Stiffened equation of state - Revision history

Dduque: p_inf in Paillère et al is p^* / gamma

2018-07-18T10:12:57Z

p_inf in Paillère et al is p^* / gamma

← Older revision		Revision as of 12:12, 18 July 2018
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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^*}{\~~rho_0~~} </math>		:<math> e = \frac{C_p}{\gamma}T + \frac{p^*}{\gamma \rho} </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).

Dduque: wrong equation, wrong units!

2018-07-18T10:07:38Z

wrong equation, wrong units!

← Older revision		Revision as of 12:07, 18 July 2018
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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^0}{p} </math>		:<math> e = \frac{C_p}{\gamma}T + \frac{p^*}{\rho_0} </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).

Carl McBride: Trivial markup change

2016-07-04T15:48:31Z

Trivial markup change

← Older revision		Revision as of 17:48, 4 July 2016
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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 $p^*$ may be computed given all the other variables.		from which the value of <math>p^*</math> 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).

Dduque: link to Cole equation of state

2015-03-05T11:35:35Z

link to Cole equation of state

← Older revision		Revision as of 13:35, 5 March 2015
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	:<math> \gamma p+p^* = (\gamma p_0 +p^* ) \left(\frac{\rho}{\rho_0}\right)^\gamma , </math>		:<math> \gamma p+p^* = (\gamma p_0 +p^* ) \left(\frac{\rho}{\rho_0}\right)^\gamma , </math>

	which, rearranged is known as the [[Cole equation of state]].		which, rearranged, is known as the [[Cole equation of state]].

	==References==		==References==

Dduque at 11:34, 5 March 2015

2015-03-05T11:34:27Z

← Older revision		Revision as of 13:34, 5 March 2015
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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> \gamma p+p^* = (\gamma p_0 +p^* ) \left(\frac{\rho}{\rho_0}\right)^\gamma </math>		:<math> \gamma p+p^* = (\gamma p_0 +p^* ) \left(\frac{\rho}{\rho_0}\right)^\gamma , </math>

			which, rearranged is known as the [[Cole equation of state]].

	==References==		==References==

Dduque: fixed typo in las formula

2013-12-03T14:11:13Z

fixed typo in las formula

← Older revision		Revision as of 16:11, 3 December 2013
Line 20:		Line 20:
	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> p+p^* = (\gamma p_0 +p^* ) \left(\frac{\rho}{\rho_0}\right)^\gamma </math>		:<math> \gamma p+p^* = (\gamma p_0 +p^* ) \left(\frac{\rho}{\rho_0}\right)^\gamma </math>

Dduque: Quite a number of chages; fixed LANL reference

2013-10-10T11:42:25Z

Quite a number of chages; fixed LANL reference

← Older revision		Revision as of 13:42, 10 October 2013
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	The '''Stiffened equation of state''' is a simplified form of the Grüneisen equation of state <ref>[http://catalog.lanl.gov/F/~~68VUJP8ILPV6YASUKIXMVALQCYF5LSLG7N58V2U9XCQ9J3QNJ8~~-~~13509~~?func=~~service~~&doc_library=LNL01&doc_number=000518513&~~line_number~~=~~0001~~&~~func_code~~=~~WEB-BRIEF~~&~~service_type~~=~~MEDIA~~ Francis H. Harlow and Anthony A. Amsden "Fluid Dynamics", Los Alamos Report Number LA-4700 page 3 (1971)]</ref>.		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-~~\gamma~~ p^0 </math>		:<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^0</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).

	~~The equation is stated in this form because~~ the [[speed of sound]] in water is given by <math>c^2=\gamma(p+p^~~0)/~~\~~rho~~</math>.		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]]

Carl McBride: Added some internal links

2012-10-22T14:16:57Z

Added some internal links

← Older revision		Revision as of 16:16, 22 October 2012
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/68VUJP8ILPV6YASUKIXMVALQCYF5LSLG7N58V2U9XCQ9J3QNJ8-13509?func=service&doc_library=LNL01&doc_number=000518513&line_number=0001&func_code=WEB-BRIEF&service_type=MEDIA Francis H. Harlow and Anthony A. Amsden "Fluid Dynamics", Los Alamos Report Number LA-4700 page 3 (1971)]</ref>.		The '''Stiffened equation of state''' is a simplified form of the Grüneisen equation of state <ref>[http://catalog.lanl.gov/F/68VUJP8ILPV6YASUKIXMVALQCYF5LSLG7N58V2U9XCQ9J3QNJ8-13509?func=service&doc_library=LNL01&doc_number=000518513&line_number=0001&func_code=WEB-BRIEF&service_type=MEDIA 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 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-\gamma p^0 </math>		:<math> p=\rho(\gamma-1)e-\gamma p^0 </math>
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^0</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^0</math> is another constant, representing the molecular attraction between [[water]] molecules. The magnitude of the later correction is about 2 gigapascals (20,000 atmospheres).

	The equation is stated in this form because the speed of sound in water is given by <math>c^2=\gamma(p+p^0)/\rho</math>.		The equation is stated in this form because the [[speed of sound]] in water is given by <math>c^2=\gamma(p+p^0)/\rho</math>.

	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.

Carl McBride: Added an equation for e

2012-10-22T13:55:20Z

Added an equation for e

← Older revision		Revision as of 15:55, 22 October 2012
Line 2:		Line 2:
	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:		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:

	:<math> p=\rho(\gamma-1)e-\gamma p^0 \,</math>		:<math> p=\rho(\gamma-1)e-\gamma p^0 </math>

	where <math>e</math> is the internal energy per unit mass, <math>\gamma</math> is an empirically determined constant typically taken to be about 6.1, and <math>p^0</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>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^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^0</math> is another constant, representing the molecular attraction between water molecules. The magnitude of the later correction is about 2 gigapascals (20,000 atmospheres).

	The equation is stated in this form because the speed of sound in water is given by <math>c^2=\gamma(p+p^0)/\rho</math>.		The equation is stated in this form because the speed of sound in water is given by <math>c^2=\gamma(p+p^0)/\rho</math>.
Line 10:		Line 14:
	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 ~~[[specific~~ 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.
	==References==		==References==
	<references/>		<references/>

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

Carl McBride: Added a reference

2012-10-22T13:37:45Z

Added a reference

← Older revision		Revision as of 15:37, 22 October 2012
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/68VUJP8ILPV6YASUKIXMVALQCYF5LSLG7N58V2U9XCQ9J3QNJ8-13509?func=service&doc_library=LNL01&doc_number=000518513&line_number=0001&func_code=WEB-BRIEF&service_type=MEDIA 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 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 equation of state is often used:

Line 10:		Line 11:

	This equation mispredicts the [[specific heat capacity]] of water but few simple alternatives are available for severely nonisentropic processes such as strong shocks.		This equation mispredicts the [[specific heat capacity]] of water but few simple alternatives are available for severely nonisentropic processes such as strong shocks.
			==References==
			<references/>

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