Separation of Flow PDF
Preview Separation of Flow
SEPARATION OF FLOW BY PAUL Κ. CHANG Professor of Mechanical Engineerings The Catholic University of America, Washington, D,C., U.S.A. 4 P E R G A M ON PRESS OXFORD . LONDON · EDINBURGH · NEW YORK TORONTO . SYDNEY · PARIS · BRAUNSCHWEIG Pergamon Press Ltd., Headington Hill Hall, Oxford 4 & 5 Fitzroy Square, London W.l Pergamon Press (Scotland) Ltd., 2 & 3 Teviot Place, Edinburgh 1 Pergamon Press Inc., Maxwell House, Fairview Park, Elmsford, New York 10523 Pergamon of Canada Ltd., 207 Queen's Quay West, Toronto 1 Pergamon Press (Aust.) Pty. Ltd., 19a Boundary Street, Rushcutters Bay, N.S.W. 2011, Australia Pergamon Press S.A.R.L., 24 rue des ίcoles, Paris 5* Vieweg & Sohn GmbH, Burgplatz 1, Braunschweig Copyright © 1970 Pergamon Press Inc. AU Rights Reserved, No part of this publication may be reproduced, stored in a retrievai system, or transmitted, in any form or by any means, electronic, mechanical, photocopying, recording or otherwise, without the prior permission of Pergamon Press Inc. First Edition 1970 Library of Congress CaUlog Card No. 72-81249 Printed in Hanguy 08 0134416 Preface THE problem of the separation of fluid flow is one of those viscous flows which is very important not only for science, but also for practical applica tion. An attempt has been made to compile references up to date mainly in the fields of basic physical processes, analyses, and experiments covering the whole area of interest, giving credit to the original source wherever possible. The content of material has been used for a graduate course at the Catholic University of America, Washington, D.C., in 1962-3, also for a special course at Escuela Tecnica Superior de Ingenieros Aeronauticos, Ciudad Universi- taria, Madrid, Spain, in 1964-5, where the author conducted classes as a Fulbright lecturer. Although the material covering the practical application is limited, this monograph is useful for engineers as a reference. The unquoted bibliographies have been added in the list of references at the end of each chapter. The author wishes to thank the US Air Force, Office of Scientific Research, for grant AF OSR 80-62, 52-63, 52-64, which made it possible to compile this monograph under the guidance of Mr. Milton Rogers, Chief of Mechan ics Division, Capt. Hyman H. Album, Capt. Lucius P. Gregg, and Major G. Stalk. The Rev. Everett F. Briggs of Monongah, West Virginia, kindly edited my manuscript. Mr. Richard M. Hartley, of the David Taylor Model Basin, has rendered most valuable assistance in reading my manuscript, in coopera tion with Mr. Michael J. Malia of the same institution. Dr. Hyok Sang Lew of the Catholic Universitv of America, and Mrs. Anne A. Ziegler, as well as Mr. Louella A. Chatfield, assisted in arranging and typing the manuscript. To these ladies and gentlemen the author wishes to express his gratitude. PAUL K. CHANG Washington, D.C. List of General Symbols A wetted area or frontal area a velocity of sound CL lift coefficient Cj^ drag coefficient Cf skin friction coefficient Cp pressure coefficient Cp specific heat at constant pressure D drag d diameter Η shape factor of boundary layer, Η = ٧*/θ h heat transfer coefficient or height Le Lewis number Μ Mach number Nu Nusselt number η index of power Pr Prandtl number ρ pressure q dynamic pressure Re Reynolds niunber Τ temperature t time u streamwise velocity component V velocity component in >;-direction w velocity component in z-direction X coordinate, streamwise direction y coordinate perpendicular to x-direction ć coordinate perpendicular to x- and >?-direction α angle of attack γ ratio of specific heats ٧ thickness of boundary layer or shear layer 0* displacement thickness of boundary layer ε eddy viscosity θ momentum thickness of boundary layer μ dynamic viscosity V kinematic viscosity C-SF 2 xvii xviii LIST OF GENER AL SYMBOLS ń density of fluid τ shear stress rp steam function Subscripts e conditions at the outer edge of boundary layer max maximum min τηίηίπΐΐιττΐ s separation wall eo undisturbed free stream CHAPTER I Introduction to the Problems of Flow Separation Symbols dm unsteady mass exchange per pulse A' height κ turbulent mixing coefficient, κ = also κ = (ό—ό* — ο)/(δ— ό*) L length of body Λ index of power ρ pressure R reattachment point S separation point stagnation temperature with no heating effective recovery temperature x' distance from neck to transition location ao ao = lim — γο tan yo = lim w/q A extent of fluctuation Ap Δρ = p-p^ λ non-dimensional function, A(|) = -21— αξ r shear stress φ angle from stagnation point 2 SEPARATON OF FLOW Subscripts Β base Ο just upstream of separation Superscripts — average The subject of fluid flow separation is one of the many aspects of viscous flow, which is very important but complicated. Because of flow separation, energy is lost. In cases of external flow at subsonic speeds, such as in airborne vehicles, the stream line deviates, the drag increases, the lift decreases, and reverse flow and stalling occur. In the transonic speed range, control and structure problems are created by flow separations. For cases of internal flow, separation can cause reduction in efläciencies. The optimum performance of fluid handling devices such as fans, turbines, pumps, compressors, etc., can only be predicted with accurate understanding of flow separation, because the separation occurs just prior to or at maximum loading. The successful opera tion of the simplest and most conunon devices can depend also upon flow separation, as, for example, the throttling action in household water faucets. Flow separation can also be useful in engineering applications. For example, a thin airfoil which is suitable for high-speed flight may be made suitable for low speeds by separation of the flow. If the flow is allowed to separate over a portion