Cole equation of state: Difference between revisions
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:<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 the [[Heat capacity#Adiabatic index | adiabatic index]] 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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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 | will be about 0.01. | ||
If the fluctuations in the density are indeed small, the | If the fluctuations in the density are indeed small, the | ||
Revision as of 12:54, 17 October 2012
The Cole equation of state [1][2] has the form
![{\displaystyle p=B\left[\left({\frac {\rho }{\rho _{0}}}\right)^{\gamma }-1\right]}](https://wikimedia.org/api/rest_v1/media/math/render/svg/e250e61d89c636d24e9bd5fb17f4140aac0989fb)
In it, Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \rho_0} is a reference density around which the density varies, Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \gamma} is the adiabatic index, and Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle 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 Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle B} is large, in the following sense. The fluctuations of the density are related to the speed of sound as
- Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \frac{\delta \rho}{\rho} = \frac{v^2}{c^2} ,}
where Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle v} is the largest velocity, and Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle c} is the speed of sound (the ratio Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle v/c} is Mach's number). The speed of sound can be seen to be
- Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle c^2 = \frac{\gamma B}{\rho_0}. }
Therefore, if Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle 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:
- Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle p = B \gamma \left[ \frac{\rho-\rho_0}{\rho_0} \right]}
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