Dynamic Similarity of Flows with Elastic Force

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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 Es is the isentropic bulk modulus of elasticity)

Thus for dynamically similar flows (cauchy)m=(cauchy)p
ie.,   

  • The velocity with which a sound wave propagates through a fluid medium equals to .
  • Hence, the term  can be written as  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)

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