of the upper surface and then reattaches and remains attached, a very thick pseudo-airfoil results. This thick airfoil is better suited for low- speed flight. Another example of desirable flow separation is that caused by placing a spike in front of a blunt body traveling at supersonic speed. Because of the presence of the spike, the flow may separate on the spike and form a cone- shaped flow region in front of the blunt nose. Because of this conical separated flow region, the shock wave will be changed from one nearly normal to an oblique one, which reduces the head drag considerably. Escape capsules and other stores, which must be ejected and recovered from high-speed vehicles, improve their performance by utilizing flow separation. Components of high Mach number vehicles, engines, nuclear reactors, re entry vehicles, etc., operate in high-temperature environments. There fore, problems of heat transfer are incorporated into the problems of hydro- and aero-dynamics. Then, heat transfer problems and flow separation prob lems, each sufficiently difficult separately, must be considered in combination. As a problem area, flow separation has been worked on by many scientists, but much work still remains to be undertaken in this portion of the field of fluid mechanics. INTRODUCTION TO THE PROBLEM OF FLOW 8ΕΡΑΙ^\Ή0Ν 3 1. Mechanics of Flow Separation The problem of flow separation is, perhaps, one of the most important hydrodynamic problems to be investigated intensively in order to find its solu tion satisfactorily. Because of the complexity of the problem, a rigorous definition of flow separation and stall should be made. The classical concept of flow separation is due to viscosity; therefore it is often expressed as "boundary layer flow separation" or "boundary layer separation". In addi tion, a necessary condition for flow separation is the adverse pressure gradient. The details involving these two factors are discussed later in this chapter. FIG. 1. Flow separation over a smooth surface FIG. 2. Flow separation over discontinuous surfaces In a general sense, according to Maskell's concept of separation, flow separation is inevitable for flow over the finite dimension. Flow will separate from the solid body surface at the trailing edge and also upstream of it if the required conditions are met there. The flow separation is not only caused by a gradual process, which is the case of flow over a smooth surface, but also by a severe discontinuity of the tangent to the surface (Figs. 1 and 2). First, the classical concept of flow separation is outlined; then the general ized definition of flow separation is discussed. 1.1 Qassical concept of onset of flow separation over a smooth curred body surface The problem of flow separation is as old as that of boundary layer theory. Ludwig Prandtl, the father of the boundary layer concept, was concerned about flow separation before he started his work on the boundary layer. As a young engineer at the Maschinen Fabrik Augsburg-Nürnberg (MAN), SEPARATION OF FLOW Prandtl found that the computed pressure recovery could not be achieved in actual diffusers. He was occupied for 3 years in figuring out why and how the flow separations and pressure losses were caused. This problem was solved finally by his new theoretical concept of the boundary layer [1]. His concept FIG. 3. Flow in a sharply diverging channel [4] FIG. 4. The boundary layer is sucked away on both walls; the flow is from left to right [4] may be referred to as classical in comparison to the modern development of separation of flow theory. The classical concept of flow separation is formed for two-dimensional and axisymmetric steady flow. Prandtl [2] states clearly that the necessary condi tion for flow separation from the wall is the increasing pressure in the stream- wise direction, i.e. positive (or adverse) pressure gradient along the flow INTRODUCTION ΤΟ THE PROBLEM OF FLOW SEPARATION 5 path (Fig. 3). This statement holds for compressible as well as incompressible flow. Therefore, it may be said that, in general, flow separation occurs under adverse pressure gradient and with laminar or turbulent viscosity effects. If one of these two factors is missing, then the flow does not separate. For example, by removing the boundary layer the viscosity effect is elimi nated. As shown in Fig. 4, Prandtl removed the boundary layer along the wall of a subsonic channel by suction. In front of the throat, the pressure decreases in the direction of flow because of the decreasing cross-sectional area of the channel. In this region, the pres sure gradient is negative (or favorable); hence, flow adheres completely to FIG. 5. Free stagnation flow without separation, as photographed by Föttinger [4] the walls. However, behind the throat, because of a large degree of divergency or a sufficiently large degree of adverse pressure gradient, the boundary layer separates from the wall and vortices are formed. But, if the boundary layer downstream of the throat is removed by suction, the flow reattaches to the surface. Föttinger [3] provides other experimental evidence to confirm the two necessary factors which cause flow separation; namely adverse pressure gradient and viscosity. Figures 5 and 6 show fluid flowing against two separate walls which were placed perpendicular to the flow direction. One wall was plain, the other equipped with a thin protruding plate. With the plain wall, there was no separation, but with the protruding plate, the flow separated. The physical phenomena may be explained as follows. In front of the stagnation point, a considerable pressure rise occurs along the direction of flow, but flow does not separate because of the absence of wall friction. Near the wall, the flow does not separate either, since the fluid
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