nearly axisymmetric, the flow conditions being similar. The three-dimensional wake is considered to the distances until it degenerates into an axisymmetric wake.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 3, pp. 148–150, May–June, 1980.We thank G. Yu. Stepanov and É. Z. Apshtein for helpful advice in a discussion of the results.     

Stability of nonisothermal couette flow

A study is made of the loss of stability of the stationary flow of a viscous fluid between two heated rotating cylinders. The linearized stability problem is studied in the Boussinesq approximation. The perturbations are assumed to be periodic in the time, and also in the axial and azimuthal directions. The neutral curves are calculated numerically. The ranges of variation of the parameters of the problem are found, and in them the most dangerous perturbations (in the class of spatially periodic perturbations) are those without rotational symmetry.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 1, pp. 167–170, January–February, 1980.I thank V. I. Yudovich for constant interest in the work.     

Circulation flow at the front surface of a sphere in a supersonic wake

Several theoretical and experimental studies of supersonic flow past a blunt body located in the wake behind another body have been made [1–7]. It has been shown that a reverse-circulation flow can occur in the shock layer at the front surface. The possibility of such a flow forming depends on the nonuniformity of the freestream flow and the Reynolds number. This paper presents new results of the theoretical study of the structure of the shock wave at the front surface of such a sphere, obtained on the basis of numerical solution of Navier-Stokes equations. It is shown that for a fixed nonuniformity of the freestream flow, an increase in the Reynolds number and cooling of the surface of the body lead to the formation of a secondary vortex in the region where the contour of the body intersects the axis of symmetry. A study is made of the variations of the drag and heat transfer parameters over the front surface of a cooled and thermally insulated sphere. The possibility of numerical simulation of the flow on the basis of the Euler equations is discussed.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 3, pp. 143–148, May–June, 1985.     

Calculation of the self-oscillations arising as a result of the loss of stability by the spiral flow of a viscous liquid in an annular tube

The spiral flow of a viscous liquid in an annular gap (formed by concentric cylinders) due to the rotation of the inner cylinder and the axial pressure gradient is considered; the stability of the flow is discussed in relation to small but finite rotationally symmetrical perturbations.Translated from Zhurnal Prikladnoi Mekhaniki i Tekhnicheskoi Fiziki, No. 3, pp. 57–63, May–June, 1976.The author wishes to thank V. I. Yudovich for interest in this work.     

Micro-scale instability in a continuously stratified fluid

Experimental observations of small-scale structures caused by flow instabilities at layers of high density gradient in the wake behind a cylinder in a fluid with a continuous salt concentration stratification are reported. In the density wake it is possible to discern a number of structures such as wedge-like structures or cusps; small-scale instabilities (breakers) in the zones of interaction of attached internal waves and the high-gradient wake envelope; small-scale instability of the density boundary layer with a complicated density gradient pattern superimposed on a smooth velocity profile, and small-scale wake structures behind attached vortices in the case of a closed (central) wake envelope.Translated from Izvestiya RossiiskoiAkademii Nauk, Mekhanika Zhidkosti i Gaza, No. 3, pp. 3–10, May–June, 1995.     

Three-dimensional wake in a stratified incompressible fluid

On the basis of the boundary layer equations in the linear approximation and the equations for the single-point second-order moments of the velocity and temperature fields closed by means of the semiempirical Kolmogorov-Rotta hypotheses, the problem of the development of a wake behind an axisymmetric body in a temperature-stratified incompressible fluid flow is solved using the integral relation method. The anisotropic nature of the expansion of the wake behind the body is established.Almaty. Translated from Izvestiya Rossiiskoi Akademii Nauk, Mekhanika Zhidkosti i Gaza, No. 4, pp. 71–77, July–August, 1996.     

Wake formation when shock wave is shed from the edge of a plate

The formation of a laminar wake in the flow behind a shock wave when the latter is shed from the trailing edge of a semi-infinite plate is investigated in this paper. It is shown that the flow on the plate and in the wake turns out to be self-similar, dependent on two dimensionless combinations of variables, and the flow on the plate, including the trailing edge, remains steady in a coordinate system coupled to the shock wave (the fact of the flow self-similarity in the wake was first noted in [1]). An analytic solution of the problem of the wake in the neighborhood of the trailing edge is obtained, from which it follows that, in contrast to [2], there is no line of singularities in the nonstationary boundary-layer equations in the flow domain. This fact is also verified by the analysis of the flow in the neighborhood of a line of tagged particles leaving the trailing edge simultaneously with the shock wave. Hence the problem under consideration is solved by the traditional numerical methods using conditions in the initial section (which is taken to be the section in the neighborhood of the trailing edge), on the wake axis, and at an infinite distance away. Approximate formulas are obtained for the longitudinal velocity profiles in the whole range of shock-wave intensities.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 4, pp. 59–66, July–August, 1978.     

