The forces on a body, moving through a fluid, depend on the body velocity (V), the fluid density (ρ), temperature, and viscosity (μ), the size of the body (l), and its shape. Using the speed of sound, a, rather than temperature (they are directly related) we can then make the following table that shows the units associated with each of the parameters. Here the numbers indicate the power to which the mass, length, or time units are raised:

F | = | f | (V, | ρ, | a, | µ, | l, | shape) |

mass | 1 | 0 | 1 | 0 | 1 | 0 | 0 | |

length | 1 | 1 | -3 | 1 | -1 | 1 | 0 | |

time | -2 | -1 | 0 | -1 | -1 | 0 | 0 |

F / (ρ V^{2} l^{2}) |
ρ V l / μ | V / a | shape |

Dimensionless Force Coefficient | Reynolds Number | Mach Number | Geometry |

The relation is: F / (ρ V^{2} l^{2}) = f (ρVl/μ, V/a, shape)

We will discuss each of these dimensionless groups in a moment, but let's
first look at the functional relationship between them. Much of applied
aerodynamics involves finding the function f, but there is a great deal
we can say, even without knowing it. For example, we can see that a wide
variety of similar flows exist. The forces on a large, slow-moving body
could be predicted from tests of a small higher-speed model as long as the
speed of sound were sufficiently high. Also, the flow around a small insect
could be represented by a large model in a very viscous fluid. The idea
behind model testing is to simulate the flow over one body by matching the
dimensionless parameters of another.

This is not always easy -- or possible. The following figure, from J. McMasters
of Boeing shows the Mach
and Reynolds number range of several wind tunnels. Why can't wind tunnels
be designed to more fully cover this range of parameters? What alternatives
exist to wind tunnel tests? (See assignments.)

Subsequent pages consider each of the dimensionless groups in a bit more detail.
First note that we could have included other fluid properties such as specific
heats. This would lead to additional dimensionless parameters such as the
Prandtl number which is important in the study of compressible boundary
layers with heat conduction. We have also left out gravity which is often
important in the flow of water around ships. This would lead to an additional
dimensionless parameter called the Froude number. There are often several
ways of combining the parameters to form dimensionless groups, but these
are commonly used in aerodynamics.