**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.

**Dimensions of few important quantities-**

Velocity potential = [L^{2} T^{-1}]

Stream function = [L^{2} T^{-1}]

Acceleration = [LT^{-2}]

Vorticity = [T^{-1}]

**Q- Try finding the dimension of dynamic viscosity?**

**Buckingham p theorem**

- It states that if all the
are described by**n-variable**, they may be grouped into**m fundamental dimensions****(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 | F_{i}/F_{V} = ρVL/µ |
Flow in closed conduit pipe |

Froude Number | (F_{i}/F_{g})^{1/2} = V/(gL)^{1/2} |
a free surface is present and gravity force is predominant |

Euler’s Number | (F_{i}/F_{p})^{1/2} = V/(p/ρ)^{1/2} |
In cavitation studies |

Mach Number | (F_{i}/F_{e})^{1/2} = V/C |
Where fluid compressibility is important |

Weber Number | (F_{i}/F_{s})^{1/2} = v/(σ/ρL) |
In capillary studies |

Where

F_{v} = Viscous Force

F_{i} = Inertia Force

F_{e} = Elastic Force

F_{p} = Pressure Force

F_{s} = 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.

**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.