Page 111 - Aerodynamics for Engineering Students

94 Aerodynamics for Engineering Students

becomes important (see Section 2.3.4), Mach number becomes a second dimension-
less quantity characterizing the flow field.
The Navier-Stokes equations are deceptively simple in form, but at high Reynolds
numbers the resulting flow fields can be exceedingly complex even for simple geo-
metries. This is basically a consequence of the behaviour of the regions of vortical
flow at high Reynolds number. Vorticity can only be created in a viscous flow and
can be regarded as a marker for regions where the effects of viscosity are important in
some sense.
For engineering applications of aerodynamics the Reynolds numbers are very
large, values well in excess of lo6 are commonplace. Accordingly, one would expect
that to a good approximation one could drop the viscous terms on the right-hand
side of the dimensionless Navier-Stokes Eqns (2.100). In general, however, this view
would be mistaken and one never achieves a flow field similar to the inviscid one no
matter how high the Reynolds number. The reason is that the regions of non-zero
vorticity where viscous effects cannot be neglected become confined to exceedingly
thin boundary layers adjacent to the body surface. As Re + oc the boundary-layer
thickness, 6 + 0. If the boundary layers remained attached to the surface they would
have little effect beyond giving rise to skin-friction drag. But in all real flows the
boundary layers separate from the surface of the body, either because of the effects of
an adverse pressure gradient or because they reach the rear of the body or its trailing
edge. When these thin regions of vortical flow separate they form complex unsteady
vortex-like structures in the wake. These take their most extreme form in turbulent
flow which is characterized by vortical structures with a wide range of length and
time scales.
As we have seen from the discussion given above, it is not necessary to solve the
Navier-Stokes equations in order to obtain useful information from them. This is
also illustrated by following example:
Example 2.3 Aerodynamic modelling
Let us suppose that we are interested carrying out tests on a model in a wind-tunnel in order to
study and determine the aerodynamic forces exerted on a motor vehicle travelling at normal
motorway speeds. In this case the speeds are sufficiently low to ensure that the effects of
compressibility are negligible. Thus for a fixed geometry the flow field will be characterized
only by Reynolds number.* In this case we can use U,, the speed at which the vehicle travels
(the air speed in the wind-tunnel working section for the model) as the reference flow speed,
and L can be the width or length of the vehicle. So the Reynolds number Re pU,L/p. For
a fixed geometry it is clear from Eqns (2.99) and (2.100) that the non-dimensional flow
variables, U, V, W, and P are functions only of the dimensionless coordinates X: Y, 2: T,
and the dimensionless quantity, Re. In a steady flow the aerodynamic force, being an overall
characteristic of the flow field, will not depend on X, Y, Z, or T. It will, in fact, depend only
on Re. Thus if we make an aerodynamic force, drag (0) say, dimensionless, by introducing
a force (i.e. drag) coefficient defined as

(2.102)

(see Section 1.5.2 and noting that here we have used Lz in place of area S) it should be clear
that
Co = F(Re) i.e. a function of Re only (2.103)

* In fact, this statement is somewhat of an over-simplification. Technically the turbulence characteristics of
the oncoming flow also influence the details of the flow field.