Chapter 8 Dimensional Analysis - Fluid Mechanics, Mechanical Engineering Mechanical Engineering Notes | EduRev
Mechanical Engineering SSC JE (Technical)
Mechanical Engineering : Chapter 8 Dimensional Analysis - Fluid Mechanics, Mechanical Engineering Mechanical Engineering Notes | EduRev
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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 π 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
| Number Equation Significance |
Reynolds No. Froude No. |
Flow in closed conduit pipe
where a free surface is present, structure
In cavitation studies.
where fluid compressibility is important.
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 :
(i) (Fr)prototype = (Fr)model
Vp / √ gp Lp = Vm / √ gm Lp
It the place of model and prototype is same, then gm = gp
Vr = √Lr
(ii) Time scale ratio
Tr = √ Lr
(iii) Acceleration scale ratio
ar = 1
(iv) Discharge scale ratio
Qr = Lr5/2
(v) Force scale ratio
Fr = ρp / ρm x (Lp / Lm)2 x (Vp / Vm)2
If the fluid used in model and prototype is same, then
Fr = Lr3
(vi) Pressure scale ratio
Pr = Lr
Applications :
- Open channels
- Notches & weirs
- Spill ways & dams
- Liquid jets from orifice
- Ship partially submerged in rough & turbulent sea
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