Similarity Principles and Dimensional Analysis

An experiment that is carried out in a laboratory under reference conditions will invariably be a scaled version of the prototype for which a mathematical model is to be developed. Data obtained from scaled experiments will be useful for the prototype only if certain conditions (called similarity principles) are satisfied. These principles have been developed for systems operating under deterministic conditions, presumably under continuous operation. These cannot be readily extended to stochastic problems, boundary-layer transition, for example. When similarity principles are followed while setting up the experiment, the data recorded in the experiment will be distortion-free . The measured data can then be linearly scaled from the experiment to the real-life process.

The principles of similarity that relate the experimental setup with the prototype are stated as:

1. Geometric similarity should be observed.

It requires that the experiments preserve the shape of the prototype, while the linear dimensions be scaled in proportion.

2. Dynamic similarity should be enforced.

It is a statement that the ratio of the relevant forces in the prototype be preserved in the model as well. This rule leads to the matching of certain dimensionless parameters.

3. Kinematic similarity should be realized.

It requires that the constant flux lines (streamlines, for example) in the experiment match those in the prototype. This requirement is most appropriate for steady state patterns since constant flux lines in unsteady problems may not be physically meaningful. If geometric and dynamic similarity are realized, kinematic similarity will be automatically satisfied.