Showing papers in "Fluid Dynamics in 1969"

Method of approximate account for the wall effect in cavitation flow around bodies in water tunnels

Abstract: One of the basic questions in the study of advanced cavitation in water tunnels of the closed-circuit type is the establishment of the correspondence between the flow patterns observed in the channel and in an unbounded stream. The objective of the study of the wall effect must be the determination of a connection between the basic characteristics of the phenomenon, i. e., the cavitation numbers, the cavity dimensions, the drag coefficients, etc., for the unbounded flow and the channel flow. A large number of works devoted to this question are known [1–7], but in the majority of them only two-dimensional flows are considered. These studies contain either exact solutions obtained with the aid of the apparatus of functions of a complex variable or solutions in the linearized formulation. At the present time there is urgent need to obtain at least approximate solutions for axisymmetric cavitation flows in a tunnel. In several studies [1, 2, 4] it has been shown that in the case of two-dimensional flows the presence of solid boundaries influences the drag coefficient only through the mechanism of a change of the magnitude of the cavitation number, while the variation of the drag coefficient itself with the cavitation number is not changed in comparison with the unbounded flow. It may be assumed that an analogous situation obtains for the axisymmetric case as well. Then the question of the wall effect may be reduced to establishing the connection between the corresponding cavitation numbers. The present paper makes an attempt to establish the correspondence between the cavitation numbers in the unbounded flow and in the tunnel for which the cavities behind the same body have the same areas of the maximal cross section.

The source-type solution in the problem of unsteady filtration in a medium with random nonuniformity

Abstract: The fundamental problems of the theory of filtration in media with random nonuniformities were formulated in [1] and methods of solution were indicated. Primary attention was devoted to the steadystate filtration processes. In the following we solve one of the most important unsteady problems and indicate the connection of the result obtained with the widely used methods of determining strata parameters from the curves of the pressure variation in nonflowing wells. We note that the interpretation of the results of such measurements is usually carried out with the aid of the solution of the corresponding problem for a homogeneous stratum or for a stratum whose nonuniformity has a regular nature (for example, [2]), which definitely limits the possibilities of the method. At the same time it is obvious that the solution of these problems for irregular media and particularly the determination of their effective characteristics requires the use of statistical methods of computation.

Asymptotic solutions of the Navier-Stokes equations in regions with large local perturbations

Abstract: There are many problems of the dynamics of viscous flows of liquids and gases at high Reynolds numbers for the solution of which the classical theory of the boundary layer cannot be used This applies, in particular, to all the problems with various sorts of local singularities in the stream-flows in the vicinity of corners, in regions of interaction of the boundary layer with an incident shock, flows near points of separation or attachment of the stream, etc The purpose of the present paper is to attempt the theoretical investigation of problems of this type on the basis of the general analysis of the asymptotic behavior of the solutions of the Navier-Stokes equations In order to do this, use is made of the familiar method of the construction and splicing of a combination of asymptotic expansions representing the solutions in the various characteristic regions of the stream with viscosity decreasing without bound [1] As an example, detailed consideration is given to the problem of viscous supersonic flow near a wall with large local curvature of the surface

Optimal shapes of bodies with attached shock waves

Abstract: We consider the problem of finding the shape of two-dimensional and axisymmetric bodies having minimal wave drag in a supersonic perfect gas flow. The solution is sought among bodies having attached shock waves. The limitations on the body contour are arbitrary: these constraints may be body dimensions, volume, area, etc. Such problems with arbitrary isoperimetric conditions may be solved by the method suggested in [1, 2]. This method involves the use of the exact equations of gasdynamics which describe the flow as additional constraints. This method was developed further in [3–6] in the solution of several problems.

Lattice of plates in an unsteady supersonic flow

Abstract: The supersonic unsteady flow of a gas past a lattice of thin, slightly curved profiles has been investigated in several studies. The paper [1] is devoted to an evaluation of the effect of wind tunnel walls on the unsteady aerodynamic characteristics of a profile, and [2] investigates the effects of the boundaries of a free jet. These cases are equivalent respectively to the anti-phase and in-phase oscillations of the profiles of an unstaggered grid. In [3] consideration is given to a more general case of gas flow past a profile in a wind tunnel with perforated walls. Flow past a lattice of profiles with stagger is studied in [4], where the magnitude of the stagger angle is limited by the condition that the lattice leading edge is located in the undisturbed stream. In the present paper we present a method of calculation of the unsteady supersonic flow of a gas past a lattice of profiles with arbitrary stagger. As an example the results are presented of the calculation of the aerodynamic forces and moments acting on an oscillating profile in a wind tunnel with solid walls and in a free jet.

