# Dimensional Analysis

## dimension_analysis.jpg

The units of measure associated with everything from distance to fluid properties are important to the field of dimensional analysis.

By Clément Bucco-Lechat (Own work) [CC-BY-SA-3.0], via Wikimedia Commons

The flow of fluids is often three-dimensional in nature and controlled by turbulent conditions. Exact solutions of the three-dimensional equations under these conditions are not possible and approximate or experimental methods are used.

Dimensional analysis is an important method used in fluid dynamics and other fields that can minimize the time and expense spent on experiments.  It also provides valuable insight into functional relationships between variables.  This insight helps gain information from the fewest number of experiments possible.

Dimensional analysis seeks to find the relationships between variables simply by making sure that consistent units or dimensions are used.  Hence, it is important that the basic dimensions used with variables encountered in fluid dynamics be understood.  Once the basic dimensions are understood, it is possible to develop systematic methods for identifying functional relationships and dimensionless parameters associated with specific fluid flow phenomena.  One of these methods is known as the Buckingham P-Method.

## Learning Objectives:

Learn reasons for using dimensional analysis.

#### Homogeneous Equations

All components of an equation that add together must have the same units. For example, before a force of 10 lb can be added to a force of 10 Newton, the forces must be converted to a single force unit.

#### Functional Relationships

The form of equations can often be deduced from an analysis of the dimensions of the variables or parameters that affect the solution to a problem.

#### Buckingham Pi-Theorem

Fortunately, the Buckingham  Theorem provides a procedure for reducing the number of variables to a fewer number of dimensionless variables or -groups.