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Model Laws or Similarity Laws | Irrigation Engineering - Civil Engineering (CE) PDF Download
For dynamic similarity ratio of corresponding forces at the corresponding points should be same. This means that the various dimensionless numbers mentioned above should be same. Based on these numbers the model or similarity laws are:
- Reynolds model law
- Froude model law
- Euler model law
- Weber model law
- Mach model law
Reynolds Model Law
When the predominant force is viscous force, Reynolds number should be same for model and prototype.
∴ (Re)m = (Re)p
or
or
or
Time scale ratio,
Acceleration scale ratio, ar = Vr/Tr
Discharge scale ratio,
Qr = ρrArVr = ρrLr2Vr
Force scale ratio Fr = mr × ar
where quantities with subscript r represent the corresponding scale ratios.
Areas where Reynolds model law can be applied are,
- Motions of air planes
- Flow of incompressible fluids in closed pipes
- Motion of Submarines completely under water, etc.
Froude Model Law
Froude model law is applied for comparison of model and prototype where the predominant force is the gravity force.
According to this,
When, gm = gp,
or
Now,
If same fluid is used, Fr = Lr3
Areas of application are:
- Flow over spill ways, sluices, etc.
- Flow of jet from nozzle or orifice
- Flow through channels
- Water bodies where surface waves are formed
Euler Model Law
When pressure forces are dominant forces, Euler’s number is the criterion for comparison. According to this,
If same fluid is used,
Areas of applications are:
- Enclosed fluid systems, where turbulence is fully developed and viscous forces are negligible.
- Where cavitation is present
- Discharge through orifices, sluices, etc.
Weber Model Law
Weber model law based on Weber’s number is used when surface tension effects predominate. According to this,
(We)m = (We)p
or
Areas of application are:
- Capillary movement of water in soils
- Flow over weirs at very small heads
- Liquid spraying systems
- Capillary rise in narrow passages, etc.
Mach Model Law
When elastic forces are predominant Mach model law is applied. According to this,
Areas of application are:
- Aerodynamic testing
- Flow of gases with supersonic velocities
- Unsteady flows with water hammer problems.
- Testing of torpedoes under water, etc.
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