Other books by C A Brebbia published by Butterworths * Finite element techniques for fluid flow,' 3rd edn (with J J Connor). 9 'Boundary element techniques in engineering (with S Walker). 6 New developments in Boundary Element Methods,' Proc of 2nd Int. Conference in B.E. Methods, Southampton 1980 (ed C A Brebbia). 'Dynamic analysis of Offshore Structures' (with S Walker). Computational Hydraulics C A. Brebbia, DipUng, PhD, Reader in Computational Mechanics, University of Southampton and A. J. Ferrante, DipUng, PhD, President of ISC and Professor of Civil Engineering, COPPE, U. Federal of Rio de Janeiro Butterworths London • Boston • Sydney • Wellington • Toronto • Durban All rights reserved. No part of this publication may be reproduced or transmitted in any form or by any means, including photocopying and recording without the written permission of the copyright holder, application for which should be addressed to the publishers. Such written permission must also be obtained before any part of this publication is stored in a retrieval system of any nature. This book is sold subject to the Standard Conditions of Sale of Net Books and may not be resold in the UK below the net price given by the Publishers in their current price list. First published 1983 © Butterworth & Co (Publishers) Ltd 1983 British Library Cataloguing in Publication Data Brebbia, C.A. Computational hydraulics. 1. Hydraulic engineering—Data processing I. Title II. Ferrante, AJ. 627.028'54 TCI 57.8 ISBN 0-408-01153-X 0-408-01322 2 Pbk Typeset in Great Britain by Mid County Press, London SW15 Printed by Thomson Litho Ltd, East Kilbride, Scotland Preface The advent of modern computers has had a profound effect in all branches of engineering, but has been more marked in structural and stress analysis applications rather than in hydraulics. This is undoubtedly due to the initial difficulties in modelling many of the complex problems frequently found in hydraulic sciences. The recent development of numerical methods and the capabilities of modern machines has changed this situation and many problems which were, up to recently, considered unsuited for numerical solution can now be solved without difficulty. Although computer codes are now available for the solution of hydraulic problems, a text does not exist which introduces both students and practicing engineers to computational methods. Hence this text has been written to present in an introductory but modern way computational techniques to hydraulic and fluid flow engineers. The book is the product of several years' experience in teaching and research at undergraduate and post-graduate level and can be used to offer a self-contained course on Computational Hydraulics for final year or M.Sc. Engineering students. The book combines classical hydraulics with new methods such as finite elements and boundary elements, which are both presented in a matrix formulation. The finite element method is now a well-known technique with many practical applications in hydraulics and fluid flow as described in Finite Element Techniques for Fluid Flow, 3rd edn, by J. J. Connor and C. A. Brebbia published by Butterworths. The boundary element method is more recent but it is presented here as an extension of the basic concepts of classical potential theory (see also Boundary Element Techniques in Engineering by C. A. Brebbia and S. Walker, published by Butterworths). The most interesting feature of the book is the integrated treatment given to the theoretical and computing aspects of numerical methods. The format which has been highly successful with structural engineers, presents a series of complete computer programs, for linear and non-linear pipe network analysis, depth flow computations,f initea nd boundary elements for Laplace equations. The programs which are written in standard FORTRAN are self-contained and easy to implement in any computer. The authors hope that this book will make practising hydraulic engineers more aware of modern computer techniques and be useful in teaching them to the next generation. C. A. Brebbia and A. J. Ferrante This book is dedicated to my former students at University of California, Irvine. Carlos Brebbia List of Programs Program for linear analysis of pipe networks 1 Main program 2 Data input (INPUT) 3 Computation of the half-bandwidth (BAND) 4 Assembling of the system matrix (ASSEM) 5 Computation of the element matrix (STIFF) 6 Addition of an element