Difference between revisions of "Cole equation of state"

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The '''Cole equation of state''' <ref>R. H. Cole "Underwater Explosions", Princeton University Press (1948) ISBN 9780691069227</ref><ref>
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The '''Cole equation of state'''
G. K. Batchelor "An introduction to fluid mechanics", Cambridge University Press (1974) ISBN  0521663962</ref>
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<ref>[http://www.archive.org/details/underwaterexplos00cole Robert H Cole "Underwater explosions", Princeton University Press, Princeton (1948)]</ref><ref>G. K. Batchelor "An introduction to fluid mechanics", Cambridge University Press (1974) ISBN  0521663962</ref><ref>[http://www.archive.org/details/supersonicflowsh00cour Richard Courant "Supersonic_flow_and_shock_waves_a_manual_on_the_mathematical_theory_of_non-linear_wave_motion", Courant Institute of Mathematical Sciences, New York University, New York (1944)]</ref>
 
has the form
 
has the form
  
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\frac{\rho-\rho_0}{\rho_0}
 
\frac{\rho-\rho_0}{\rho_0}
 
  \right]</math>
 
  \right]</math>
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It is quite common that the name "[[Tait equation of state]]" is improperly used for this EOS. This perhaps stems for the classic text by Cole calling this equation a "modified Tait equation" (p. 39).
  
  

Revision as of 12:55, 5 March 2015

The Cole equation of state [1][2][3] 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 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]


It is quite common that the name "Tait equation of state" is improperly used for this EOS. This perhaps stems for the classic text by Cole calling this equation a "modified Tait equation" (p. 39).


References