THREE DIMENSIONAL INCOMPRESSIBLE BOUNDARY LAYERS

Abstract

The present thesis is an attempt to study some of the problems of unsteady three dimensional incompressible boundary layer flow past different bodies. It deals with the effects due to the curvatures of the body and that of the stream lines in the main stream. It contains four chapters dealing with axisymmetric flow, parabolic flow, flow past a Yawed infinite wedge and Yawed infinite circular cylinder, and also the thermal boundary layer when there is a parabolic flow over a moving flat plate. It deals with the problems of general character under which many particular problems come as special cases. The stream function, the velocity vector, the temperature, the skin friction and the heat transfer, the resultant skin friction distribution, the angle of deflection of the velocity vector and the displacement thickness are the important quantities associated with a boundary layer which are obtained in the general analysis of the problems considered. In the general methods given in various chapters of the thesis, we obtain systems of general differential equations in a single variable. For the complete study of a particular problem, we have solved these equations by numerical integration for various values of the parameters and numerical results are tabulated in the appendices given at the end of each chapter. In most of the chapters the techniques used are the extensions of the techniques followed by Lighthill and Sarma who have solved the unsteady two dimensional boundary layer equations. Each chapter begins with its introduction and contains different sections dealing with particular types of problems coming under the chapter. The first chapter is on axisymmetric flow. (ii) In the present age of high speed flight, the study of purely meridional flow over axially symmetric bodies has acquired considerable practical significance. Furthermore this type of flow offers the simplest example of a three dimensional effect. As is known already, when the outer flow stream lines coincide with the surface geodesies, as in this case, no secondary flow occurs. The only three dimensional effect is one of flow divergence*, that is the surface geometry causes a stretching (or contraction) in width of the individual stream tubes which results in thinning or thickening of the boundary layer. This chapter is divided into four sections. In section (la), the introduction of the chapter and in section (lb) the general analysis of steady and unsteady equations are given and solutions of steady equations are found. In secticnOc), the unsteady boundary layer on an axisymmetric body moving in the direction of its axis is studied when the main stream velocity perturbation is zero and in section (Id), the unsteady boundary layer on an axisymmetric body, is studied when the main stream velocity perturbation is taken as the power law and the body is at rest. In this chapter a general analysis is made and two particular problems, the flow past a cone and the flow past a sphere are studied in detail. In sections (Ic) and (Id), the solutions are developed for large as well as for small times. The skin friction coefficients and the dimensionless displacement thicknesses are calculated and their variations with angle and perturbation parameters and time are analysed. The ranges of validity of the solutions for small and large times are determined graphically. In case of sphere the analysis is made for the two values of the main stream velocity, one for the theoretical value as done by (iii) Schlichting and the other for the experimental value as done by Tomotika and Imai. A part of the work of this chapter contained in section (Ic) is accepted for publication in'Proceedings of the National Institute of Sciences of India' and the work of section(Id) is under publication in 'Proceedings of the National Institute of Sciences of India'. The second chapter deals with parabolic flow and contains five sections. In this chapter we see that the three dimensional effect arises when the outer flow stream lines are curved and forms a system of parabolic translates on a flat plate at zero incidence. There is a spanwise pressure gradient in the outer flow which results in cross flow and giving rise to the curvature to the outer flow stream lines. The effects of cross flow in the boundary layer on a flat plate are studied in two cases, one when the potential flow stream lines are straight (Yawed flat plate), and the other when the stream lines are parabolic and concave with respect to the points on the chordwise direction. In section (lla) the introduction of the chapter and in section (lib) the basic equations and the general analysis of steady and unsteady equations are given. In section (lie) the solution of unsteady boundary layer equations for flow past a flat plate which is in motion in the chordwise direction with curved stream lines in the main stream is found. In section (lid), unsteady boundary layer on a flat plate is studied when the main stream perturbation is varying as the power law in the chordwise direction and the plate is at rest. In section (lie) solutions are found when the main stream velocity component in the chordwise direction is in unsteady motion with wave disturbance and the plate being at rest. The magnitude and deflection of the . resultant skin friction and the dimensionless displacement thickness are the main features studied in this chapter. The range of (iv) validity of the solutions for large as well as for small times are determined graphically. In section (lie) i.e. when the main stream velocity component in the chordwise direction is in wave motion, the amplitudes and phase leads of skin friction distributions along chord and spanwise directions are found for small and large times. A part of the work of this chapter contained in section (lie) is accepted for publication in'Roorkee University Research Journal'. The third chapter deals with the flow past a Yawed infinite wedge and Yawed infinite circular cylinder. A general theory is given under which the problems of Yawed infinite wedge and Yawed infinite circular cylinder come as the special cases by giving different values to the constants. As is known already that the general three dimensional incompressible boundary layer flow problem reduce to a special class of three dimensional boundary layer problems for the case of an infinite swept wing. These simplified equations reduce to an independent chordwise equation and a apanwise equation depend on this. The boundary layer may be considered known if the spanwise equation (the cross flow equation) is solved. The effects of cross flow when the potential flow is defined by a chordwise velocity component in powers of the chordwise length measured from the stagnation point and a constant velocity component in the spanwise direction, are considered in this chapter and the chapter is divided into six sections. The sections (Ilia) and (Illb) cover the introduction of the chapter and contain the basic equations and solution of steady equations. In sections (IIIc) and (llld),the solutions of unsteady three dimensional incompressible boundary layer equations are found for large as well as for small times when the cylinder is in unsteady motion in the chordwise direction and stream is at rest. In sections (Ille) and (Illf), the solutions of (v) unsteady three dimensional incompressible boundary layer equations are found for large as well as for small times when the main stream velocity is in unsteady motion with power law perturbation and the cylinder is at rest. The magnitude and deflection of resultant skin friction, and dimensionless displacement thickness are found in each section and the results are interpretted graphically for large as well as for small times. The last chapter deals with thermal boundary layers when there is a parabolic flow over a moving flat plate. It contains two problems. The first problem deals with the solution of three dimensional energy equation when the temperature of the main stream is a constant and that of the plate is prescribed and second problem deals with the solution when the temperature of the main stream is a constant and the temperature gradient at the plate is prescribed. The results of the unsteady velocity boundary layer over a moving flat plate given in chapter II are used to analyse the steady and unsteady thermal boundary layers considered in this chapter. The chapter is divided into two parts and the problem one is considered in part 1 and problem second Is considered in part II. In each part two sections are given. The sections (IVa) and (IVb) contain the introduction of the chapter and general analysis of steady and unsteady energy equations. In sections (IVc) and (IVe), the solutions of steady energy equations are found for both the problems. In sections (IVd) and (IVf), the solutions of unsteady energy equations are found for both the problems respectively. Two solutions in each problem are developed, one for large and the other for small times in sections (IVd) and (IVf). The temperature and the temperature gradient are determined at the plate in both the problems. The (vi) effects of cross flow on the amount of heat transfer from the plate and on the temperature of the plate are the main deductions of this chapter. In whole of the thesis, the most of numerical work has been carried out at IBM 1620 computer which is available at the Structural Engineering Research Centre,Roorkee and part of the numerical work has been carried out at 7044 computer at Indian Institute of Technology, Kanpur. The programs for the computer were prepared by the author himself. Whatsoever the author has investigated in this thesis is interpreted graphically with the aid of numerical results given In various appendices at the end of each chapter.