Environmental Engineering Reference
In-Depth Information
taken not to draw conclusions from the model that are
beyond its validity (Lane and Richards, 2001), which
raises difficult questions over validation and calibration
(Beven and Binley, 1992; Beven, 2002; Lane
et al
., 2005;
Odoni and Lane, 2010) (see also Chapter 2).
and
+
∂
(
ρ
u
i
u
j
)
∂
∂
(
ρ
u
i
)
=−
∂
p
∂
∂
µ
∂
u
i
x
i
+
+
S
i
,
∂
t
x
j
∂
x
j
∂
x
j
(6.2)
where
x
i
(
i
1, 2, 3) are the three co-ordinate directions,
u
i
are the components of the velocity [L T
−
1
]inthese
directions,
p
is the pressure [ML
−
1
T
−
2
],
=
6.2 CFD fundamentals
ρ
is the density
[ML
−
3
]and
µ
is the dynamic viscosity [ML
−
1
T
−
1
].
Equation 6.1 encapsulates conservation of mass, while
Equation 6.2 is a momentum equation where the first
term on the left-hand side is the time variation and the
second is the convection term and on the right-hand
side the terms are, respectively, a pressure gradient, a
diffusion term and the source term, which can add in
the contribution from a body force (Coriolis, wind shear,
gravity, etc.). In most cases, these equations cannot be
solved analytically to give an exact solution. CFD opens
up the possibility of providing approximate solutions at
a finite number of discrete points.
6.2.1 Overview
At a fundamental level, CFD takes the physical laws of
fluid flow and produces solutions in particular situations
with the aid of computers. Soon after the advent of com-
puters, mathematicians began to investigate their use in
solving partial differential equations and by the 1970s the
use of computers to solve the nonlinear partial differential
equations governing fluid flow was under active investi-
gation by researchers. During the 1980s CFD became a
viable commercial tool and a number of companies began
to market CFD software. Over the intervening decades,
CFD has become an accepted analysis and design tool
across a range of engineering disciplines and is now an
integral part of computer-aided design.
Whilst in general the term CFD is taken to imply a full,
three-dimensional calculation of a turbulent flow field,
in environmental calculations it is often possible to use
calculations that are either one- or two-dimensional, or
include significant simplifications of the flowfield in other
ways. Thus whilst in this section CFD is described in its
conventional three-dimensional form, many of the appli-
cations in Section 6.3 use models that are significantly
simpler, which gives significant advantages in terms of
practicality and ease of use.
There are many topics devoted to CFD and it is not
possible to cover the field in a short chapter. Consequently
this chapter highlights the major issues in environmental
applications. Readers are referred to other texts: Versteeg
and Malalasekera (2007) for a general treatment and
Abbott and Basco (1989) for applications to the shallow-
water equations.
6.2.3 Gridstructure
Initial investigations into the solutions of partial differ-
ential equations (PDEs) used the finite difference method
and this approach was carried through into early CFD.
The finite difference method approximates a derivative at
a point in terms of values at that point and adjacent points
through the use of a Taylor-series approximation. It is
simple to calculate and implement for basic cases. How-
ever, it was found to have limitations in application to
fluid-flow problems. Engineering CFD researchers devel-
oped and favoured an alternative, more versatile method
called the finite volume method. This method divides
the area of interest (or domain) into a large number of
small control volumes, which together are described as
a mesh (or grid). The physical laws are integrated over
each control volume to give an equation for each law
in terms of values on the face of each control volume.
These face values, and the fluxes through them, are then
calculated from adjacent values by interpolation. This
method ensures that mass is conserved in the discrete
form of the equations just as it is in the physical situation.
The finite-volume technique is the most widely used in
commercial CFD software.
Constructing a suitablemesh is often themost demand-
ing part of a CFD simulation both in terms of operator
time and expertise. Adequate details of the physical sit-
uation must be included in the mesh, but too great a
level of detail will lead to a mesh that contains more
6.2.2 Equationsofmotion
CFD is based on solving the physical laws of Conservation
of Mass and Newton's Second Law as applied to a fluid.
For an incompressible fluid, these laws are encapsulated
in the Navier-Stokes equations:
∂
u
i
∂
x
i
=
0,
(6.1)
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