Method of calculating supersonic three-dimensional flow past smooth bodies

A method is suggested in [1] for calculating supersonic flow past smooth bodies that uses an analytic approximation of the gasdynamic functions on layers and the method of characteristics for calculating the flow parameters at the nodes of a fixed grid. In the present paper this method is discussed for three-dimensional flows of a perfect gas in general form for cylindrical and spherical coordinate systems; relations are presented for calculating the flow parameters at the layer nodes, results are given for the calculation of the flow for specific bodies, and results are shown for a numerical analysis of the suggested method. Three-dimensional steady flows with plane symmetry are considered. In the relations presented in the article all geometric quantities are referred to the characteristic dimension L, the velocity components u, v, w and the sonic velocitya are referred to the characteristic velocity W, the density is referred to the density of the free stream, and the pressure p is referred to w².     

Effect of a longitudinal magnetic field on the characteristics of a stabilized channel arc

The results of calculations of the temperature profiles and volt-ampere characteristics of a long cylindrical argon arc in a longitudinal uniform magnetic field are presented. The calculation was made for the following parameters: pressure p =0.1–10.0 atm; temperatures T = 1000-20,000°K; magnetic field induction B =0-10 T; diameter of cylindrical channel d = 1.0 cm. It is shown that for strongly radiating arcs (p1.0 atm) the temperature profiles become more inflated with an increase in the magnetic field, while for weakly radiating arcs (p 0.1 atm) the appearance of loops in the volt-ampere characteristics is typical for certain conditions (14,000T20,000°K, B1.0 T), indicating the impossibility of arcing under these conditions.Translated from Zhurnal Prikladnoi Mekhaniki i Tekhnicheskoi Fiziki, No. 2, pp. 147–153, March–April, 1975.     

Invariant solutions of the equations of a multicomponent charged-particle beam

The concept of the invariant-group solution (H-solution) was introduced and a general method for obtaining it was developed in [1–3]. The group properties of the equations of a monoenergetic charged-particle beam with the same value and sign of the specific charge, assuming univalency of the velocity vector V, were studied in [4–6], where all essentially different H-solutions were also constructed. Below, the results of [4–6] are extended to the case of a beam in the presence of a fixed background of density ₀ (§1), and also to the case of multivelocity (V is an s-valued function) and multicomponent beams (i.e., beams formed by particles of several kinds) (§2). A number of analytic solutions that describe some nonstationary processes in devices with plane, cylindrical, and spherical geometry —among them a continuous periodic solution for a plane diode with a period determined by the background density -are obtained in §1. A transformation that contains arbitrary functions of time and preserves Vlasov's equations is given (§2). The equations studied can be treated as the equations of a rarefied plasma in the magnetohydrodynamic approximation, when the pressure gradients are negligible as compared with forces of electromagnetic origin.     

Numerical Investigation of Off-Design Regimes of Flow Past Power-Law Waveriders Based on the Flows Behind Plane Shocks

The results of a numerical investigation of supersonic off-design flow past waveriders at the freestream Mach numbers M = 4 and 8 are presented. Flow regimes with M both greater and smaller than the design value M _d are analyzed. Configurations based on the flows behind plane shocks and described by power-law functions are considered. The results are obtained by the finite-volume solution of the Euler equations using higher-order TVD Runge-Kutta schemes.     

Calculating fluid pressure on piston with constant expansion rate

We consider the problem of the expansion at a constant rate of a planar, cylindrical, or spherical piston in a compressible fluid and calculate the fluid pressure on the piston as a function of its velocity. We consider the solution of the self-similar problem of piston expansion in a compressible fluid at a constant velocity. A similar problem has been solved by Kochina and Mel'nikova in On the expansion of a piston in water, PMM, vol. 23, no. 1, 1959. Results are presented of the numerical solution of this problem for certain values of the parameters characterizing the problem, and the variation of the pressure on the piston as a function of the piston velocity is approximated by several empirical formulas. For the cases of the cylindrical and spherical pistons the approximate analytic relations for the pressure on the piston as a function of the velocity are compared with the numerical solution of the self-similar problem of piston expansion.     

