DIMENSIONLESS NUMBERS AND A PROTOTYPE

 

https://www.mesubjects.net/wp-admin/post.php?post=7698&action=edit       FM introduction-1

https:// www.mesubjects.net/wp-admin/post.php?post=10636&action=edit      FM introduction-2

https://www.mesubjects.net/wp-admin/post.php?post=3528&action=edit      Types of fluid flow-2

https://www.mesubjects.net/wp-admin/post.php?post=3518&action=edit      Types of fluid flow-1

https://www.mesubjects.net/wp-admin/post.php?post=3419&action=edit       Fluid properties & forces

https://www.mesubjects.net/wp-admin/post.php?post=7515&action=edit       Drag

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https://www.mesubjects.net/wp-admin/post.php?post=4814&action=edit        Pressure Drop-2

https://www.mesubjects.net/wp-admin/post.php?post=2638&action=edit        Pressure Drop-1

https://www.mesubjects.net/wp-admin/post.php?post=6099&action=edit          Q. Ans. Turbulent flow

DIMENSIONLESS NUMBERS

AND A PROTOTYPE

Dimensionless numbers do not have

any units. Reynolds number is used

to know laminar, turbulent or transition

flow. In every dimensionless number,

Numerator is always inertia force and

Denominator is some other force.

         These numbers are used in the development of a model of a prototype. A prototype means an actual object. A prototype can be a big object like an airplane, a dam and a turbine. A model in case of a big object is  small. This model is made and tested for geometric, kinematic and dynamic similarity in terms of dimensionless numbers. These numbers help to achieve predicted performance for the actual prototype. Then the prototype is made and put to practical use. Hence a model can be smaller or bigger than the actual object (Prototype).

TABLE: Dimensionless numbers

Sr. No.

Dimensionless number

Definition

Symbol

Formula

Practical application

1.
Reynolds
number
Inertia force/viscous force
Re
ρ  v L / µ
Flow of viscous
fluids in pipes
and over a flat
plate
2.
Froude
number
(Inertia force/Gravity force)0.5
Fr
v/(Lg)0.5
Flow in open
channels
3.
Euler
number
(Inertia force/pressure force)0.5
Eu
v/(P/ρ)0.5
Flow in pipes/
tubes
4.
Weber
number
(Inertia force/Surface tension force)0.5
We
v/(σ/ρL)0.5
Flow through
capillary tube
5.
Mach
number
(Inertia force/Elastic force)0.5
M
(v/(σ/ρ))0.5
High speed
flow of fluids
6.
Schmidt
number
momentum diffusivity/mass diffusivity
Sc
μ / ρD
Schmidt number
is analogous to
Prandtl number
Every dimensionless number in fluid flow is the ratio of inertia force to some other  force as described below. There are in all six forces. Accordingly, there are five dimensionless numbers in fluid mechanics.
1. Reynolds number
It is a ratio of inertia force to viscous force.
Inertia force = mass x acceleration
                     = ρ Vol v/t =  ρ (Vol/t) v
                     = ρ A v v= ρ A v2
Viscous force = Shear stress x area
                       = τ x A = µdu/dy A = µ (v/L) A
Re =  ρA v2 / µ (v/L) A = ρ v L / µ
Reynolds number signifies that viscous forces play a significant role or the fluid is viscous.
It is used to analyze different types of flow namely Laminar, Turbulent, or Transition.
(i) When Viscous forces are dominant, it is a laminar flow
(ii) If Inertial forces are dominant, it is a Turbulent flow.
Practical applications of viscous flow are
(a) Low velocity fluid flow around the airplanes and automobiles
(b) Motion of fluid around a completely submerged submarine
(c) Flow in a high speed centrifugal compressor
(d) In-compressible fluid flow in a small diameter pipe (capillary tube)
2. Froude number
It is square root of the ratio of inertia force to gravity force.
Fr =(Fi / Fg) 0.5 = ( ρA v2/m g)0.5 = v/ (Lg)0.5
Froude number is applicable where gravity forces play a significant role.
Practical applications where gravity force play a significant role are
(a)  Spillway of a dam
(b) Notches and weirs
(c) Fluid flow in open channels
(d) Motion of a fluid around a ship in the ocean
3. Euler number
It is square root of ratio of inertia force to pressure force.
Eu =(Fi / Fp )0.5 = (ρA v2/p A)0.5 = v/(P/ρ)0.5
It is applicable where pressure forces are predominant.
Such practical applications are
(a) Flow through pipes, orifices, mouthpieces and sluice gates
(b) Water hammer phenomenon in a pen-stock
(c) Pressure rise due to sudden closure of valves
4. Weber number
It is the square root of the ratio of inertia force to surface tension force.
We = (Fi / Fs)0.5 = (( ρA v2/σ L)0.5 = v/(σ/ρL)0.5
Its practical applications are
(a) Flow in capillary tube
(b)  Blood flow in veins
5. Mach number
Square root of ratio of inertia force to the elastic force.
M = (Fi /Fe)0.5 = ((ρA v2/σ x A)0.5 = v/( σ/ρ)0.5
σ/ρ = velocity of sound in the fluid
Practical applications for elastic force are
(a) Compressible fluid flow problems at high velocities
(b) High velocity fluid flow in pipes
(c) Fluid motion across high speed projectiles and missiles