Difference between revisions of "Cole equation of state"

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m (Added link to the adiabatic index)
m (Slight rewrital)
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The '''Cole equation of state''' <ref>R. H. Cole "Underwater Explosions", Princeton University Press (1948) ISBN 9780691069227</ref><ref>
 
The '''Cole equation of state''' <ref>R. H. Cole "Underwater Explosions", Princeton University Press (1948) ISBN 9780691069227</ref><ref>
 
G. K. Batchelor "An introduction to fluid mechanics", Cambridge University Press (1974) ISBN  0521663962</ref>
 
G. K. Batchelor "An introduction to fluid mechanics", Cambridge University Press (1974) ISBN  0521663962</ref>
can be written, when atmospheric pressure is negligible, has the form
+
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
<math>\gamma</math> is an exponent and <math>B</math> is a pressure parameter.
+
<math>\gamma</math> is the [[Heat capacity#Adiabatic index | adiabatic index]] and <math>B</math> is a pressure parameter.
  
 
Usually, the equation is used to model a nearly incompressible system. In this case,
 
Usually, the equation is used to model a nearly incompressible system. In this case,
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:<math>c^2 = \frac{\gamma B}{\rho_0}. </math>
 
:<math>c^2 = \frac{\gamma B}{\rho_0}. </math>
  
where <math>\gamma</math> is the [[Heat capacity#Adiabatic index | adiabatic index]].
 
 
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
 
If the fluctuations in the density are indeed small, the
[[Equations of state | equation of state]] may be rewritten thus:
+
[[Equations of state | equation of state]] may be approximated by the simpler:
  
 
:<math>p = B \gamma \left[
 
:<math>p = B \gamma \left[

Revision as of 13:52, 17 October 2012

The Cole equation of state [1][2] 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 the adiabatic index 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 approximated by the simpler:

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