Page 27 - Introduction to Computational Fluid Dynamics

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INTRODUCTION
formed by invoking a similarity variable. In such circumstances, often the solution
is in the form of a series. We assume, of course, that the reader is familiar with the
restrictive circumstances (often of significant practical consequence) under which
such analytical solutions are constructed.
Analytical solutions obtained in the manner described here are termed exact
solutions. They are applicable to every point of the time and/or space domain. The
solutions are also called continuous solutions. All the aforementioned solutions are
well covered in an undergraduate curriculum and in textbooks (see, for example,
[34, 80, 88]).
Unlike analytical solutions, numerical solutions are obtained at a few chosen
points within the domain. They are therefore called discrete solutions. Numerical
solutions are obtained by employing numerical methods. The latter are really an
intermediary between the physics embodied in the transport equations and the
computers that can unravel them by generating numerical solutions. The process
of arriving at numerical solutions is thus quite different from the process by which
analytical solutions are developed.
Before describing the essence of numerical methods, it is important to note
that these methods, in principle, can overcome all three aforementioned imped-
iments to obtaining analytical solutions. In fact, the history of CFD shows that
numerical methods have been evolved precisely to overcome the impediments in
the order of their mention. Thus, the earliest numerical methods dealt with one-
dimensional equations for which analytical solutions may or may not be possible.
Methods for two-dimensional transport equations, however, had to incorporate sub-
stantially new features. In spite of these new features, many methods applicable to
two-dimensional coupled equations could not be extended to three-dimensional
equations. Similarly, the earlier methods were derived for transport equations cast
in only orthogonal co-ordinates (Cartesian, cylindrical polar, or spherical). Later,
however, as computations over complex domains were attempted, the equations
were cast in completely arbitrary curvilinear (ξ 1 , ξ 2 , ξ 3 ) coordinates. This led to
development of an important branch of CFD, namely, numerical grid generation.
With this development, domains of arbitrary shape could be mapped such that
the coordinate lines followed the shape of the domain boundary. Today, complex
domains are mapped by yet another development called unstructured mesh gener-
ation. In this, the domain can be mapped by a completely arbitrary distribution of
points. When the points are connected by straight lines, one obtains polygons (in
two dimensions) and polyhedra (in three dimensions). Several methods (as well as
packages) for unstructured mesh generation are now available.

1.4 Main Task

It is now appropriate to list the main steps involved in arriving at numerical solutions
to the transport equation. To enhance understanding, an example of an idealised