Dimensional Analysis and its Application Notes

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Dimensional Analysis Definition

  • 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

Read Also: Non-Newtonian Fluids

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
NumberEquationsSignificance
Reynolds NumberFi/FV = ρVL/µFlow in closed conduit pipe
Froude Number(Fi/Fg)1/2 = V/(gL)1/2a free surface is present and gravity force is predominant
Euler’s Number(Fi/Fp)1/2 = V/(p/ρ)1/2In cavitation studies
Mach Number(Fi/Fe)1/2 = V/CWhere 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.

Click Here For: Fluid Mechanics: Objective Questions

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