An approximate analytical solution of the laminar boundary layer equations

Using the pressure gradient as the new variable instead of.the ordinarylongitudinal coordinate x,Liu transformed the ordinary laminar boundary equationsinto a new form.On this base Liu obtained the frictional stress factor by using thegraphical method.In this paper the same variable replacement as in[1]is used and an approximateanalytical solution of the laminar boundary layer equations by the series method isobtained.The author also obtains a formula of frictional stress factor.For the case ofthe main function without the term of constant,the author makes a furthersimplification.The error of the frictional stress factor obtained by the author is stillless than 10％,compared with that of[1].     

Rayleigh waves in an isotropic elastic half-space coated by a thin isotropic elastic layer with smooth contact

In the present paper, we are interested in the propagation of Rayleigh waves in an isotropic elastic half-space coated with a thin isotropic elastic layer. The contact between the layer and the half space is assumed to be smooth. The main purpose of the paper is to establish an approximate secular equation of the wave. By using the effective boundary condition method, an approximate, yet highly accurate secular equation of fourth-order in terms of the dimensionless thickness of the layer is derived. From the secular equation obtained, an approximate formula of third-order for the velocity of Rayleigh waves is established. The approximate secular equation and the formula for the velocity obtained in this paper are potentially useful in many practical applications.     

Self-similar thermal boundary layers on plane and axisymmetric bodies

This paper proposes an approximate solution procedure for the prediction of the forced convection heat transfer through self-similar laminar boundary layers. The differential equations governing the viscous and thermal boundary layers have been reduced to a pair of algebraic equations for the boundary layer shape factor and the boundary layer thickness ratio. The local Nusselt number predicted under various pressure gradients turns out to be in excellent agreement with that of the exact solution over a wide range of the Prandtl number.     

A new approximate analytical solution of the ideal potential flow around a circular cylinder between two parallel flat plates

In ref. [1], Lin obtained an approximate analytical solution of the ideal potential flow around a circular cylinder between two parallel flat flates.In this paper, the author shows that one may obtain the result coinciding with that obtained in ref. [1] by making use of the Shvez's method^[2]. Morever, we can obtain a more accurate result than that obtained in ref. [1], if we make use of the improved Shvez's method^[2]. Some calculating examples are presented.     

Choice of a self-similar solution in boundary-layer theory

If the speed of the outer flow at the edge of the boundary layer does not depend on the time and is specified in the form of a power-law function of the longitudinal coordinate, then a self-similar solution of the boundary-layer equations can be found by integrating a third-order ordinary differential equation (see [1–3]). When the exponent of the power in the outerflow velocity distribution is negative, a self-similar solution satisfying the equations and the usually posed boundary conditions is not uniquely determinable [4], A similar result was obtained in [5] for flows of a conducting fluid in a magnetic field. In the present paper we study the behavior of non-self-similar perturbations of a self-similar solution, enabling us to provide a basis for the choice of a self-similar solution.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 4, pp. 42–46, July–August, 1974.     

The laminar boundary layer near a body corner point

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.     

Lift force exerted on a spherical particle in a laminar boundary layer

A spherical particle moving in an unbounded viscous shear flow is acted upon by a lift force [1, 2] which results from taking the inertial terms into account in the equations of motion. When the particle moves at the bottom of a laminar boundary layer the magnitude of the force differs from that obtained in [1, 2], The problem of determining the lift force exerted on the particle as a function of its distance from the wall has been solved by matched asymptotic expansions. The magnitude of the force is expressed in terms of a multiple integral which can be evaluated numerically.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 5, pp. 66–71, September–October, 1989.In conclusion, the author wishes to thank M. N. Kogan, N. K. Makashev, and A. Yu. Boris for useful discussions.     

Boundary condition formulation for viscous fluid flow problems

In the solution of the Navier-Stokes equations by difference methods in infinite regions, the question arises as to the nature of the approximate boundary conditions at those portions of the computational region boundary where these conditions are not determined directly by the formulation of the basic problem. In certain cases of practical importance, these boundary conditions may be obtained by coupling the N-S equations with equations which are similar to the boundary-layer equations.In the present paper, we propose boundary conditions for the case of viscous incompressible fluid flow. Their application is illustrated for the problem of flow past the leading edge of a semi-infinite flat plate.The author wishes to thank I. Yu. Brailovskaya and L. A. Chudov for helpful suggestions in the course of this investigation.     

