COMPRESSIBLE BOUNDARY LAYERS

Abstract

In the present thesis an attempt has been made to study three dimensional compressible boundary layer flows. The study of compressible boundary layer flow is very important because of its direct application in various practical fields. The unsteady compressible boundary layers play a vital role in the stability of missiles and reentry vehicles. Small fluctuations of the angle of attack, gas injection through the skin, perturbation of temperature and its gradient,vibrations of the vehicle are few boundary layer phenomena that may have catastrophic effects on the stability of the body. Such boundary layer phenomena have been studied quite extensively in flutter, helicopter blades, turbomachinery cascades etc. The study of the two dimensional unsteady incompressible boundary layers was initiated by Lighthill [l] and subsequently extended by Rott and Rosenzweig [2], Lam and Rott [3], Ackerberg and Phillips [if] and Brown and Stewartson [5,6]. In these studies, the unsteadiness is due to the small fluctuation in the velocity of the main stream. Sarma [8] studied the compressible boundary layers with arbitrary Prandtl number, unsteadiness being due to the perturbations in the temperature or temperature gradient or velocity in the main stream or velocity of the wall. ii. The present thesis deals with the flows past yawed cylinders. The following problems have been investigated in detail. (1) Heat transfer when the temperature gradient at the wall is prescribed. (2) Heat transfer when the temperature at the wall is prescri bed and less than that of the main stream (Cooled wall). (3) Heat transfer when the prescribed temperature at the wall is greater than that of the main stream (Heated wall). In the present work Stewartson transformation [9] has been applied to elliminate density variable unlike Sarma [8] who has applied Howarth transformation. In general the equations are not exactly transformed to incompressible form but to an almost incom pressible form because of the presence of the (a) Mach number (b) Prandtl number (c) Viscosity - temperature law (d) general boundary conditions. For a detailed study of the flow the differential equations are linearized by the technique suggested by Lightfaill [l]. They are later on solved by assuming a series solutions in powers of Mach number. The mathematical technique adopted is perturbation methods and asymptotic solutions. The present thesis is divided into five chapters. In Chapter I, an analytical study of three dimensional oscillatory iii. compressible laminar boundary layer flow past infinite yawed cylinders which are kept at a given temperature gradient is made. The temperature gradient at the wall oscillatory in time is assumed to be perturbed about a zero mean. General analysis is given to study the effects of (1) compressibility (2) the shape of the cylinder (3) the orientation of the cylinder (if) the Prandtl number (5) the prescribed heat transfer at the wall, on the behaviour of the temperature and skin friction components. The equations are first subjected to Stewartson transformation and then linearised as by Lighthill [l]« And finally following Sarma [8] and Ackerberg and Phillips [hi asymptotic and composite solutions are found for small and large frequencies. In Chapter II a numerical study of the behaviours of tem perature and skin friction coefficients with dimensionless freq uency parameter has been made for small and large frequencies. For small frequencies numerical solutions are obtained by fourth order Runge-Kutta method. The main difficulty met in the numeri cal analysis is that all the boundary values are not defined either at zero or at infinity. This difficulty is overcome by the method suggested by Reshotko and Beckwith [15] in which the unknown boundary values are adjusted iteratively. In the other method given by Moore [16] this difficulty is overcome by spilliting two point problem into two single point problems. Assuming that the flow in the X,Y-plane defined by the Stewartson variables [9] is just a flow past a wedge shaped cylinder, three iv. distinct problems are studied in detail (1) yawed flat plate (2) yawed wedge with an included angle being equal to a right angle (3) yawed flat plate with a stagnation point. The beha viours of the magnitudes and the tangent of the arguments of the temperature at the wall and the skin friction coefficients are represented graphically. In Chapter III, an analytical study of three dimensional oscillatory compressible laminar boundary layer flow past infi nite yawed cylinders which are kept at a given temperature is made. The temperature at the wall cscillatory in time is assumed to be perturbed about a steady mean. General analysis is given to study the effects of (1) compressibility (2) the shape of the cylinder (3) the orientation of the cylinder (If) the Prandtl number and (5) the prescribed temperature at the wall on the behaviour of the temperature gradient and skin friction components. The mathematical technique adopted for solutions of differential equations was same as in Chapter I. In Chapter IV, based on the analytical study described in Chapter III, a numerical study of the problem when the prescribed temperature at the wall is less than that of the main stream (Cooled wall) is presented. The numerical technique adopted in this study has been described in Chapter II. Three distinct problems are studied in detail (1) yawed flat plate (2) yawed wedge with an included angle of 90° (3) yawed flat plate with a stagnation point. The results of these studies have been l tabulated and represented graphically. V. In Chapter V, the problem of heated wall ie when the prescribed temperature at the wall is greater than that of the main stream is analysed. As in Chapter IV three different problems are investigated in detail. Die graphs presented in Chapters II, IV and Vshow that in some of the cases the curves for small and large frequencies do net join smoothly but only show a tendency to join. This may be an indication that there may be some eigen solutions similar to those given by Brown and Stewartson [5,6] but no attempt has been made to model out these complicated eigen solutions in the present thesis. However it is felt that this study would be very important and interesting.