Multiparameter boundary layer method for slender bodies of revolution

The problem of the laminar and steady incompressible boundary layer on elongated slender bodies of revolution in an axisymmetric flow is examined in those cases in which the ratio of the boundary layer thickness to the radius of the body's cross section should not be neglected. For this purpose the boundary layer equation for slender bodies of revolution is universalized on the basis of Loitsyanskii's method.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 6, pp. 22–27, November–December, 1991.     

Acoustic radiation from slender bodies

A general approach is proposed to compute the acoustic field radiated by axisymmetric slender bodies at medium (Ka 1) and low (Ka 1) frequencies. A multiple-scales technique is developed to determine the sound pressure in the near field. The far field solution is built up as a continuous distribution of sectorial spherical harmonics and their densities are deduced by matching with the near field solution. Some applications are presented to support the analytical results.     

On configurational balance in slender bodies

We propose a new derivation of the evolution equation of a sharp, coherent interface in a two-phase body having elongated shape, a body which we regard as a one-dimensional micropolar continuum. To this aim, we introduce a system of forces acting at the interface, and we apply the method of virtual powers to derive a balance law involving these forces. By exploiting the dissipation inequality, we manage to write this balance law in terms of a scalar field whose form is reminiscent of a well-known expression for the configurational stress in three dimensional micropolar continua.     

Aerodynamic interference between high-speed slender bodies

Under supersonic flow conditions, slender bodies in close proximity induce aerodynamic interference effects. This paper aims to quantify the magnitude of the resulting interference loads and to understand the underlying flow-physics mechanisms that cause them. A pair of identical slender bodies are investigated through a series of wind-tunnel experiments and supporting computational fluid dynamics (CFD) predictions. The bodies induce a complex interference flowfield, which tends to be bespoke to each configuration. The flow features include impinging shock and expansion waves, conical shock reflections, strong skewing of the boundary-layer flows and shock diffraction. The effects of axial stagger, lateral separation and the strength of the primary disturbance flow field are evaluated. The interference loads are found to be most sensitive to the initial location of the primary disturbance but are also affected by its strength. In addition, maximum interference loads which equate to an effective incidence of up to 6° are observed. Finally, very good agreement is found between the measurements and the CFD predicted normal force and pitching moment.     

Wave forces on stationary slender bodies

A theory is presented for the prediction of the wave forces on ships and the pressure field on slender bodies vibrating in an acoustic medium. In both radiation and diffraction the flow in the near field is approximated by a sequence of two-dimensional problems supplemented with homogeneous components which account for longitudinal flow interactions. These are matched to three-dimensional far-field approximations represented by axial source distributions and two integral equations are solved for their strengths. The theory is valid from the incompressible long-wavelength limit to wavelengths comparable to the body beam. Comparisons of wave forces and the acoustic radiation impedance pressure are in very good agreement with exact solutions. It is shown that the asymptotic matching conserves energy.     

Flutter of slender bodies under axial stress

The equations of motion of flexible slender bodies with constant body sections immersed in a uniform axial flow are discussed and used to derive some simple results for the divergence speed and the flutter speed. The results are compared with a classical waving flag result in two-dimensional flow. The slender body result for the flutter speed is compared with values obtained from wind tunnel experiments for some low budget paper strips.     

Flutter of slender bodies under axial stress

The equations of motion of flexible slender bodies with constant body sections immersed in a uniform axial flow are discussed and used to derive some simple results for the divergence speed and the flutter speed. The results are compared with a classical waving flag result in two-dimensional flow. The slender body result for the flutter speed is compared with values obtained from wind tunnel experiments for some low budget paper strips.     

Method of curved bodies in problems of unsteady hypersonic flow past slender bodies

The method of curved bodies involves replacing the unsteady flow past a body by steady flow past a different body obtained from the original body by suitable curvature of its form. The idea of the method was proposed by Vetchinkin in 1918 and was first carried out in [1]. Here the authors started from the assumption that the pressure on the body surface is determined only by its local angle of attack.We know that this method is justified only for circular motion of a slender body with constant velocity within the framework of subsonic or supersonic linearized theory.It will be shown below that the method of curved bodies is rigorously justified for hypersonic unsteady flow past slender pointed bodies within the framework of the law of plane sections, which is often used to study unsteady flows, for example [2, 3]. Here the idea of the method involves the selection of a body of form such that for uniform translational motion its wake in a stationary, normally intersected plane coincides in time with the wake of the original body.The general theory is presented for arbitrary bodies, in particular for bodies of the type of slender oscillating wings, but attention is devoted primarily to the motion of a rigid body of rotation. In this case, in the hypersonic approximation (of the type of [4, 5]) the method also extends to slender blunted bodies.In the general case this method reduces the four-dimensional unsteady problem to a three-dimensional steady problem, which presents no particular difficulty in view of the existence of suitable methods and programs (for example [6]). Here, in contrast with the classical version of the method [1], in the general case the original body is replaced at very moment of time by a one-parameter (with parameter t₀) family of curved bodies.In the case which is most often encountered in practice of slow oscillation of the body surface, when the unsteady component of the solution is small in comparison with the steady compoent, the small-parameter method is used, which allows us to represent the solution in a simple form with an explicit linear dependence on the parameter t₀.The basic notation L body length - ₀ body characteristic relative thickness or angle of attack - ₀ characteristic Strouhal number - r₀ maximal radius of the blunt nose - ,a undisturbed medium density and speed of sound - V and M velocity and Mach number of the center of rotation or of the point x₀ - T₀ characteristic time of the unsteady motion (for example, the period of the oscillation) - T=L/V time for the body to pass a fixed plane - V²p pressure The author wishes to thank A. V. Antonets and Yu. M. Lipnitskii for carrying out the calculations and analyzing their results.     

