Dimensional Analysis?

  • 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 = [L2 T-1]

Stream function = [L2 T-1]

Acceleration = [LT-2]

Vorticity = [T-1]

Q- Try finding the dimension of dynamic viscosity?

Buckingham π theorem

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

Equation

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/Fσ)1/2 = v/(σ/ρL)

In capillary studies

Where Fi = Inertia Force

Fv = Viscous Force

Fe = Elastic Force

Fp = Pressure Force

Fσ = Surface Tension Force

Applications of Reynolds’s Model Law

  • Flow through small sized pipes.
  • Low-velocity motion around automobiles and aeroplanes.
  • Submarines completely under water.
  • Flow through low-speed turbo machines.

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.

  • Flow of aeroplanes 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.
  • Very thin sheet of liquid flowing over a surface.