Structure of the wake behind a cylinder in a fluid with a variable buoyancy frequency,studied with the help of echo sounding and optical observation

Acoustic sounding of the wake behind a circular cylinder and observation of the shadow pattern indicate that for a density gradient varying on distances of the order of the cylinder's vertical dimension two parametrically different vortex systems may coexist in the wake. In the particular case when the profile of the buoyancy frequency is symmetric and its axis coincides with the path along which the cylinder's axis travels, the vortex systems on both sides of the wake are nearly identical. However, if the axes do not coincide, then the structures of the upper and the lower parts of the wake are rather different.If the Froude number Fr=u/NR is based on the flow velocityu, the radius of the cylinderR, and the current value of the buoyancy frequencyN, then the observed flow pattern corresponds to known regime diagrams presented by several authors. In the present experimentsN was found using the characteristic periods of oscillation of the intensity of volume ultrasonic scattering.The vertical profiles of the coefficient of backscattering of ultrasound and the synchronous shadow flow patterns in the wake behind the body are subjected to simultaneous analysis.Moscow. Translated from Izvestiya Rossiiskoi Akademii Nauk, Mekhanika Zhidkosti i Gaza, No. 6, pp. 3–12, November–December, 1995.     

Investigation of the wake of a hypersonic sphere in air by means of an open microwave resonator

The use of an open microwave resonator with plane-parallel mirrors for plasma diagnostics was first proposed in [1]. A resonator with spherical mirrors, which provides better spatial resolution in addition to high sensitivity, was used later [2, 3] to investigate the wake flow of models moving through air at hypersonic velocities. The presence of free electrons in the flow field is caused by ionization processes behind the bow shock and in the model boundary layer in this case. However, only the results of measurements of the density of electrons are presented in [2, 3], and no information is given on another important plasma parameter: the effective collision frequency of electrons with heavy particles. In the present study we use a microwave (8-mm range) resonator for an experimental study of the flow of gas in the wake of a polymer (Kaprolon) sphere traveling through air at hypersonic velocity. The flow is visualized by the schlieren technique. The electron densities and effective collision frequency of electrons with heavy particles are determined as a function of the distance behind the sphere.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 4, pp. 156–160, July–August, 1978.     

Numerical solution for the flow of viscoelastic fluids around an inclined circular cylinder.

Finite difference solutions have been obtained by the perturbation method to investigate the influence of shear thinning and elasticity on the flow around an inclined circular cylinder of finite length in a uniform flow. In this numerical analysis a generalized upper-convected Maxwell model, in which the viscosity changes according to the Cross model, has been used.The local flow over the cylinder is only slightly deflected. However, in the wake flow behind the cylinder the particle path is remarkably influenced by the axial flow and rapidly flows up parallel to the cylinder's axis. Then it gradually rejoins direction of the incoming flow. It is found that viscoelastic fluids are prone to flow axially in the vicinity of the cylinder. The numerical predictions generally agree with the flow visualization results.The numerical solutions also demonstrate that elasticity has a strong effect on the velocity profile especially around both ends of the cylinder; elasticity increases the asymmetric profiles of both circumferential velocity and axial velocity with respect to equal to 90° and decreases a difference in the circumferential velocity between the windward end and the leeward end.For non-Newtonian fluids, the length of the wake flow is influenced by not only the Reynolds number but also the cylinder diameter and it is larger for the cylinder with the smaller diameter at the same Reynolds number.Partly presented at the 9th Australasian Fluid Mechanics Conference, University of Auckland, New Zealand, 8–12 December, 1986     

Experimental and theoretical study of temperature fluctuations in the wake of a linear heat source

The temperature and temperature fluctuations, the thermal intermittency, and the fluctuation spectra of the velocity and temperature are measured in the wake behind a linear heat source (a heated wire whose diameter is much smaller than the turbulence scale of the incident flow). A self-similar solution for the far region of the wake is found numerically and the initial non-self-similar part of the wake is calculated taking into account rheological effects and the experimentally observed enhanced dissipation.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 4, pp. 39–45, July–August, 1978.The authors are indebted to V. R. Kuznetsov for help in solving the self-similar problem, to I. P. Smirnova for writing the program for calculating the non-self-similar flow, and to V. N. Novikov for help with the experiments.     

Pressure pulsations on a rigid wall caused by a vortex street

A topic of obvious interest is the interaction between the vortex wake behind bluff bodies and nearby objects, in particular plane surfaces. At present the most detailed studies in flows such as these have been on the characteristics of the velocity field of the vortices [1, 2]. The main topic of attention in what follows is pressure pulsations on a wall caused by separation of vortices from an adjacent cylinder. This problem is especially important for the determination of vibrations and noises due to bodies located on the rigid boundary of the flow.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 2, pp. 167–169, March–April, 1986.The author wishes to express his appreciation to M. I. Rabinovich for discussion of the results, and to P. R. Gromov and A. B. Zobnin for help in conducting the experiment.     

Investigation of gas blowing into a base cavity

The results of an experimental investigation of various methods of influencing the flow in the near wake by injecting a system of jets and varying the rear-end configuration for a Mach number M=3 are presented. The structure of the base flow and the pressure fluctuations behind an axisymmetric body with a base cavity when circular and annular gas jets are injected is investigated. The results are compared with the data for bodies with a flat base.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 4, pp. 170–172, July–August, 1989.     

