Dynamic Similarity Pi Theorem

Assume, a physical phenomenon is described by m number of independent variables like x₁, x₂, x₃, ..., x_m

The phenomenon may be expressed analytically by an implicit functional relationship of the controlling variables as
 
ƒ (x₁, x₂, x₃, ......., x_m) = 0              (1)
         
Now if n be the number of fundamental dimensions like mass, length, time, temperature etc. involved in these m variables, then according to Buckingham's p theorem.

The phenomenon can be described in terms of (m – n) independent dimensionless groups like π₁,π₂, ..., π_m–n, where p terms, represent the dimensionless parameters and consist of different combinations of a number of dimensional variables out of the m independent variables defining the problem.

Therefore. the analytical version of the phenomenon given by Eq. (1) can be reduced to

F(π₁, π₂,...., π_m-n) = 0                   (2)

According to Buckingham's pi theorem

This physically implies that the phenomenon which is basically described by m independent dimensional variables is ultimately controlled by (m–n) independent dimensionless parameters known as π terms.
 
Alternative Mathematical Description of (π) Pi Theorem

A physical problem described by m number of variables involving n number of fundamental dimensions (n < m) leads to a system of n linear algebraic equations with m variables of the form
     

 
 
Determination of π terms

A group of n (n = number of fundamental dimensions) variables out of m (m = total number of independent variables defining the problem) variables is first chosen to form a basis so that all n dimensions are represented. These n variables are referred to as repeating variables.

Then the p terms are formed by the product of these repeating variables raised to arbitrary unknown integer exponents and anyone of the excluded (m – n) variables.

For example, if x₁ x₂ ...x_n are taken as the repeating variables. Then
 

 
• The sets of integer exponents’ a₁, a₂ . . . a_n are different for each p term.

• Since p terms are dimensionless, it requires that when all the variables in any p term are expressed in terms of their fundamental dimensions, the exponent of all the fundamental dimensions must be zero.

• This leads to a system of n linear equations in a, a₂ . . . a_n which gives a unique solution for the exponents. This gives the values of a₁ a₂ . . . a_n for each p term  and hence the p terms are uniquely defined.

In selecting the repeating variables, the following points have to be considered:

1. The repeating variables must include among them all the n fundamental dimensions, not necessarily in each one but collectively.

2. The dependent variable or the output parameter of the physical phenomenon should not be included in the repeating variables.

No physical phenomena is represented when –

• m < n; because there is no solution, and

• m = n; because there is a unique solution of the variables involved and hence all the parameters have fixed values.

Therefore all feasible phenomena are defined with m > n .

When m = n + 1, then, according to the Pi theorem, the number of pi term is one and the phenomenon can be expressed as

ƒ(π₁) = 0
     
Where, the non-dimensional term π₁ is some specific combination of n + 1 variables involved in the problem.

When m > n+ 1,

1. The numbers of π terms are more than one.

2. A number of choices regarding the repeating variables arise in this case.

Again, it is true that if one of the repeating variables is changed, it results in a different set of π terms. Therefore the interesting question is which set of repeating variables is to be chosen, to arrive at the correct set of π terms to describe the problem. The answer to this question lies in the fact that different sets of π terms resulting from the use of different sets of repeating variables are not independent. Thus, anyone of such interdependent sets is meaningful in describing the same physical phenomenon.

From any set of such π terms, one can obtain the other meaningful sets from some combination of the π terms of the existing set without altering their total numbers (m–n) as fixed by the Pi theorem.

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