An unsteady-state thermal boundary layer on a flat isothermal plate

The Kármán-Polhausen integral method is used to investigate the problem of an unsteady-state thermal boundary layer on an isothermal plate with a stepwise change in the conditions of flow around the plate; analytical expressions are obtained for the thickness of the thermal boundary layer. A dependence is found for the rate of movement of the boundary between the steady-state and unsteady-state regions of the solution on the Prandtl number. A similar problem was solved in [1, 2] for a dynamic layer, Goodman [3] discusses the more partial problem of an unsteady-state thermal boundary layer under steady-state flow conditions. Rozenshtok [4] considers the problem in an adequate statement but, unfortunately, he permitted errors of principle to enter into the writing of the system of characteristic equations; this led to absolutely invalid results. In an evaluation of the advantages and shortcomings of the integral method under consideration, given in [4], it must only be added that the method is applicable to problems in which the initial conditions differ from zero since, in this case, approximation of the velocity and temperature profiles by polynomials is not admissible.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 4, pp. 64–69, July–August, 1970.     

Unsteady Radiative–Convective Heat Transfer in a Flow Emitting‐Absorbing and Scattering Medium around an Ablating Plate

A conjugate problem of radiative–convective heat transfer in a turbulent hightemperature gasdisperse flow around a thermally thin ablating plate is considered. The plate experiences intense radiative heating by an external source, which is a blackbody. The temperature fields and the distributions of heat fluxes along the plate under unsteady conditions are calculated. The data gained make it possible to examine the effect of the Stark number and phasetransition heat in the plate material on the time evolution of the thermal state of the boundarylayer medium and the plate itself being heated by a hightemperature radiation source.     

Stability and Transition on a Swept Cylinder in a Supersonic Flow

Results of experimental investigations of the evolution of natural disturbances and laminar–turbulent transition in a supersonic boundary layer on the attachment line of a circular cylinder with a sweep angle of 68° and a freestream Mach number M = 2 are presented. The experimental studies are supplemented by calculations of the mean flow and stability characteristics. Flow regimes in the boundary layer on the attachment line are determined by a hotwire technique as functions of the Reynolds number and height of twodimensional roughness elements. The results are compared with NASA (Ames) experiments.     

Spectral and correlation characteristics of the turbulent boundary layer on an extended flexible cylinder

In marine geophysical seismological prospecting extensive use is made of towed receiving systems consisting of extended flexible cylinders containing acoustic sensors over which the water flows in the longitudinal direction. The boundary layer pressure fluctuations on the cylinder surface are picked up by the sensors as hydrodynamic noise. This paper is concerned with the study of the energy and spacetime characteristics of the pressure fluctuations in the turbulent boundary layer on an extended flexible cylinder in a longitudinal flow. The pressure fluctuations on the cylinder surface have been investigated experimentally for Re_X=(2–4)·10⁷, a dimensionless diameter of the pressure fluctuation sensors d _p ⁺ =d_pu^*/=500, where d_p is the sensor diameter, u^* the dynamic viscosity, and the viscosity coefficient, and frequencies 0.02311.259 (=^*/U). The spectral and correlation characteristics of the pressure fluctuations on the surface of the flexible cylinder are found to differ from the corresponding characteristics for a rigid cylinder [1–4].Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i aza, No, 5, pp. 49–54, September–October, 1989.     

Combined effect of longitudinal riblets and LEBU-devices on turbulent friction on a plate

The possibility of reducing turbulent friction with the help of large-eddy-breakup devices (LEBUs) and riblets is studied experimentally. The tests were conducted in a low-turbulence wind tunnel on a flat plate for 2·10⁶ Re 7·10⁶. The local friction coefficient was measured using internal strain-gauge balances, and the total drag was estimated by the momentum-transfer method. It is shown that a combination of LEBUs and riblets makes it possible to reduce the total turbulent friction drag of a flat plate 1800 mm long by 16%. The effects of the length of a ribbed surface on the efficiency of friction reduction and of LEBUs and riblets on the structure of a turbulent boundary layer are analyzed.Translated from Izvestiya Rossiiskoi Akademii Nauk, Mekhanika Zhidkosti i Gaza, No. 3, pp. 39–46, May–June, 1995.     