matrix to the system matrix (ELASS) 7 Introduction of boundary conditions (BOUND) 8 Solution of the system of equations (SLBSI) 9 Evaluation of results (RESUL) 10 Result output (OUTPT) Program for non-linear analysis of pipe networks 11 Main program 12 Data input (INPUT) 13 Evaluation of the initial element coefficient (INCOE) 14 Computation of the element matrix (STIFF) 15 Multiplication of a symmetric banded matrix by a vector (MULTI) 16 Evaluation of results (RESUL) Program for computation of depth of flow 17 Main program for testing subroutine CHDEP 18 Evaluation for depth of flow for channel Finite element program for the solution of the laplace equation 19 Main program 20 Input program (INPUT) 21 Computation of the element matrices (STIFF) 22 Routine for calculating the nodal variables' derivatives (RESUL) 23 Output program (OUTPT) Boundary element program for the solution of the Laplace equation 24 Main program 25 Input program (INPUT) 26 Assembling of the total system of equations and introduction of the boundary conditions (ASBOU) 27 Computation of the off-diagonal terms of the H and G matrices (OFFDGT) 28 Computation of the diagonal terms of matrix H (DIAGT) 29 Solution of the system of equations: non-positive-definite case (SLNPD) 30 Computation of internal results (RESUL) 31 Output of the results (OUTPT) Note. A computer tape containing copies of these programs is available from: Computational Mechanics Centre Ashurst Lodge, Ashurst, Hants S04 2AA, England Chapter 1 Properties of fluids 1.1 Introduction All physical matter is found in nature either as a solid, a liquid or a gas. These forms of matter have chemically stable thermodynamic phases. Depending on parameters such as temperature, pressure, etc., a given substance may adopt any of the three basic forms of matter. A solid is a substance that does not flow perceptibly under moderate stress, and has a well defined shape and volume. A liquid is a substance characterised by the free movement of its molecules, but without tendency to separate. A liquid has a definite volume but more or less readily takes the shape of its container. When placed in an open container a liquid will be bound by a free surface. A liquid is slightly compressible. A gas is a substance that has neither an independent shape nor a volume, but which will tend to expand indefinitely. When placed in a closed container, a gas will tend to fill all the available space. A gas is highly compressible. Due to their similar behaviour under dynamic conditions, both liquids and gases are known as fluids. All fluids have a certain amount of viscosity and they deform continuously under a shearing stress. Viscosity is a measure of the resistance to deformation of the fluid and generates internal viscous forces. In practice there are many fluid problems where viscous forces are negligible compared with other acting forces. In such cases it is consistent to assume that the fluid has a null viscosity. These fluids are called inviscid fluids. When af luidi s inviscid tangential stresses cannot be applied to a fluid layer by other layers or by solid walls. Two adjacent layers of an inviscid fluid can move with different velocities, without any interaction derived from internal friction. Thus, any layer of an inviscid fluid can be replaced by a solid contour having the same shape, without the movement of the other layers being altered. Under the assumption of zero viscosity the flow of a fluid in a pipe will present the distribution shown in Figure 1.1(a). For a real fluid, however, the actual velocity field will show a parabolic distribution, as indicated in Figure 1.1(b), where the fluid tends not to slip along the solid surface of the pipe. In spite of this obvious drawback, the behaviour of many fluids can be accurately 1 2 Properties of fluids Figure 1.1 Velocity distributions for (a) inviscid and (b) viscous fluids described as inviscid. Thesef luidsa re sometimes called idealf luids,a s opposed to real fluids, which have non-zero viscosity. Another property of fluids which is sometimes neglected is compressibility, or the ability of a substance to change volume when subjected to direct pressure. Liquids, in particular, are usually considered to be incompressible substances. 1.2 Basic definitions Mass, denoted by M, is the quantity of matter; it is a function of the internal structure, and the dimensions of a substance. In the SI system the unit used to define mass is the kilogram, kg. Density, denoted by p, is defined as the mass per unit volume of a substance. 