Dimensional Analysis in Fluid Mechanics Click here for discussion of similitude.

Key Concept: All terms in equations describing the response of phenomena must each have the

same dimensions (called the principle of dimensional homogeneity). It is possible to group the

variables appearing in these equations into a reduced set of dimensionless products called Pi terms.

These dimensionless products can also be used to extrapolate performance between models and

prototypes (similitude) leading to design decisions of the prototype based on laboratory models.

In a Nut Shell: By the Buckingham Pi Theorem the number of Pi terms (dimensionless products)

equals the number of variables minus the number of dimensions describing the problem.

To start identify all the pertinent variables describing the problem. This is a critical first step. The

idea is to select the fewest number of relevant variables and to then form the dimensionless products.

If important variable are excluded, then incorrect results will be obtained. If extraneous variables are included, then too many Pi terms appear in the final solution and are difficult to eliminate. Application

of dimensional analysis applies to any field of engineering, not just fluid mechanics.

Strategy in performing Dimensional Analysis using Method of Repeating Variables

1.

Identify variables governing the problem. (Critical and essential first step)

2.

List dimensions for each variable. You have a choice between sets of dimensions:

F, L, T

or

M, L, T

F = force, M = mass, L = length, T = Time

3.

Total number of dimensionless products (Pi terms) =

number of variables – number of dimensions

4.

Select the dependent variable, the non- repeating variables, and the repeating variables.

5.

Identify and use the repeating variables to form dimensionless products of the dependent variable and non-repeating variables. Each repeating variable will have one dimension associated with it and will be used to eliminate that dimension from the dependent and

non-repeating variables. Click here for strategies to calculate Pi Terms.

6.

Form the Pi terms by inspection (appropriate multiplication and division of variables)

Note: Since Pi terms are dimensionless the products of pi terms are also dimensionless.

i.e. If you have Pi₁ and Pi₂, then Pi₃ = Pi₁ x Pi₂

Click here to continue with discussion.

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