Experimental investigation of separation of turbulent boundary layer in vicinity of a step

The results of an experimental investigation of the separation of a turbulent boundary layer in the vicinity of a step on a flat plate at M = 2 and 3, and Re = U/v = (26–66)·10⁶ m^–1 are given. The step height was varied from 3 to 16 mm, which corresponded to the range of relative heights 1.1 h/ 7.6, where is the thickness of the boundary layer at the point at which the pressure starts to increase in front of the step. The obtained data for the pressure distribution in front of the step, and on its face and top surface, and the results of probe measurements in the separation and adjacent regions provide a more accurate scheme of the flow. The obtained data are compared with the results of other investigations.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 5, pp. 73–80, September–October, 1977.We express our thanks to A. M. Kharitonov for valuable comments made during the discussion of this work, and also to M. A. Gol'dfel'd for kindly providing the experimental data for axisytnmetric steps.     

Asymptotic theory of the turbulent boundary layer as a problem of singular perturbations

We consider an asymptotic theory of the turbulent boundary layer [1,2]. In this paper we make an attempt to further develop the mathematical aspects of this theory. We demonstrate the features of this theory by applying it to a problem which is close to the so-called equilibrium turbulent boundary layer with a pressure gradient and blowing.Notation x, y coordinates, parallel and perpendicular to the wall - u velocity component in the x direction - p, ',v pressure, density, and kinematic viscosity coefficient - l' scale of turbulence - tangential stress - u speed at the outer edge of the boundary layer - thickness of the boundary layer - * displacement thickness - ** momentum loss thickness - Cf coefficient of friction - R Reynolds number Translated from Zhurnal Prikladnoi Mekhaniki i Tekhnicheskoi Fiziki, No. 5, pp. 86–95, September–October, 1971.The author thanks S. S. Kutateladze, A. I. Leont'ev, and G. V. Aronovich for their interest in this effort.     

Heat transfer process in turbulent flows

A system of differential equations is proposed to describe turbulent flows of incompressible fluid boundary layer type with constant thermophysical characteristics A turbulent temperature conductivity is introduced which is expressed in terms of the energy and scale of turbulence, the dimensionless gradients of the mean velocity and turbulence energy, and the dimensionless distance, to the surface being streamlined. This system is integrated on an electronc computer by the mesh method for the flow in a flat-plate boundary layer with different Prandtl numbers (0.2P100) For air (P=0.71) the system is integrated for nonzero values of the transverse mean velocity component on the streamlined surface (0v_W/U0.0045).Moscow. Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 4, pp. 18–24, July–August, 1972.     

Structure of a collisionless boundary layer and turbulent damping of ions

The flow of a low-pressure plasma in a MHD channel is unstable in a number of cases. The instability can be caused by a current flowing across the magnetic field. In this study we investigate an unstable, turbulent flow of a rarefied plasma near the magnetized electrodes, representing plane magnetic dipoles. Owing to the growth of microscopic turbulence near the electrodes, the maximum density of the current that is induced in the plasma is localized and turbulent damping of the incoming flow occurs. The energy of damping goes into the turbulent heating of the plasma. Under these conditions a structure of the boundary layer is found for a stationary flow. The characteristic transverse dimension of the boundary layer is considerably less than the particle mean free path; therefore, such a boundary layer can be called collisionless.Translated from Zhurnal Prikladnoi Mekhaniki i Tekhnicheskoi Fiziki, No. 5, pp. 17–28, September–October, 1972.     

