Buckingham Pi-Theorem

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The Reynolds number that is developed in the Example Problem is named after Osborne Reynolds.  The Reynolds number is a non-dimensional number that is frequently encountered in fluid dynamics and is equal to the ratio of inertial to viscous forces.  

By John CollierSeanwal111111 at en.wikipedia [Public domain or Public domain], from Wikimedia Commons

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.



 

Learning Objectives:

Learn how to apply the Buckingham Π-theorem.