Dimensional Analysis | Civil Engineering SSC JE (Technical) - Civil Engineering (CE) PDF Download

 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 

Number	Equation	Significance
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 :

(F_r) m = (F_r) 

(i) Time ratio
T_r = L_r^1/2 g_r^–1/2 = L_r^1/2

(ii) Acceleration ratio a_r = g_r = 1

(iii) Force ratio

(iv) Power ratio

(v) Discharge ratio

Applications : 

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