Page 34 - Introduction to Computational Fluid Dynamics

P2: IWV
P1: JYD/GKJ
0 521 85326 5
CB908/Date
May 20, 2005
0521853265c01
EXERCISES
Chapter 5 deals with solution of complete transport equations on Cartesian grids. 12:20 13
Only 2D flow situations that may involve regions of fluid recirculation are consid-
ered. The transport equations now take the elliptic form. In essence, this chapter
introduces all ingredients required to understand CFD practice. In this sense, the
chapter provides a firm foundation for development of solution procedures employ-
ing curvilinear and unstructured grids. The latter developments are described in
Chapter 6.
Chapters 7–9 deal with special topics in CFD. In Chapter 7, the reader is in-
troduced to the topic of phase change. In engineering practice, heat and mass
transfer are often accompanied by solid-to-liquid, liquid-to-vapour, and/or solid-to-
vapour (and vice versa) transformations. This chapter, however, deals only with
solidification/melting phenomena in one dimension to develop understanding of
the main difficulties associated with obtaining numerical solutions. Chapter 8 deals
with the topic of numerical grid generation andmethods for curvilinear and unstruc-
tured grid generation are introduced. Finally, in Chapter 9, methods for enhancing
the rate of convergence of iterative numerical procedures are introduced.
There are three appendices. Appendix A provides the derivation of the transport
equations. In Appendix B, a computer code for solving 1D conduction problems is
given. This code is based on material of Chapter 2. Appendix C provides a computer
code for 2D conduction–convection problems in Cartesian coordinates. This code
is based on material of Chapter 5. Familiarity with the use of these codes, it is
hoped, will provide readers with sufficient exposure to enable development of their
own codes for boundary layer flows (Chapter 4) , for employing curvilinear and
unstructured grids (Chapter 6), for phase change (Chapter 7), and for numerical
grid generation (Chapter 8).
At the end of each chapter, exercise problems are given to enhance learning.
Also, in each chapter, sample problems are solved and results are presented to aid
their interpretation.

EXERCISES
terms in Equation 1.3 for i = 1, 2, and 3. Show
1. Express full forms of the S u i
= 0.
that if µ and ρ are constant then, for an incompressible fluid, S u i
2. Consider Equations 1.1–1.5. Assuming SI units, verify that units of each term
in a given equation are identical.
3. Show that summing of each term in Equation 1.2 over all species of the mixture
results in the mass conservation equation (1.1) for the mixture.
4. Consider the plug-flow thermo-chemical reactor (PFTCR) shown in Figure 1.4.
To analyse such a reactor, the following assumptions are made: (a) All s vary
only along the length (say, x) of the reactor. (b) Axial diffusion and conduction