Showing papers in "Fluid Dynamics in 1971"

Approximate kinetic equations in rarefied gas theory

Abstract: The fundamental kinetic equation of gas theory, the Boltzniann equation, is a complex integrodiffcrential equation. The difficulties associated with its solution are the result not only of the large number of independent variables, seven in the general case, but also of the very complicated structure of the collision integral. However, for the mechanics of rarefied gases the primary interest lies not in the distribution function itself, which satisfies the Boltzmann equation, but rather in its first few moments, i.e., the averaged characteristics. This circumstance suggests the possibility of obtaining the averaged quantities by a simpler way than the direct method of direct solution of the Boltzmann equation with subsequent calculation of the integrals. It is well known that if a distribution function satisfies the Boltzmann equation, then its moments satisfy an infinite system of moment equations. Consequently, if we wish to obtain with satisfactory accuracy some number of first moments, then we must require that these moments satisfy the exact system of moment equations. However, this does not mean that to determine the moments of interest to us we must solve this system, particularly since the system of moment equations is not closed. The closure of the system by specifying the form of the distribution function (method of moments) can be considered only as a rough approximate method of solving problems. First, in this case it is not possible to satisfy all the equations and we must limit ourselves to certain of the equations; second, generally speaking, we do not know which equation the selected distribution function satisfies, and, consequently, we do not know to what degree it has the properties of the distribution function which satisfies the Boltzmann equation. A more reliable technique for solving the problems of rarefied gasdynamics is that based on the approximation of the Boltzmann equation, more precisely, the approximation of the collision integral. The idea of replacing the collision integral by a simpler expression is not new [1–4]. The kinetic equations obtained as a result of this replacement are usually termed model equations, since their derivation is usually based on physical arguments and not on the direct use of the properties of the Boltzmann collision integral. In this connection we do not know to what degree the solutions of the Boltzmann equation and the model equations are close, particularly since the latter do not yield the possibility of refining the solution. Exceptions are the kinetic model for the linearized Boltzmann equation [5] and the sequence of model equations of [6], constructed by a method which is to some degree analogous with that of [5]. In the present paper we suggest for the simplification of the solution of rarefied gas mechanics problems a technique for constructing a sequence of approximate kinetic equations which is based on an approximation of the collision integral. For each approximate equation (i.e., equation with an approximate collision operator) the first few moment equations coincide with the exact moment equations. It is assumed that the accuracy of the approximate equation increases with increase of the number of exact moment equations. Concretely, the approximation for the collision integral consists of a suitable approximation of the reverse collision integral and the collision frequency. The reverse collision integral is represented in the form of the product of the collision frequency and a function which characterizes the molecular velocity distribution resulting from the collisions, where the latter is selected in the form of a locally Maxwellian function multiplied by a polynomial in terms of the components of the molecular proper velocities. The collision frequency is approximated by a suitable expression which depends on the problem conditions. For the majority of problems it may obviously be taken equal to the collision frequency calculated from the locally Maxwellian distribution function; if necessary the error resulting from the inexact calculation of the collision frequency may be reduced by iterations. To illustrate the method, we solve the simplest problem of rarefied gas theory-the problem on the relaxation of an initially homogeneous and isotropic distribution in an unbounded space to an equilibrium distribution.

Slow steady flows of a conducting fluid at large Hartmann numbers

Abstract: We consider slow steady flows of a conducting fluid at large values of the Hartmann number and small values of the magnetic Reynolds number in an inhomogeneous magnetic field. The general solution is obtained in explicit form for the basic portion (core) of the flow, where the inertia and viscous forces may be neglected. The boundary conditions which this solution must satisfy at the outer edges of the boundary layers which develop at the walls are considered. Possible types of discontinuity surfaces and other singularities in the flow core are examined. An exact solution is obtained for the problem of conducting fluid flow in a tube of arbitrary section in an inhomogeneous magnetic field. The content of this paper is a generalization of some results on flows in a homogeneous magnetic field, obtained in [1–8], to the case of arbitrary flows in an inhomogeneous magnetic field. The author's interest in the problems considered in this study was attracted by a report presented by Professor Shercliff at the Institute of Mechanics, Moscow State University, in May 1967, on the studies of English scientists on conducting fluid flows in a strong uniform magnetic field.

Wave-flow theory for a thin viscous liquid layer

Abstract: On the basis of a simplified system of equations we study the process of development and stability of wave flows in a thin layer of a viscous liquid. Any unstable disturbance of the laminar flow grows and leads to the establishment of the wave regime. The time to establish the flow changes little for large flow rates, but increases sharply with reduction of the flow rate. Given the same amplitudes of the initial disturbances, the optimum regimes which provide the greatest flow rate in a layer of given average thickness develop more rapidly than the other regimes. All the wave regimes are unstable to disturbances in the form of traveling waves. With moderate flow rates, the optimum regimes will be most stable to near-by disturbances. Strictly periodic wave flows in a thin layer of a viscous liquid under the influence of the gravity force were calculated in [1], Various flow wave regimes which differ in wavelength can theoretically be established for a given liquid flow rate. In particular, there is a wavelength for which the flowing layer exhibits minimum average thickness (and maximum flow rate for a given average thickness). These optimum regimes correspond closely to the experimental data [2]; however, with regard to calculation technique these regimes are no different from the others. In the following we make a comparison of the wave regimes on the basis of the nature of their development and stability.

Artificial cavitation flow behind a slender wedge on the lower surface of a horizontal wall

Abstract: 1. The problem, considered in [1], is posed of cavitation flow past a slender wedge (bump) consisting of a flat plate mounted at the small angle a to a wall. In contrast with [1], the ratio of the cavity length L to the wedge lengtha 1 is considered to be arbitrary. To study this flow we use a modification of the Ryabushinskii scheme (Fig. 1, curve 1), in which the length b1 of the fictitious wedge which closes the cavity is assumed small in comparison with the bump lengtha 1. Here the angleβ at which the surface of the fictitious wedge approaches the horizontal wall is not specified ahead of time.

Acoustic modification of the aerodynamic characteristics of a turbulent jet

Abstract: In r ecen t years, a long wi th the e x p e r i m e n t a l and t heo re t i c a l s tudies of the a e r o d y n a m i c charac te r i s t i cs of tu rbulent je t flows the search for methods of ac t ive , purposive mod i f i ca t i on of these charac te r i s t ics has b e c o m e impor tant . This mod i f i ca t i on may be accompl i shed by means of vary ing the i n i t i a l nonuni formi ty [1] or the i n i t i a l tu rbu lence [2] of the flow, and also by means of impos ing on the average s teadys ta te flow a low f requency pulsa t ing flow [3], which a long with other effects leads to an increase of the i n i t i a l flow turbulence . Since the turbulent boundary l aye r and, in par t icular , the turbulent je t flows are noise generators, there is some basis to suggest tha t by means of acous t i ca l mod i f i ca t i on we can, in turn, a l t e r somewhat the a e r o d y n a m i c pa ramete r s of these sort of flows. According to the Lighth i l l theory [4] the acous t ic dis turbances gene ra t ed by the turbulent flows do not have any reverse ef fect on the a e r o d y n a m i c charac te r i s t ics of these flows. Hence i t is c l ea r tha t for any marked change of the f low ae rodynamic charac te r i s t i cs i t is necessary that the in tens i ty of the a r t i f i c i a l acous t ic dis turbances exceed s ign i f i can t ly the in t en s i ty of the natura l noise of the o r ig ina l flow. Severa l studies have been publ ished on the inves t iga t ion of the ef fect of acous t ic dis turbances on the boundary l aye r [5, 6] and on j e t flows [7 -10] . These works are devoted b a s i c a l l y to studying the effect of acous t ic disturbances on the loss of s t ab i l i t y and t rans i t ion of l a m inar flow into turbulent . Thus, in [5, 8] i t is shown that impos i t ion of an acous t ic f ie ld may e i the r f a c i l i t a t e ea r l i e r t rans i t ion (for m e d i u m frequencies) or, conversely, may de l ay t rans i t ion (for h igh f requencies); low frequency acoust ic disturbances have no ef fec t on t ransi t ion.

Double Mach reflection of strong shock waves

Abstract: The Mach reflection of shock waves in those cases in which the gas ideality condition is satisfied with high accuracy is well-known. The effects associated with the excitation of the internal degrees of freedom for the molecules lead to a qualitative change in the reflection pattern. The present study is an extension of [1, 2], devoted to the study of the Mach reflection of shock waves from a wedge under conditions in which the physical and chemical transformations in the gas heated by the shock wave play a significant role.

Self-similar solutions of non-Newtonian fluid boundary layer in MHD

Abstract: The self-similar solutions of the boundary layer for a non-Newtonian fluid in MHD were considered in [1, 2] for a power-law velocity distribution along the outer edge of the layer and constant electrical conductivity through the entire flow. However, the MHD flows of many conducting media, which are solutions or molten metals, cannot be described by the MHD equations for non-Newtonian fluids. The self-similar solutions of the boundary layer for a non-Newtonian fluid without account for interaction with the electromagnetic field were studied in [3]. In the following we present the self-similar solutions for the boundary layer of pseudoplastic and dilatant fluids with account for the interaction with an electromagnetic field for the case of a power-law velocity distribution along the outer edge of the layer, when the conductivity of the fluid is constant throughout the flow and the magnetic Reynolds number is small.

Solution of the equations of motion for a selectively radiating gas in a shock layer

Abstract: Numerical solutions are obtained for the system of integro-differential equations describing the flow of a viscous, heat-conducting, selectively radiating gas in the region between the shock wave and a blunt body. The calculations are made for bodies of radius from 0.1 to 3 m with stagnation temperature from 6000° to 15 000° K. As a result of the calculations the convective and radiative thermal fluxes in the vicinity of the stagnation point are obtained. The effect of injection on convective and radiative heat transfer is studied.

Rarefied gas flow near the forward stagnation point of a blunt body at hypersonic speeds

Abstract: With reduction of the density in a hypersonic stream the transition of the flow from continuum to free molecule takes place gradually The transition region may be divided into several regimes, in each of which a definite physical phenomenon is most significant For the case of the flow in the vicinity of the forward stagnation point of a blunt body these phenomena include increase of the thickness of the detached shock wave and of the boundary layer, the presence of viscous flow in the entire disturbed layer ahead of the blunt body, reduction of the number of collisions between molecules and the associated relaxation effects, the increasing role of the interaction of the stream molecules with the surface, and the phenomena of slip and temperature jump

Boltzmann equation and moment equations in curvilinear coordinates

Abstract: Various forms of writing the Boltzmann equation in an arbitrary orthogonal curvilinear coordinate system are discussed. The derivation is presented of a general transport equation and moment equations containing moments of the distribution function no higher than the fourth. For a gas of Maxwellian molecules it is shown that the system of moment equations for flows which differ little from equilibrium flows transforms into the system of hydrodynamic equations. The resulting equations may be useful in solving problems on motions of a rarefied gas by the moment methods. The results are valid for both the Boltzmann equation and model kinetic equations.

Effect of high-molecular formations on turbulence in dilute polymer solutions

Abstract: The marked reduction of hydrodynamic resistance with addition of certain soluble high polymers to a liquid, discovered by Toms in 1948 [1], later attracted considerable attention by researchers. The Toms effect is observed both under conditions of the internal problem (flow in pipes) and the external problem (flow past bodies). The Toms phenomenon was discovered using a solution of polymethyl methacrylate in monochlorbenzine. Later many experiments were made in aqueous solutions of the sodium salt of carboxymethylcellulose with a molecular weight of about 70 000, where a marked effect was obtained with concentrations of order 10−4. Attention to this effect increased sharply after it was found (apparently first published by Hoyt and Fabula [2]) that there are far more effective polymers, such as polyoxyethylene; the addition of a few dozen parts per million permits reduction of hydrodynamic frictional resistance in pipe flow by about a factor of three.

Calculation of supersonic flow past blunt bodies using the complete Navier-Stokes equations

Abstract: A numerical method is presented for solving the steady problem of supersonic flow of a low density gas past blunt bodies. In this flow regime the effects of compressibility, viscosity, and heat conduction of the gas are significant. The study of the flow field is made using the two-dimensional Navier-Stokes equations for a compressible gas, which are integrated in a finite region near a blunt body. The solution of the boundary value problem is sought by the asymptotic method. An explicit difference scheme [1] is used to approximate the unsteady NavierStokes system. The primary objective of the study is to check on the computation method. The nature of the asymptotic approach to the steady-state solution and the peculiarities of the calculation using the selected difference scheme are clarified; the effect on some surfaces within the flow is analyzed; the computation accuracy is evaluated.

Supersonic flow of combustible gas mixture past sphere with account for ignition delay time

Abstract: Assume an axisymmetric blunt body or a symmetric profile is located in a uniform supersonic combustible gas mixture stream with the parameters M1, p1, and T1. A detached shock is formed ahead of the body and the mixture passing through the, shock is subjected to compression and heating. Various flow regimes behind the shock wave may be realized, depending on the freestream conditions. For low velocities, temperatures, or pressures in the free stream, the mixture heating may not be sufficient for its ignition, and the usual adiabatic flow about the body will take place. In the other limiting case the temperature behind the adiabatic shock and the degree of gas compression in the shock are so great that the mixture ignites instantaneously and burns directly behind the shock wave in an infinitesimally thin zone, i. e., a detonation wave is formed. The intermediate case corresponds to the regime in which the width of the reaction zone is comparable with the characteristic linear dimension of the problem, for example, the radius of curvature of the body at the stagnation point.

Results of a calculation of gas-particle flow in axisymmetric nozzles by the method of characteristics and comparison with results of the one-dimensional approximation

Abstract: The majority of the published works on the calculation of flows of a gas with foreign (solid or liquid) particles in nozzles are based on the use of the one-dimensionai approximation; certain exceptions are calculations of the flow in axisymmetric nozzles made by the method of ctlaracteristics by Kliegei and Nickerson [1-8] and by Hoffman and Lorenc [1-6] . The basic results of the present work are also obtained by the method of characteristics. Here the examination of a broader range of variation of the nozzle expansion ratio and the relative particle flow rates has made it possible to find new, previously unknown characteristic features of the flow. A comparison is made of the results obtained by rile method of characteristics with analogous results of the one-dimensional approximation, which showed that the errors of the one-dimensional approximation are very high, particularly for high relative particle flow rates. w The motion of a mixture of gas with foreign particles, if certain conditions are satisfied, which are specified, for example, in [6], may be described with adequate accuracy with the aid of the model of a two-velocity (or mult i -veloci ty) medium. In so doing, the tea! phenomenon is replaced by the murually penetrating flow of two (or more) continuous media: the gas proper and the \"gas\" (or \"gases\") of particles, which has no self-pressure. The interaction between these media is caused by the viscosity and ttmrmal conductivity of the gas and is realized through the medium of the force f with which the gas acts on the particles and through the heat flux q from the particles to the gas, where f and q shall refer to quantities obtained by referring the forces and heat flux for a single particle to its mass. We will assume that f and q are known functions of the f!ow parameters, having the form

Drag of a rotating sphere

Abstract: We consider the problem of steady incompressible viscous fluid flow about a rotating sphere, with the flow specified on a sphere of finite radius, which reduces to the solution of the complete Navier-Stokes equations.

Exact and approximate formulations of problems for unsteady flows of conducting fluids in MHD channels

Abstract: The exact formulation of problems for the unsteady flows of viscous incompressible conducting fluids in MHD channels with arbitrary wall conductivity envisions the joint solution of the equations for the fluid and for the surrounding medium, connected by the conditions at the interface, where the electric and magnetic fields must be continuous [1, 2]. If the side walls of the channel are made from highly conductive material and are connected with the external circuit, then these equations in the general case must be supplemented by the system of equations for the external circuit, written in accordance with Kirchhoff's second law. The solution of such problems in the exact formulation presents extreme difficulties. Moreover, in many particular cases which are of practical interest the problem formulation may be simplified, and solutions may be constructed in closed form. In the following we consider the possibilities of such simplification in studying unsteady flows of a fluid of high conductivity in planar MHD channels with an external electrical circuit.

Dfag measurements on star-shaped body at m ≈ 6 and 8

Abstract: Results have been obtained in recent years which make it possible to get an idea of the optimal shape of a three-dimensional body at high supersonic speeds. It has been shown [1–6] that bodies with a cross section in the form of a star with certain limitations have the least wave drag and remain optimal with respect to total drag with approximate account for the friction forces. The transition from the optimal body of revolution to the star-shaped body of equivalent volume and length makes a several-fold drag reduction. These theoretical results, initially obtained on the basis of the Newton drag law, were then confirmed by the exact solution [7] for bodies which were close in form to the optimal. Subsequent experimental studies investigated the flow pattern between two lobes representing an element of the star over a wide range of included angles. The experiments showed that there actually exists a flow between the rays corresponding to the solution [7], that this flow is stable, and that the wave drag calculated from the pressure distribution over the body surface is several fold less than for the equivalent cone. Although these results are encouraging, they do not prove the advantages of the star-shaped form for practical use. The point is that the “star” has considerably more wetted area; therefore the effect of the marked reduction of the wave drag may be compensated by an increase of the friction drag. The references above to the theory which considers friction are not convincing, since the friction estimates are approximate, while real friction is complicated by the presence of shock waves within the flow, the possibility of a turbulent boundary layer, separation, etc. Not all these factors are amenable to calculation, and it is clear that conclusions can be drawn on “star” drag only after making direct measurements of the total force acting on a model in a flow.

One-dimensional nonequilibrium air flow

Abstract: A solution is found for the problem of steady quasi-one-dimensional air flow in a stream tube with nonequilibrium chemical reactions, ionization reactions, and nonequilibrium excitation of the vibrational degrees of freedom in the molecular components. We consider the inverse problem: for a given pressure distribution find the distributions of all the other gas-thermodynamic quantities and the streamtube sections. The use of an implicit scheme for approximating the equations makes it possible to carry out the calculations over the entire range of variation in the degree of nonequilibrium — from the frozen state to equilibrium. We discuss the nature of the variation in temperature, vibrational energies, and component concentrations along the stream tube. A numerical analysis is made of the transition to equilibrium flow.

Structure of filtration pseudoturbulence

Abstract: The problem of large-scale (“pseudoturbulent”) motion of a fluid in a nonuniform porous medium was formulated in [1]. Since in practice the local porosity e(x) is unknown, it may be considered a continuous random point function. The difference of the local values of the porosity e(x) from the mean value e∘ for the medium as a whole leads to the occurrence of random pseudoturbulent motions of the filtering fluid, which are superposed on the mean filtration flow. The characteristics of the large-scale filtration motion in a medium with this sort of random porosity were considered in detail in [1], where the formal solution is presented for the resulting equations for two-point correlations, based on the use of considerations of spatial invariance. Also presented is a qualitative discussion of the effect of pseudoturbulence of the filtering medium on the transport processes in the medium.

Nonequilibrium ionizatlon of air behind a shock wave front at speeds of 5–10 km/sec

Abstract: The solution of the problem of nonequilibrium ionization of dissociating air in a shock wave propagating with a speed of 5–10 km/sec has shown that the electron concentration distribution has a maximum behind the wave front for speeds below 9 km/sec. The formation of this maximum is caused by the high associative ionization rate in comparison with the nitrogen dissociation rate. In the nonequilibrium region behind the shock wave front there is formed a considerable concentration peak of the molecular ions which are formed as a result of associative ionization and charge exchange.

The laminar boundary layer near a body corner point

Abstract: A method is developed for calculating the characteristics of a laminar boundary layer near a body contour corner point, in the vicinity of which the outer supersonic stream passes through a rarefaction flow. In the study we use the asymptotic solution of the Navier-Stokes equations in the region with large longitudinal gradients of the flow functions for large values of the Reynolds number, the general form of which was used in [1]. The pressure, heat flux, and friction distributions along the body surface are obtained. For small pressure differentials near the corner the solution of the corresponding equations for small disturbances is obtained in analytic form. The conventional method for studying viscous gas flow near body surfaces for large values of the Reynolds number is the use of the Prandtl boundary layer theory. Far from the body the asymptotic solution of the Navier-Stokes equations in the first approximation reduces to the solution of the Euler equations, while near the body it reduces to the solution of the Prandtl boundary layer equations. The characteristic feature of the boundary layer region is the small variation of the flow functions in the longitudinal direction in comparison with their variation in the transverse direction. However, in many cases this condition is violated. The necessity arises for constructing additional asymptotic expansions for the region in which the longitudinal and transverse variations of the flow functions are quantities of the same order. The general method for constructing asymptotic solutions for such flows with the use of the known method of outer and inner expansions is presented in [1]. In the following we consider the flow in a laminar boundary layer for the case of a viscous supersonic gas stream in the vicinity of a body corner point. Behind the corner the flow separates from the body surface and flows around a stagnant zone, in which the pressure differs by a specified amount from the pressure in the undisturbed flow ahead of the point of separation. A pressure (rarefaction) disturbance propagates in the subsonic portion of the boundary layer upstream for a distance which in order of magnitude is equal to several boundary layer thicknesses. In the disturbed region of the boundary layer the longitudinal and transverse pressure and velocity disturbances are quantities of the same order. In this study we construct additional asymptotic expansions in the first approximation and calculate the distributions of the pressure, friction stress, and thermal flux along the body surface.