Showing papers in "Fluid Dynamics in 1977"

Hydrodynamics of wetting

Abstract: Fluid motion along a smooth, solid surface is examined when the free surface forms a final visible angle with the solid boundary. The dependence of this angle on the velocity with allowance for capillary forces is determined. The Reynolds number is small. The problem of the motion of the phase interface in a capillary and the spreading of viscous fluid drops on solid surfaces are solved. Experimental results are explained. Up to now, not only were analytical results lacking in this field, but also there was not even a precise formulation of the problem (see the review in [1]).

Flow structure in motion of a spherical drop in a fluid medium at intermediate Reynolds numbers

Abstract: The problem of flow of a viscous fluid around a spherical drop has been examined for the limiting case of small and large Reynolds numbers in several investigations (see [1–3], for instance; there is a detailed review of various approximate solutions in [4]). For the intermediate range of Reynolds numbers (approximately 1≤Re≤100), where numerical integration of the complete Navier-Stokes equations is necessary, there are solutions of special cases of the problem —flow of air around a solid sphere [5–7], a gas bubble [8, 9], and water drops [10]. The present paper deals with flow around a spherical drop at intermediate Reynolds numbers up to Re=200 for arbitrary values of the ratio of dynamic viscosities Μ=Μ1/Μ2 inside and outside the drop. It is shown that a return flow can arise behind the drop in flow without separation. In such conditions the circulatory flow inside the drop breaks up. An approximate formula for the drag coefficient of the drop is given.

Dynamics of vorticity at a sphere

Abstract: The article discusses the two-dimensional flow of an incompressible liquid between two infinitely close concentric spheres, due to an initial distribution of the vorticity differing from zero. The concept of point singularities (vortices, sources, and sinks) at a sphere is introduced. Equations of motion are obtained for point vortices, as well as invariants of the motion, known for the plane case [1]. The simplest case of the mutual motion of a pair of vortices is considered. Equations are obtained for the motion of point vortices at a rotating sphere. Integral invariants for the continuous distribution of the vorticity are obtained, having the dynamic sense of the total kinetic energy and the momentum of the liquid at the sphere. The effect of the topology of the sphere on the dynamics of the vorticity is noted, and a comparison is made with the plane case.

Solitary waves in a layer of viscous liquid

Abstract: Solitary waves in a thin layer of viscous liquid which is running down a vertical surface under the action of gravity are investigated. The existence of such waves was demonstrated in the experiments of [1, 2]. The difficulties that must be faced in a theoretical computation were also noted in these studies. Below a solution of the problem of stationary waves is obtained by the method of expansion in the small parameter in two regions with subsequent matching and also by a numerical integration method. It is shown that in each case a solution of solitary wave type exists along with the single-parameter family of periodic solutions (parameter—the wave number α). On decreasing the wave number, the periodic waves go over into a succession of solitary waves.

Interaction between a supersonic boundary layer and acoustic disturbances

Abstract: The development of disturbances in a boundary layer that have been induced by an external acoustic field are investigated. The problem is considered in the linear formulation. It is shown that the oscillations inside the supersonic boundary layer can have several times the intensity of the external disturbances. The susceptibility of the boundary layer to the acoustic disturbances increases with increasing Mach number. Cooling of the surface leads to a small decrease in the intensity of the longitudinal velocity oscillations in the layer. The effect of the parameters of the acoustic wave is considered, i.e., the effect of the frequency and phase velocity on the development of the disturbances.

Nonsteady supersonic flow over spiked bodies

Abstract: In longitudinal supersonic flow over spiked cylinders nonsteady regimes can occur in which a separation zone is periodically generated at the spike, grows vigorously in size, and then vanishes. Several authors [1–6] have investigated the physical pattern of flow with separation zone fluctuations (using shadowgraphs) and have determined the boundaries of existence of the nonsteady regime as a function of the ratio between the spike length and diameter of the cylinder. The authors, however, did not systematically study the dependence of the pulsation frequency on the freestream Mach and Reynolds numbers or on the relative diameter and tip angle of the spike. We have undertaken such an investigation. We are concerned primarily with the influence of the dimensionless parameters on the Strouhal number Sh of the separation zone pulsations at a spike attached to the front of a flat-ended cylinder. Earlier investigations [4–6] have been carried out using motion pictures with film speeds up to 32·103 frames/sec. In the present study we used high-speed motion pictures with a speed of 6.25· 105 frames/sec. This speed allowed us to determine the precise sequence of phases of the pulsations and their relative durations, as well as the speed at which the boundaries of the separation zone move.

