Editing Stiffened equation of state
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where <math>e</math> is the internal energy per unit mass, given by (Eq. 15 in <ref>[http://dx.doi.org/10.1016/S0045-7930(02)00021-X H. Paillère, C. Corre, and J. R. Garcı́a Cascales "On the extension of the AUSM+ scheme to compressible two-fluid models", Computers & Fluids '''32''' pp. 891-916 (2003)]</ref>): | where <math>e</math> is the internal energy per unit mass, given by (Eq. 15 in <ref>[http://dx.doi.org/10.1016/S0045-7930(02)00021-X H. Paillère, C. Corre, and J. R. Garcı́a Cascales "On the extension of the AUSM+ scheme to compressible two-fluid models", Computers & Fluids '''32''' pp. 891-916 (2003)]</ref>): | ||
:<math> e = \frac{C_p}{\gamma}T + \frac{p^ | :<math> e = \frac{C_p}{\gamma}T + \frac{p^0}{p} </math> | ||
where <math>C_p</math> is the [[heat capacity]] at constant [[pressure]]. <math>\gamma</math> is an empirically determined constant typically taken to be about 6.1, and <math>p^*</math> is another constant, representing the molecular attraction between [[water]] molecules. The magnitude of the later correction is about 2 gigapascals (20,000 atmospheres). | where <math>C_p</math> is the [[heat capacity]] at constant [[pressure]]. <math>\gamma</math> is an empirically determined constant typically taken to be about 6.1, and <math>p^*</math> is another constant, representing the molecular attraction between [[water]] molecules. The magnitude of the later correction is about 2 gigapascals (20,000 atmospheres). | ||
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It can be shown, by linearizing the [[Euler equations]], that the [[speed of sound]] in water is given by | It can be shown, by linearizing the [[Euler equations]], that the [[speed of sound]] in water is given by | ||
:<math>c^2=\frac{\gamma p+p^* }{\rho_0}</math>, | :<math>c^2=\frac{\gamma p+p^* }{\rho_0}</math>, | ||
from which the value of | from which the value of $p^*$ may be computed given all the other variables. | ||
Thus water behaves as though it is an [[ideal gas]] that is ''already'' under about 20,000 atmospheres (2 GPa) pressure, and explains why water is commonly assumed to be incompressible: when the external pressure changes from 1 atmosphere to 2 atmospheres (100 kPa to 200 kPa), the water behaves as an ideal gas would when changing from 20,001 to 20,002 atmospheres (2000.1 MPa to 2000.2 MPa). | Thus water behaves as though it is an [[ideal gas]] that is ''already'' under about 20,000 atmospheres (2 GPa) pressure, and explains why water is commonly assumed to be incompressible: when the external pressure changes from 1 atmosphere to 2 atmospheres (100 kPa to 200 kPa), the water behaves as an ideal gas would when changing from 20,001 to 20,002 atmospheres (2000.1 MPa to 2000.2 MPa). | ||
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It is useful to notice that, given this equation of state, the [[adiabatic]] law is modified from its ideal form: | It is useful to notice that, given this equation of state, the [[adiabatic]] law is modified from its ideal form: | ||
:<math> | :<math> p+p^* = (\gamma p_0 +p^* ) \left(\frac{\rho}{\rho_0}\right)^\gamma </math> | ||
==References== | ==References== |