Page 57 - Computational Fluid Dynamics for Engineers

42 2. Conservation Equations

Depending on the flow conditions, it is appropriate and sometimes neces-
sary to use the reduced forms of the Navier-Stokes equations as discussed in
Section 2.4. These simplified equations reduce the complexity of solving the full
Navier-Stokes equations, provide substantial savings on computer time, and
in some situations permit accurate analytical and numerical solutions to the
conservation equations.
The prediction of transition from laminar to turbulent flow remains one of
the unsolved problems of fluid mechanics. The subject is important in many
applications, as discussed in Chapter 1. The only approach that may provide a
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general prediction method in the near future is the e -method based on the solu-
tion of the linearized stability equations. In Section 2.5 we discuss the derivation
n
of the stability equations, while discussing their solutions and the e -method
later in Chapter 8.
The Navier-Stokes equations and their reduced forms are partial-differential
equations. Before using numerical methods to solve these equations, one must
know whether they are hyperbolic, elliptic or parabolic. In Section 2.6 we discuss
the classification of the partial-differential equations, and the general principles
one must consider to solve each class of equations subject to the boundary
conditions discussed in Section 2.7.
Except for Chapters 10 and 12 in this book, the treatment of the compu-
tational fluid dynamics equations are for incompressible flows. For this reason,
the derivations and discussions will concentrate mostly on incompressible flows.
In some instances, and whenever appropriate and necessary, the conservation
equations for compressible flows will also be given in order to pave the way for
the discussion of the numerical solution of Euler and Navier-Stokes equations
for compressible flows in Chapters 10 and 12.

2.2 Navier-Stokes Equations

The Navier-Stokes equations may be obtained by using infinitesimal or finite
control volume approaches, and the governing equations can be expressed in
differential or integral forms. In subsection 2.2.1 these equations are presented
in differential form, and in subsection 2.2.2 in integral form. For a detailed
discussion of the derivation of these equations in either form, the reader is
referred to Anderson [4,5].

2.2.1 Navier-Stokes Equations: Differential Form
The Navier-Stokes equations in differential form can be derived by using an
infinitesimal control volume either fixed in space with the fluid moving through
it or moving along a streamline with a velocity vector V = (u, v, w) T equal to
the flow velocity at each point. Here we follow the second choice, discussed in [2],