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216 7. Boundary-Layer Equations

7.3 Numerical Method for the Standard Problem

In this section the momentum and continuity equations are considered and their
solutions for the standard problem corresponding to an external flow in which
the boundary conditions are given by Eq. (2.7.3) are discussed; the boundary
conditions are renumbered for convenience

y = 0, u = 0, v = v w(x) (7.3.1a)
y = <5, u — u e(x) (7.3.1b)

Since the boundary-layer equations are parabolic, their solution procedure em-
ploys a marching scheme similar to the unsteady heat conduction equation
discussed in Section 4.4. It begins with initial conditions, say at x = XQ and
proceeds in the positive x-direction with dependent variables determined in se-
quence normal to the flow at each x-station subject to the boundary conditions.
The continuity and momentum equations given by Eqs. (7.1.3) and (7.1.1)
can be solved in the form they are expressed. They can also be solved after
they are expressed as a third order equation by using the definition of stream
function ij;(x,y). Noting that

Eqs. (7.1.3) and (7.1.1), with a prime denoting differentiation with respect to y
can be written as
"WT^'g^i (-3-3)

In either form, for given initial conditions, say at x = XQ and eddy viscosity
distribution, these equations are solved subject to their boundary conditions in
the interval 0 to 6 at each specified x-location greater than XQ. The boundary-
layer thickness <5(x), however, increases with increasing downstream distance x
for both laminar and turbulent flows; to maintain computational accuracy, it is
necessary to take small steps in the streamwise direction to maintain computa-
tional accuracy.
Transformed coordinates employing similarity variables are advantageous
in the solution of boundary-layer equations since they can reduce the growth
of 6(x) and thus allow larger steps to be taken in the streamwise direction.
Furthermore, and as shown below, for laminar flows they can also be used to
generate initial conditions.
There are several transformations that can be used for this purpose, and
even though most of them have been developed for laminar flows, they can also
be used for turbulent flows. The Falkner-Skan transformation discussed in [4] is
a convenient choice. In this transformation, a dimensionless similarity variable
rj and a dimensionless stream function f(x,rj) are defined by