In the Dimensional Analysis section, methods to identify dimensionless parameters governing a particular model were presented. The discussion in this section is focused on how to relate a model to an actual prototype (i.e., the real full scale structure). In other words, it is desired to find out under what conditions will testing or experiments done on a model accurately represent or predict the actual phenomena.

Wind Tunnel Testing

In the study of fluid mechanics, models are frequently used for testing and development purposes in laboratories before a full scale prototype is built. The model can be either smaller than the prototype (e.g., design of dam, airplane and automobiles) or larger than the prototype (e.g., study of interaction between red blood cells and the vessel wall).

Take the aircraft industry for example. It is a common practice to design and develop an aircraft by testing the model in a wind tunnel. This way, the expenses of product development will be reduced considerably compared to conducting the design and development process on a full scale aircraft.

The next question would be under what conditions will the model accurately portray the actual full scale prototype. In order to yield useful information from the model being tested, the model and prototype should fulfill the requirement of geometric, kinematic and dynamic similitude, as discussed in the subsequent paragraphs.

Geometric Similitude
The first requirement is for the model and prototype to be geometrically similar. The model and prototype should have the same shape, and their dimensions should be scaled proportionally. In addition, their orientation and surface roughness should also have the same scale.

Kinematic Similitude
To obtain useful information from the model testing, the second requirement is for the model and prototype to be kinematically similar. In order to satisfy this condition, the flow conditions should be the same. In other words, the velocities and accelerations at corresponding points should have the same directions and scaled magnitudes. Model and prototype that are kinematic similar are also geometrically similar.

Dynamic Similitude
In order to achieve dynamic similitude, all forces (e.g., pressure, shear, viscous, and surface tension forces) at corresponding points of the model and prototype should have the same directions and scaled magnitudes. Model and prototype that are dynamically similar are both geometrically and kinematically similar.

Flow Past a Sphere: Geometric and
Kinematic Similitude

Flow Past a Sphere: Dynamic Similitude

Take flow past a sphere for example. A prototype and model satisfying geometric similitude are shown in the figure. The diameters of the spheres are proportional. Based on dimensional analysis, it can be determined that the drag force (D) can be modeled as,

     D/(d²ρV²) = function (Re)

where Re is the Reynolds number, and is given by

     Re = ρVd/μ

Both the model and prototype should follow the same relationship. In order to satisfy the dynamic similitude requirement, the Reynolds number and dimensionless drag for the model and prototype should be the same:

     Re_m = Re_p    and    

     D_m/(d_m² ρ_mV_m²) = D_p/(d_p² ρ_pV_p²)

where the subscripts m and p refer to the model and prototype, respectively.

From this example, it is shown that for the model and prototype to fulfill the requirement of similitude, all the corresponding pi terms should be equal. In other applications, the Froude number will be the same for both the model and prototype when free surface flow is involved. The Weber number will be the same for model and prototype when surface tension force is important.