The effect of background particulates on the delayed transition of a heated 9:1 ellipsoid

An experimental study was done to quantify the effects of a variety of background particulates on the delayed laminar-turbulent transition of a thermally stabilized boundary layer in water. A Laser-Doppler Velocimeter system was used to measure the location of boundary layer transition on a 50 mm diameter, 9:1 fineness ratio ellipsoid. The ellipsoid had a 0.15 m RMS surface finish. Boundary layer transition locations were determined for length Reynolds numbers ranging from 3.0 × 10⁶ to 7.5 × 106. The ellipsoid was tested in three different heating conditions in water seeded with particles of four distinct size ranges. For each level of boundary layer heating, measurements of transition were made for clean water and subsequently, water seeded with 12.5 m, 38.9 m, 85.5 m and 123.2 m particles, alternately. The three surface heating conditions tested were no heating, T = 10°C and T = 15°C where T is the difference between the inlet model heating water temperature, T _i, and free stream water temperature, T . The effects of particle concentration were studied for 85.5 m and 123.2 m particulates.The results of the study can be summarized as follows. The 12.5 m and 38.9 m particles has no measurable effect on transition for any of the test conditions. However, transition was significantly affected by the 85.5 m and 123.2 m particles. Above a length Reynolds number of 4 × 10⁶ the boundary layer transition location moved forward on the body due to the effect of the 85.5 m particles for all heating conditions. The largest percentage changes in transition location from clean water, were observed for 85.5 m particles seeded water.Transition measurements made with varied concentrations of background particulates indicated that the effect of the 85.5 m particles on the transition of the model reached a plateau between 2.65 particulates/ml concentration and 4.2 particles/ml. Measurements made with 123.3 m particles at concentrations up to 0.3 part/ml indicated no similar plateau.     

Analysis of turbulent boundary layer interaction with an external supersonic flow behind a step

An integral method of analyzing turbulent flow behind plane and axisymmetric steps is proposed, which will permit calculation of the pressure distribution, the displacement thickness, the momentum-loss thickness, and the friction in the zone of boundary layer interaction with an external ideal flow. The characteristics of an incompressible turbulent equilibrium boundary layer are used to analyze the flow behind the step, and the parameters of the compressible boundary layer flow are connected with the parameters of the incompressible boundary layer flow by using the Cowles-Crocco transformation.A large number of theoretical and experimental papers devoted to this topic can be mentioned. Let us consider just two [1, 2], which are similar to the method proposed herein, wherein the parameter distribution of the flow of a plane nearby turbulent wake is analyzed. The flow behind the body in these papers is separated into a zone of isobaric flow and a zone of boundary layer interaction with an external ideal flow. The jet boundary layer in the interaction zone is analyzed by the method of integral relations.The flow behind plane and axisymmetric steps is analyzed on the basis of a scheme of boundary layer interaction with an external ideal supersonic stream. The results of the analysis by the method proposed are compared with known experimental data.Notation x, y longitudinal and transverse coordinates - X, Y transformed longitudinal and transverse coordinates - , ^*, ^** boundary layer thickness, displacement thickness, momentum-loss thickness of a boundary layer - , ^*, ^** layer thickness, displacement thickness, momentum-loss thickness of an incompressible boundary layer - u, velocity and density of a compressible boundary layer - U, velocity and density of the incompressible boundary layer - , stream function of the compressible and incompressible boundary layers - , dynamic coefficient of viscosity of the compressible and incompressible boundary layers - r₁ radius of the base part of an axisymmetric body - r radius - R transformed radius - M Mach number - friction stress - p pressure - a speed of sound - s enthalpy - v Prandtl-Mayer angle - P Prandtl number - P_t turbulent Prandtl number - r₂ radius of the base sting - b step depth - =0 for plane flow - =1 for axisymmetric flow Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 3, pp. 33–40, May–June, 1971.In conclusion, the authors are grateful to M. Ya. Yudelovich and E. N. Bondarev for useful comments and discussions.     