Investigation of the microstructure a turbulent jet in a wake

Abstract: The study of the characteristics of the turbulence in the boundary layer and in free jets is one of the most important problems of the aerodynamics of viscous fluids. The accumulation of information on the pulsation characteristics of jet flows and the establishment of the corresponding governing laws may serve to verify the basic hypotheses of the semiempirical theories of turbulence, and also for the development of more advanced computational methods. In many cases the measurement of the pulsation characteristics of turbulent jets is of practical interest. The studies made up till now [1–5] of the microstructure of turbulent flow in the primary region of submerged axisymmetric jets have made it possible to obtain several interesting results. In particular, in addition to the average velocity profiles, hot-wire anemometric equipment has been used to measure the normal and tangential Reynolds stresses and also the intermittency factor in cross sections of the jet, the distribution of the intensity of the longitudinal and lateral velocity pulsations along the axis, the correlation coefficients and the corresponding integral turbulence scales, etc. These measurements have made it possible to draw several important conclusions on the mechanism of turbulent exchange, on the order of the terms omitted in the equation of motion, and on the semiempirical theories of turbulence [6–9]. The common deficiency of the studies mentioned above is that near the boundary of a submerged jet, where the average velocity is practically equal to zero, the intensity of the pulsations is so great that it makes the reliability of the results obtained by means of the hotwire anemometer questionable. In this connection Townsend [6] indicated the advisability of studying the microstructure of a turbulent jet issuing into a low-velocity ambient flow. The present study had as its objective the investigation of the microstructure of the primary region of an axisymmetric jet in a wake flow over quite a broad range of the flow ratio parameter m=uδ/u0;here u0 is the average velocity at the nozzle exit, uδ is the velocity of the ambient stream. For various values of the parameter m in the primary region of the jet measurements were made of the profiles of the three components of the pulsation velocity and the Reynolds shear stresses, and also the values of the average velocity and two components of the pulsation velocity at a large number of points on the jet axis. The measured profiles of the Reynolds shear stresses were compared with the corresponding profiles calculated on the basis of the boundary layer equations from the experimentally determined average velocity profiles. For two values of the parameter m, in one of the sections of the jet measurements were made of the correlation coefficients of the longitudinal components of the pulsation velocity and the variation across the jet of the integral turbulence scale was determined. The results obtained give an idea of the influence of the parameter m on the characteristics of the turbulent jet in an ambient stream.

Two-fluid hydrodynamics of a fluidized bed

Abstract: Prior to the formulation of specific problems on the motion or processes of transport in a fluidized bed it is necessary to answer two fundamental questions, without the satisfactory resolution of which consistent theoretical study of the fluidized bed is quite impossible. The first question is related to the formulation of the governing equations-the equations of motion of the fluidized bed as a two-phase continuous medium-the second question is concerned with the construction of some model of the interaction of the fluidized bed with the solid surfaces bounding the bed and the consideration of the boundary conditions which must be imposed on the solution of these equations.

Heat transfer and equilibrium temperature of a sphere in a supersonic rarefied gas flow

Abstract: Some results are presented of experimental studies of the equilibrium temperature and heat transfer of a sphere in a supersonic rarefied air flow.

Hypersonic similarity in the flow of an imperfect gas around slender blunt bodies of complex shape

Abstract: The hypersonic similarity laws for flow around slender blunt bodies [1–3] are generalized to bodies with nonsmooth lateral surface, in particular, those having corners (under the condition of unseparated supersonic flow). The similarity conditions are considered for a gas which is imperfect throughout the entire disturbed region.

Stability of a laminar boundary layer in an incompressible fluid with variable physical properties

Abstract: The stability of plane-parallel flows of an incompressible fluid with variable kinematic viscosity in the presence of solid walls has been discussed in [1–5]. The stability of Couette flow is considered in [1]. The method of solution, which is the same as that used in [2], differs from the Tollmien-Schlichting method, since the expansion of the solutions in powers of αR which is used assumes the smallness of this quantity. A general formulation of the problem of stability of a nonuniform fluid is presented in [3]. Di Prima and Dunn [4] used the Galerkin method to study the stability of the boundary layer relative to vortex-like disturbances in the case of variable kinematic viscosity. Since the development of this sort of disturbance depends only weakly on the form of the velocity profile in the boundary layer, a marked change of the viscosity had little effect on the critical Reynolds number. This same problem is considered in [5], The present author was not able to find in the literature any references to study of the stability of the laminar boundary layer in an incompressible fluid with variable kinematic viscosity relative to disturbances of the Tollmien-Schlichting type, with the exception of mention in [4] of an unpublished work of MacIntosh which showed the essential dependence of the critical Reynolds number on the viscosity gradient. In Section 1 an analysis is made of the effect of variability of the kinematic viscosity on the stability of the boundary layer relative to Tollmien-Schlichting waves under the condition of constant fluid density. Two approaches are possible to the study of the development of disturbances in a heterogeneous fluid. On the one hand, we can assume that the displacement of the fluid particles does not cause changes in the distribution of p(y) and υ*(y), i.e., the velocity pulsations are not accompanied by pulsations of ρ and υ*. This will be the case if a particle which is characterized by the quantities ρ1,υ 1 entering a layer with the different values ρ2 υ 2, instantaneously alters its properties so that its density becomes equal to ρ2, and its kinematic viscosity becomes equal to υ2. On the other hand, we can consider that a fluid particle moving from layer 1 into layer 2 fully retains the properties which it had in layer 1. In this case the velocity pulsations naturally lead to pulsations of the quantities ρ and υ*. In actuality, the phenomenon develops along some intermediate scheme, since the particle alters its properties as it moves in a heterogeneous fluid. The degree of approximation of the process to the first or second scheme depends on the rate of these changes. The analyses below are based on the first scheme.