Asymptotic investigation of two-phase couette flow

We consider plane and cylindrical Couette flow for a two-phase medium. The motion of the medium is described by the equations obtained in [1]. Collisions between the particles are disregarded, and their motion, in addition to the inertial forces, is determined by the pressure gradient of the carrying phase and the forces of viscous interaction between the carrying phase and the particles. We obtain simple asymptotic solutions of the indicated problems for small and large values of the dimensionless determining parameters. In a number of cases the solution has the nature of a boundary layer on solid walls.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 4, pp. 67–73, July–August, 1978.     

THE ASYMPTOTIC SOLUTION OF THE UNSTEADY PLANAR PARALLEL GAS FLOW FOR THE CASE THAT THE ADIABATIC INDEX NUMBER γ IS BIG

In ref I, under the condition that the components of velocity are only the functions of time and polar angle θ, Drornikov solved eqss. (1.1) (1.3) of the ideal gas unsteady planar parallel potential flow. It was pointed out in ref. [1] that in general cases, the evident solutions could not he obtained. Only for two especial cases, the evident solutions were obtained.In this paper, the author studies the same prohlein as that in ref. [1]. In the first section we obtain the evident solution of equations (1.1)-(1.3) under the condition that the sonic velocity is restricted by some complemental conditions. In the second section, we obtain the first-order approximate solutions of the fundamental equation for the case that γ>>1     

Computation of the viscous shock layer on blunted cones

A numerical method is described for solving the equations of the compressible viscous shock layer on smooth spherically blunted axisymmetric cones at zero angle of attack and flow of a perfect gas. Effective use is made of the scheme of separating the original system of equations into parabolic (second order) and inviscid (first order) subsystems, which are solved by intrinsic methods. The results of the computations are presented. The method is capable of natural generalization to the case of nonequilibrium physical and chemical processes and diffusion. In most published papers dealing with computation of the compressible shock layer, the authors examine either the vicinity of the stagnation point or a certain region of spherical blunting [1–5]. In all the papers except [4, 5], a number of simplified assumptions have been made regarding the flow picture. Very few papers [6–8] have calculated the viscous shock layer on the forward surface of blunted bodies. In [6, 7] an approximate examination was made only of hyperboloids and paraboloids of revolution, which have very favorable geometry. Reference [8] used a approximate Karman—Polhausen integral method for a very simple system of equations. The method proposed here is essentially an accurate numerical method for solution of the viscous shock layer equations.     

Viscous shock layer near the stagnation point on a rotating body with unsteady injection and surface cooling

Unsteady supersonic flow regimes in the neighborhood of a stagnation point are investigated on the basis of a system of viscous shock layer equations [10] containing all the terms of the Euler equations and the boundary layer equations. An analytic solution of the unsteady equations valid near the surface of the body is found in the case of strong injection. The unsteady equations of the viscous shock layer are solved numerically on the basis of a divergent implicit scheme of the second order of approximation across the shock layer, using Newtonian linearization and vector sweep methods with allowance for the boundary relations on the surface of the body and at the isolated bow shock. Certain calculation results illustrating the effect of injection, surface cooling, the swirl of the external flow and the angular velocity of the body on the structure of the steady and unsteady viscous shock layer are presented.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 5, pp. 114–122, September–October, 1987.     

Developed steady wave motions of a liquid film falling along a vertical plane

The falling of a thin viscous fluid layer (film) along a vertical plane under the effect of gravity is accompanied by wave motions in which capillary forces play an essential part. An equation for the film thickness h(x, t) is used extensively in analyses of these motions. This equation, obtained from the Navier—Stokes equations and the boundary conditions under different assumptions, reduces to an ordinary third-order nonlinear differential equation [1–7] for steady plane motions. Periodic solutions of this equation were sought by the methods of asymptotic expansions in the amplitude or by Fourier series expansions [1–7], which assumes a sequential accounting of the nonlinearity as a small perturbation. This limits the validity of the results obtained to the domain of small amplitudes. The case of arbitrary amplitudes is considered in this paper. A solution of the problem, based on an asymptotic expansion in the parameter ε is constructed. In this expansion the equation for the first approximation remains nonlinear but admits of integration, which discloses the class of bounded periodic solutions. Moreover, strict integral relations (for any ε) are obtained, and a variational problem about seeking the lower bound of values of the mean film thickness and other characteristics of the ultimately developed optimal motions is formulated and solved on their basis. The results obtained agree with experiments.     

A numerical simulation of boundary layer effects in a shock tube

A numerical scheme is used to investigate boundary layer effects in a shock tube. The method consists of a mixture of Roe's approximate Riemann solver and central differences for the convective fluxes and central differences for the viscous fluxes and is implicit in one space dimension. Comparisons are made with experimental data and with solutions obtained via boundary layer equations. Examination of the calculated flow field explains the observed behaviour and highlights the approximate nature of boundary layer solutions.     

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.     

