Difference between revisions of "Cole equation of state"

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Therefore, if <math>B=100 \rho_0 v^2 / \gamma</math>, the relative density fluctuations
 
Therefore, if <math>B=100 \rho_0 v^2 / \gamma</math>, the relative density fluctuations
 
will be of about 0.01.
 
will be of about 0.01.
 +
 +
If the fluctuations in the density are indeed small, the
 +
EOS may be rewritten thus:
 +
 +
:<math>p = B \gamma \left[
 +
\frac{\rho-\rho_0}{\rho_0}
 +
\right]</math>
 +
  
 
==References==
 
==References==
 
<references/>
 
<references/>
 
[[category: equations of state]]
 
[[category: equations of state]]

Revision as of 12:59, 23 May 2012

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.

If the fluctuations in the density are indeed small, the EOS may be rewritten thus:

p = B \gamma \left[
\frac{\rho-\rho_0}{\rho_0}
 \right]


References

  1. R. H. Cole "Underwater Explosions", Princeton University Press (1948) ISBN 9780691069227
  2. G. K. Batchelor "An introduction to fluid mechanics", Cambridge University Press (1974) ISBN 0521663962