Asymptotic theory of the flow near the rear end of a slender axisymmetric body

Steady high-Reynolds-number flow of a viscous incompressible fluid past a slender axisymmetric body is considered. The structure of the near wake and the boundary layer in the vicinity of the rear end of the body is studied. The relationship between the maximum relative body thickness and the rearend shape giving a local boundary layer — potential flow interaction zone in a small neighborhood of the rear end is found. The boundary value problem for this region is solved numerically.Translated from Izvestiya Rossiiskoi Akademii Nauk, Mekhanika Zhidkosti i Gaza, No. 5, pp. 68–77, September–October, 1993.     

The pressure and temperature behind bodies with flat bases in a supersonic flow when inert and active gases are introduced into the base region

The effect of introducing active and inert gases into the base region behind two-dimensional and axisymmetric bodies on the pressure and temperature in that region is studied in the absence of heat transfer between the body and the base region. The flow in the mixing zone is assumed to be turbulent. To investigate the effect of introducing active gases on the parameters in the base region, a model of diffusion combustion is used. The base pressure is calculated by Korst's method [1]. For the velocity profile in the mixing zone an expression is used which results from integrating the equation of motion in von Mises variables. The temperature and concentrations of the components in the base region are determined from integral equations for the conservation of enthalpy and concentration, which make it possible to calculate the parameters in the base region for arbitrary flow rates of the active gas, including arbitrarily small rates. The results are given of calculating the base pressure when hydrogen and argon are blown into the base region. In the latter case computational and experimental results are compared.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 2, pp. 48–57, March–April, 1971.     

Combined method of calculating supersonic ideal-gas flow over a wing with a supersonic blunt leading edge

A combined numerical method, based on the successive calculation of the flow regions near the blunt leading edge and center of a wing, is proposed on the assumption that the angle of attack and the relative thickness and bluntness radius of the leading edge are small. The flow in the neighborhood of the leading edge of the wing is assumed to be identical to that on the windward surface of a slender body coinciding in shape with the surface of the blunt nose of the wing and is numerically determined in accordance with [1]. The flow parameters near the center of the wing are calculated within the framework of the law of plane sections [2]. In both regions the equations of motion of the gas are integrated by the Godunov method. The flow fields around elliptic cones are obtained within the framework of the combined method and the method of [3], A comparative analysis of the results of the calculations makes it possible to estimate the region of applicability of the method proposed.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 1, pp. 159–164, January–February, 1989.The authors wish to express their gratitude to A. A. Gladkov for discussing their work, and to G. P. Voskresenskii, O. V. Ivanov, and V. A. Stebunov for making available a program for calculating supersonic flow over a wing with a detached shock.     

Stability of two-dimensional flow near a stagnation point in the presence of massive blowing

In the framework of linear theory, a study is made of the stability of laminar flow in the boundary layer on a flat porous plate near a stagnation point in the presence of massive blowing. The problem is solved in the inviscid approximation. The neutral curves are calculated, and the values of the critical parameters and the position of the point at which stability is lost are determined. It is found that the extension of the region of laminar flow is directly proportional to the blowing velocity.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 2, pp. 157–158, March–April, 1982.     

Supersonic flow around the trailing edge of a thin profile

The method of mergeable asymptotic expansions has recently been used effectively in investigations devoted to the study of boundary layer interaction with an external inviscid flow at high subcritical Reynolds numbers Re. The asymptotic analysis permits obtaining a limit pattern of the flow around a solid as Re þ, and determining the similarity and quantitative regularity laws which are in good agreement with experimental results. Thus by using the method of mergeable asymptotic expansions it is shown in [1–4] that near sites with high local curvature of the body contour and flow separation and attachment points, an interaction domain appears that has a small length on the order of Re^-3/8. In this flow domain, which has a three-layer structure, the pressure distribution in a first approximation already depends on the change in boundary-layer displacement thickness, while the induced pressure gradient, in turn, influences the flow in the boundary layer. An analogous situation occurs in the neighborhood of the trailing edge of a flat plate where an interaction domain also appears [5, 6]. The flow in the neighborhood of the trailing edge of a flat plate around which a supersonic viscous gas flows was examined in [7]. Numerical results in this paper show that the friction stress on the plate surface remains positive everywhere in the interaction domain, and grows on approaching the trailing edge. The supersonic flow around the trailing edge of a flat plate at a small angle of attack was investigated in [8, 9], Supersonic flow of a viscous gas in the neighborhood of the trailing edge of a flat plate at zero angle of attack is examined in [10], but with different velocity values in the inviscid part of the flow on the upper and lower sides of the plate. The more general problem of the flow around the trailing edge of a profile with small relative thickness is investigated in this paper.Translated from Zhurnal Prikladnoi Mekhaniki i Tekhnicheskoi Fiziki, No. 3, pp. 36–42, May–June, 1981.     

