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

The document Chapter 8 Dimensional Analysis - Fluid Mechanics, Mechanical Engineering Mechanical Engineering Notes | EduRev is a part of the Mechanical Engineering Course Mechanical Engineering SSC JE (Technical).
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Chapter 8

DIMENSIONAL ANALYSIS

  • Velocity potential = [LT–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.                         Chapter 8 Dimensional Analysis - Fluid Mechanics, Mechanical Engineering Mechanical Engineering Notes | EduRev                Flow in closed conduit pipe

Froude No.                        Chapter 8 Dimensional Analysis - Fluid Mechanics, Mechanical Engineering Mechanical Engineering Notes | EduRev                where a free surface is present, structure
Eulers No.                         Chapter 8 Dimensional Analysis - Fluid Mechanics, Mechanical Engineering Mechanical Engineering Notes | EduRev                  In cavitation studies.
Mach No.                           Chapter 8 Dimensional Analysis - Fluid Mechanics, Mechanical Engineering Mechanical Engineering Notes | EduRev                    where fluid compressibility is important.
Weber No.                         Chapter 8 Dimensional Analysis - Fluid Mechanics, Mechanical Engineering Mechanical Engineering Notes | EduRev             In capillary studies.

 

 Reynolds Model Law :

Chapter 8 Dimensional Analysis - Fluid Mechanics, Mechanical Engineering Mechanical Engineering Notes | EduRev

(i) Velocity ratio

Chapter 8 Dimensional Analysis - Fluid Mechanics, Mechanical Engineering Mechanical Engineering Notes | EduRev

(ii) Time ratio

Chapter 8 Dimensional Analysis - Fluid Mechanics, Mechanical Engineering Mechanical Engineering Notes | EduRev

(iii) Acceleration ratio,

Chapter 8 Dimensional Analysis - Fluid Mechanics, Mechanical Engineering Mechanical Engineering Notes | EduRev

(iv) Force ratio

Chapter 8 Dimensional Analysis - Fluid Mechanics, Mechanical Engineering Mechanical Engineering Notes | EduRev

(v) Power ratio

Chapter 8 Dimensional Analysis - Fluid Mechanics, Mechanical Engineering Mechanical Engineering Notes | EduRev

(vi) Discharge ratio

Chapter 8 Dimensional Analysis - Fluid Mechanics, Mechanical Engineering Mechanical Engineering Notes | EduRev

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 / √ gL= Vm / √ gm Lp
It the place of model and prototype is same, then gm = gp
V =  √Lr
(ii) Time scale ratio
Tr  =  √ Lr
(iii) Acceleration scale ratio
ar  = 1
(iv) Discharge scale ratio
Qr  = Lr5/2
(v) Force scale ratio
F = ρp / ρm x (L/ Lm)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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