Flows in the initial sections of channels with permeable walls

Abstract: Calculations of two types of flows in the initial sections of channels with permeable walls are carried out on the basis of semiempirical turbulence theories during fluid injection only through the walls and during interaction of the external flow with the injected fluid. Experimental studies of the first type [1–3] show that at least within the limits of the lengths L/h 100 (Re0=v0h/Ν or Re0=v0a/Ν, where v0 is the injection velocity), and small fluid compressibility, the axial velocity component is described by the relations for ideal eddying motion: u=(π/2)x× cos (πy/2) in a flat channel and u=πx cos (πy2/2) in atube (the characteristic values for the coordinates are, respectively, h anda). Measurements indicate the existence of a segment of laminar flow; its length depends on the Reynolds number of the injection [3]. In the turbulent regime the maximum generation of turbulent energy occurs significantly farther from the wall than in parallel flow. Flows of the second type in tubes were studied in [5–7]. These studies disclosed that for Reynolds numbers of the flow at the entrance to the porous part of the tube Re=u0a/Ν 0.01 leads to suppression of turbu lence in the initial section of the tube. An analogous phenomenon was observed in the boundary layer with v0/u0>0.023 [8, 9]. Laminar-turbulent transition in flows with injection was explained in [10, 11] on the basis of hydrodynamic instability theory, taking into account the non-parallel character of these flows. The mechanisms for the development of turbulence and reverse transition in channels with permeable walls are not theoretically explained. Simple semiempirical turbulence theories apparently are insufficient for this purpose. In the present work results are given of calculations with two-parameter turbulence models proposed in [12, 13] for describing complex flows. Due to the sharp changes of turbulent energy along the channel length, a numerical solution of the complete system of equations of motion was carried out by the finite-difference method [14].

Dynamics of vapor-gas bubbles

Abstract: The nonlinear problem of thermal, mass, and dynamic interaction between a vapor-gas bubble and a liquid is considered. The results of numerical solution of the problem of radial motion of the bubble caused by a sudden pressure change in the liquid, illustrating the behavior of vapor-gas bubbles in compression and rarefaction waves, are presented. The corresponding problem for single-component gas and vapor bubbles was considered in [1, 2].

Stability of a supersonic boundary layer on a permeable surface with heat transfer

Abstract: The effect of cooling of a permeable surface on the stability of a supersonic boundary layer on it is investigated. As distinct from the case of an impermeable surface, deep cooling can reduce the critical Reynolds number. Common points of the continuous and discrete spectra are found in the region of the disturbance parameters.

Thermophoresis and photophoresis in a rarefied gas

Abstract: The present article discusses the problem of the force acting on a spherical particle in a heated rarefied gas (thermophoresis) and the problem of the force acting on such a particle in an isothermal rarefied gas, heated by an external heat flux (photophoresis). Both problems are solved in a linear statement, i.e., under the assumption of the smallness of the temperature gradient of the gas and of the external heat flux, respectively. The rising interest in these problems is due to problems of atmospheric contamination, the physics of clouds, etc.

Mass transfer of particles located on the flow axis at large peclet number

Abstract: Diffusional influx of a substance dissolved in a medium into absorbent particles moving relative to one another in a viscous incompressible liquid is examined. An approximate analytical expression is obtained for the differential and integral flux of material into the surface of each particle, accounting for variations in the velocity and concentration fields due to the presence of the other particles. The results obtained can be applied to a lattice of spheres washed directly and uniformly in an infinite flow and located at distance 1 ≪l ≪ P1/3 relative to each other. It is shown that the diffusional flux of material into the first sphere is almost twice as large as into the other, and for a large number of spheres k the total diffusion flux tends to zero inversely as the 1/3 power of k.

