Dimensionless Groups

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
The Buckingham pi theorem states that the number of dimensionless parameters is equal to the number of parameters minus the rank of the above matrix. In this case 7 - 3 = 4. So, there exists a functional relationship among the four dimensionless groups. We can express the force on a body, for instance, by a relationship between the following four dimensionless parameters:

F / (ρ V2 l2) ρ V l / μ V / ashape
Dimensionless Force Coefficient Reynolds Number Mach NumberGeometry

The relation is: F / (ρ V2 l2) = 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.