Boundary layer transition

Abstract

Transition phenomena commonly occur in nature. These arise either due to structural or behavioral changes in the medium. Examples for these abound in all applied sciences and to mention a few of these, we have, boundary layer, elastic-plastic deformation, and shocks. The present work is devoted to the study of laminar boundary layer transition. In this case, transition from the region near the surface of the body to the main stream takes place within a thin layer called the boundary layer. Although the basic properties of the fluid remain the same its behavior changes appreciably fro: the surface of the body to the main stream. Owing to the presence of spin, rotation or vorticity effects, the transition phenomenon is nonlinear, irreversible and non-conservative and hence it cannot be treated satisfactorily by superposition or perturbation techniques. In this thesis an attempt is made to study the transition as an asymptotic phenomenon from the boundary layer. The flow in the presence of any body is divided into two regions, (a) boundary layer, (b) all the region excepting the boundary layer, called the transition region. The classical boundary layer theory due to Prandtl, is based on his main assumptions that (a) in the boundary layer, the viscous and inertial forces are of the same order, (b) the transverse velocity in the case of a flat plate is taken of the same order as that of the transverse coordinate, (c) the variation of pressure in the boundary layer is negligible. On careful examination, it becomes clear that the above assumptions are not quite reasonable. In the present investigation the boundary layer thickness is estimated without making any of these assumptions since the ratio of the viscous to the inertial forces varies continuously from infinity near the boundary to zero at the outer edge of the boundary layer. Also the order of the transverse velocity need not be the same as that of the transverse coordinate and the continuity of pressure across the boundary layer comes out from the transition analysis and therefore it is not necessary to assume it. By making an order of magnitude analysis, the boundary layer thickness for two dimensional flow is estimated in terms of two parameters. One of these parameters depends upon the relative order of magnitude of the viscous and inertia forces at the outer edge of the boundary layer and the other depends upon the order of vorticity allowable at the outer edge of the boundary layer. The transition phenomenon in boundary flow is treated as an asymptotic phenomenon from the boundary layer. In order to study the transition region, a limiting form of the Navier-Stokes equations in three dimensions is obtained, which is called the transition equation. Owing to the importance of vorticity in the transition region, the transition equation is solved for the vorticity. The form of vorticity shows that in general the functions which govern the transition region are either subharmonic or superharmonic functions. In classical two dimensional flow the study of cylindrical vortex is made by employing matching techniques. There does not exist any mathematical treatment of the spiral formation which exists in case of flow past a body at large Reynolds number. In the present thesis a study of two dimensional flow past a body at large Reynolds number is undertaken on the basis of transition analysis, thus obtaining a satisfactory mathematical treatment of various phenomena that occur in the boundary layer flow. The transition equations for axisymmetric and two dimensional flow are also obtained. Besides other known results, transition equation in two dimensions gives the stagnation points and the formation of spirals which is noticed in the flow of a real fluid past any body at large Reynolds number. Transition equation also gives the formation of cylindrical vortices. These vortices are given by the limiting form of the stream function and come out from the transition equation itself without the use of any matching process as is done in current literature. Hence it can be concluded that the transition equation is a global representation of different phenomena which exist in fluid flow past a body at large Reynolds number. The transition concept is also extended to magnetohydrodynamics. A formula for the magnetohydrodynamic boundary layer thickness is obtained in terms of two parameters on the basis of a magnitude analysis. The transition equation for two dimensional magnetohydrodynamic case is also obtained, and its solution gives the spiral formations.