Cole equation of state: Difference between revisions
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The '''Cole equation of state''' <ref> | The '''Cole equation of state''' | ||
G. K. Batchelor "An introduction to fluid mechanics", Cambridge University Press (1974) ISBN 0521663962</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 | has the form | ||
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\frac{\rho-\rho_0}{\rho_0} | \frac{\rho-\rho_0}{\rho_0} | ||
\right]</math> | \right]</math> | ||
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 11:55, 5 March 2015
The Cole equation of state [1][2][3] 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, 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 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]}
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)