Difference between revisions of "Cole equation of state"

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(Derivation --- work in progress)
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The '''Cole equation of state'''
 
The '''Cole equation of state'''
 
<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>
 
<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>
is the adiabatic version of the [[stiffened equation of state]].
+
is the adiabatic version of the [[stiffened equation of state]]. (See ''Derivation'', below.)
 
It has the form
 
It has the form
  
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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).
 
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).
  
 +
==Derivation==
 +
 +
Let us write the stiffened EOS as
 +
 +
:<math>p+ p^* = (\gamma -1) e \rho = (\gamma -1) E / V ,</math>
 +
 +
where ''E'' is the internal energy. In an adiabatic process, the work is the only responsible of a change in internal energy. Hence the
 +
first law reads
 +
 +
:<math>  dW= -p dV  = dE</math>
 +
 +
...
  
 
==References==
 
==References==
 
<references/>
 
<references/>
 
[[category: equations of state]]
 
[[category: equations of state]]

Revision as of 15:10, 6 March 2015

The Cole equation of state [1][2][3] is the adiabatic version of the stiffened equation of state. (See Derivation, below.) It 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).

Derivation

Let us write the stiffened EOS as

p+ p^* = (\gamma -1) e \rho = (\gamma -1) E / V ,

where E is the internal energy. In an adiabatic process, the work is the only responsible of a change in internal energy. Hence the first law reads

  dW= -p dV  = dE

...

References