Difference between revisions of "Cole equation of state"

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m (Added a couple of internal links)
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can be written, when atmospheric pressure is negligible, has the form
 
can be written, when atmospheric pressure is negligible, has the form
  
:<math>p = B \left[ \left( \frac{\rho}{\rho_0} \right)^\gamma  -1 \right]</math>.
+
:<math>p = B \left[ \left( \frac{\rho}{\rho_0} \right)^\gamma  -1 \right]</math>
  
 
In it, <math>\rho_0</math> is a reference density around which the density varies
 
In it, <math>\rho_0</math> is a reference density around which the density varies
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where <math>v</math> is the largest velocity, and <math>c</math> is the speed of
 
where <math>v</math> is the largest velocity, and <math>c</math> is the speed of
sound (the ratio <math>v/c</math> is [[Mach's number]]). The speed of sound can
+
sound (the ratio <math>v/c</math> is [[Mach's number]]). The [[speed of sound]] can
 
be seen to be
 
be seen to be
  
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If the fluctuations in the density are indeed small, the
 
If the fluctuations in the density are indeed small, the
EOS may be rewritten thus:
+
[[Equations of state | equation of state]] may be rewritten thus:
  
 
:<math>p = B \gamma \left[
 
:<math>p = B \gamma \left[

Revision as of 14:37, 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 equation of state 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