SIMILARITY OF FLOWS

Another interesting aspect of the process of nondimensionalizing the equations in the previous section is that two different flows are considered to be similar if the nondimensional numbers of Eq. (1.60) are the same. For most practical cases, where gravity and unsteady effects are negligible, only the Reynolds and the Mach numbers need to be matched. A possible implementation of this principle is in water or wind-tunnel testing, where the scale of the model differs from that of the actual flow conditions.

For example, many airplanes are tested in small scale first (e. g., l/5th scale). In order to keep the Reynolds number the same then either the airspeed or the air density must be increased (e. g., by a factor of 5). This is a typical conflict that test engineers face, since increasing the airspeed 5 times will bring the Mach number to an unreasonably high range. The second alternative of reducing the kinematic viscosity v by compressing the air is possible in only a very few wind tunnels, and in most cases matching both of these nondimensional numbers is difficult.

Another possibility of applying the similarity principle is to exchange fluids between the actual and the test conditions (e. g., water with air where the ratio of kinematic viscosity is about 1:15). Thus a 1/15-scale model of a submarine can be tested in a wind tunnel at true speed conditions. Usually it is better to increase the speed in the wind tunnel and then even a smaller scale model can be tested (of course the Mach number is not always matched but for such low Mach number applications this is less critical).