Diffusion and adsorption of a radioactive gas in a porous medium

Abstract: The results of solving the one-dimensional problem of the motion of a pulse of radioactive gas, carried through a porous medium by a stream of inert carrier with constant velocity, are generalized by the case of taking diffusion processes into account. For a delta-shaped input pulse, the solution is obtained of a system of equations which describes the migration of the pulse, taking diffusion washout and nonequilibrium adsorption into account. It is shown that in the case of equilibrium adsorption the time of appearance of the concentration maximum at the adsorber outlet depends on the decay constant and the coefficient of diffusion. Approximate solutions for strong- and weak-nonequilibrium adsorption and in the case of weak diffusion are considered. An estimate is carried out of the maximum magnitude of the diffusion coefficient, when its effect can be neglected.

Velocity field of a liquid stream in a spiral channel of rectangular cross section

Abstract: The equations for the velocity distribution of an ideal incompressible liquid in a rectangular tube twisted along a spiral line are established within the framework of the theory of a uniform spiral stream [1]. On the basis of the equations obtained, a numerical calculation is made of the velocity field in a hydroseparator [2] designed for the enrichment of valuable minerals. The results obtained are compared with the available experimental data, indicating their satisfactory qualitative agreement. Secondary flows develop during the flow of a liquid in curvilinear channels. The liquid near the upper and lower walls flows toward the inner side wall while the liquid in the central part of the stream flows from the inner to the outer walls. A qualitative explanation of this phenomenon is given in [3, 4]. Despite the fact that viscous forces play an important role in the formation of the secondary flow, a correct description of the flow structure can be obtained within the framework of the Gromeka equation. Good qualitative and quantitative agreement between experimental data and theoretical calculations of vortex flows made on the basis of the Gromeka equation has been obtained in many reports ([1, 5], for example).

Averaged temperature fields in asymmetrical turbulent streams over localized heat sources

Abstract: A modification of the shadow method which can be used to measure the three-dimensional averaged fields of the refractive index in turbulent flows is described. The method is applied to the measurement of a temperature field in a turbulent convective flow. A stream of heated fluid flowing slowly out of a circular opening is used as the heat source. The measurement results are compared with asymptotic dependences for buoyant convective fluids [1].

Outflow of an unexpanded jet into counter supersonic and subsonic flows

Abstract: The article gives the results of an experimental investigation of the geometric structure of an opposing unexpanded jet. It discusses flow conditions with interaction between the jet and sub- and supersonic flows. It is shown that, with the outflow of an unexpanded jet counter to a supersonic flow, there are unstable flow conditions. For stable flow conditions with one roll, dependences are proposed determining the form of a jet in a supersonic opposing flow. A generalized dependence is obtained for the distribution of the pressure at the surface of a body with a jet, flowing out counter to a subsonic flow. The range of change in the determining parameters are the following: Mach numbers at outlet cross section of nozzle, M a = 1 and 3; Mach numbers of opposing flow, M∞ = 0.6−0.9 and 2.9; degree of effectiveness of jet, n = p a /p∞ = 0.5−800 (p a and p∞ are the static pressures at the outlet cross section of the nozzle and in the opposing flow); the ratios of the specific heat capacities, γ a = γ∞ = 1.4; the drag temperatures of the jet and the flow, To∞ = Toa = 290°K.

Branching of secondary modes in the flow between rotating cylinders

Abstract: The steady secondary flows (Taylor vortices) of a viscous incompressible fluid between concentric rotating cylinders are studied. The range of wave numbers a and Reynolds numbers R in which there are several secondary flow modes is determined. The branching of these modes is studied.

