SOME PROBLEMS ON UNSTEADY INCOMPRESSIBLE BOUNDARY LAYERS

Abstract

The thesis has been devoted to the study of soma problems on unsteady two-dimanaional and three-dimensional in-coapressible boundary layer flows past different bodies. Most of the work is on three-dimensional flows and there is only one chapter on two-dimen sional flows. It deals with the effects due to the curvature of different bodies, of the stream lines in the main stream, and of the rotation of body or main stream. The thesis has been divided into four chapters dealing with three-dimensional flows with and without swirl, heat transfer in three-dimensional flows, and two-dimensional flows. It deals with problems of a general character covering many particular problems as special cases. The velocity,the temperature, the skin friction, the resultant skin friction, the heat transfer, the angle of deflection of the velocity vector and the displacement thickness are the important quantities associated with a boundary layer which have been obtained in the general analysis of the problems considered. As a general raethod adopted here, the author expanded the unknown functions in the form of single or double series, and conver ted the partial differential equations Into a set of ordinary differential equations assuming a non-dimensional similarity vari able, tor the completeness of the particular problems, these equ ations are numerically integrated using Runge-Kutta method , by (ii) converting the boundary value problems into initial value problems, for various values of the parameters and the necessary numerical results are tabulated inside each ehapter. Most of the results are interpreted graphically. The techniques adopted are the extension [18J [33 , 341 of the techniques followed by M. J.Lighthill and 0. if. Sanaa who have solved the unsteady two-dimensional boundary layer equations. Each chapter contains three or four sections the first bein*, that of introduction of the chapter itself and the remaining sections deal with different problems of different flow structures and body shapes. In the first chapter the problems on unsteady three-dimen sional flows without swirl are considered. It contains four sect ions the first one, i.e. (la), is that of introduction of the chapter. The second section, i.e. (lb), is devoted to the study of the effects of disturbances at the entrance section of the flow within a circular pipe. In the third section, i.e. (Ic), the unsteady flow past a body of revolution is studied. The steady flows are disturbed only in magnitude (to keep the symmetry undisturbed) but not in direction. These flows are three-dimensional due to the lateral curvature of the bodies. The fourth section, i.e. (Id), is devoted to the study of the effects of yaw of an infinite wedge in unsteady flow. The velo city components of the main strea are assumed to be such that the yaw of the wedge is the same in both steady and unsteady flows. The main stream is perturbed in magnitude as well as in direction. This (ill) is also a three-dimensional problem due to tha secondary flow in the main stream. The work of this section haa bean accepted for publication in • Indian Journal of Pure and Applied Mathematics" in 1971. It is under publication with title " Unsteady Boundary Layer on an Infinite Ifawed wedge". In the second chapter the problems on unsteady threedimensional flows with swirl are considered, it contains three sections the first one, i.e. (Ha), is that of introduction of the chapter. In section (lib) the solution of unsteady flow paat a sphere rotating about a diameter (axis of symmetry), using the steady solution given In section (Io), is obtained. In section (He) the problem of swirl atomiser is studied by perturbing the steady swirl ing main stream flow only in magnitude (to keep the symmetry undis turbed) but not in direction. Hie third chapter deals with tha thermal boundary layers when there is (1) a flow over an infinite yawed wedge, and (11) a parabolic flow over a flat plate. The results for the unsteady velo city distribution obtained in section (Id) for flow over an infinite [9] yawed wadge and tha results obtained by T.,*. Gupta for parabolic flow over a flat plate are used here to analyae the steady and the unsteady thermal boundary layers considered in this chapter. It contains four seetions the first one, i.e. (Ilia), is that of Introduction of the chapter. The three-dimensional energy equation, when the temperature of the main stream is a constant, is solved in all the remaining (iv) throe sections. In section (IHb), the temperature gradient at the surface of a wedge is assumed to be prescribed.In sections(HIc)and (Hid) the temperatures of awedge and that of flat plate respectively are assumed to be prescribed. The effects of cross flow on the amount of heat transfer from the surface of awedge and aflat plate, and on the temperature of these bodies are the main deductions of this chapter. The work of the section(IIIb) has been published in "Proceedings of the Indian Academy of Sciences" 77,5,A, 239-p4, 1973, with the title "Unsteady Thermal Boundary Layer on an Infinite Yawed Wedge whose Temperature Gradient is prescribed". The work of the section(HIc) has been •••IfIII for publication in "Indian Journal of Pure and Applied Mathematics" in 1973. It is under publication with title »Local Heat Transfer from a Tawed Wedge". In the last chapter the problems on unsteady two-dimensional flows are considered.lt contains three sections tk^first one,i.e.(IVa), is that of introduction of the chapter.The section (IVb) deals with the unsteady two-dimensional flow between parallel walls followed by acon vergent or a divergent section. Velocity distribution, displacement thick ness and skin friction on the walls are calculated. In section(IVc) a problem on unsteady two-dimensional Jet emerging from along narrow slit is attempted.Velocity distribution,mass rate of discharge and displacement thickness are calculated. In all the problems considered, two types of solutions are developed one for large times and the other for small times,except the two (v) dimensional jet problems considered in section (I?o) where the author has developed a solution true for aaymptotically large times. The Unsteady part of the momentum flux along the axis of the jet varies with the distance alon# the axis unlike the steady part. This varia tion with the distance, for small times, can not be maintained, as it seems, with a purely time dependent perturbation in momentum flux which the author is assuming to be prescribed. So we could not develop a solution for small times. Among the two asymptotic solutions obtained for large and small times, the small time solutions are true at the surface of bodies and do not satisfy the conditions at the edge of the boundary layer, to failed to find a solution for small times satisfying all the boundary conditions simply because the diffusion dominates the convection and the secondary boundary layer formed at the surface interacts with the main stream giving rise to an additional velocity in the main stream. So the solution for small times diverges as we go away from the surface. The rate of divergence Can be checked by vanishing the arbitary cons tant, the coefficient of exponentially increasing term in the complimen tary function. Thus all the solutions for small times in the thesis do not increase exponentially but may do so algebraically. The numerieal work has been extensively carried out on the IBM 1620 Computer available at the Structural Engineering Research Cent Centre,Roorkee and a part of the numerical work was done on the (vi) TDC 12 Computer at the O.B. Pant University of Agriculture and Technology,Pantnagar. The programs for the numerical work were prepared by the author himself.