Three-dimensional Separated Flow Topology E-Book

Jean Délery

• Herausgeber: John Wiley & Sons
• Kategorie: Wissenschaft und neue Technologien
• Sprache: Englisch

Seitenzahl: 143

Veröffentlichungsjahr: 2013

Beschreibung

This book develops concepts and a methodology for a rational description of the organization of three-dimensional flows considering, in particular, the case where the flow is the place of separations. The descriptive analysis based on the critical point theory of Poincaré develops conventional but rather unfamiliar considerations from aerodynamicists, who face the understanding of complex flows including multiple separation lines and vortices. These problems concern industrial sectors where aerodynamics plays a key role, such as aerospace, ground vehicles, buildings, etc. Contents 1. Skin Friction Lines Pattern and Critical Points. 2. Separation Streamsurfaces and Vortex Structures. 3. Separated Flow on a Body. 4. Vortex Wake of Wings and Slender Bodies. 5. Separation Induced by an Obstacle or a Blunt Body. 6. Reconsideration of the Two-Dimensional Separation. 7. Concluding Remarks. About the Authors Jean Délery is a Supaero (French National Higher School of Aeronautics and Space) engineer who has worked at Onera (French national aerospace research center) since 1964. He has participated in several major French and European aerospace programs, is the author of many scientific publications, and has occupied various teaching positions particularly at Supaero, the University of Versailles-Saint-Quentin, Ecole polytechnique in France and "La Sapienza" University in Rome, Italy. He is currently emeritus adviser at Onera.

Leseprobe

Contents

Introduction

1 Skin Friction Lines Pattern and Critical Points

1.1. Basic properties of the three-dimensional boundary layer

1.2. Skin friction lines and surface flow pattern

1.3. Critical points of the skin friction line pattern

1.4. Critical points of the wall vorticity lines

2 Separation Streamsurfaces and Vortex Structures

2.1. Generalization to the flow field and three-dimensional critical points

2.2. Separation and attachment lines

2.3. Streamsurfaces of separation and attachment

2.4. Vortical structures

2.5. Some properties of a vortical structure

3 Separated Flow on a Body

3.1. Basic rules and definitions

3.2. General definition: the basic separated structures

3.3. Field associated with a separation with one saddle point and three nodes: the horseshoe vortex

3.4. Field associated with a separation with one saddle point and two foci: the tornado-like vortex

4 Vortex Wake of Wings and Slender Bodies

4.1. Vortical structures over a delta wing

4.2. Vortical flow over a slender body

4.3. Vortex wake of a classical wing

5 Separation Induced by an Obstacle or a Blunt Body

5.1. Separation in front of an obstacle

5.2. Flow induced by an obstacle of finite height or protuberance

5.3. Separation on a non-propelled afterbody

5.4. The flow past an automobile

6 Reconsideration of the Two-Dimensional Separation

6.1. Some definitions: a reminder

6.2. Two-dimensional separation

6.3. Special critical points

6.4. Three-dimensional structure of a two-dimensional separated flow

6.5. Axisymmetric afterbody

7 Concluding Remarks

Bibliography

List of Symbols

Index

First published 2013 in Great Britain and the United States by ISTE Ltd and John Wiley & Sons, Inc.

Apart from any fair dealing for the purposes of research or private study, or criticism or review, as permitted under the Copyright, Designs and Patents Act 1988, this publication may only be reproduced, stored or transmitted, in any form or by any means, with the prior permission in writing of the publishers, or in the case of reprographic reproduction in accordance with the terms and licenses issued by the CLA. Enquiries concerning reproduction outside these terms should be sent to the publishers at the undermentioned address:

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© ISTE Ltd 2013

The rights of Jean Délery to be identified as the author of this work have been asserted by him in accordance with the Copyright, Designs and Patents Act 1988.

Library of Congress Control Number: 2012955534

British Library Cataloguing-in-Publication Data

A CIP record for this book is available from the British Library

ISSN: 2051-2481 (Print)

ISSN: 2051-249X (Online)

ISBN: 978-1-84821-450-7

Introduction

The main goal of this book is primarily didactic. It does not present a complete analysis on separated three-dimensional flows, but some basic concepts to be used to arrive at a rational description of three-dimensional flows containing what are called separations. This explains the detail of some chapters, those devoted to the critical point theory in particular, and the absence of references in the text. However, a bibliography giving the basic publications is provided.

The passage from the familiar two-dimensional to the mysterious three-dimensional requires a complete reconsideration of apparently obvious concepts (separation and reattachment points, separated bubbles, recirculation zone, limiting streamline, etc.) which are inappropriate and even dangerous to use in three-dimensional flows. Such a situation entails a risk of misunderstanding, hence leading to sterile actions to interpret facts from inadequate concepts.