Waves in wood: axisymmetric waves in slender solids of revolution

Low-frequency axisymmetric waves in slender axisymmetric anisotropic columns are studied. The governing equation of motion is solved using power series with a stretched radial coordinate, leading to a variety of ordinary differential equations in the longitudinal direction. Detailed results for cylindrically orthotropic columns are given.     

Radiation and scattering from loaded bodies of revolution

The problem of electromagnetic radiation and scattering from loaded bodies of revolution of arbitrary shape is considered. The analysis assumes the existence of an impedance function relating the tangential electric field to the surface current on the body. A solution is obtained by the method of moments applied to the potential integral formulation of the problem. The results are expressed in terms of generalized network parameters, using formulas previously obtained for unloaded bodies. Representative computations are given for plane-wave scattering and radiation from apertures in loaded cylinders and hemispheres. A general computer program for arbitrary bodies of revolution is available.     

Calculation of the viscosity effect on the aerodynamic characteristics of slender bodies at hypersonic flow velocities

Approximating dependences of the local coefficients of friction, heat transfer, and pressure induced by a boundary layer on the generalized similarity parameters, including the inviscid flow characteristics, are obtained on the basis of the results of a numerical calculation of hypersonic flow past a number of plane and axisymmetric bodies. If the inviscid flow characteristics are known, these relations can be used to take the viscosity approximately into account under conditions of interaction between the laminar boundary layer and the hypersonic inviscid stream [1].Moscow. Translated from Izvestiya Rossiiskoi Akademii Nauk, Mekhanika Zhidkosti i Gaza, No. 4, pp. 142–150, July–August, 1995.     

Investigation of transonic bounded gas flow past three-dimensional slender bodies

The pressure coefficient distributions induced in the process of a bounded gas flow past a schematized vehicle of the Khotol type are obtained for various transonic regimes in a wind tunnel with a perforated working section as a result of simultaneously solving the boundary-value problems for the near and far flow fields. These distributions are compared with the pressure coefficient distributions in the case of unbounded flow past the body. The additional pressure coefficient induced by the tunnel walls is calculated for subsonic flow past a vehicle in a cylindrical wind tunnel of large radius. The calculation results are consistent with the asymptotic Malmuth theory for subsonic velocities.Moscow. Translated from Izvestiya Rossiiskoi Akademii Nauk, Mekhanika Zhidkosti i Gaza, No. 2, pp. 149–158, March–April, 1995.     

Hypersonic flow past ducted blunt bodies of revolution

Chernyi [1, 2] has examined the problem of hypersonic flow past a ducted cone with sharp leading edge. In the following we present an analysis of the characteristic features of this problem in the case of a blunt leading edge. We use hypersonic theory for flow past slender bodies with nose blunting of relatively small dimensions [1, 3, 4], based on replacing the nose by a concentrated force and use of the nonsteady analogy. It has been shown in [4, 5] that within the framework of this theory the effect of the violation of the law of plane sections and also the effect of the chemical and physical transformations of the gas in the high-entropy layer is qualitatively equivalent to a change in the drag coefficient of the nose. This approach makes it possible to establish useful similarity laws. The development of these ideas in the direction of the study of the flow structure behind the bow shock wave and analysis of the parameters defining this structure is given in [6–8] in which, in particular, the role of the entropy distribution with respect to the streamlines in the transitional section between the nose and the side surface was clarified and the important practical empirical result was established that this distribution is universal for noses of any form for given flow conditions. In the following these results are extended to blunt bodies of revolution with a duct in the nose. We examine the flow region which is external to the duct under the assumption that the external flow regime corresponds to maximum flow rate through the duct. A characteristic feature of the problem is associated with the additional characteristic linear dimension r₀, which determines the gas mass lost through the duct.     

Prediction of asymmetric vortical flows around slender bodies using Navier-Stokes equations

Steady and unsteady asymmetric vortical flows around slender bodies at high angles of attack are solved using the unsteady, compressible, this-layer Navier-Stokes equations. An implicit, upwind-biased, flux-difference splitting, finite-volume scheme is used for the numerical computations. For supersonic flows past point cones, the locally conical flow assumption has been used for efficient computational studies of this phenomenon. Asymmetric flows past a 5° semiapex-angle circular cone at different angles of attack, free-stream Mach numbers, and Reynolds numbers has been studied in responses to different sources of disturbances. The effects of grid fineness and computational domain size have also been investigated. Next, the responses of three-dimensional supersonic asymmetric flow around a 5° circular cone at different angles of attack and Reynolds numbers to short-duration sideslip disturbances are presented. The results show that flow asymmetry becomes stronger as the Reynolds number and angles of attack are increased. The asymmetric solutions show spatial vortex shedding which is qualitatively similar to the temporal vortex shedding of the unsteady locally conical flow. A cylindrical afterbody is also added to the same cone to study the effect of a cylindrical part on the flow asymmetry. One of the cases of flow over a cone-cylinder configuration is validated fairly well by experimental data.