Cole equation of state

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The Cole equation of state [1][2] can be written, when atmospheric pressure is negligible, has the form

p = B \left[ \left( \frac{\rho}{\rho_0} \right)^\gamma  -1 \right].

In it, \rho_0 is a reference density around which the density varies \gamma is an exponent and B is a pressure parameter.

Usually, the equation is used to model a nearly incompressible system. In this case, the exponent is often set to a value of 7, and B is large, in the following sense. The fluctuations of the density are related to the speed of sound as

\frac{\delta \rho}{\rho} = \frac{v^2}{c^2} ,

where v is the largest velocity, and c is the speed of sound (the ratio v/c is Mach's number). The speed of sound can be seen to be

c^2 = \frac{\gamma B}{\rho_0}.

Therefore, if B=100 \rho_0 v^2 / \gamma, the relative density fluctuations will be of about 0.01.

References

  1. R.H. Cole, Underwater Explosions. Princeton University Press 1948
  2. G.K. Batchelor, An introduction to fluid mechanics. Cambridge University Press 1974