Formation of an intense charged-particle beam in the neighborhood of a curved emitter

A local study is made of the flow region and the charge-free region for an axisymmetric regular beam (the normal component of the magnetic field is zero at the emitter). The study is made within the context of hydrodynamic theory. The equation of the beam boundary and the beam potential and normal derivative on it are determined. A solution is obtained for Laplace's equation in the neighborhood of the emitter surface and the equation of the zero-potential shaping electrode is derived. The cases of space-charged-limited (-mode), temperature (T-mode), and nonzero-initial-velocity emission are investigated. The emitting surface and the Cauchy conditions on it are assumed to be defined by analytic functions. A similar problem was solved in [1] for emission in the p-mode and zero magnetic field. The results of [2–4] are utilized. Note that [5] also dealt with solution of the beam equations in the neighborhood of a curved emitter.     

Experimental investigation of the resistance to the flow of a conducting fluid in flat insulated ducts in the presence of a transverse magnetic field with allowance for fringe effects and wall roughness

The distribution of pressure, velocity, and electrical potential has been investigated for a mercury flow in insulated rectangular ducts with a large side ratio (Hartmann-type flow). The ranges of variation of the Reynolds, Hartmann, and Stewart numbers were 7·10²R5·10⁵, 0H490, and 0N24, respectively. Special attention is given to the sections of the channel where the flow enters and leaves the magnetic field. In these zones the pressure is sharply nonuniform and the velocity profiles in a plane perpendicular to the field acquire an M shape. A relation is established between the length of the entrance section, where the flow is three-dimensional, and the MHD similarity criteria. It is shown that ducts which are hydraulically smooth in the absence of a magnetic field become increasingly rough as the field grows stronger. Data are obtained on the resistance coefficient for a stabilized flow measured in a magnetic field and on the dependence of the critical Reynolds number on the Hartmann number.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 4, pp. 10–21, July–August, 1971.     

Chapman-Jouguet waves in a dusty gas

An analytic solution is obtained of the problem of flow of a two-phase medium, representing a mixture of gas and solid or liquid particles behind plane, cylindrical, and spherical Chapman-Jouguet detonation waves. It is assumed that all the particles are the same, are chemically inert, have a true density much greater than the density of the gas, and that their volume concentration a is low. The interaction of the particles and the influence of Brownian motion on them are disregarded. The gas is assumed to be perfect. On the detonation wave, the particle parameters are assumed to be continuous, and the usual gas-dynamical relations on the detonation wave have been applied for the gas parameters because is low. Behind the detonation front, the phases interact through interphase forces and heat transfer. It has been found that the dust content of the combustible gas qualitatively changes the character of flows with Chapman-Jouguet (C-J) waves. It is shown that a plane C-J wave is an envelope of one of the acoustic families of characteristics, and not a characteristic, as occurs in a pure gas [1]. In view of this, only two solutions of the problem of flow behind a plane C-J wave are possible: one solution corresponds to a rarefaction flow and the other to a compression flow. In a pure gas such a problem has a nondenumerable set of solutions: an arbitrary Riemann rarefaction wave can adjoin the plane C-J wave. It is found that in a dusty gas there are converging cylindrical and spherical C-J waves. In a pure gas, there are no converging C-J waves [2, 3]. An expression is found for the distance r_* from the axis (center) of symmetry on which the converging cylindrical (spherical) C-J wave changes into a supercompressed detonation wave. It has been found that r_* d/₀, = 1, 2 for the cylindrical and spherical waves, respectively, d is the particle diameter, ₀ is their initial volume concentration, and the proportionality factor decreases together with d. For the detonating mixture 2H₂ + O₂ the calculations of r_* are given in a number of cases.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 4, pp. 111–118, July–August, 1985.The author wishes to thank V. A. Levin for his interest in the work and his useful discussion of the results.     

