Numerical investigations of unsteady incompressible boundary - layer flows

Abstract

This thesis presents a numerical study of some problems in unsteady, laminar incompressible boundary-layer theory. It is divided into five chapters, the first being a general introduction to boundary-layer theory with special reference to unsteady flows, convective heat transfer, and to various methods available for solving boundary-layer flow problems. The remaining four chapters are constituted of the specific problems investigated by the author. Each chapter commences with an introduction briefly summarizing earlier work relevant to the problem, followed by the statement of the problem. The mathematical formulation, governing equations, with initial and boundary conditions and the solution procedure are then delineated. The results are discussed in detail, with the aid of graphs and tables collectively placed at the end of each chapter. A brief conclusion sums up the chapter. For convenience, books and papers referred to in the text of this thesis are enlisted at the end of each chapter in spite of having some repetition of them at different chapters. The various symbols used are defined as and when they occur. Nevertheless, the most commonly occurring among them are also provided at the end of the thesis. The second chapter deals with the problem of unsteady, laminar incompressible mixed convection flow at a three-dimensional stagnation point with large injection rates. Semi-similar and locally self-similar solutions have been obtained for both nodal and saddle point regions. A formal asymptotic solution for the locally self-similar nodal point flow has also been obtained using an approximate method. The governing nonlinear partial differential equations are solved using an implicit finite-difference scheme in combination with quasilinearization technique in the nodal point region with variable step size. Due to the occurrence of reverse flow in one of the velocity components, many methods including the present one failed to work in the saddle point region. This difficulty has been overcome by using an implicit finite-difference scheme with parametric differentiation. Semi-similar solutions have been obtained assuming the free stream velocity distribution varying arbitrarily with time, while the locally self-similar solutions were obtained when the free stream velocity and square of the surface mass transfer vary inversely as a linear function of time. The results indicate that the unsteadiness in the free stream and large rates of injection have a pronounced effect on both skin friction and heat transfer. The velocity and temperature fields are significantly influenced by the buoyancy force and large injection rates. Dual solutions for the locally self-similar steady flow are found to exist for both buoyancy assisting and buoyancy opposing flows. The location of the dividing streamline is pushed away from the boundary due to large rates of injection, but the buoyancy force tends to bring it nearer the boundary. The asymptotic solution, obtained by an approximate method, is found to be in good agreement with the numerical solution only when the rate of injection is large. The third chapter is a study of the effects of the temperature-dependent viscosity and Prandtl number on the forced convection in unsteady, laminar nonsimilar two-dimensional and axisymmetric water boundary layers. The fact that the viscosity and the Prandtl number of water vary inversely with temperature has been incorporated in the study through two empirical formulae. The governing nonlinear partial differential equations with three independent variables have been solved using an implicit finite-difference scheme in conjunction with quasilinearization technique from the starting point of the streamwise coordinate to the point of zero skin friction. Computations have been carried out for three different unsteady free stream velocity distributions, namely accelerating/decelerating flow and the oscillating flow. It is observed that, in the presence of variable viscosity and Prandtl number, the skin friction and heat transfer strongly respond to the unsteady free stream velocity distributions. The unsteadiness and injection cause the point of zero skin friction to move upstream. However, the effect of the variable fluid properties is to move it downstream. The heat transfer is found to depend strongly on viscous dissipation, but the skin friction is little affected by it. In general, it is found that the results pertaining to variable fluid properties differ significantly from those of constant fluid properties. This vindicates our claim, in this problem, that the consideration of temperature-dependent viscosity and Prandtl number is quantitatively important in estimating the skin friction factor and heat transfer rate in the water boundary-layer flow over two-dimensional and axisymmetric bodies. In the fourth chapter, an analysis is performed to study the unsteady, laminar incompressible MHD boundary-layer flow with heat transfer due to a point sink. It consists of two subdivisions, namely, Part-A and Part-B. In Part-A, unsteady motion of the electrically conducting fluid, due to the free stream velocity distribution varying continuously with time, has been analyzed when the strength of the point sink and wall temperature are prescribed. Semi-similar solution has been obtained for three forms (namely, accelerating, decelerating and exponentially decelerating forms) of the time-dependent free stream velocity distributions. In Part-B, the transient motion, arising due to an impulsive change either in the strength of the point sink or in the wall temperature, has been examined until the development of the ultimate steady state. Both in Part-A and Part-B, the nonlinear partial differential equations governing the motion are solved using an implicit finite-difference scheme along with a quasilinearization technique. Since the flow initially is assumed to be steady in both Part-A and Part-B, the initial conditions necessary for the computation of unsteady/transient flow are obtained by solving the steady state equations using the same method. It is found that both the types of unsteadiness in the flow have a significant effect on the skin friction and heat transfer. The magnetic field increases the skin friction but reduces the heat transfer. The heat transfer and the temperature field are strongly influenced by the viscous dissipation and Prandtl number. The transient nature of the flow is active for short time or long time depending, respectively, on the impulse imparted on the strength of the point sink or on the wall temperature. The viscous dissipation prolongs the transient behaviour of the flow. The fifth and final chapter considers the problem of unsteady, nonsimilar forced convection laminar boundary-layer flow over a moving longitudinal cylinder. The unsteadiness is due to the free stream velocity, cylinder velocity, surface temperature of the cylinder and the mass transfer, and nonsimilarity is due to the transverse curvature. The cylinder is assumed to move in the same or in the opposite direction to the free stream. The transformed partial differential equations with three independent variables are solved using an implicit finite-difference scheme with a quasilinearization technique. Nonsimilar Solutions have been obtained by considering the effects of various parameters, including that of the cylinder velocity. It is observed that the results of the problem are crucially dependent on the nondimensional parameter, which is the ratio of the cylinder velocity to the free stream velocity. Indeed, in the case of a downstream moving cylinder, the skin friction decreases with the increase of this nondimensional parameter, but heat transfer increases. For the case of an upstream moving cylinder, it is found that the solutions exist only for a small range of this nondimensional parameter and further they are found to be non-unique in this region. The time-dependent free stream velocity distributions have a pronounced effect on both skin friction and heat transfer. Also, skin friction and heat transfer are found to increase as the transverse curvature or the suction increases, but the effect of injection is opposite. The heat transfer is significantly affected by the viscous dissipation and variation of the surface temperature with time. The thesis is partly based on the following publication: “Unsteady forced convection laminar boundary-layer flow over a moving longitudinal cylinder”, (with G. Nath). Acta Mechanica, 93, 13, 1992. Papers based on the remaining work reported in the thesis will be communicated for publication shortly.