Magnitudes of Different Forces - 2 | Fluid Mechanics for Mechanical Engineering PDF Download

Dynamic Similarity of Flows with Gravity, Pressure and Inertia Forces

A flow of the type in which significant forces are gravity force, pressure force and inertia force, is found when a free surface is present.

  Examples can be

1.   the flow of a liquid in an open channel.

2.  the wave motion caused by the passage of a ship through water.

3.   the flows over weirs and spillways.

The condition for dynamic similarity of such flows requires

•  the equality of the Euler number Eu (the magnitude ratio of pressure to inertia force),

                                                                                   and

• the equality of the magnitude ratio of gravity to inertia force at corresponding points in the systems being compared. 
         Thus ,
      (18.2e)

• In practice, it is often convenient to use the square root of this ratio  so to deal with the first power of the velocity.

• From a physical point of view, equality of  (1g)^1/2 /v implies equality of 1g /v² as regard to the concept of dynamic similarity.

The reciprocal of the term  (1g)^1/2 /v is known as Froude number ( after William Froude who first suggested the use of this number in the study of naval architecture. )

Hence Froude number, Fr = V /(1g)^1/2.

Therefore, the primary requirement for dynamic similarity between the prototype and the model involving flow of fluid with gravity as the significant force, is the equality of Froude number, Fr, i.e..,
   (18.2f)


Dynamic Similarity of Flows with Surface Tension as the Dominant Force

Surface tension forces are important in certain classes of practical problems such as ,

1. flows in which capillary waves appear
2. flows of small jets and thin sheets of liquid injected by a nozzle in air
3. flow of a thin sheet of liquid over a solid surface.

Here the significant parameter for dynamic similarity is the magnitude ratio of the surface tension force to the inertia force.

 This can be written as  


The term    is usually known as Weber number, Wb (after the German naval architect Moritz Weber who first suggested the use of this term as a relevant
parameter. )

Thus for dynamically similar flows (Wb)_m =(Wb)_p



Dynamic Similarity of Flows with Elastic Force

When the compressibility of fluid in the course of its flow becomes significant, the elastic force along with the pressure and inertia forces has to be considered.

Therefore, the magnitude ratio of inertia to elastic force becomes a relevant parameter for dynamic similarity under this situation.

Thus we can write,
    (18.2h)

The parameter   is known as Cauchy number ,( after the French mathematician A.L. Cauchy)

If we consider the flow to be isentropic , then it can be written
    (18.2i)

(where E_s is the isentropic bulk modulus of elasticity)

 Thus for dynamically similar flows (cauchy)_m=(cauchy)_p

 

• The velocity with which a sound wave propagates through a fluid medium equals to   .

• Hence, the term    can be written as V² / a² where a is the acoustic velocity in the fluid medium.

The ratio V/a is known as Mach number, Ma ( after an Austrian physicist Earnst Mach)

It has been shown in Chapter 1  that the effects of compressibility become important when the Mach number exceeds 0.33.

The situation arises in the flow of air past high-speed aircraft, missiles, propellers and rotory compressors. In these cases equality of Mach number is a condition for dynamic similarity. 
Therefore,

  (Ma)_p=(Ma)_m

i.e.

   (18.2j)

Ratios of Forces for Different Situations of Flow