Interaction of a supersonic jet exhausting into vacuum with an obstacle

A study is made of the interaction between an axisymmetric supersonic jet exhausting into vacuum and an obstacle of a fairly complicated configuration and positioned relative to the nozzle in such a way that in the interaction region behind the detached shock wave there is a three-dimensional flow possessing a symmetry plane. The flow in the interaction region is described by the system of equations of motion of an inviscid perfect gas with boundary conditions on the shock wave (Rankine-Hugoniot relation) and on the surface of the obstacle (no-flow condition). The other boundaries of the region are the symmetry plane of the flow and an arbitrarily chosen surface in the supersonic part of the flow.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti Gaza, No. 1, pp. 156–161, January–February, 1981.     

A study of the dimensions of the wake behind a cone travelling at hypersonic speed

In a series of papers (see, for example, [1–5]) numerous results have appeared detailing experimentally determined dimensions of wakes behind spheres or spherically blunted cylinders travelling with hypersonic speed in air. During this same period substantially less attention has been paid to the study of the parameters of the wake behind a cone, in particular, a cone at an angle of attack. In the present paper we present the results of measuring the mean width of the wake, and the mean-square deviation of the wake boundary, for a spherically blunted cone of 10 half-angle with nose radius 6% of the base diameter, travelling in air at Mach number M=12 and Reynolds number Re=0.3·10⁶, and with an angle of attack varying from 12 to 24.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 6, pp. 161–163, November–December, 1976.The author expresses his thanks to N. N. Baulin for his help In the experimental work.     

Solution of Navier-Stokes equations near the trailing edge of a plate

The appearance of a local singularity in the solutions for the neighborhood of the trailing edge of a plate in a sub- or supersonic flow makes it necessary to consider the flow in the local region which is described in the first approximation by the Navier-Stokes equations for an incompressible gas. In this paper numerical solutions are obtained for such a region for both a thin plate and a plate with thickness. The streamline patterns and the distributions of the flow functions over the surface of the plate and in the wake behind it are presented. For the plate with finite thickness, a numerical solution is obtained in a wide range of variation of the local Reynolds number.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 4, pp. 173–176, July–August, 1985.     

Calculation of hysteresis and flow-rate oscillations of base pressure in supersonic annular nozzles

The interaction of the turbulent axisymmetric near wake behind the face of the central body of an annular nozzle with the supersonic annular jet discharging from this nozzle is analyzed. The flow in the monoparametric near wake is calculated by the integral method [1] while the flow in the nonviscous jet is calculated by the method of through calculation using a monotonic explicit difference system of the first order of accuracy [2]. The interaction between the nonviscous and turbulent streams is determined by the displacement thickness of the wake. The initial conditions of the wake are determined from the integral conditions of attachment with the mixing flow in the isobaric base region. The interaction flow is described by the particular solution of the equations which passes through the singular saddle point — the throat of the wake. The near wake and base pressure in different modes of discharge from an annular nozzle at the exit cross section of which the ratio of outer and inner radii is y₂/y₁ = 1.3 and the Mach number is M = 2.54 are calculated as an example. The region of hysteresis of the base pressure, connected with the ambiguity of the interaction flow owing to the formation of the throat of the wake within the first or second barrel of the jet, and the parameters of the low-frequency flow-rate oscillations of base pressure in this region are determined. The results of the calculations are in satisfactory agreement with experimental data.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 1, pp. 125–130, January–February, 1977.     

Calculation of flow around a lifting surface in the case of large deformations

Many studies have been made of the nonstationary flow of an ideal incompressible fluid around a lifting surface. The present state of the numerical methods of solution of this problem is reviewed in [1]. The present paper studies three-dimensional nonstationary flow around a lifting surface which undergoes deformation and behind which a wake vortex surface is formed. The lifting and wake vortex surfaces are represented in parametric form. The metrics of these surfaces are used, and the introduced vortex function is approximated by bicubic splines. For the convenient application of the theory developed here to the flapping flight of insects, for which it is sometimes difficult to distinguish the lateral and trailing edges of the wings, the following terminology is introduced. The part of the edge of the lifting surface from which the wake vortex surface is shed is called the trailing edge. The remaining part is called the leading edge. On the leading edge, the velocity has a singularity. Test calculations have demonstrated the effectiveness of the method.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 4, pp. 72–79, July–August, 1980.     

Method of calculating subsonic gas flow with separation past wings

The steady nonlinear problem of subsonic compressible gas flow past a wing of arbitrary shape in plan is considered. A numerical method was devized for solving the problem; this is a further development of the method of discrete vortices. The surface of the body and the vortex wake behind it are simulated by systems of discrete vortex sections, but, in contrast to the case of an incompressible medium, it is necessary in this case for the sources to be distributed outside the wing. The circulations of the attached vortices, the strengths of the sources, and the shape of the wake are determined by iterations.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 4, pp. 140–147, July–August, 1984.