Ensuring the widespread applicability of experimental results is crucial. The concept of similitude, which involves establishing a relationship between a model and a real system, serves as a valuable tool in achieving this goal. This process is meticulously executed through the application of dimensional analysis and the Buckingham Pi Theorem during the planning and execution of experiments.
When many variables are present in an experiment, it may be difficult to carry out or determine results. The Buckingham Pi Theorem allows for the original variables to be collected into fewer, non-dimensional combinations.
For the Buckingham Pi Theorem, an equation with $k$ variables that is dimensionally homogeneous can be reduced to an equation with $k-r$ independent dimensionless variables. $r$ is the minimum number of reference dimensions required to describe the variables. This is a 7-step process.
Step 1 involves compiling a comprehensive list of all dimensionless variables relevant to the problem at hand.
Express each of the variables from the previous step in basic dimensional terms, either using the MLT system or FLT system.
Knowing $k$ = number of variables and $r$ = number of fundamental dimensions, use $k-r$ to find the required number of pi terms.
Step 4 requires the selection of a set of $r$ independent dimensional variables. This set of ‘repeating variables’ must include all the primary dimensions. Dimensionless variables and the dependent variable must not be a part of this set. Likewise, all the required basic dimensions must be present in this group (MLT or FLT).
$$ \Pi_i=u_iu_1^au_2^bu_3^c $$
To determine each pi term, multiply each nonrepeating variable by the product of each of the repeating variables, each of which raised to a variable exponent that will be solved for. To solve, each pi term must be dimensionless.
This step is used to check that the pi terms calculated in Step 5 are dimensionless.
The final step expresses the final form of the relationship in terms of the pi terms, as seen below:
$$ \Pi_1=\phi(\Pi_2,\Pi_3,...,\Pi_{k-r}) $$