Determination of pi Terms

Several methods can be used to form dimensionless products or pi terms that arise in dimensional analysis. But, there is a systematic procedure called method of repeating variables that allows in deciding the dimensionless and independent pi terms . For a given problem, following distinct steps are followed.

Step I: List out all the variables that are involved in the problem. The ‘variable' is any quantity including dimensional and non-dimensional constants in a physical situation under investigation. Typically, these variables are those that are necessary to describe the “geometry” of the system (diameter, length etc.), to define fluid properties (density, viscosity etc.) and to indicate the external effects influencing the system (force, pressure etc.). All the variables must be independent in nature so as to minimize the number of variables required to describe the complete system.

Step II: Express each variable in terms of basic dimensions. Typically, for fluid mechanics problems, the basic dimensions will be either or . Dimensionally, these two sets are related through Newton 's second law so that e.g. or . It should be noted that these basic dimensions should not be mixed.

Step III: Decide the required number of pi terms . It can be determined by using Buckingham pi theorem which indicates that the number of pi terms is equal to , where k is the number of variables in the problem (determined from Step I) and r is the number of reference dimensions required to describe these variables (determined from Step II).

Step IV: Amongst the original list of variables, select those variables that can be combined to form pi terms . These are called as repeating variables . The required number of repeating variables is equal to the number of reference dimensions . Each repeating variable must be dimensionally independent of the others, i.e. they cannot be combined themselves to form any dimensionless product. Since there is a possibility of repeating variables to appear in more than one pi term , so dependent variables should not be chosen as one of the repeating variable.

Step V: Essentially, the pi terms are formed by multiplying one of the non-repeating variables by the product of the repeating variables each raised to an exponent that will make the combination dimensionless. It usually takes the form of where the exponents a, b and c are determined so that the combination is dimensionless.

Step VI : Repeat the ‘Step V' for each of the remaining non-repeating variables. The resulting set of pi terms will correspond to the required number obtained from Step III.

Step VII: After obtaining the required number of pi terms , m ake sure that all the pi terms are dimensionless. It can be checked by simply substituting the basic dimension of the variables into the pi terms .

Step VIII: Typically, the final form of relationship among the pi terms can be written in the form of Eq. (6.1.2) where, would contain the dependent variable in the numerator. The actual functional relationship among pi terms is determined from experiment.