# Difference between revisions of "Cole equation of state"

Carl McBride (talk | contribs) m (Added link to the adiabatic index) |
m (Slight rewrital) |
||

Line 1: | Line 1: | ||

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> | ||

− | + | 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 | + | <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, | ||

Line 20: | Line 20: | ||

:<math>c^2 = \frac{\gamma B}{\rho_0}. </math> | :<math>c^2 = \frac{\gamma B}{\rho_0}. </math> | ||

− | |||

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 | + | [[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

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 of about 0.01.

If the fluctuations in the density are indeed small, the equation of state may be approximated by the simpler:

## References

- ↑ R. H. Cole "Underwater Explosions", Princeton University Press (1948) ISBN 9780691069227
- ↑ G. K. Batchelor "An introduction to ﬂuid mechanics", Cambridge University Press (1974) ISBN 0521663962