# Difference between revisions of "Stiffened equation of state"

Carl McBride (talk | contribs) m (Added some internal links) |
(Quite a number of chages; fixed LANL reference) |
||

Line 1: | Line 1: | ||

− | The '''Stiffened equation of state''' is a simplified form of the Grüneisen equation of state <ref>[http://catalog.lanl.gov/F/ | + | The '''Stiffened equation of state''' is a simplified form of the Grüneisen equation of state <ref>[http://catalog.lanl.gov/F/YJK296TXG13BGDB17QSADFQLGAAKTSYRFYFSA24M48868X7V31-25707?func=item-global&doc_library=LNL01&doc_number=000518513&year=&volume=&sub_library=LANL Francis H. Harlow and Anthony A. Amsden "Fluid Dynamics", Los Alamos Report Number LA-4700 page 3 (1971)]</ref>. |

When considering water under very high pressures (typical applications are underwater explosions, extracorporeal shock wave lithotripsy, and sonoluminescence) the stiffened [[Equations of state|equation of state]] is often used: | When considering water under very high pressures (typical applications are underwater explosions, extracorporeal shock wave lithotripsy, and sonoluminescence) the stiffened [[Equations of state|equation of state]] is often used: | ||

− | :<math> p=\rho(\gamma-1)e- | + | :<math> p=\rho(\gamma-1)e- p^* </math> |

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>): | ||

Line 8: | Line 8: | ||

:<math> e = \frac{C_p}{\gamma}T + \frac{p^0}{p} </math> | :<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^ | + | 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). |

− | + | 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>, | ||

+ | 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). | ||

This equation mispredicts the heat capacity of water but few simple alternatives are available for severely nonisentropic processes such as strong shocks. | This equation mispredicts the heat capacity of water but few simple alternatives are available for severely nonisentropic processes such as strong shocks. | ||

+ | |||

+ | It is useful to notice that, given this equation of state, the [[adiabatic]] law is modified from its ideal form: | ||

+ | |||

+ | :<math> p+p^* = (\gamma p_0 +p^* ) \left(\frac{\rho}{\rho_0}\right)^\gamma </math> | ||

+ | |||

+ | |||

==References== | ==References== | ||

<references/> | <references/> | ||

[[Category:equations of state]] | [[Category:equations of state]] |

## Revision as of 13:42, 10 October 2013

The **Stiffened equation of state** is a simplified form of the Grüneisen equation of state ^{[1]}.
When considering water under very high pressures (typical applications are underwater explosions, extracorporeal shock wave lithotripsy, and sonoluminescence) the stiffened equation of state is often used:

where is the internal energy per unit mass, given by (Eq. 15 in ^{[2]}):

where is the heat capacity at constant pressure. is an empirically determined constant typically taken to be about 6.1, and is another constant, representing the molecular attraction between water molecules. The magnitude of the later correction is about 2 gigapascals (20,000 atmospheres).

It can be shown, by linearizing the Euler equations, that the speed of sound in water is given by

- ,

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).

This equation mispredicts the heat capacity of water but few simple alternatives are available for severely nonisentropic processes such as strong shocks.

It is useful to notice that, given this equation of state, the adiabatic law is modified from its ideal form: