Incompressible fluid flow past a circular cylinder with linear relation between the vorticity and the stream function

Incompressible fluid flow with a linear relationship between the vorticity and the stream function past a circular cylinder is studied.Vortical flows about profiles have been considered in several studies [1–15], but in all these studies with the exception of [15] a constant vorticity was assumed (in [15] an approximate solution is found of the problem of incompressible fluid flow about a Zhukovskii profile with parabolic distribution of the velocities in the approaching stream).A freestream velocity profile similar to that considered below occurs, for example, in a planar jet (laminar or turbulent), in the wake behind a bluff body, in the boundary layer along an infinite plane [4,13], in turbulent jet flows with reverse fluid currents [16]. A similar situation also arises in the flow past an array of cylinders with large spacing which is located in the wake of another array.The author wishes to thank V. E. Davidson for posing the problem and for guidance in its solution.     

2D Slightly Compressible Ideal Flow in an Exterior Domain

We consider the Euler equations of barotropic inviscid compressible fluids in the exterior domain. It is well known that, as the Mach number goes to zero, the compressible flows approximate the solution of the equations of motion of inviscid, incompressible fluids. In dimension 2 such limit solution exists on any arbitrary time interval, with no restriction on the size of the initial data. It is then natural to expect the same for the compressible solution, if the Mach number is sufficiently small. First we study the life span of smooth irrotational solutions, i.e. the largest time interval of existence of classical solutions, when the initial data are a small perturbation of size from a constant state. Then, we study the nonlinear interaction between the irrotational part and the incompressible part of a general solution. This analysis yields the existence of smooth compressible flow on any arbitrary time interval and with no restriction on the size of the initial velocity, for any Mach number sufficiently small. Finally, the approach is applied to the study of the incompressible limit. For the proofs we use a combination of energy estimates and a decay estimate for the irrotational part.     

On the Singular Incompressible Limit of Inviscid Compressible Fluids

We consider the Euler equations of barotropic inviscid compressible fluids in a bounded domain. It is well known that, as the Mach number goes to zero, the compressible flows approximate the solution of the equations of motion of inviscid, incompressible fluids. In this paper we discuss, for the boundary case, the different kinds of convergence under various assumptions on the data, in particular the weak convergence in the case of uniformly bounded initial data and the strong convergence in the norm of the data space.     

On Slightly Compressible Ideal Flow¶in the Half-Plane

We consider the Euler equations of barotropic inviscid compressible fluids in the half-plane. It is well known that, as the Mach number goes to zero, the compressible flows approximate the solution of the equations of motion of inviscid, incompressible fluids. In 2D (two dimensions) such limit solution exists on any arbitrary time interval, with no restriction on the size of the initial data. It is then natural to expect the same for the compressible solution, if the Mach number is sufficiently small. We decompose the solution as the sum of the irrotational part, the incompressible part and the remainder, which describes the interaction between the first two components. First we study the life span of smooth irrotational solutions, i.e., the largest time interval T(?) of existence of classical solutions, when the initial data are a small perturbation of size ? from a constant state. Related to this is a decay property for the irrotational part. Then, we study the interaction between the two components and show the existence on any arbitrary time interval, for any Mach number sufficiently small. This yields the existence of smooth compressible flow on any arbitrary time interval. For the proofs we use a combination of energy and decay estimates.     

Application of unsteady mixing equations to some aerodynamic problems

Tangential discontinuities [1] are introduced in solving several transient and steady-state problems of gas dynamics. These discontinuities are unstable [2] as a result of the effects of viscosity and thermal conductivity. Therefore it is advisable to replace the tangential discontinuity by a mixing region and account for its interaction with the inviscid flows, establishing on the boundaries of this region the conditions of vanishing friction stress and equality of the velocity and temperature components to the corresponding velocity and temperature components of the inviscid flows. This formulation improves the accuracy of the solution of such problems by posing them as problems with irregular reflection and intersection of shock waves [1].The consideration of the interaction of unsteady turbulent mixing regions with the inviscid flow also permits the formulation of several problems in which the effects of viscosity lead to complete rearrangement of the flow pattern (the lambda-configuration) with the interaction of the reflected shock wave with the boundary layer in the shock tube [3,4], the formation of zones of developed separation ahead of obstacles, etc.).In this connection, §1 presents an analysis of the self-similar solutions of the unsteady turbulent mixing equations (a corresponding analysis of the laminar mixing equations which coincide with the boundary layer equations is presented in [1]). It is shown that these self-similar solutions describe, along with the several problems noted above, the problems of the formation of steady jets and mixing zones in the base wake.As an example, §2 presents, within the framework of the proposed schematization, an approximate solution of the problem of the interaction of a shock wave reflected from a semi-infinite wall with the boundary layer on a horizontal plate behind the incident shock wave. The results obtained are applied to the analysis of reflection in a shock tube. Computational results are presented which are in qualitative agreement with experiment [3, 4].     

Use of the equation for turbulent viscosity to describe the flow near a rough surface

An investigation of the flow at a rough surface, as well as in pipes and channels with rough walls, is one of the most important problems of applied hydrodynamics. Results of classical investigations, in which the most important flow properties near a rough surface are clarified, are generalized in [1–3]. These investigations are the basis for the construction of numerous semiempirical theories using the “mixing path∝ model of L. Prandtl (for instance, [4–6]). However, despite their simplicity these methods possess all the disadvantages inherent in the Prandtl theory: They are not universal, they describe the transition from the laminar to the turbulent mode poorly, and they are not applicable for the computation of complex non-self-similar flows. Meanwhile, an analysis of the experimental results obtained in [7], for example, indicates an extremely complex flow structure both in the neighborhood of the rough surface and far away from it. Models using the differential equation of the kinetic energy of turbulence have recently been developed to describe turbulent flow near a rough surface [8]. The possibilities of applying a model using the equation for turbulent viscosity to close the problem [9] are analyzed in this paper in an example of a steady turbulent incompressible fluid flow in a circular pipe with rough walls.     

Equations of hydromechanics and compression shock in two-velocity and two-temperature continuum with phase transformations

The equations of motion of multiphase mixtures have been considered in [1–10] and several other studies. In [1] it is proposed that the mixture motion be considered as an interpenetrating motion of several continua when velocity, pressure, mean density, concentration, etc., fields for each phase are introduced in the flowfield. The equations of motion are written separately for each phase, and the force effect of the other components is considered by introducing the interaction forces, which for the entire system are internal. The assumption of component barotropy is used to close the system.The energy equations are used in [2, 3] in place of the component barotropy assumption. Moreover, mixtures without phase transformations are considered. In [4] an analysis is made of the equations of turbulent motion with account for viscous forces for a two-velocity, but single-temperature medium in which equilibrium phase transformations are assumed, i. e., a two-phase medium is considered in which the phase temperatures are the same, the composition is equilibrium, but the phase velocities are different. In [5] the equations are written on the interface in a multicomponent medium consisting of barotropic fluids. A discontinuity classification is also presented here. In the aforementioned work [3] the equations on the shock are written for a continuum with particles without the use of the property of barotropy of the carrier fluid. Various different aspects of the motion of multiphase mixtures are considered in [6–11], for example, the effect of particle collisions with one another, the effect of the volume occupied by the particles on the parameters stream, shock waves, etc. In [7] a study is made of the force effect of an agitated medium on a particle on the basis of the Basset-Boussinesq-Oseen equation.In the following we derive the equations of motion of a two-velocity and two-temperature continuum with drops or particles with nonequilibrium phase transformations, i. e., a medium in which the phase velocities and temperatures are different and the composition may be nonequilibrium. In addition, we study the effect of the presence of particles or drops on the gas parameters behind a shock. Further, the equations obtained here are used to study compression waves, and in particular shock waves.The author wishes to thank Kh. A. Rakhmatulin, S. S. Grigoryan, and Yu. A. Buevich for helpful discussions and valuable comments.     

Turbulent mixing of accompanying and opposing flows

The solution of the problem of the edge of a turbulent jet, obtained in [1] starting from a model of the vortical motion of an ideal liquid, is generalized for the case of the turbulent mixing of two plane semibounded (accompanying or opposing) flows of an incompressible liquid. It is shown that the results of calculation are in qualitative agreement with experimental data and that they are close to them quantitatively. Some of the special characteristics of the method are discussed.Translated from Zhurnal Prikladnoi Mekhaniki i Tekhnicheskoi Fiziki, No. 6, pp. 92–101, November–December, 1973.     

Behavior of perturbations of solitary and periodic waves on the surface of a heavy liquid

In [1] a system of equations was obtained for the case of a potential motion of an ideal incompressible homogeneous fluid; the system described the propagation of a train of waves in a medium with slowly varying properties, the motion in the train being characterized by a wave vector and a frequency. A solitary wave is a particular case of a wave train in which the length of the waves in the train is large. In [2, 3] a quasilinear system of partial differential equations was obtained which described two-dimensional and three-dimensional motion of a solitary wave in a layer of liquid of variable depth. It follows from this system that if the unperturbed state of the liquid is the quiescent state, then some integral quantity (the average wave energy [2–4]), referred to an element of the front, is preserved during the course of the motion. This fact is also valid for a train of waves, and can be demonstrated to be so upon applying the formalism of [1] to a Lagrangian similar to that used in [2]. In the present paper we obtain, for the case of a layer of liquid of constant depth, a solution in the form of simple waves for a system, equivalent to the system obtained in [3], describing the motion of a solitary wave and also the motion of a train of waves. We show that it is possible to have tilting of simple waves, leading in the case considered here to the formation of corner points on the wave front. We consider several examples of initial perturbations, and we obtain their asymptotics as t→∞. We make our presentation for the solitary wave case; however, in view of our statement above, the results automatically carry over to the case of a train of waves.     

Development of steady-state mixing in an aerodynamic wake

We consider the unsteady, isothermal flow in the wake behind a body on which a shock wave has impinged. This flow is studied for sufficiently large distance from the body, when the effect of the latter on the external flow, as in the corresponding steady problem [1, 2], may be neglected. The analysis is performed for laminar and turbulent mixing with planar and axial symmetry.In § 1 we use the characteristics of the system of equations consisting of the momentum equation written on the wake centerline and an integral equation to determine approximately the boundary separating the steady and unsteady flows. It is noted that the unsteady flow takes place between this boundary and the tagged particle line, which at the moment of detachment of the shock wave from the body coincided with it.In §2 we show the possibility of calculating the flow in this region using the characteristics of the system of equations noted above. Some particular solutions of the latter are obtained.Then, in §§ 3 and 4 the exact solutions corresponding to the flow regions identified in §§1 and 2 are presented.Section 3 presents the exact solutions of the linear system of equations for large values of the time, when the velocity in the wake is close to the velocity of the particles behind the shock wave.On the basis of the analysis made in §2, and previously in [3], §4 obtains the exact solution of the nonlinear system of equations which describes the development of the mixing near the tagged particle line for planar turbulent flow. This solution is compared with the corresponding approximate solution of §2. The accuracy of the latter is noted. Comparison of the horizontal component velocity profiles with the profiles obtained in the preceding section and used in [4] in the method of integral relations shows satisfactory agreement between them     

Multicomponent Reynolds-averaged Navier–Stokes simulations of reshocked Richtmyer–Meshkov instability-induced mixing