Motion of an ideal fluid of constant density in the presence of sinks

Abstract: The problem under consideration is the unsteady motion of an ideal fluid with constant density in an unbounded volume when the velocity divergence is nonzero and is specified by the sink density a which depends on the coordinates r and the time t. It is well known that the introduction of such idealized hydrodynamic objects as a point vortex, a source, or a sink and the related studies of fluid flows are useful in solving a number of specific hydrodynamic problems [1, 2]. There have been many studies of point vortices, and some of the earliest are reviewed in [3], whereas the motion of free point sinks or sources has not been studied. The reason for this situation is that it is hard to find the appropriate hydrodynamic counterparts. The aim of the present paper is to study the basic laws governing the motion of a system of sinks and sources, both point and distributed, and then apply the results obtained to a simulation of thermal convection in a plane horizontal fluid layer consisting, for example, of periodic convective cells. Special attention is given to the asymptotic behavior of σ as t→∞. Conservation laws for a system of N point sinks are derived and discussed. The qualitative behavior of the system for large t is investigated. Under the assumption of a frozen sink density in the velocity field of the fluid, an evolution equation for σ is obtained for an arbitrary initial distribution of the velocity divergence. In the case of a finite integrated intensity of the sink density in an unbounded volume, an exact solution of the evolution equation is given for a cylindrically symmetric initial distribution. The asymptotic behavior of this solution as t→∞ is studied in three qualitatively different cases. Finally, a steady-state solution of the evolution equation is obtained.

Parametric excitation of waves at an interface

Abstract: Experiments on the parametric excitation of waves at a fluid interface show a strong disagreement with theoretical results [1–3], since the latter do not take into account the influence of the second medium. This proves to be especially important at low frequencies. Thus, for a water-air interface with an excitation frequency ω = 60 sec−1 the contribution amounts to ∿10%,and with ω = 30 sec−1, even 20%. In this paper the stability of the interface of two viscous, incompressible fluids of finite depth in a variable gravity field is considered. The problem is put in the linear form by making an expansion with respect to the small viscosity and is solved by taking the Laplace transform with respect to time. A second-order integrodifferential equation with periodic coefficients is obtained for the deviation of the interface from the equilibrium position; its solution is sought by the method of averaging [4]. It is shown that the presence of the second fluid significantly raises the threshold of instability.

Calculation of a turbulent two-phase isobaric jet

Abstract: A numerical calculation is carried out by the finite-difference method based on proposed equations for a turbulent submerged jet containing an admixture of solid particles. The relative longitudinal particle velocity and the influence of particles on the turbulence intensity are taken into account. The calculated results adequately agree with available experimental data. A turbulent two-phase jet is examined in [1] on the basis of the theory for a variable density jet, assuming equal mean velocities for the gas and particles and not considering the influence of particles on the turbulence intensity. Particles are analogously taken into account by a noninertial gas mixture in [2, 3], and a particle Schmidt number of 1.1 is assumed in [4]. A model is proposed in [5] which takes into account the influence of particles on the turbulence intensity of the gas phase. Problems concerning the initial and main sections of a submerged jet were solved in [6] by the integral method on the basis of this model and the assumed equality of the mean velocities of the gas and particles. Turbulent mixing of homogeneous two-phase flows with allowance made for dynamic nonequilibrium of the phases is considered in [7]. However, the neglect of turbulent transfer of particle mass and momentum led to a physically unrealistic solution for the particle concentration in the far field of the mixture. A two-phase jet is considered in the present work on the basis of the theory of a two-velocity continuous medium [8, 9] with allowance made for turbulent transfer of particle mass and momentum. The influence of particles on the turbulence intensity of the gas phase is taken into account with the model of [5].

Investigation of nonequilibrium two-phase flows in axisymmetric laval nozzles

Abstract: The singularities of two-phase flows in Laval nozzles were investigated within the framework of the model of a two-fluid continuous medium [1, 2] mainly in a quasi-one-dimensional approximation ([3] and the bibliography therein). Two-dimensional computations of such flows were performed only recently by using the method of buildup [4–7]. However, systematic computations to clarify the influence of the second phase on such fundamental nozzle characteristics as the magnitude of the specific impulse, its losses, and discharge coefficient were performed only in the quasi-one-dimensional approximation [8, 9] and only for the supersonic parts of the nozzle in the two-dimensional approximation under the assumption of uniform flow in the throat [10, 3]. Such an investigation is performed in this paper in the two-dimensional case for the nozzle as a whole, including the sub-, trans-, and supersonic flow domains, and a comparative analysis is given of the magnitudes of the loss of a unit pulse obtained in the quasi-one-dimensional approximation [8].

