Page 116 - Aerodynamics for Engineering Students

Governing equations of fluid mechanics 99

Simplifying these two equations gives

(2.118)
1
f f‘ = -a2F’ - vf” (2.119)
2
where use has been made of the definition of kinematic viscosity (v = p/p). Evidently
the assumptions made above were acceptable, since we have succeeded in the aim
of reducing the Navier-Stokes equations to ordinary differential equations. Also
note that the second Eqn (2.1 19) is only required to determine the pressure field,
Eqn (2.118) on its own can be solved for f , thus determining the velocity field.
The boundary conditions at the wall are straightforward, namely
u= v=O at y=O implying f =f’=O at y=O (2.120)

As y -+ m the velocity will tend to its form in the corresponding potential flow. Thus
u-ax as y-+ co implying f’=a as y-+ 00 (2.121)
In its present form Eqn (2.118) contains both a and v, so that f depends on these
parameters as well as being a function of y. It is desirable to derive a universal form
of Eqn (2.118), so that we only need to solve it once and for all. We attempt to
achieve this by scaling the variables f(y) and y, i.e. by writing
f(Y) = P$(r]), 71 = aY (2.122)

where a and ,B are constants to be determined by substituting Eqn (2.122) into
Eqn (2.118). Noting that

Eqn (2.1 18) thereby becomes
QZp2$’2 - &@q5$“ = a2 + ya3pq5“‘ (2.123)
Thus providing
a2p2 = 2 = m3p, implying a = m, p = (2.124)

they can be cancelled as common factors and Eqn (2.124) reduces to the universal
form:

q5’” + qkj” - $12 + 1 = 0 (2.125)
with boundary conditions
$(O) = q5’(0) = 0, qqco) = 1
In fact, 4‘ = u/U, where U, = ax the velocity in the corresponding potential flow
found when r] -+ cc. It is plotted in Fig. 2.33. We can regard the point at which
$’ = 0.99 as marking the edge of the viscous region. This occurs at r] E 2.4. This
viscous region can be regarded as the boundary layer in the vicinity of the stagnation
point (note, though, no approximation was made to obtain the solution). Its thick-
ness does not vary with x and is given by
S N 2.4G (2.126)