Scale Effect

From our studies on similarity analysis in fluid mechanics, we know that, for dynamic similarity between the forces acting on an actual (or full-scale) machine and a scaled-down model used for testing (usually wind tunnel tests), the actual machine and the scale model must satisfy geometric and kinematic similar­ities. Thus, the test model and the actual machine should be geometrically similar, and if the model tests give an aerodynamic coefficient Cad, m for a test conducted at a Reynolds number Rem, the scale effect on the aerodynamic force coefficient Cad of the actual machine is given by:

Cad f (Re)

Cad, m f (Rem)

where Re is the Reynolds number of the flow around the actual machine and Rem is the Reynolds number of the flow around the model. The model tests will give aerodynamic coefficient (Cad = Cad, m) directly, if Re = Rem. If the viscosity p and density p are kept the same in the flow fields of the actual machine and its scale model, then both the flow velocity V and the characteristic length (for example, chord for an aerofoil) should be adjusted in such a way to keep Re = Rem. But the characteristic length lm for the model will be, usually, smaller than the l for the actual machine. Therefore, the test speed for the model has to be greater than the speed of the actual machine.

If there is provision to use compressed air wind tunnel, then the density p also can be increased to adjust the model Reynolds number to match the Reynolds number of the actual machine. In this kind of studies, it is essential to make a statement about the length scale used for calculating the Reynolds number.

Example 1.3

An aircraft wing profile has to be tested in a wind tunnel. If the actual wing of mean chord 1.2 m has to fly at an altitude, where the pressure and temperature are 50 kPa and 2 °C, respectively, with a speed of 250 km/h. Determine the chord of the wing model to be tested in the wind tunnel, ensuring dynamic similarity, if the test-section conditions are 90 m/s, p = 100 kPa, T = 22 °C.

Solution

Let the subscripts p and m refer to the prototype (actual) wing and the wing model to be tested in the wind tunnel, respectively.