An analysis on entrance region effect of the laminar radial flow between two parallel disks

In this paper, B. B. Golubef method^[1] is used for calculating the radial diffuse flow between two parallel disks for the first step. The momentum integral equation together with the energy integral equation is derived from the boundary layer momentum equation, and the expression of secondary approximation explicit function in which the channel length of entrance region varies with the boundary layer thickness can be obtained by using Picard iteration[2] in the solution of the energy integral equation. Therefore, this has made it possible to analyze directly and analytically the coefficients of the entrance region effect. In particular, when the outer diameter of disk is smaller than the entrance region length, the advantage of this method can be prominently manifest.Only because the energy integral equation is employed, the terms in the pressure loss coefficient can be independently derived theoretically. The computable value of the pressure loss coefficient presented in this paper is nearer to the testing value than that in ref. [3] when the entrance correction Reynolds number Re<100. Therefore the results in this paper within Re<100 are both reliable and simple.     

Approximate analytical solutions and approximate value of skin friction coefficient for boundary layer of power law fluids

This paper presents a theoretical analysis for laminar boundary layer flow in a power law non-Newtonian fluids.The Adomian analytical decomposition technique is presented and an approximate analytical solution is obtained.The approximate analytical solution can be expressed in terms of a rapid convergent power series with easily computable terms.Reliability and efficiency of the approximate solution are verified by comparing with numerical solutions in the literature.Moreover,the approximate solution can be successfully applied to provide values for the skin friction coefficient of the laminar boundary layer flow in power law non-Newtonian fluids.     

Fundamentals of physics of fluid flow reconsidered

Recently, two significant defects of the state-of-the-art of theoretical fluid mechanics have been evidenced, namely the ambiguity of the Navier Stokes equations with respect to the viscous stress tensor, and the inconsistency of the Reynolds stress approach with the symmetry of the inertial tensor. Therefore, by means of two classical topics of fluid mechanics, namely the universal velocity profile and the laminar boundary layer, the physics of fluid flow are scrutinized. The validity of the so obtained results is tested at the example of an exact solution of the Navier Stokes equations, namely the diffusion of the circulation of an eddy.     

BOUNDARY ELEMENT METHOD FOR SOLVING DYNAMICALRESPONSE OF VISCOELASTIC THIN PLATE(II)──-THEORETICAL ANALYSIS

I.IntroductionInpaper[l],thedynamicresponseoftheviscoelasticthinplate-onrectangledomainisdiscussedbytheusingofthetwokindsofapproximatefundamentalsolutionsandthecorrespondingboundaryelementmethod(BEM)inLaplacespace.Theanalysisofthenumericalexamplesshowsthatonlywhenthetruncatednumberkintheapproximatefundamentalsolutionandthemaximummeshsizehsatisfycertainrelation,itcouldbeguaranteedthattheapproximatesolutionhaveenoughaccuracy.Inpresentpaper,theerrorestimationfortheapproximateboundaryelemelltmet…     

Uniqueness of the self-similar solutions of three-dimensional laminar boundary-layer equations

The equations of the three-dimensional laminar boundary layer on lines of flow outflow and inflow are studied for conical outer flow under the assumption that the Prandtl number and the productρμ are constant. It is shown that in the case of a positive velocity gradient of the secondary flow (α₁>0) the additional conditions which result from the physical flow pattern determine a unique solution of the system of boundary-layer equations. For a negative velocity gradient of the secondary flow (α₁≤0) these conditions are satisfied by two solutions. An approximate solution is obtained for the boundary layer equations which is in rather good agreement with the numerical integration results. Compressible gas flow in a three-dimensional laminar boundary layer is described by a system of nonlinear differential equations whose solution is not unique for given boundary conditions. Therefore additional conditions resulting from the physical pattern of the gas flow are imposed on the resulting solution. In the solution of problems with a negative pressure gradient these additional conditions are sufficient for a unique selection of the solution of the boundary-layer equations. However, in the case of a positive pressure gradient the solution of the boundary-layer equations satisfying the boundary and additional conditions may not be unique. In particular, in [1] in a study of a three-dimensional laminar boundary layer in the vicinity of the stagnation point it was shown that for $$c = {{\frac{{\partial v_e }}{{\partial y}}} \mathord{\left/ {\vphantom {{\frac{{\partial v_e }}{{\partial y}}} {\frac{{\partial u_e }}{{\partial x}}}}} \right. \kern-\nulldelimiterspace} {\frac{{\partial u_e }}{{\partial x}}}} > 0$$ the solution is unique, while for c<0 there are two solutions. In the present paper we study the question of the uniqueness of the self-similar solution of the three-dimensional laminar boundary-layer equations on lines of flow outflow and inflow for a conical outer flow.