# Dimensional Analysis and its Application Notes

## Dimensional Analysis Definition

Contents
• All engineering quantities can be defined in four basic or fundamental dimensions- Mass (M), Length (L), Time (T), and Temperature (θ).
• Dimensional Analysis is the analysis of the relationships between different physical quantities by identifying their fundamental dimensions and units of measure.
• It is also known as the factor-label method or unit-factor method.
• Total no. of variables influencing the problem is equal to the no. of independent variables plus one, one being the no. of the dependent variable.

### Important Quantities Dimension

Velocity potential = [L2 T-1]

Stream function = [L2 T-1]

Acceleration = [LT-2]

Vorticity = [T-1]

Homework- Try finding the dimension of dynamic viscosity?

### Buckingham p theorem

• It states that if all the n-variable are described by m fundamental dimensions, they may be grouped into (n –m) dimensionless p terms.

Ex- Simple pendulum Case- We wish to find its time period (T), then we find there are three other quantities involved (variables) i.e. length, mass, and gravity.

So, n= 4 (time period, length, mass, gravity)

And m = 3 (Mass (M), Length (L), Time (T))

Hence, no. of dimensionless numbers = n-m = 3 – 2 = 1

### Model Testing

• In order to predict the performance of the real thing, we test a model.
• Geometric similarity – Similarity of shape
• Kinematic similarity – Similarity of motion
• Dynamic similarity – Similarity of forces
 Number Equations Significance Reynolds Number Fi/FV = ρVL/µ Flow in closed conduit pipe Froude Number (Fi/Fg)1/2 = V/(gL)1/2 a free surface is present and gravity force is predominant Euler’s Number (Fi/Fp)1/2 = V/(p/ρ)1/2 In cavitation studies Mach Number (Fi/Fe)1/2 = V/C Where fluid compressibility is important Weber Number (Fi/Fs)1/2 = v/(σ/ρL) In capillary studies

Where

Fv = Viscous Force

Fi = Inertia Force

Fe = Elastic Force

Fp = Pressure Force

Fs = Surface Tension Force

### Applications of Reynolds’s Model Law

• Flow-through small-sized pipes.
• Low-velocity motion around automobiles and airplanes.
• Submarines completely underwater.
• Flow-through low-speed turbomachines.

Also Read: Basic Properties of Fluids notes

### Applications of Froude’s Model Law

• Open channels.
• Spillways.
• Liquid jets from an orifice.
• Notches and weirs.
• Ship partially submerged in the rough & turbulent sea.

### Applications of Euler Model Law

• Water hammer phenomenon
• Phenomenon of cavitation
• Fully turbulent flow in case closed pipe
• Models of aerofoil’s fan blades.
• Pressure distribution on a ship.

### Applications of Mach Model Law

The law in which models are based on Mach number.

• The flow of airplanes of supersonic speed
• Underwater testing of torpedoes.
• Aerodynamic testing
• The flow of missiles, rockets.
• Water hammer problem.

### Applications of Weber Model Law

When surface tension effects predominate in additional inertia force the pertinent similitude law is obtained by equating the Weber number for its model and prototype.

• Capillary waves in the channel.
• A very thin sheet of liquid flowing over a surface.