Chan - A note on a similarity transformation for three-dimensional compressible laminar boundary layer equations

THE LAMINAR BOUNDARY LAYER EQUATIONS

N. Curle

DOVER PUBLICATIONS, INC.

Mineola, New York

Bibliographical Note

This Dover edition, first published in 2017, is an unabridged republication of the work originally published in the Oxford Mathematical Monographs series by The Clarendon Press, Oxford, in 1962.

Library of Congress Cataloging-in-Publication Data

Names: Curle, N. (Newby), author.

Title: The laminar boundary layer equations / N. Curle.

Description: Mineola, New York : Dover Publications, Inc., 2016. | The Laminar Boundary Layer Equations, first published by Dover Publications, Inc., in 2016, is a reprint of material published under the same name by Oxford: The Clarendon Press (London) in 1962Title page verso.

Identifiers: LCCN 2016040432| ISBN 9780486812397 | ISBN 0486812391

Subjects: LCSH: Boundary layer.

Classification: LCC TL574.B6 C8 2016 | DDC 629.132/37dc23 LC record available at https://lccn.loc.gov/2016040432

Manufactured in the United States by LSC Communications

81239101 2017

www.doverpublications.com

PREFACE

T HE concept of the boundary layer, introduced by Ludwig Prandtl in 1904, has been a particularly fruitful one. Research on this topic has now reached the stage where there is a certain body of fundamental definitive information which is unlikely to be superseded to any great extent. This relates mainly to the steady incompressible laminar boundary layer in two dimensions. Work proceeds, however, and a considerable number of papers are being published, on unsteady boundary layers, on three-dimensional boundary layers in incompressible flow, and upon such topics as boundary-layer stability. In compressible flow, too, where additional important parameters arise, much is being done and more remains.

This monograph is one of a series, each of which is being written by an author on the general field in which his own research interests lie. It is inevitable, therefore, that there will be a certain amount of bias in the choice of material. I have tried to make the book reasonably self-sufficient, though lack of space has led to the omission of a number of very interesting problems of boundary layers. The topics so axed include unsteady boundary layers and boundary-layer stability, boundary layers on porous walls with suction or blowing, boundary layers in three dimensions (including axi-symmetric flow) and boundary layers with vorticity in the mainstream. For a discussion of these topics reference may be made to more encyclopaedic works, such as Modern Developments in Fluid Dynamics (Oxford, edited by S. Goldstein), the companion volumes on High Speed Flow (edited by L. Howarth), Laminar Boundary Layers (Oxford, edited by L. Rosenhead), and Volumes III to V of the series High Speed Aerodynamics and Jet Propulsion.

The purpose of ) of some aspects of the problem of the interaction between shock waves and laminar boundary layers.

It is my hope that this book will be of value to a wide variety of workers. In the first place I have tried to present the material in a sufficiently ordered and logical manner as to make it of value as an introduction to boundary-layer theory for young research workers who are new to the subject, or to undergraduates who are familiar with the elements of classical inviscid fluid dynamics. Secondly, the book should be of some value to research workers in this field, since one of the things which has governed my choice of material has been the question of whether a particular piece of work has been an end in itself or whether it has assisted in opening up the way for further advances. Finally, I have borne in mind the needs of practising engineers, and have tried where possible to indicate the limitations, the likely accuracy, and the practical complexity of the methods described for calculating the various properties of laminar boundary layers.

In conclusion, I have great pleasure in expressing my thanks to the many people who have helped me, directly or indirectly, in the writing of this book. To Dr. M. J. Lighthill, F.R.S., Director of the Royal Aircraft Establishment, whose student I was at Manchester University, for the wise counsel he gave me then in so many branches of fluid dynamics, and whose influence is, I hope, evident in this book. To my colleagues at the National Physical Laboratory, for the stimulating discussions I have had with them at various times, and most particularly Dr. J. T. Stuart and Dr. G. E. Gadd. These two colleagues have been good enough to offer useful comments on a first draft of this book, although the responsibility for its deficiencies remains entirely my own. To Mrs. M. E. M. Sayer, for her patient and careful typing of the manuscript, and her cheerful approach to the difficult task of reading my writing. To Professor G. Temple, F.R.S., editor of this series, and the staff of the Oxford University Press for the courteous way they have dealt with the various problems which have arisen. To Sir Gordon Sutherland, F.R.S., Director of the National Physical Laboratory, for permission to write this book. The writing has not in fact been part of my official duties, and the views expressed are entirely my own. Thanks are also due to various bodies for permission to use copyright material such as charts and tables. These bodies include

the Aeronautical Research Council,

the Clarendon Press, Oxford,

the Controller, H.M. Stationery Office, Prof. L. Crocco,

the Institute of the Aero/Space Sciences,

the Editor, Journal of Fluid Mechanics,

the Director, National Physical Laboratory,

the Royal Aeronautical Society,

the Royal Society,

the United States Air Force,

the Editor, Zeitschrift fr angew. Math, und Mech.

Finally, to my wife and, though they know it not, to my children, for so ordering their lives as to make the task of writing this book much less difficult. To all these people go my sincere thanks for their much appreciated help.

Hanworth, Middlesex

June 1961

N. C.

Hawker-Siddeley Reader Department of Aeronautics and Astronautics University of Southampton (Formerly Principal Scientific Officer Aerodynamics Division, National Physical Laboratory Teddington, Middlesex)

CONTENTS

INTRODUCTION

T HIS book is about solutions of the laminar-boundary-layer equations. The concept of the boundary layer, one of the corner-stones of modern fluid dynamics, was introduced by Prandtl (1904) in an attempt to account for the sometimes considerable discrepancies between the predictions of classical inviscid incompressible fluid dynamics and the results of experimental observations. As an example, we may remark that according to inviscid theory any body moving uniformly through an unbounded homogeneous fluid will experience zero drag!

Now the classical inviscid theories assume that the viscous forces in a fluid may be neglected in comparison with the inertia forces. This, indeed, would seem a reasonable approximation, since the viscosity of many fluids (and of air in particular) is extremely small. However, in certain regions of flow, fortunately often limited, the viscous forces can still be locally important, as Prandtl observed. The reason for this is that a typical viscous stress is of magnitude (u/y), where is the viscosity, u is the velocity measured in a direction parallel to that of the stress, and y is distance measured normal thereto, so that when the velocity gradient (or shear) u/y is large the viscous stress can become important even though itself is small. It was Prandtl who remarked that in flow past a streamlined body, the region in which viscous forces are important is often confined to a thin layer adjacent to the body, and to a thin wake behind it. This thin layer is referred to as the boundary layer. When this condition holds the equations governing the motion of the fluid within the boundary layer take a form considerably simpler than the full viscous-flow equations, though less simple than the inviscid equations, and it is the solution of these equations with which we shall be presently concerned.