Page 92 - Introduction to Computational Fluid Dynamics

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4 2D Boundary Layers May 25, 2005 11:7

4.1 Governing Equations

It will be fair to say that the early developments in CFD and heat and mass transfer
began with calculation of boundary layers. The term boundary layer is applied to
long and thin flows: long in the streamwise direction and thin in the transverse
direction. The term applies equally to flows attached to a solid boundary (wall
boundary layers) as well as to jets or wakes ( free-shear layers).
Calculation of boundary layer phenomena received a considerable boost follow-
ing the development of a robust numerical procedure by Patankar and Spalding [50].
This made phenomena that were either impossible or too cumbersome to calculate
by means of earlier methods (similarity, nonsimilarity, and integral) amenable to
fast and economic computation. The procedure, for example, permitted use of vari-
able properties, allowed for completely arbitrary variations of boundary conditions
in the streamwise direction, and led to several new explorations of diffusion and
source laws. Thus, calculation of free or forced flames or wall fires could be carried
out by considering the detailed chemistry of chemical reactions. Similarly, cal-
culation of turbulent flows (and development of turbulence models, in particular)
could be brought to a substantial level of maturity through newer explorations of
diffusion and source laws governing transport of variables that characterise turbu-
lence. Computer programs based on the Patankar–Spalding procedure are available
in [50, 77, 10]. There are also other methods, for example, the Keller–Box method
described in [35].
The emphasis in this chapter is on describing the Patankar–Spalding procedure
using simple notation. The procedure generalises all two-dimensional boundary
layer phenomena by introducing the coordinate system shown in Figure 4.1. This
system permits consideration of

1. axisymmetric as well as plane flows,
2. wall boundary layers as well as free-shear layers, and
3. internal (or ducted) as well as external boundary layers.

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