Dimensional Analysis, Similitude & Modeling

Dimensional Analysis & Similarity:

The main purposes of dimensional analysis (To test before making any prototype) are:

• To generate non-dimensional parameters that help in the design of experiment and in the reporting of experimental results.
• To obtain scaling laws so that prototype performance can be predicted from model performance.
• To predicts trends in the relation between parameters.

Principle Of Similarity:

There are three principles of similarity which are described below:

1. Geometric Similarity:

The model must be of the same shape as that of prototype but may be scaled by some constant scale factor. Just keep in mind only dimensions are to be scaled not angle.

2. Kinematic Similarity:

The velocity at any point in the model flow must be proportional to the velocity at the corresponding point in the prototype.

3. Dynamic Similarity:

It is achieved when all forces in the model flow scale by a constant factor to the corresponding forces in the prototype flow.

Buckingham Pi Theorem:

There are two types of question arises:

1. Can we always reduce the number of variables by dimensional analysis?
2. How many dimensionless products are required to replace the original list of variables?

The answer to the following above questions is: "If an equation involving k variables is dimensionally homogeneous, it can be reduced to a relation among k-r independent dimensionless products, where 'r' is the minimum number of reference dimensions required to describe the variables".

Determination Of Pi Terms Using Buckingham Pi Theorem (Method Of Repeating Variables):

There are seven steps involve in this method of repeating variables which are described ahead:

1. List down all parameters (dependent, independent, dimensional, non-dimensional and constants) and count them. Let say 'n' number of parameters.
2. List down the primary dimensions (length, mass, time) of all parameters.
3. Guess the reduction factor 'j'. The total number of Pi terms will be k = n - j.
4. Choose repeating variables j.
5. Generate Pi terms by grouping the repeating variables with one of the remaining parameters.
6. Check all Pi terms are indeed dimensionless.
7. Write the final expression like 

Selection Of Repeating Variables:

One of the most difficult step in Buckingham Pi Theorem is the selection of repeating variables. Variables are classified as Geometric, Material or Fluid Property, Flow or External Effects.

1. Geometric Property:

The geometric characteristics ca usually be described by a series of lengths and angle.

2. Flow Property:

It includes the property of fluid to flow like velocity, acceleration and force which is doing that.

3. Fluid or Material Property:

It is the response of a system to the applied external effects. It includes modulus of elasticity, density, viscosity, weight.

Some of the final result of the last step of Buckingham Pi Theorem are:

References:

• Materials From Class Lectures + Own Knowledge + Book named Fundamentals of Fluid Mechanics by Munson, Young and Okiishi's (8th Edition).
• Photos from Google Images.
• Videos from YouTube + Google.