Parameters of spatial flow past a hydrofoil near the free surface of a ponderable liquid

An approach is presented to the investigation of spatial flow past an submerged hydrofoil in the presence of a ponderable liquid. The approach uses the method of singularities, which represents the influence of the free surface of the water by the influence of an unbounded layer of sources. The linear solution is completely found for the problem of determining the wave production and downwash behind an submerged hydrofoil with given parameters of the eddy system. An estimate is made of the influence on the flow characteristics of nonlinear factors associated with the change produced by the wave production in the form of the free eddies leaving the hydrofoil. The distribution of the downwash in the flow behind the hydrofoil is found and recommendations given for the arrangement of a hydrofoil system with positive interaction of the elements. The results of the calculation are compared with experiment.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza., No. 2, pp. 54–62, March–April, 1979.     

Turbulent flow of a fluid in the neighborhood of the trailing edge of a plate at zero angle of attack

The laminar flow regime of an incompressible fluid at the trailing edge of a plate was studied by Stewartson and Messiter [1, 2] by means of the method of matched asymptotic expansions. In. the present paper, this method is used to analyze the same problem, but in the case of turbulent flow in the boundary layer and the wake. A system of linear equations of elliptic type with variable coefficients is obtained for the averaged values of the flow parameters in the main part of the boundary layer and the wake that is responsible for the change in the displacement thickness. A solution of this system is constructed by the Fourier method in the case of a power law of the velocity in front of the interaction region.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 6, pp. 17–23, November–December, 1983.     

Hypersonic flow of gas over a slender wing of variable shape

In a formulation analogous to [1–3], a study is made of the flow of a uniform homogeneous hypersonic ideal gas over the windward side of a slender wing whose surface profile depends on the time. The problem is solved by the thin shock layer method [4]. The bow shock is assumed to be attached to the leading edge of the wing at at least one point. The corrections of the first approximation to the main Newtonian flow are found. For wings of finite aspect ratio, when the bow shock is attached along the whole of the leading edge of the wing, computational formulas are obtained for determining the parameters of the gas in the shock layer.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 4, pp. 94–101, July–August, 1979.     

Linear stability of channel entrance flow

The spatial stability of two dimensional, steady channel flow is investigated in the downstream entry zone for both exponentially and algebraically growing disturbances. A model based on previous work is presented for the base flow which represents a small deformation of plane Poiseuille flow. The base flow evolution towards the fully developed state comes from the experimental and theoretical study of M. Asai and J.M. Floryan [M. Asai, J.M. Floryan, Certain aspects of channel entrance flow, Phys. Fluids 16 (2004) 1160–1163]. This flow is found to be more stable than the parabolic Poiseuille flow. The most destabilizing base flow defect is then calculated using a variational method. The compromise between the destabilizing effect of the defect, which diffuses downstream, and the instability growth is found to be insufficient to provoke transition in the downstream laminar flow.     

Parameters of flow near the axis in the shock layer in the interaction of a supersonic jet with a barrier

The axisymmetric interaction between a supersonic jet with a finite expansion ratio and a barrier is accompanied by the formation of complex sub- and supersonic flow in a shock layer whose thickness depends on the parameters of the jet and the position of the barrier. The main relationships of the interaction process have been established experimentally ([1–3] and others) and individual results of numerical calculations of such flows are known [4]. An analytical investigation of the parameters in the shock layer formed ahead of a plane barrier when an underexpanded jet impinges on it is presented below. The results of [5], where the region near the axis of a shock layer of arbitrary thickness is analyzed within the framework of a model of flow with a constant density, is placed at the basis of the analysis.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 5, pp. 63–70, September–October, 1978.The author thanks Yu. M. Tsirkunov for useful discussions.     

