Chapter 8 Dimensional Analysis
- Velocity potential = [L2 T–1]
Stream function = [L2 T–1]
Acceleration = [LT–2]
Vorticity = [T–1]
- Total no. of variables influencing the problem is equal to the no. of independent variables plus one, one being the no. of dependent variable.
- Buckingham p theorem states that if all the n-variable are described by m fundamental dimensions, they may be grouped into (n - m) dimensions p terms.
- Selection of 3 repeating variables from the geometry of flow, fluid properties and fluid motion.
- Geometric similarity - similarity of shape
Kinematic similarity - similarity of motion
dynamic similarity - similarity of forces
|Reynolds No.||Flow in closed conduit pipe|
|Froude No||where a free surface is present,structure eg. weirs spillway, channels, etc. w here gravity force is predominant.|
|Eulers No.||In cavitation studies.|
|Mach No.||where fluid compressibility is important.|
|Weber No|| ||In capillary studies.|
Reynolds Model Law :
(i) Velocity ratio
(ii) Time ratio
(iii) Acceleration ratio,
(iv) Force ratio
(v) Power ratio
(vi) Discharge ratio
Applications of Reynold’s Model Law :-
- Flow through small sized pipes
- Low velocity motion around automobiles and aeroplane.
- Submarines completely under water.
- Flow through low speed trubo machines.
- Froude’s Model law :
(Fr) m = (Fr)
(i) Time ratio
Tr = Lr1/2 gr–1/2 = Lr1/2
(ii) Acceleration ratio ar = gr = 1
(iii) Force ratio
(iv) Power ratio
(v) Discharge ratio
- Open channels
- Notches & weirs
- Spill ways & dams
- Liquid jets from orifice
- Ship partially submerged in rough & turbulent sea