Steady flow of a liquid metal in a rectangular MHD channel

Abstract: The known experimental studies of steady flows of a liquid metal in magnetohydrodynamic (MHD) channels of rectangular section [1–4] were performed only for a few values of the Reynolds number, which does not permit a clear delineation of the fundamental governing laws of the flow in the zone of transition from laminar to turbulent flow. In addition, the study of turbulent MHD flows has been limited to two-dimensional channels.

Unsteady supersonic flow around blunt bodies with detached shock wave

Abstract: The numerical method of calculating the supersonic three-dimensional flow about blunt bodies with detached shock wave presented in [1–3] is applied to the case of unsteady flow. The formulation of the unsteady problem is analogous to that of [4], which assumes smallness of the unsteady disturbances. The paper presents some results of a study of the unsteady flow about blunt bodies over a wide range of variation of the Mach number M∞=1.50−∞ and dimensionless oscillation frequency ωl/V∞=0−1.0. A comparison is made with the results obtained from the Newton theory.

Method of local similarity for a three-dimensional laminar boundary layer with pressure gradient

Abstract: The hypersonic boundary layer has several unique features, one of which is its relatively slight sensitivity to a negative pressure gradient. This is associated with the presence in the boundary layer of a near-wall region of comparatively cold gas with high density. Therefore, following [1], for the description of such flows we may make use of the method of local similarity with the use of the known Blasius solution for a plate.

Vertical impact of a sphere half submerged in a liquid of finite depth

Abstract: As is known, the study of impact in an ideal incompressible liquid in the classical formulation reduces to mixed problems of potential theory.

Unsteady diffusion of a magnetic field into a plasma cooled by solid walls

Abstract: In recent y e a r s exper iments have been conducted with very s t rong shock waves (M ~ 100). The essence of the exper iments is the following (Fig. 1). At the ini t ia l instant of t ime a d i scharge takes place in the d i scharge chamber 1 which is f i l led with the t e s t gas and which communica tes with tube 2. The d i scharge p l a s m a is e jec ted into tube 2 forming a shock wave which propagates in the gas. Ionized gas , which s e r v e s as the tes t medium, t r ave l s behind the shock wave. One of the methods of studying p lasma is that of the d i sp laced magnet ic flux [1, 2]. The e s sence of this method follows. The shock wave and the a s s o c ia ted p l a sma t r a v e l in the longitudinal magnet ic f ield c r ea t ed by the coil 3. The p l a sma has high e l e c t r i c conductivity; the re fore as it moves it d i sp laces the magnet ic field. However, the p lasma conductivity is f ini te , the re fore the magnet ic f ield diffuses into the p l a sma , and the magnet ic flux, d isp laced by the p l a s m a , d iminishes with t ime. The ra te of p e n e t r a t ion of the magnet ic f ield is r eco rded by the m e a s u r ing coils 4, 5. From the known rate of diffusion of the magnetic field we can judge the plasma conductivity and from that we can judge the temperature, if we consider the plasma to be fully ionized. This procedure was followed basically in [I, 2]. The present study was made to evaluate the role of thermal conductivity in the process of diffusion of a magnet ic f ield into a p lasma . F i r s t the l i n e a r p rob l em of the diffusion of a m a g net ic f ie ld into a cy l ind r i ca l conductor without account for t he rma l conductivity is solved. This p rob lem is solved using the method of in tegra l re la t ions and the method of expanding the unknown functions in s e r i e s , with the l a t t e r method being used to evaluate the accu racy of the f i r s t . In this case there is no a s s u r ance of uniform convergence of the method of s e p a r a t ion of the v a r i a b l e s s ince the in i t ia l condit ions a re not coupled with the boundary condit ions. Then the quest ion is analyzed of the cooling of a p l a s m a moving in a tube. Es t ima tes a re made of the max ima l p e r m i s s i b l e t imes for p r o c e s s e s in which the cooling does not affect the p l a s m a moving in the cen te r of the tube. F ina l ly , the p rob lem of the diffusion of the magnet ic f ie ld into a p l a s m a which is cooled by the wal ls is solved. All the analyses are carried out under the assumption of one-dimensionality of the processes, i.e., the diffusion and thermal layers are assumed quite thin in comparison with the characteristic linear dimension of the plasma slug. The plasma is considered as a solid conductor with the conductivity ff and the coefficient of thermal conduction k, calculated using the corresponding equations for a fully ionized gas.