Probability distribution and conditional averaging in turbulent flows

Abstract: The results of an experimental investigation of the turbulence characteristics in the plane mixing layer and in the wake behind a cylinder are given. Measurements are made of the distribution of the velocity and temperature probabilities, the intermittency coefficient, and the conditionally averaged values of the square of the velocity and temperature derivatives.

Self-oscillations in an underexpanded jet flowing into a barrier

Abstract: The distribution of the phases and amplitudes of the static pressure fluctuations with self-oscillations of an underexpanded jet flowing into a barrier is obtained experimentally in the present paper. The distribution of the Mach number in the compressed layer and in the subsonic flow in front of the barrier is shown. The results of the measurements of the characteristics of the self-oscillation process are discussed.

Law of transverse sections for the three-dimensional boundary layer on a thin wing in a hypersonic stream

Abstract: The investigation of three-dimensional flows in boundary layers is important to determine the aerodynamic characteristics of wings such as the heat fluxes and friction drag. However, the circumstance that interaction of the boundary layer and the wake with an inviscid stream can play a governing role for the formation of the flow diagram as a whole is more important. The three-dimensional flow on a thin delta wing in a hypersonic stream is investigated in this paper. An important singularity of hypersonic flow is the low value of the gas density in the boundary layer as compared with the density on its outer boundary. It is shown that in the general case when the pressure in the wing span direction varies mainly by an order, high transverse velocities originate because of the smallness of the density within the boundary layer. This circumstance permits expansion of the solution for smallspan wings in a series in an appropriate small parameter. The equations in each approximation depend on two variables, while the third—longitudinal—variable enters as a parameter. The zero approximation can be considered as the formulation of the law of transverse plane sections for a three-dimensional boundary layer. As a comparison with the exact solutions calculated for delta wings with power-law distributions of the wing thickness has shown, the first approximation yields a very good approximation. Furthermore, flow modes with a different direction of parabolicity on the whole wing, as well as zones in which interaction with the external stream should absolutely be taken into account, are found.

Simple stationary waves in an inclined magnetic field

Abstract: One component of the solution to the problem of flow around a corner within the scope of magnetohydrodynamics, with the interception or stationary reflection of magnetohydrodynamic shock waves, and also steady-state problems comprising an ionizing shock wave, is the steady-state solution of the equations of magnetohydrodynamics, independent of length but depending on a combination of space variables, for example, on the angle. The flows described by these solutions are called stationary simple waves; they were considered for the first time in [1], where the behavior of the flow was investigated in stationary rotary simple waves, in which no change of density occurs. For a magnetic wave, of parallel velocity, the first integrals were found and the solution was reduced to a quadrature. The investigations and the applications of the solutions obtained for a qualitative construction of the problems of streamline flow were continued in [2–8]. In particular, problems were solved concerning flow around thin bodies of a conducting ideal gas. The general solution of the problem of streamline flow or the intersection of shock waves was not found because stationary simple waves with the magnetic field not parallel to the flow velocity were not investigated. The necessity for the calculation of such a flow may arise during the interpretation of the experimental results [9] in relation to the flow of an ionized gas. In the present paper, we consider stationary simple waves with the magnetic field not parallel to the flow velocity. A system of three nonlinear differential equations, describing fast and slow simple waves, is investigated qualitatively. On the basis of the pattern constructed of the behavior of the integral curves, the change of density, magnetic field, and velocity are found and a classification of the waves is undertaken, according to the nature of the change in their physical quantities. The relation between waves with outgoing and incoming characteristics is explained. A qualitative difference is discovered for the flow investigated from the flow in a magnetic field parallel to the flow velocity.