Page 112 - Aerodynamics for Engineering Students

Governing equations of fluid mechanics 95

If we wish the model tests to produce useful information about general characteristics of the
prototype’s flow field, in particular estimates for its aerodynamic drag, it is necessary for the
model and prototype to be dynamically similar, i.e. for the forces to be scale invariant. It can be
seen from Eqn (2.103) that this can only be achieved provided

Re, = Re,, (2.104)

where suffices m and p denote model and prototype respectively.
It is not usually practicable to use any other fluid but air for the model tests. For standard
wind-tunnels the air properties in the wind-tunnel are not greatly different from those experi-
enced by the prototype. Accordingly, Eqn (2.104) implies that

u, =-up (2.105)
LP
L,
Thus, if we use a 1/5-scale model, Eqn (2.105) implies that U, = 5UP. So a prototype speed of
100 km/hr (c. 30 m/s) implies a model speed of 500 km/hr (c. 150 m/s). At such a model speed
compressibility effects are no longer negligible. This illustrative example suggests that, in
practice, it is rarely possible to achieve dynamic similarity in aerodynamic model tests using
standard wind-tunnels. In fact, dynamic similarity can usually only be achieved in aerody-
namics by using very large and expensive facilities where the dynamic similarity is achieved by
compressing the air (thereby increasing its density) and using large models.
In this example we have briefly revisited the material covered in Section 1.4. The objective
was to show how the dimensional analysis of the Navier-Stokes equations (effectively the exact
governing equations of the flow field) could establish more rigorously the concepts introduced
in Section 1.4.

2.10 Exact solutions of the Navier-Stokes
equations

Few physically realizable exact solutions of the Navier-Stokes equations exist. Even
fewer are of much interest in Engineering. Here we will present the two simplest
solutions, namely Couette flow (simple shear flow) and plane Poiseuille flow (channel
flow). These are useful for engineering applications, although not for the aerody-
namics of wings and bodies. The third exact solution represents the flow in the
vicinity of a stagnation point. This is important for calculating the flow around
wings and bodies. It also illustrates a common and, at first sight, puzzling feature.
Namely, that if the dimensionless Navier-Stokes equations can be reduced to an
ordinary differential equation, this is regarded as tantamount to an exact solution.
This is because the essentials of the flow field can be represented in terms of one or
two curves plotted on a single graph. Also numerical solutions to ordinary differ-
ential equations can be obtained to any desired accuracy.

2.10.1 Couette flow - simple shear flow

This is the simplest exact solution. It corresponds to the flow field created between
two infinite, plane, parallel surfaces; the upper one moving tangentially at speed UT,
the lower one being stationary (see Fig. 2.30). Since the flow is steady and two-
dimensional, derivatives with respect to z and t are zero, and w = 0. The streamlines