Wave resistance of a viscous liquid

Abstract: In this paper we examine the resistance encountered by a system of normal stresses during its rectilinear motion along the surface of a viscous liquid of infinite depth. The problem is solved in the linear formulation, i.e., it is assumed that amplitudes of the waves which arise are small and the waves are shallow. The solution for the two-and three-dimensional problems is obtained in the general case in closed form. In the two-dimensional case a detailed study is made of the case when a constant pressure p0, moving with the constant velocity U, is given on a segment of length 2l. In the three-dimen-sional problem the case is studied when the normal stress is concentrated on a segment of a straight line of length 2l, which can replace a ship moving along a straight course with the constant velocity U. The integrals obtained in both cases are studied using the stationary phase method, the application of which for the three-dimensional integrals with respect to a volume with boundaries is justified in §1 of the paper. As a result we obtain equations for the wave resistance in the two- (§2) and three-dimensional (§3) cases.

Influx of two fluids to a completed gallery and well in stratified soils

Abstract: It often happens that waters from different water-bearing horizons have different mineralization. Salt accumulation is usually found in clayey interlayers. Therefore during pumping from one horizon water may enter that horizon from two neighboring strata or clayey interlayers, and the water pumped out may be a mixture of waters with differing mineralization.

Unsteady motion of systems of thin hydrofoils

Abstract: I. Let us consider the unsteady motion of a thin profile immersed under the free surface of a liquid at the depth h (Fig. 1). We shall consider that the profile is a flat plate with the chord ( -a , a), inclined to the stream at the small angle of attack 3. The profile moves with the velocity u0 m the positive direction of the ~eal axis, and also has vertical and rotational perturbations with the small velocities y' and w~ respectively.

Some unique features of supersonic unsteady gas flow around bodies

Abstract: We consider unsteady supersonic gas flow about bodies for small Strouhal numbers. The amplitude of the angles of attack is assumed arbitrary under the condition that the bow shock remains attached, and the flow behind the shock is supersonic. A criterion is formulated which permits the comparison of the damping characteristic for small and large amplitudes of the disturbed motion. A comparison is made, using the wedge as an example, of the damping characteristics calculated by various theories, and the hypersonic similarity law is verified. Simple asymptotic equations are presented for the rotary derivatives of the thin wedge in a hypersonic gas stream.

General expressions for unsteady forces in a lattice of profiles

Abstract: A large number of studies have been devoted to the unsteady flow of a viscid incompressible fluid past a lattice of thin profiles and the determination of the resulting aerodynamic forces and moments. For example, in the particular case of the motion of a lattice with stagger γ with zero phase shift α of the oscillations between neighboring profiles, Haskind [1] determined the unsteady lift force and moment. Popescu [2] suggested expressions for the force and moment in the case when α=0 and γ=0, using the method of conformal mapping. Samoilovich [3] obtained equations for the unsteady lift force and moment by the method of the acceleration potential for phase shift α=0 and α=π of the oscillations between neighboring profiles. Musatov [4] used an electronic digital computer to calculate the overall unsteady aerodynamic characteristics of a grid by the vortex method, taking into account the amplitude of the oscillations and the initial circulation for α=mπ (∥m∥≤1). Gorelov [5] determined the coefficients of the over-all unsteady aerodynamic force and moment of a profile in a lattice with the stagger γ and any value of α=mπ. He used a method based on the unsteady flow past an isolated profile with subsequent account for the interference of the profiles in the lattice.

Investigation of basic characteristics of the MHD bearing

Abstract: The load-carrying capacity and the friction torque of a magnetohydrodynamic bearing are calculated The calculation is made for specified boundaries of the lubricating layer (0, π) with a small magnitude of the radial gap It is shown that the presence of an external magnetic field directed along the axis of the bearing and a radial electric field leads to a considerable increase of the load-carrying capacity of the electrically conductive lubricating layer The condition for which the friction torque on the journal of an MHD bearing vanishes is found