Approximate solution of the boundary problem for the equations of a stationary electrostatic charged-particle beam

The solution is given of the equations of a three-dimensional stationary electrostatic beam of charged particles of like sign filling the region between two nearby curvilinear surfaces. We assume that the flow is nonrotational and nonrelativistic and that the velocity vector is a single-valued function. The solution is constructed in the form of an asymptotic series in powers of the small parameter , which is the ratio of the characteristic transverse (a) and longitudinal (l) dimensions of the problem. The first dimension is taken to be the distance between the electrodes, andl defines the scale at which the geometric and physical parameters (emitter curvature, electric field E on the emitter, and the emission current density J) change noticeably. The emission regimes limited by the space charge (-regime), temperature (T-regime), and the case of nonzero initial velocity (U-regime) are studied. The asymptotic behavior is given by the formulas for the corresponding one-dimensional flow between parallel surface.The solution of the boundary problem for emission in the-regime reduces to determination of the emission current density J for fixed electrode geometry and given accelerating voltage. The corresponding formulas are presented, retaining terms of order ³.Two approximations with respect to are performed for the T- and U-regimes. Here the unknown quantity for given properties of the emitting surface (J) will be the electric field E.The results provided by the constructed expansions are compared with the exact solution for flow from a planar emitter along circular trajectories [1]. As an example we examine the two-dimensional problem of flow between two nearby circular cylindrical electrodes with disruption of the coaxiality.The conventional tensor notations are used.     

Some properties of homogeneous flows of rarefied gases

A generalization of the existence conditions for homogeneous flows of a rarefied monatomic gas mixture [2, 3] to the case where external forces are present is presented in [1]. Below we obtain for this case the solution of the Cauchy problem for the Boltzmann equation under free molecular (collisionless) conditions, when the collision integrals may be neglected (Knudsen number K 1). On the basis of this solution we construct a general solution for the equations of the kinetic moments of a Maxwellian monatomic gas mixture in the form of a series in inverse powers of K. Some additional remarks are made concerning the properties of the solutions of the second-order kinetic moment equations, and on the applicability of the Grad 13-moment equations and the Chapman-Enskog method [in particular, for the calculation of slow (Stokesian) motions of a gas mixture].The authors wish to thank M. N. Kogan and A. A. Nikol'skii for their comments.     

Turbulent boundary layer on a permeable surface

Several theoretical [1–4] and experimental [5–7] studies have been devoted to the study of the effect of distributed injection of a gaseous substance on the characteristics of the turbulent boundary layer. The primary study has been made of flow past a flat plate with gas injection. The theoretical methods are based primarily on the semiempirical theories of Prandtl [1] and Karman [2].In contrast with the previous studies, the present paper proposes a power law for the mixing length; this makes it possible to obtain velocity profiles which degenerate to the known power profiles [8] in the case of flow without blowing and heat transfer. This approach yields analytic results for flows with moderate pressure gradient.Notation x, y coordinates - U, V velocity components - density - T temperature - h enthalpy - H total enthalpy - c mass concentration - , , D coefficients of molecular viscosity, thermal conductivity, diffusion - c_p specific heat - adiabatic exponent - r distance from axis of symmetry to surface - boundary layer thickness - U velocity in stream core - friction - c_f friction coefficient - P Prandtl number - S Schmidt number - S_t Stanton number - M Mach number - j=0 plane case - j=1 axisymmetric case The indices 1 injected gas - 2 mainstream gas - w quantities at the wall - core of boundary layer - 0 flow of incompressible gas without injection - v=0 flow of compressible gas without injection - ^* quantities at the edge of the laminar sublayer - quantities at the initial section - turbulent transport coefficients     

On the steady simple shear flows of the one-mode Giesekus fluid

Analytical solutions for the plane Couette flow and the plane Poiseuille flow of the one-mode Giesekus fluid without any retardation time have been obtained by considering the domain of definition for each of the two branch solutions which arise due to the presence of the quadratic stress terms in the constitutive equations. For each fixed value of the mobility parametera, the limiting value of the Weissenberg number for the upper branch solution, i.e., the physically realistic solution is determined in terms of the corresponding dimensionless shear stress for the plane Couette flow and in terms of the corresponding dimensionless pressure gradient for the plane Poiseuille flow. In the case of the plane Couette flow, it is shown that fora falling in the range 0a1/2 only the physically realistic solution exists while for 1/2<a 1 a nonphysical solution coexists with the realistic one. In the case of the plane Poiseuille flow, it is shown that the non-physical solution cannot even exist around the center plane of the channel, and the effects of the mobility parameter and the dimensionless pressure gradient on the flow variables are investigated. Possible extensions of the present approach to other steady simple shear flows with and without the introduction of the retardation time are also discussed.     

