Overview

Many practical flow problems of different nature can be solved by using equations and analytical procedures, as discussed in the previous modules. However, solutions of some real flow problems depend heavily on experimental data and the refinements in the analysis are made, based on the measurements. Sometimes, the experimental work in the laboratory is not only time-consuming, but also expensive. So, the dimensional analysis is an important tool that helps in correlating analytical results with experimental data for such unknown flow problems. Also, some dimensionless parameters and scaling laws can be framed in order to predict the prototype behavior from the measurements on the model. The important terms used in this module may be defined as below;

Dimensional Analysis: The systematic procedure of identifying the variables in a physical phenomena and correlating them to form a set of dimensionless group is known as dimensional analysis.

Dimensional Homogeneity: If an equation truly expresses a proper relationship among variables in a physical process, then it will be dimensionally homogeneous. The equations are correct for any system of units and consequently each group of terms in the equation must have the same dimensional representation. This is also known as the law of dimensional homogeneity.

Dimensional variables: These are the quantities, which actually vary during a given case and can be plotted against each other.

Dimensional constants: These are normally held constant during a given run. But, they may vary from case to case.

Pure constants: They have no dimensions, but, while performing the mathematical manipulation, they can arise.

Let us explain these terms from the following examples:

- Displacement of a free falling body is given as, , where, is the initial velocity, g is the acceleration due to gravity, t is the time, are the final and initial distances, respectively. Each term in this equation has the dimension of length and hence it is dimensionally homogeneous. Here, are the dimensional variables, are the dimensional constants and arises due to mathematical manipulation and is the pure constant .

- Bernoulli's equation for incompressible flow is written as, . Here, p is the pressure, V is the velocity, z is the distance, ρ is the density and g is the acceleration due to gravity. In this case, the dimensional variables are , the dimensional constants are and is the pure constant. Each term in this equation including the constant has dimension of and hence it is dimensionally homogeneous.