Experimental study of heat and mass transfer from a horizontal cylinder in downward air-water mist flow with blockage effect

An experimental study was made on convective heat and mass transfer from a horizontal heated cylinder in a downward flow of air-water mist at a blockage ratio of 0.4. The measured local heat transfer coefficients agree fairly well with the authors' numerical solutions obtained previously for the front surface of a cylinder over the ranges mass flow ratio 0–4.5×10⁻², a temperature difference between the cylinder and air 10–43 K, gas Reynolds number (7.9–23)×10³, Rosin-Rammler size parameter 105–168 μm, and dispersion parameter 3.4–3.7. Heat transfer augmentation, two-pahse to single-phase of greater than 19 was attained at the forward stagnation point. For heat transfer in the rear part of the cylinder, an empirical formula is derived by taking into account the dimensionless governing variables, that is, coolant-feed and evaporation parameters.     

Hypersonic viscous gas flow over a delta wing in the intermediate interaction regime with allowance for wake flow

The results of calculating the hypersonic flow over a plane delta wing of finite length with allowance for wake flow in the intermediate interaction regime are presented. A comparison is made with the data for flow over a delta wing with given pressure at the trailing edge.Translated from Izvestiya Rossiiskoi Akademii Nauk, Mekhanika Zhidkosti i Gaza, No. 5, pp. 142–149, September–October, 1993.     

On turbulent separated flow past a truncated body of revolution

Results of testing a series of truncated bodies of revolution with convergent afterbodies in a hydrodynamic tunnel are presented. It is shown that the base pressure can be substantially raised and hence the total drag reduced by varying the shape and convergence of the afterbodies. This effect is caused by intense reverse jets formed as a result of the collision of flow particles moving toward the axis of symmetry.The turbulent flow past the bodies is calculated using the method of viscous-inviscid interaction. The formulas derived for the base pressure and drag coefficients agree satisfactorily with the experimental data.Moscow. Translated from Izvestiya Rossiiskoi Akademii Nauk, Mekhanika Zhidkosti i Gaza, No. 6, pp. 50–55, November–December, 1996.     

Subsonic flow of a compressible gas around a semiinfinite flat plate in a pipe

The subsonic flow of an ideal compressible gas around the rear end of a semiinfinite flat plate in a pipe is considered. The flow pattern is similar to that assumed by Efros [1, 2] with a return stream for the cavitational flows. Fal'kovich's method [3] is used to solve the problem and this makes it possible to obtain the solution to the problems of the gas streams at several typical velocities. The method is a generalization of that of Chaplygin [4] for flow problems at one typical velocity.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 4, pp. 101–108, July–August, 1970.     

Fluid motion near the surface of a cylinder in a cross flow

An attempt is made to reconstruct the instantaneous flow structures near the surface of a cylinder from measurements of the various characteristics in the precritical self-similar regime, at the moment of crisis and beyond. Information concerning these structures is of decisive importance in connection with the construction of numerical flow models [1–4] and the reliability and accuracy of the calculation results.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 4, pp. 65–71, July–August, 1989.     

Approximation of the Base Pressure and Determination of the Optimal Height of the Rear Face of a Two-Dimensional Afterbody

On the basis of the available experimental and calculated data, approximate relations for determining the base pressure behind the rear face of a two-dimensional body in Mach 0 to 4 flow are derived, the relative thickness of the turbulent boundary layer on the body ranging from 0 to . Using these relations, the optimum afterbody contours giving a two-dimensional body maximum thrust are determined. The rear face heights of these contours are determined for arbitrary afterbody lengths and boundary layer thicknesses at M = 1–4.     

