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**Chapter 8**

**DIMENSIONAL ANALYSIS**

**Velocity potential**= [L^{2 }T^{–1}]**Stream function**= [L^{2}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. Flow in closed conduit pipe Froude No. where a free surface is present, structure |

__ 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) (F _{r})_{prototype} = (F_{r})**

V

It the place of model and prototype is same, then g

V

T

a

Q

F

If the fluid used in model and prototype is same, then

F

P

**Applications :**

- Open channels
- Notches & weirs
- Spill ways & dams
- Liquid jets from orifice
- Ship partially submerged in rough & turbulent sea

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