01-08-2012, 12:24 PM
Computational Fluid Dynamics
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Introduction to Computational Fluid Dynamics
We have been using the idea of distributions of singularities on surfaces to study the
aerodynamics of airfoils and wings. This approach was very powerful, and provided us with
methods which could be used easily on PCs to solve real problems. Considerable insight into
aerodynamics was obtained using these methods. However, the class of effects that could be
examined was somewhat restricted. In particular, practical methods for computing fundamentally
nonlinear flow effects were excluded. This includes both inviscid transonic and boundary layer
flows.
In this chapter we examine the basic ideas behind the direct numerical solution of differential
equations. This approach leads to methods that can handle nonlinear equations. The simplest
methods to understand are developed using numerical approximations to the derivative terms in
the partial differential equation (PDE) form of the governing equations. Direct numerical
solutions of the partial differential equations of fluid mechanics constitute the field of
computational fluid dynamics (CFD). Although the field is still developing, a number of books
have been written.1,2,3,4,5,6 In particular, the book by Tannehill et al,1 which appeared in 1997 as a
revision of the original 1984 text, covers most of the aspects of CFD theory used in current codes
and reviewed here in Chapter 14. Fundamental concepts for solving partial differential equations
in general using numerical methods are presented in a number of basic texts. Smith7 and Ames8
are good references.
The basic idea is to model the derivatives by finite differences. When this approach is used
the entire flowfield must be discretized, with the field around the vehicle defined in terms of a
mesh of grid points. We need to find the flowfield values at every mesh (or grid) point by writing
down the discretized form of the governing equation at each mesh point. Discretizing the
equations leads to a system of simultaneous algebraic equations. A large number of mesh points
is usually required to accurately obtain the details of the flowfield, and this leads to a very large
system of equations. Especially in three dimensions, this generates demanding requirements for
computational resources. To obtain the solution over a complete three dimensional aerodynamic
configuration millions of grid points are required!
Approximations to partial derivatives
There are many ways to obtain finite difference representations of derivatives. Figure 8-1
illustrates the approach intuitively. Suppose that we use the values of f at a point x0 and a point a
distance Dx away. Then we can approximate the slope at x0 by taking the slope between these
points. The sketch illustrates the difference between this simple slope approximation and the
actual slope at the point x0. Clearly, accurate slope estimation dependents on the method used to
estimate the slope and the use of suitably small values of Dx.
Finite difference representation of Partial Differential Equations (PDE's)
We can use the approximations to the derivatives obtained above to replace the individual
terms in partial differential equations. The following figure provides a schematic of the steps
required, and some of the key terms used to ensure that the results obtained are in fact the
solution of the original partial differential equation. We will define each of these new terms
below.