Page 39 - Aerodynamics for Engineering Students

22 Aerodynamics for Engineering Students

1.4.2 Dimensional analysis applied to aerodynamic force
In discussing aerodynamic force it is necessary to know how the dependent variables, aero-
dynamic force and moment, vary with the independent variables thought to be relevant.
Assume, then, that the aerodynamic force, or one of its components, is denoted by
F and when fully immersed depends on the following quantities: fluid density p, fluid
kinematic viscosity v, stream speed V, and fluid bulk elasticity K. The force and
moment will also depend on the shape and size of the body, and its orientation to the
stream. If, however, attention is confined to geometrically similar bodies, e.g.
spheres, or models of a given aeroplane to different scales, the effects of shape as
such will be eliminated, and the size of the body can be represented by a single typical
dimension; e.g. the sphere diameter, or the wing span of the model aeroplane,
denoted by D. Then, following the method above

(1.39)

In dimensional form this becomes

Equating indices of mass, length and time separately leads to the three equations:
(Mass) l=c+e (1.40a)

(Length) 1 =a+b-3c+2d-e (1.40b)
(Time) - -2 = -a - d -2e (1.40~)

With five unknowns and three equations it is impossible to determine completely all
unknowns, and two must be left undetermined. These will be d and e. The variables
whose indices are solved here represent the most important characteristic of the body
(the diameter), the most important characteristic of the fluid (the density), and the
speed. These variables are known as repeated variables because they appear in each
dimensionless group formed.
The Eqns (1.40) may then be solved for a, b and c in terms of d and e giving

a = 2 -d - 2e
b=2-d
c=l-e

Substituting these in Eqn (1.39) gives
F = v2-d-2eg2-d 1-e d e
P vK
K
= pV2D2 (&r (-) e (1.41)
P V2
The speed of sound is given by Eqns (1.6b,d) namely,
d=-=- K
7P
P P