Simulation by a source of a thin jet from a body

An exact analytic solution is found to the following plane hydrodynamic problem. An unbounded flow of an ideal incompressible fluid flows around a plate BB' placed at right angles to the velocity vector of the flow at infinity. The pressure on the free boundary P is equal to the pressure in the flow. From an opening in the center of the plate, a jet with flow rate Q from a cavity with pressure P₀ encounters the flow head-on. As a result of the solution, it is found that for fixed width of the opening the values of Q allowed by the scheme are limited. In the limiting case Q = 0 Chaplygin's flow is obtained with stagnation region at the front [1], and in the limiting case Q = Q_max a jet out of a cavity with pressure P₀ into a cavity with pressure P. As Q varies in this interval, the total drag, regarded as the drag of the plate and the chamber from which the jet emerges, takes a minimal value at a certain point. If the width of the opening tends to the length of the slab, the problem of the collision of two jets is obtained; if the width of the opening tends to zero (Q o), the problem of jet flow past a slab with a source is obtained. It is shown that the replacement of the jet by the source gives a good approximation in both the sense of the force characteristics and in the sense of the behavior of the free streamlines.Translated from Izvestiya Akademii Nauk SSSR, Hekhanika Shidkosti i Gaza, No. 5, pp. 47–54, September–October, 1979.We thank L. I. Sedov for his interest in the work and G. Yu. Stepanov for proposing the method of solution and for a helpful discussion.     

Numerical Investigation of the Natural Convection of Water in the Neighborhood of the Density Inversion Point For Grashof Numbers up to 106

The natural convection of fresh water in a square cell is considered at a temperature close to the density inversion temperature for Grashof numbers 2.9 · 10⁴ Gr 10⁶. As a result of the numerical investigation, one steady-state and three self-oscillating regimes are found in addition to the three steady-state flows previously detected earlier and described for low Grashof numbers ( 0 Gr 2 · 10⁵). The basic characteristics of the unsteady flows are analyzed by means of the Fourier method, the fundamental oscillation frequencies are found, and the flow evolution and the variation of the oscillation characteristics with increase in the Gr number are considered.     

Hypersonic flow-past of plane blunt bodies by a nonviscous radiating gas

An analytic solution is obtained in the work in a Newtonian approximation [1] for the flow-past problem for a plane blunt body by a steady-state uniform hypersonic inviscous space-radiating gas flow. The hypersonic flow-past problem for axisymmetrical blunt bodies by a nonviscous space-radiating gas has been previously considered [2–4]. In this case a satisfactory solution of the problem was obtained even in a zero-th approximation by decomposing the unknown values in terms of a parameter equal to the ratio of gas densities before and after passage of the shock wave. The solution of the problem in a zero-th approximation with respect to in the case of flow-past of plane blunt bodies does not turn out to be satisfactory, since the departure of the shock and the radiant flux to the body as gas flows into the shock layer turns out to be strongly overstated under nearly adiabatic conditions. Freeman [5] demonstrated that results may be significantly improved for flow-past of a plane blunt body by a nonradiating gas if a more precise expression is used for the tangential velocity component expressed in a new approximation with respect to the parameter . This refinement is applied in this work for solving the flow-past problem for a plane blunt body by a space-radiating gas. The distribution of the gasdynamic parameters in the shock layer, the departure of the shock wave, and the radiant heat flux to the surface of the body are found. The solution obtained is analyzed in detail for the example of flow-past regarding a circular cylinder.Translated from Zhurnal Prikladnoi Mekhanikii Tekhnicheskoi Fiziki, No. 3, 68–73, May–June, 1975.     

