Table Of ContentOther 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
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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