A third-order weighted essentially nonoscillatory (WENO) finite-difference implementation of a two-equation K–ε multicomponent Reynolds-averaged Navier–Stokes (RANS) model is used to simulate reshocked Richtmyer–Meshkov turbulent mixing of air and sulfur hexafluoride at incident shock Mach numbers Ma_s = 1.24, 1.50, 1.98 with Atwood number At = 0.67 and Ma_s = 1.45 with At = ?0.67. The predicted mixing layer width evolutions are compared with experimental measurements of the width before and after reshock [M. Vetter, B. Sturtevant, Shock Waves 4 (1995) 247; F. Poggi, M.H. Thorembey, G. Rodriguez, Phys. Fluids 10 (1998) 2698] and with the analytical self-similar power-law solution of the simplified model equations before reshock. A new procedure is introduced for the specification of the initial turbulent kinetic energy and its dissipation rate, in which these quantities are related by the linear instability growth rate. The predicted mixing layer widths before reshock are shown to be sensitive to changes in the initial turbulent kinetic energy and its dissipation rate, while the widths after reshock are sensitive to changes in the model coefficients C_ε0 and σ_ρ appearing in the buoyancy (shock) production terms in the turbulent kinetic energy and dissipation rate equations. A set of model coefficients and initial conditions is shown to predict mixing layer widths in generally good agreement with the pre-reshock experimental data, and very good agreement with the post-reshock data for all cases. Budgets of the turbulent kinetic energy equation just before and after reshock for the Ma_s = 1.24 case are used to identify the principal physical mechanisms generating turbulence in reshocked Richtmyer–Meshkov instability: buoyancy production (pressure work) and shear production. Numerical convergence of the mixing layer widths under spatial grid refinement is also demonstrated for each of the Mach numbers considered.     

Energy-balance equation for turbulent pulsations in the theory of the free turbulent boundary layer

Semiempirical expressions are proposed for the coefficient of turbulent viscosity and for the scale of turbulence in the equations for the free turbulent boundary layer in an incompressible fluid, these equations consisting of the equation of continuity, the equations of motion, and the equation for the average energy balance in the turbulent pulsations. The advantage of the expressions over the existing ones is that the two empirical constants in the equations have nearly the same values for circular and plane turbulent streams and also for a turbulent boundary layer at the edge of a semiinfinite homogeneous flow with a stationary fluid. The mean-energy distribution and the mean energy of the turbulent pulsations computed in this paper agree well with the experimental values.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 6, pp. 75–79, November–December, 1970.     

Optimization of body shape at small reynolds numbers

The problem of the optimization of the shape of a body in a stream of viscous liquid or gas was treated in [1–5]. The necessary conditions for a body to offer minimum resistance to the flow of a viscous gas past it were derived in [1], The necessary optimality conditions when the motion of the fluid is described by the approximate Stokes equations were derived in [2], The shape of a body of minimum resistance was found numerically in [3] in the Stokes approximation. The optimality conditions when the motion of the fluid is described by the Navier—Stokes equations were derived in [4, 5], and in [4] these conditions were extended to the case of a fluid whose motion is described in the boundary-layer approximation. The necessary optimality conditions when the motion of the fluid is described by the approximate Oseen equations were derived in [5] and an asymptotic analysis of the behavior of the optimum shape near the critical points was performed for arbitrary Reynolds numbers.Translated from Zhurnal Prikladnoi Mekhaniki i Tekhnicheskoi Fiziki, No. 1, pp, 87–93, January–February, 1978.     

Effect of particles suspended in a fluid flow on the decay of isotropic turbulence