Effect of Streamwise Streaky Structures on Turbulization of a Circular Jet

Experimental investigations of the influence of streamwise streaky structures on turbulization of a circular laminar jet are described. The qualitative characteristics of jet evolution are studied by smoke visualization of the flow pattern in the jet and by filming the transverse and longitudinal sections of the jet illuminated by the laser sheet with image stroboscopy. It is shown that the streaky structures can be generated directly at the nozzle exit, and their interaction with the Kelvin–Helmholtz ring vortices leads to emergence of azimuthal beams ( structures) by a mechanism similar to threedimensional distortion of the twodimensional Tollmien–Schlichting wave at the nonlinear stage of the classical transition in nearwall flows. The effect of the jetexhaustion velocity and acoustic action on jet turbulization is considered.     

Flow Structure in the Separation Zone in the Interaction between Shock Waves and the Boundary Layer on a Plate with Slip

A model which makes it possible to calculate the reverse-flow parameters in the separation zone is constructed on the basis of the results of an integrated experimental study of the characteristics of the separated flow developed in the transition from free to non-free interaction between plane shock waves and the boundary layer on a plate with slip. The effect of the Mach number of the reverse flow in the separation zone on the properties of inner boundary layer separation is analyzed. Features of the interference flow due to boundary layer transition are described. The present study is a continuation of investigations [1–3] devoted to the study of a new steady-state type of interaction between shock waves and the boundary layer on a plate with slip in which the separation line formed would propagate upstream beyond the sharp leading edge if no leading edge was present, i.e., so-called non-free interaction.     

Asymptotic theory for calculating heat fluxes near the corner of a body

A method is presented for calculating the distribution of the thermal fluxes, friction stresses, and pressure near the corner point of a body contour in whose vicinity the outer supersonic flow passes through an expansion wave. The method is based on a study of the asymptotic solutions of the Navier-Stokes equations as the Reynolds number R approaches infinity for the flow region in which the longitudinal gradients of the flow functions are large, invalidating conventional boundary layer theory. This problem was examined in part in [1], in which the distribution of the friction and pressure in a region with length on the order of a few thicknesses of the approaching boundary layer was obtained in the first approximation. The leading term of the expansion for the thermal flux to the surface of the body vanishes for a value of the Prandtl number equal to unity and for other values of the Prandtl number does not match directly with its value in the undisturbed boundary layer.The thermal-flux distribution is obtained for values of the Prandtl number approaching unity. For this purpose it was necessary to consider a more general double passage to the limit as 1 and 0 for a finite value of the parameter B=[(–1)/] [–ln ^1/4/]^1/4 characterizing the ratio of the effects of thermal conduction, viscous dissipation, and convection. The solution obtained previously [1] corresponds to the particular case B and therefore for actual values of R=10⁴–10⁶, ~ 0.7 overestimates considerably the effect of the dissipative term on heat transfer, although even in first approximation it describes the pressure distribution well and the friction distribution satisfactorily. For smooth matching of the solutions with the corresponding flow functions in the undisturbed boundary layer it was necessary to introduce a flow region with free interaction for the expansion flow. Equations and boundary conditions which describe the flow as a whole are presented. Examples are given of numerical calculations and comparison with experiment.     

Interpolation formulas for maxwell viscosity of certain metals as a function of shear-strain intensity and temperature

The purpose of this study is the construction of interpolation formulas for the dependence of Maxwell viscosity, a quantity which is the reciprocal of shear-strain relaxation time , on shear-strain intensity and temperature for several metals: iron, aluminum, copper, and lead. This function was interpolated in various temperature and deformation velocity ranges in accordance with available experimental data for iron (0 10⁷ sec^–1, 200 ° T 1500 °); aluminum (0 10⁷ sec^–1, 300 ° T 900 °); copper (0 10⁵ sec^–1, 300 ° T 1300 °); lead (0 10⁶ sec^–1, 90 ° T 400 °); temperatures in °K.Translated from Zhurnal Prikladnoi Mekhaniki i Tekhnicheskoi Fiziki, No. 4, pp. 114–118, July–August, 1974.     

