Page 30 - Introduction to Computational Fluid Dynamics

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1.5 A NOTE ON NAVIER–STOKES EQUATIONS
7. “Interpret the solution:” The numerical solution results in values of each 12:20 9
at each node. Such a field provides the distribution of over the domain.
The task now is to interpret the solution to retrieve quantities of engineering
interest such as the friction factor, a Nusselt number at the wall, or average
concentrations of CO, fuel, and NO x at the exit from a combustion chamber.
Sometimes the field may be curve-fitted to take the appearance of an analytical
solution. Similarly, the derived quantities may also be curve-fitted to take the
appearance of an experimentally derived correlation for ready use in further
design work.
8. “Display of results:” Since a numerical solution is obtained at discrete points,
the solution comprises numbers that can be printed in tabular forms. The in-
convenience of reading numbers can be circumvented by plotting results on a
graph or by displaying the fields by means of contour or vector plots. Fortu-
nately, such graphic displays can now be made using computers. This activity
is called postprocessing of results. The commercial success of computer codes
often depends on the quality and flexibility of their postprocessors.

The primary focus of this book is to explain procedures for executing these
steps. Computer code developers and researchers adopt a variety of practices to
implement the procedures depending on their background, familiarity, and notions
of convenience. Clearly biases are involved.
In this book, emphasis is laid on physical principles. In fact, the attitude is one
of relearning fluid mechanics and heat and mass transfer by obtaining numerical
(as opposed to restrictive analytical) solutions. The book is not intended to provide
a survey of all numerical methods; rather, the objective is to introduce the reader
to a few specific methods and procedures that have been found to be robust in a
wide variety of situations of a specific class. The emphasis is on skill development,
skills required for problem formulation, computer code writing, and interpretation
of results.

1.5 A Note on Navier–Stokes Equations

The law of conservation of mass for the bulk fluid together with Newton’s second
law of motion constitutes the main laws governing fluid motion. As shown in
Appendix A, the equations of motion are written in differential form and, therefore,
assume existence of a fluid continuum. In this section, attention is drawn to an
often overlooked requirement that assumes considerable importance in the context
of CFD in which numerical solutions are obtained at discrete points rather than at
every point in space as in a continuum.
Attention is focussed primarily on the normal stress expressions given in
Appendix A (see Equations A.15). As presented in Schlichting [65], the normal