Dynamic equations have been obtained for the two-point double correlations of the fluctuation velocities of a fluid and the particles suspended in it at low volume concentrations of the solid phase. In the case of uniform isotropic turbulence these equations can be considerably simplified. The final period of decay of isotropic turbulence has been studied in detail. At this stage in the case of high-inertia particles the inhomogeneous-fluid turbulence is similar to the turbulence of a homogeneous fluid (without particles) in the sense that the presence of the particles affects only the fluctuation energy but leaves unchanged the spatial scales of turbulence and the spatial energy spectrum function. The suspended particles lead to exponential damping of the turbulent pulsations.Little theoretical information is available on the hydrodynamics of a suspension of fine particles in a turbulent liquid or gas. Research has been mainly confined to the behavior of the individual particles in a given turbulence field [1]. The problem of the turbulent motion of the mixture as a whole has been examined by Barenblatt [2], who derived the equations of motion of the mixture, using Kolmogorov's hypothesis to close them. Hinze [3] has also attempted to derive equations for turbulent pulsations of the mixture. However, as Murray showed [4], Hinze' s equations contradict Newton' s third law.The effect of suspended particles on the turbulence of a two-phase flow is governed by the noncorrespondence of the local velocities of the particles and the medium. The forces of resistance to the motion of the particles relative to the fluid lead to additional dissipation of fluctuation energy and decay of turbulence [2]. On the other hand, if the averaged velocities of particles and medium do not correspond, the suspended particles may also have a destabilizing effect [5, 6], causing energy transfer from the averaged to the pulsating motion. Below we shall consider the case where the averaged velocities of the two phases coincide, i.e., we shall deal only with the first of the two above-mentioned effects.The authors thank G.I. Barenblatt for his useful advice.     

Propagation of a laminar jet of electrically conducting fluid in a uniform magnetic field

Several papers [1–4] have considered the propagation of a plane laminar jet of incompressible conducting fluid in a uniform magnetic field for magnetic Reynolds numbers much less than unity. These papers have investigated the flow of a free jet in a transverse magnetic field for small values of the magnetic interaction parameter. Equations for the first approximations were obtained in [1, 2] by a series expansion in the small interaction parameter close to the ordinary solution (without magnetic field) for the jet. The equations for the zero-th and first approximations were integrated in [3]. The same author also found a similar solution for a turbulent jet, the turbulent transfer coefficient being chosen according to Prandtl's method. As regards the solution found in [4], it suffers from the defect that the constant of integration which connects the real velocity profiles with those found in the paper remains undetermined. The present paper gives an approximate solution of the same dynamic problem of the propagation of a free plane jet in a uniform field, no assumption being made as to the smallness of the interaction parameter. In order to do this the integral method of solution, common in ordinary hydrodynamics [5, 6] is employed. The solution of the problem is generalized to include the case of a finite value of the Hall parameter.     

Method of outer and inner asymptotic expansions in the theory of Brownian motion of aerosol particles

We examine the Brownian motion of particles in a gaseous medium, complicated by the influence of inertial forces. The equation for the distribution function in phase space describing motion of this type was obtained in [1]. Also presented in [1] are the solutions of this equation for certain simple particular cases. The approximate equations of motion of aerosol particles in coordinate space were first obtained in [2] and solved for certain concrete problems in [3,4]. More exact equations of motion in coordinate space, and also the limits of applicability of the equations of [2], are presented in [5].     

Some problems in the theory of plane jets of conducting fluid

Using the boundary-layer equations as a basis, the author considers the propagation of plane jets of conducting fluid in a transverse magnetic field (noninductive approximation).The propagation of plane jets of conducting fluid is considered in several studies [1–12]. In the first few studies jet flow in a nonuniform magnetic field is considered; here the field strength distribution along the jet axis was chosen in order to obtain self-similar solutions. The solution to such a problem given a constant conductivity of the medium is given in [1–3] for a free jet and in [4] for a semibounded jet; reference [5] contains a solution to the problem of a free jet allowing for the dependence of conductivity on temperature. References [6–8] attempt an exact solution to the problem of jet propagation in any magnetic field. An approximate solution to problems of this type can be obtained by using the integral method. References [9–10] contain the solution obtained by this method for a free jet propagating in a uniform magnetic field.The last study [10] also gives a comparison of the exact solution obtained in [3] with the solution obtained by the integral method using as an example the propagation of a jet in a nonuniform magnetic field. It is shown that for scale values of the jet velocity and thickness the integral method yields almost-exact values. In this study [10], the propagation of a free jet is considered allowing for conduction anisotropy. The solution to the problem of a free jet within the asymptotic boundary layer is obtained in [1] by applying the expansion method to the small magnetic-interaction parameter. With this method, the problem of a turbulent jet is considered in terms of the Prandtl scheme. The Boussinesq formula for the turbulent-viscosity coefficient is used in [12].This study considers the dynamic and thermal problems involved with a laminar free and semibounded jet within the asymptotic boundary layer, propagating in a magnetic field with any distribution. A system of ordinary differential equations and the integral condition are obtained from the initial partial differential equations. The solution of the derived equations is illustrated by the example of jet propagation in a uniform magnetic field. A similar solution is obtained for a turbulent free jet with the turbulent-exchange coefficient defined by the Prandtl scheme.     