Calculation of heat transfer accompanying hypersonic three-dimensional flow of air over slender blunt-nosed cones

The effective length method [1, 2] has been used to make systematic calculations of the heat transfer for laminar and turbulent boundary layers on slender blunt-nosed cones at small angles of attack ( + 5° in a separationless hypersonic air stream dissociating in equilibrium (half-angles of the cones 0 20°, angles of attack 0 15°, Mach numbers 5 M 25). The parameters of the gas at the outer edge of the boundary layer were taken equal to the inviscid parameters on the surface of the cones. Analysis of the results leads to simple approximate dependences for the heat transfer coefficients.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 5, pp. 173–177, September–October, 1981.     

Hypersonic flow over a narrow delta plate at large angles of attack

The problem of hypersonic flow over a flat delta plate with a high sweepback anglex at angles of attack close to /2 is solved using a numerical algorithm based on transition to the conical solution. The existence of conical flow at /2 with the velocity vector directed towards the apex of the plate is established. Values ofC _p/sin² and the thickness of the shock layer in the plane of symmetry of the plate are given as functions of the hypersonic similarity parameterk=tan tanx. A comparison of the calculated and experimental data shows that they are in good agreement.Translated from Izvestiya Rossiiskoi Akademii Nauk, Mekhanika Zhidkosti i Gaza, No.5, pp. 183–185, September–October, 1992.     

Effect of Dynamic Prehistory on Aerodynamics of a Laminar Separated Flow in a Channel Behind a Rectangular Backward‐Facing Step

The effect of dynamic prehistory of the flow and the channelexpansion ratio on aerodynamics of a steady separated laminar flow behind a rectangular backwardfacing step located in a planeparallel channel is numerically studied. It is shown that the boundary layer upstream of the flow separation exerts a strong effect on flow characteristics behind the step. A decrease in the boundarylayer thickness in the cross section of the step leads to a decrease in the separationregion length, and an increase in the channelexpansion ratio with a fixed initial boundarylayer thickness and Reynolds number leads to an increase in the separationregion length.     

Instability of the boundary layer on a curved surface

The excitation of instability of the boundary layer on a curved surface in the limit R (R is the Reynolds number) is analyzed.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 1, pp. 176–179, January–February, 1990.I am grateful to O. S. Ryzhov for his interest in the work.     

On laminar flow through a uniformly porous pipe

Numerous investigations ([1] and [4–9]) have been made of laminar flow in a uniformly porous circular pipe with constant suction or injection applied at the wall. The object of this paper is to give a complete analysis of the numerical and theoretical solutions of this problem. It is shown that two solutions exist for all values of injection as well as the dual solutions for suction which had been noted by previous investigators. Analytical solutions are derived for large suction and injection; for large suction a viscous layer occurs at the wall while for large injection one solution has a viscous layer at the centre of the channel and the other has no viscous layer anywhere. Approximate analytic solutions are also given for small values of suction and injection.

Nomenclature

General r distance measured radially - z distance measured along axis of pipe - u velocity component in direction of z increasing - v velocity component in direction of r increasing - p pressure - density - coefficient of kinematic viscosity - a radius of pipe - V velocity of suction at the wall - r ²/a ² - R wall or suction Reynolds number, Va/ - f() similarity function defined in (6) - u ₀() eigensolution - U(0) a velocity at z=0 - K an arbitrary constant - B _K Bernoulli numbers Particular Section 5 perturbation parameter, –2/R - ² a constant, –K - x / - g(x) f()/ Section 6 perturbation parameter, –R/2 - ² a constant, –K - g() f() - g _c ()=g() near centre of pipe - * point where g()=0 Section 7 2/R - ² K - t (1–)/ - w(t, ) [1–f(t)]/ - ₀, ₁ constants - g() f()– ₀ - ₀/ - ₀ a constant - * point where f()=0