Robert Legendre recognized this fact in the early 1950s when he studied the flow past delta wings and other three-dimensional bodies. He introduced a precise definition of the basic concepts on which we have to reason to describe such a flow; then he applied to these concepts rigorous results inspired from the works of the mathematician Henri Poincaré. Hence, the introduction of notions such as skin friction lines, critical points, separation (or re-attachment) lines, separation (or re-attachment) surfaces and topological rules that allowed a consistent description of flow fields. The work of Legendre on three-dimensional separation was in great part inspired and confirmed by the visualizations performed at the same time by Henri Werlé in his water tunnels at Onera.

The present theory of three-dimensional separation is not a predictive theory in that it does not predict separation on an obstacle in any given condition. For this, the Navier–Stokes equations must be solved. It is a descriptive theory reasoning on the properties of a given vector field (skin friction and velocity) provided by experimentation or calculation. It is a tool to give a rational description and interpretation of an observation made on a model in a wind tunnel or on the screen of a workstation, displaying computed results. Even with its limitations, the presented theory is of great help in analyzing complex flows, if one considers the numerous mistakes made when speaking about three-dimensional separated flows: at separation, the skin friction vanishes, the separation is open or closed, or worst it is of the bubble type. As will be seen, on a three-dimensional vehicle, whose flow is separated, the skin friction is nearly always non-zero, most separated zones are open and the bubble notion applies only to very singular mathematical situations, hence highly improbable in reality.

It is not possible to discuss separation in three-dimensional flows without introducing the critical point theory. It is to the great credit of Robert Legendre to have realized that this mathematical theory is the only rational tool that allows the understanding of the organization – or topology – of three-dimensional separated flows. The critical point theory comes from the works of Poincaré on the singular points of the system of differential equations. In general, it applies to a space of n dimensions and leads to major conclusions and results on the behavior of dynamical systems, including the concepts of strange attractors, the notions of ill- and well-posed problems, unpredictability, and ultimately chaos theory. In what follows, we restrict ourselves to a very narrow aspect of this theory, which will be sufficient for our purpose and which, in fact, was only considered by Legendre. Critical points are found in many disciplines where they can have a different denomination and physical meaning. They are present in electromagnetism, in the theory of dynamics system, in the stability theories, etc.

The first articles published in 1952 by Robert Legendre had, in France, a limited echo mainly because of their extreme conciseness and their apparently too mathematical approach. However, it seemed that these notions were too abstract – or fundamental, not to say academic! – to be applied to real configurations, engineers being then familiar with two-dimensional analyses. However, Legendre’s publications were not ignored abroad where the critical point theory was revisited by Lighthill in 1963 to depict the separation of a three-dimensional boundary layer, which, in fact, was a non-justified restriction. The critical point theory applied to three-dimensional separation led to a series of publications: in the United States where Murray Tobak published a very interesting paper in 1982, and in the United Kingdom and in Germany. We also have to cite the essential contribution of Henri Werlé whose water tunnel visualizations have constituted the physical basis, and the first motivation, of Legendre’s publications.

The considerations that follow apply to a non-fluctuating flow, a laminar flow – rarely achieved in aeronautical applications – as well as to time-averaged flow introduced to describe and compute turbulent flows. In this case, as far as the organization of the flow is concerned, there are no major differences between laminar and turbulent situations (the governing equations are almost the same) except for the characteristic scales. In addition, we consider the flow as steady, although the theory can be applied to the instantaneous field of a time-dependent flow.

In many circumstances, three-dimensional separation is such a catastrophic and overwhelming phenomenon that it is nearly independent of the Reynolds number, which is, in fact, the correct scaling parameter in boundary-layer-like situations where viscous effects are confined within thin layers. The Mach number also is not a determining parameter for largely separated flows, the accompanying shock system being in reality an epiphenomenon. Thus, if we exclude exceptional circumstances where the Reynolds number is extremely low, the organization of the flow over a delta wing, downstream of the base of a missile, in front of a blunt obstacle, or past an automobile, is basically the same as in a small-size water tunnel at low velocity of a few centimeters per second. To pretend that in most cases there is no Reynolds number effect would thus be an exaggeration, but the flow physics does not depend critically on this parameter. If we exclude laminar to turbulent transition, the Reynolds number has its most direct influence on the location of the boundary-layer separation. In situations where this separation is imposed by a local singularity, like a sharp edge, the Reynolds number plays a nearly negligible role.

In Chapter 1, the critical point theory is presented, showing the origin of the nodes, saddle points and foci observed in surface flow visualizations. In Chapter 2, we introduce the basic notions of separation and attachment lines, separation and attachment surfaces, and vortical structures. This allows the rational definition in Chapter 3, of a separated flow and the description of the associated field with the formation of horseshoe and tornado-like vortices. Then, the elements and grammar rules thus established are used to describe the organization, or topology, of typical separated flows. Chapter 4 analyzes the separation over slender bodies and wings (delta and classical wings), showing how the wing vortex-wake forms. Chapter 5 deals with separation induced by an obstacle and on a blunt body. This includes the description of the flow induced by protuberances, past an afterbody and an automobile. In Chapter 6, the so-called two-dimensional (planar or axisymmetric) case is reconsidered in the light of the concepts introduced by the three-dimensional theory.

1

Skin Friction Lines Pattern and Critical Points