2.4 Components of a Numerical Solution Method
25
Visualization of numerical solutions using vector, contour or other kinds
of plots or movies (videos) of unsteady flows is important for the interpretation of results. It is far and away the most effective means of interpreting
the huge amount of data produced by a calculation. However, there is the
danger that an erroneous solution may look good but may not correspond
to the actual boundary conditions, fluid properties etc.! The authors have
encountered incorrect numerically produced flow features that could be and
have been interpreted as physical phenomena. Industrial users of commercial CFD codes should especially be careful, as the optimism of salesmen is
legendary. Wonderful color pictures make a great impression but are of no
value if they are not quantitatively correct. Results must be examined very
critically before they are believed.
2.4 Components of a Numerical Solution Method
Since this book is meant not only for users of commercial codes but also for
young researchers developing new codes, we shall present the important ingredients of a numerical solution method here. More details will be presented
in the following chapters.
2.4.1 Mathematical Model
The starting point of any numerical method is the mathematical model, i.e.
the set of partial differential or integro-differential equations and boundary
conditions. Some sets of equations used for flow prediction were presented in
Chap. 1. One chooses an appropriate model for the target application (incompressible, inviscid, turbulent; two- or three-dimensional, etc.). As already
mentioned, this model may include simplifications of the exact conservation
laws. A solution method is usually designed for a particular set of equations.
Trying to produce a general purpose solution method, i.e. one which is applicable to all flows, is impractical, if not impossible and, as with most general
purpose tools, they are usually not optimum for any one application.
2.4.2 Discretization Method
After selecting the mathematical model, one has to choose a suitable discretization method, i.e. a method of approximating the differential equations
by a system of algebraic equations for the variables at some set of discrete
locations in space and time. There are many approaches, but the most important of which are: finite difference (FD), finite volume (FV) and finite
element (FE) methods. Important features of these three kinds of discretization methods are described a t the end of this chapter. Other methods, like
spectral schemes, boundary element methods, and cellular automata are used
in CFD but their use is limited t o special classes of problems.
25
Visualization of numerical solutions using vector, contour or other kinds
of plots or movies (videos) of unsteady flows is important for the interpretation of results. It is far and away the most effective means of interpreting
the huge amount of data produced by a calculation. However, there is the
danger that an erroneous solution may look good but may not correspond
to the actual boundary conditions, fluid properties etc.! The authors have
encountered incorrect numerically produced flow features that could be and
have been interpreted as physical phenomena. Industrial users of commercial CFD codes should especially be careful, as the optimism of salesmen is
legendary. Wonderful color pictures make a great impression but are of no
value if they are not quantitatively correct. Results must be examined very
critically before they are believed.
2.4 Components of a Numerical Solution Method
Since this book is meant not only for users of commercial codes but also for
young researchers developing new codes, we shall present the important ingredients of a numerical solution method here. More details will be presented
in the following chapters.
2.4.1 Mathematical Model
The starting point of any numerical method is the mathematical model, i.e.
the set of partial differential or integro-differential equations and boundary
conditions. Some sets of equations used for flow prediction were presented in
Chap. 1. One chooses an appropriate model for the target application (incompressible, inviscid, turbulent; two- or three-dimensional, etc.). As already
mentioned, this model may include simplifications of the exact conservation
laws. A solution method is usually designed for a particular set of equations.
Trying to produce a general purpose solution method, i.e. one which is applicable to all flows, is impractical, if not impossible and, as with most general
purpose tools, they are usually not optimum for any one application.
2.4.2 Discretization Method
After selecting the mathematical model, one has to choose a suitable discretization method, i.e. a method of approximating the differential equations
by a system of algebraic equations for the variables at some set of discrete
locations in space and time. There are many approaches, but the most important of which are: finite difference (FD), finite volume (FV) and finite
element (FE) methods. Important features of these three kinds of discretization methods are described a t the end of this chapter. Other methods, like
spectral schemes, boundary element methods, and cellular automata are used
in CFD but their use is limited t o special classes of problems.