# Difference between revisions of "Cole equation of state"

(Added ref and a link to Tait EOS) |
(added link to the stiffened EOS) |
||

Line 1: | Line 1: | ||

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> | ||

− | has the form | + | is the adiabatic version of the [[stiffened equation of state]]. |

+ | It 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> |

## Revision as of 12:55, 5 March 2015

The **Cole equation of state**
^{[1]}^{[2]}^{[3]}
is the adiabatic version of the stiffened equation of state.
It has the form

In it, is a reference density around which the density varies, is the adiabatic index, and 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 is large, in the following sense. The fluctuations of the density are related to the speed of sound as

where is the largest velocity, and is the speed of sound (the ratio is Mach's number). The speed of sound can be seen to be

Therefore, if , 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:

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

- ↑ Robert H Cole "Underwater explosions", Princeton University Press, Princeton (1948)
- ↑ G. K. Batchelor "An introduction to ﬂuid mechanics", Cambridge University Press (1974) ISBN 0521663962
- ↑ 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)