The Rayleigh Method demonstrated in the previous section works only if the number of exponents equals the number of basic dimensions. If the number of exponents exceeds the number of basic dimensions it is not possible to solve the exponent equations.
Fortunately, the Buckingham Π Theorem provides a procedure for reducing the number of variables to a fewer number of dimensionless variables or Π-groups. According to the Buckingham Π Theorem, the number of Π-groups (dimensionless quantities) is equal to (n-k) where n is the number of independent parameters involved and k is the number of basic dimensions involved. The functional relationship between the Π-groups can be written as
Finding the (n-k) Π-groups can be accomplished by following a series of steps.
Step 1 – Identify Parameters
The first step is to identify all independent parameters for the system. These parameters generally include fluid properties (e.g., density, viscosity and surface tension), system geometry (e.g., length, area and volume) or flow conditions (e.g., velocity, pressure change and applied force).
Step 2 – Identify Basic Dimensions Involved
The second step is to determine the number of basic dimensions involved.
Step 3 – Determine the Number of Π-Groups
The next step is to determine the number of dimensionless parameters (denoted by Π) using the Buckingham P theorem.
Step 4 – Select Repeating Parameters
From the list of parameters determined in Step 1, select k number of repeating parameters. These repeating parameters must include all the basic dimensions, but they cannot be dimensionless nor have the same basic dimensions.
Step 5 – Write and Solve Exponent Equations
The -Groups are formed by multiplying the remaining parameters by the parameters chosen in Step 4 raised to an exponent.
Step 6 – Write Functional Relationship
The final step is simply writing the functional relationship between the Π-Groups.
Learn how to apply the Buckingham Π-theorem.