Bifurcation and transition to turbulence in the gap between rotating and stationary parallel disks

The flow in the gap between rotating and stationary parallel disks is an attractive object for studying the transition characteristics in three-dimensional internal flows. Firstly, in this case a large region of the basic motion is satisfactorily described by a self-similar solution to the Navier-Stokes equations [1]; secondly, as the parameter = h²/v ( is the. angular velocity of rotation of one of the disks and h is the gap width) varies, there is an evolution of the basic motion, so that it is easy to produce different types of initial and subsequent instabilities. The basic steady regime for axially symmetric flow has been studied by many authors (see [1, 2]). Questions of the transition in the gap between disks have been considered [3, 4]. This paper presents a methodology and the results of experimental investigations for different types of initial and subsequent instabilities in the gap between disks enclosed by a cylindrical cover. It was found that as a result of the loss of stability of the basic regime one of two steady vortex regimes is developed depending on the value of the relative gap width. The subsequent stages of soft excitation of the turbulent regime are described and the corresponding boundaries established. It is shown that in very narrow gaps the excitation of turbulence has a hard nature of the type realized in Couette flow. The stability limit for a laminarized boundary layer on a rotating disk and the boundary for complete turbulence of the layer were determined for relatively wide gaps. A comparison was made with known data for an unenclosed rotating disk.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 5, pp. 28–36, September–October, 1984.     

Jet-Cavitation Flow Past “Fluid Cylinders”

A new class of plane steady-state flows of an inviscid incompressible weightless fluid in the presence of point singularities inside the flow and constant-pressure regions is studied. Solutions of the problems of jet and cavitation flow past the atmospheres of these singularities are constructed. At positive cavitation numbers, the singular-point method of Chaplygin and the Efros scheme are used for cavity closure. The case of negative cavitation numbers is also considered. A parametric and numerical analysis of the solutions obtained is carried out. The studied flows can be treated as either jet or circulation flow past curvilinear contours of special shape. They can also be used for constructing new schemes for the closure of developed cavitation zones.     

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.     

Conical flows in the neighborhood of breaks in the edges of surfaces separating cocurrent supersonic streams of perfect gas

An investigation is made into the conical flows which occur when a perfect (inviscid and nonheat conducting) gas flows over the terminal edges of surfaces with breaks separating an external and an internal flow with velocity vectors parallel to the line of intersection of the surfaces. Such flows are observed, in particular, in the neighborhood of breaks in the outlet edge of a nozzle of rectangular cross section with a straight or skewed exit plane under conditions of underexpanded flow of a supersonic jet into a cocurrent supersonic stream. By means of a linear analysis flow patterns corresponding to various flow interaction regimes and edge geometries are constructed and a law of similarity is formulated. The validity of the results thus obtained is confirmed by examples of the numerical solution of the complete nonlinear system of Euler equations. In this connection, within the framework of the approach outlined in [1], as a rule, together with the shocks and characteristic surfaces bounding the conical flow in question, the shear discontinuity separating the external and internal streams is constructed.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 1, pp. 119–127, January–February, 1935.     

Nearly singular two-dimensional Kolmogorov flows for large Reynolds numbers

We study the convergence of two-dimensional stationary Kolmogorov flows as the Reynolds number increases to infinity. Since the flows considered are stationary solutions of Navier-Stokes equations, they are smooth whatever the Reynolds number may be. However, in the limit of an infinite Reynolds number, they can, at least theoretically, converge to a nonsmooth function. Through numerical experiments, we show that, under a certain condition, some smooth solutions of the Navier-Stokes equations converge to a nonsmooth solution of the Euler equations and develop internal layers. Therefore the Navier-Stokes flows are nearly singular for large Reynolds numbers. In view of this nearly singular solution, we propose a possible scenario of turbulence, which is of an intermediate nature between Leray's and Ruelle-Taken's scenarios.