Structure of the flow of a viscous gas near the trailing edge of a flat plate

Solution of the complete system of Navier-Stokes equations forms the basis for a study of the nature of flow of a viscous heat-conducting gas in the neighborhood of a trailing edge of a flat plate. The problem was solved in accordance with a difference scheme of the third order of accuracy [1]. The calculation was carried out under the same conditions as the experiment of [2], in which a plate of finite dimensions (L = 12 cm) had supersonic M = 2, Re, = 1000 gas flow round it. In order to obtain a thickness of the boundary layer which was acceptable for the purpose of making the measurements (of the order of 2 cm), the unperturbed gas was slightly rarefied. In the study of such problems [3–7] it is necessary to use the complete system of Navier-Stokes equations, since in the immediate neighborhood of the trailing edge one of the important assumptions in the theory of the boundary layer, ²u/y^{2 2}u/x², does not hold. As a result the flow upstream near the trailing edge of the plate will depend on the flow immediately behind the edge, since the perturbations propagate both upstream and downstream in this case. The rarefaction of the gas creates additional difficulties in the formulation of the boundary conditions on the plate with flow round it when this problem is studied numerically.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 2, pp. 27–30, March–April, 1987.     

Flow in the base region of flat bodies

A considerable number of theoretical and experimental papers has been devoted in recent years to studying the flow details in the base region which were not included in the Chapman-Korst scheme. The authors obtained new data supplementing the known facts, or altering the conception about the substance of certain phenomena (the origination of the edge shock, for example). Results are presented below of experimental investigations of the flow configuration in the base region of flat bodies, and the fundamental parameters governing the difference between the flow behind pointed and blunt bodies are examined. The experiments were conducted in the wind tunnel of the Moscow State University Institute of Mechanics in the M=0.3–3.8 Mach number range. The Reynolds numbers, referred to 0.1 m and computed by means of the free-stream parameters, varied between 2·10⁶ and 3·10⁶. The flow behind a wedge (with a 45° vertex angle and 0.025 m altitude) mounted on a plate 0.2 m wide and 0.4 m long was investigated. The flow was studied in the near wake on wedges withapex half-angles =15, 20, and 30°, width 0.2, altitude of the rear section 0.05 m, and a cylinder of the same width and 0.05 m diameter. The models were mounted in the center of the working section by using slender lateral pylons fastened to the perforated walls of the tunnel. A strip of emery paper was glued to turbulize the boundary layer on the models. The pressure in the base domain was measured by using total and static pressure detectors (1.2 mm outer diameter) mounted on a traversing gear.Institute of Mechanics, Moscow State University. Moscow. Translated from Izvestiya Akademii Nauk SSSR. Mekhanika Zhidkosti i Gaza, No. 6, pp. 61–70, November–December, 1972.     

Calculation of base pressure and enthalpy behind a blunt trailing edge separating two supersonic streams with allowance for boundary layers and heat fluxes

A relatively simple method of calculating the parameters of the flow behind a blunt trailing edge separating two supersonic streams is developed. The method is based on the use of the boundary layer approximation and integral laws of mass and energy conservation (viscous-inviscid interaction model). It makes it possible to determine the base pressure and base enthalpy with allowance for the effect of Mach numbers, Reynolds numbers, initial boundary layer thicknesses, specific heat ratios and wall enthalpies for various ratios of the total pressures and enthalpies of the two streams.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 5, pp. 167–176, September–October, 1991.     

Disturbance evolution in the leading-edge region of a blunt flat plate in supersonic flow

The spatio-temporal dynamics of small disturbances in viscous supersonic flow over a blunt flat plate at freestream Mach number M_∞=2.5 is numerically simulated using a spectral approximation to the Navier–Stokes equations. The unsteady solutions are computed by imposing weak acoustic waves onto the steady base flow. In addition, the unsteady response of the flow to velocity perturbations introduced by local suction and blowing through a slot in the body surface is investigated. The results indicate distinct disturbance/shock-wave interactions in the subsonic region around the leading edge for both types of forcing. While the disturbance amplitudes on the wall retain a constant level for the acoustic perturbation, those generated by local suction and blowing experience a strong decay downstream of the slot. Furthermore, the results prove the importance of the shock in the distribution of perturbations, which have their origin in the leading-edge region. These disturbance waves may enter the boundary layer further downstream to excite instability modes.