Page 40 - Aerodynamics for Engineering Students

Basic concepts and definitions 23

Then
K pa2 2

and V/a is the Mach number, M, of the free stream. Therefore Eqn (1.41) may be
written as
(3
F = pV2D2g - h(M) (1.42)

where g(VD/v) and h(M) are undetermined functions of the stated compound vari-
ables. Thus it can be concluded that the aerodynamic forces acting on a family of
geometrically similar bodies (the similarity including the orientation to the stream),
obey the law
F
-- (1.43)
pVD2

This relationship is sometimes known as Rayleigh‘s equation.
The term VD/v may also be written, from the definition of v, as pVD/p, as above in
the problem relating to the eddy frequency in the flow behind a circular cylinder. It is
a very important parameter in fluid flows, and is called the Reynolds number.
Now consider any parameter representing the geometry of the flow round the
bodies at any point relative to the bodies. If this parameter is expressed in a suitable
non-dimensional form, it can easily be shown by dimensional analysis that this
non-dimensional parameter is a function of the Reynolds number and the Mach
number only. If, therefore, the values of Re (a common symbol for Reynolds
number) and M are the same for a number of flows round geometrically similar
bodies, it follows that all the flows are geometrically similar in all respects, differing only in
geometric scale and/or speed. This is true even though some of the fluids may be gaseous
and the others liquid. Flows that obey these conditions are said to be dynamically similar,
and the concept of dynamic similarity is essential in wind-tunnel experiments.
It has been found, for most flows of aeronautical interest, that the effects of
compressibility can be disregarded for Mach numbers less than 0.3 to 0.5, and in
cases where this limit is not exceeded, Reynolds number may be used as the only
criterion of dynamic similarity.

Example 1.1 An aircraft and some scale models of it are tested under various conditions:
given below. Which cases are dynamically similar to the aircraft in flight, given as case (A)?

Case (A) Case (B) Case (C) Case (D) Case (E) Case (F)
span (m) 15 3 3 1.5 1.5 3
Relative density 0.533 1 3 1 10 10
Temperature (“C) -24.6 +15 +15 +15 +15 +15
Speed (TAS) (ms-’) 100 100 100 75 54 54

Case (A) represents the full-size aircraft at 6000 m. The other cases represent models under test
in various types of wind-tunnel. Cases (C), (E) and (F), where the relative density is greater
than unity, represent a special type of tunnel, the compressed-air tunnel, which may be
operated at static pressures in excess of atmospheric.