3 In the SI system, as mass is defined in kg and volume in cubic metres, m , density is defined in kilograms per cubic metre, i.e. (1.1) Densities of different liquids are given in Table 1.1. From the previous definitions it follows that the total mass M can be expressed as the product of the density p times the volume V, (1.2) A mass will be at rest or under uniform motion, unless acted upon by a force. TABLE 1.1. Density of various liquids (After Hodgman, C. (Ed.), Handbook of Chemistry and Physics, 37th edn, Chemical Rubber Publishing Co., Cleveland (1955).) 3 3 3 3 Liquid Temperature (°C) DDenesnistyit y(1 (01~0~ k gk/gm/m) ) Alcohol Ethyl 20 0.792 Methyl 20 0.791 Glycerin 0 1.260 Gasoline 0.6-0.69 Mercury 13.6 Sea water 15 1.025 Water 4 1.0 Basic definitions 3 Thus, to put in motion a mass which is at rest, or to change the nature of the motion of a moving mass the application of a force is required. The weight, W, of an object is the force applied to that object by the action oí gravity. In the SI system the acceleration due to gravity, denoted by g, is equal to 2 9 = 9.81 m s" (1.3) where the unit of time is the second, s. Then the weight of a mass is equal to the product of the acceleration of gravity times the mass: W=gM=gpV (1.4) A dimensional analysis of expression (1.4) will show that, in the SI system, weight is defined by kilograms times metres, divided by square seconds, i.e. (1.5) That unit is called a newton, with symbol N, and, 2 1 N = l newton = 1 m kg s" (1.6) When using grams, g, and centimetres, cm, i.e. the c.g.s. system, rather than kilograms and metres, the unit for force is the dyne, dyn, given by 2 5 ldyn = l g c m s - = 10" N (1.7) Pressure, denoted by p, is a force per unit area. From its definition it follows that (1.8) To define velocity let us consider a fluid particle, that is the fluid contained in an infinitesimal volume, which is at a position P at initial time t0. As shown in Figure 1.2, the particle position can be defined by vector F, which goes from the reference origin O to the particle position P. At time tl the particle has moved to a new position, F , which is now identified by vector ?', going from points O to F . The difference between the initial position P and the final position P' is given by the vector Af, also shown in Figure 1.2. Figure 1.2 Position of particle The velocity v of the particle is defined by the limit of the quotient of A r divided by Aí, with Aí equal to tx - i 0 , when tx tends to i0, or tl ->i0, (1.9) 4 Properties of fluids Figure 1.3 Fluid deformation Thus, the velocity is equal to the derivative of the position vector nvith regard to time. Therefore, in the SI system, the velocity will be defined in metres per second. Clearly, the velocity of a system of particles is a function of position and time. The viscosity of a fluid, as suggested in the introduction of this chapter, is a measure of the resistance of the fluid to deformation caused by shearing stresses. Although in practice a null viscosity situation is sometimes accepted, all real fluids have some viscosity. In order to clarify further the meaning of a non-null viscosity let us consider the case shown in Figure 1.3, which indicates a fluid filling the space between two parallel plates. It is assumed that the only part of the upper body in contact with the fluid is the face of area A. When a horizontal force F is applied to the upper body, the body starts to move with a certain velocity vA. This velocity is transmitted to the fluid, for real fluids, in such a way that at face A the fluid velocity is equal to vA and decreases linearly in the vertical direction, until it becomes zero at the bottom plate. The relationship between the shearing stress T, at surface A, and the velocity v at any horizontal layer in the fluid is given by Newton's viscosity law, (1.10) where ¡x is called dynamic viscosity, and is equal to the ratio of the shear stress to the velocity gradient. Clearly, the shear stress will be null for a stationary fluid, which accounts for n being called 'dynamic' viscosity. Fluids for which the dynamic viscosity \i is a constant are called Newtonian fluids. For ideal fluids the dynamic viscosity is equal to zero. For solids the shear stress is independent of the velocity gradient. These cases are illustrated in Figure 1.4. Figure 1.4 Newtonian fluids