Possibility of forming an inverted distribution over the rotational levels of nitrogen in an expanding gas stream

A number of theoretical papers have been devoted to an investigation of the relaxation kinetics of the population of a system of rotational levels of molecules in a stream of gas freely expanding from a sonic nozzle [1–3]. The complexity of the task of constructing models of relaxation and of collisions consistent in accuracy, however, as well as the difficulties in solving the resulting system of kinetic and gas-dynamic equations, lead to the necessity of using substantial approximations. Some disagreement between the experimental data and calculated results [1, 2] requires an evaluation of the accuracy of the various approximations used and further refinement of the theoretical models. In contrast to [1], in order to bring out the possible mutual influence of nonequilibrium energy exchange between the degrees of freedom of nitrogen molecules and the variation of the gas-dynamic parameters, the calculation presented below is based on a numerical solution of a self-consistent system of kinetic and gas-dynamic equations for the populations of rotational states and the temperature, density, and velocity of gas in the stream. Collisional probabilities of rotational transitions, calculated with allowance for the long-range part of the potential of the interaction between molecules [4], are used for this.Translated from Zhurnal Prikladnoi Mekhaniki i Tekhnicheskoi Fiziki, No. 3, pp. 9–16, May–June, 1986.     

Averaging of stochastic evolution equations of transport in porous media

A study is made of the problem of averaging the simplest one-dimensional evolution equations of stochastic transport in a porous medium. A number of exact functional equations corresponding to distributions of the random parameters of a special form is obtained. In some cases, the functional equations can be localized and reduced to differential equations of fairly high order. The first part of the paper (Secs. 1–6) considers the process of transport of a neutral admixture in porous media. The functional approach and technique for decoupling the correlations explained by Klyatskin [4] is used. The second part of the paper studies the process of transport in porous media of two immiscible incompressible fluids in the framework of the Buckley—Leverett model. A linear equation is obtained for the joint probability density of the solution of the stochastic quasilinear transport equation and its derivative. An infinite chain of equations for the moments of the solution is obtained. A scheme of approximate closure is proposed, and the solution of the approximate equations for the mean concentration is compared with the exactly averaged concentration.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 5, pp. 127–136, September–October, 1985.We are grateful to A. I. Shnirel'man for pointing out the possibility of obtaining an averaged equation in the case of a velocity distribution in accordance with a Cauchy law.     

Similarity solutions of vertical plane wall plume based on finite analytic method

The turbulent flow of vertical plane wall plume with concentration variation was studied with the finite analytical method. The k-epsilon model with the effect of buoyancy on turbulent kinetic energy and its dissipation rate was adopted. There were similarity solutions in the uniform environment for the system of equations including the equation of continuity, the equation of momentum along the flow direction and concentration, and equations of k, epsilon. The finite analytic method was applied to obtain the similarity solution. The calculated data of velocity, relative density difference, the kinetic energy of turbulence and its dissipation rate distribution for vertical plane plumes are in good agreement with the experimental data at the turbulent Schmidt number equal to 1.0. The variations of their maximum value along the direction of main flow were also given. It shows that the present model is good, i.e., the effect of buoyancy on turbulent kinetic energy and its dissipation rate should be taken into account, and the finite analytic method is effective.