en>Hhhippo: removed Category:Materials testing; added Category:Materials science using HotCat

2013-03-28T18:05:52Z

removed Category:Materials testing; added Category:Materials science using HotCat

← Older revision		Revision as of 20:05, 28 March 2013
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	The ~~writer~~'s ~~title~~ is ~~Andera~~ and ~~she believes it seems quite good psychic~~ [[~~http~~://~~www~~.~~taehyuna~~.~~net~~/xe/~~?document_srl~~=~~78721~~ http://www.~~taehyuna~~.~~net]]~~. ~~For a whilst I've been in Mississippi but now I'm contemplating other choices~~. ~~Credit authorising is how he makes money~~. ~~To play lacross is the factor I adore most of all~~.
			[[Image:SodShockTubeTest Regions.png\|thumb\|300px\|right\|Density Plot after time evolution of t=0.2[-]]]
			The '''Sod shock tube''' problem, named after Gary A. Sod, is a common test for the accuracy of [[Computational fluid dynamics\|computational fluid codes]], like [[Riemann solver]]s, and was heavily investigated by Sod in 1978.

			The test consists of a one dimensional [[Riemann problem]] with the following parameters, for left and right states of an [[ideal gas]].

			<center>
			<math>
			\left( \begin{array}{c}\rho_L\\P_L\\v_L\end{array}\right)
			=
			\left( \begin{array}{c}1.0\\1.0\\0.0\end{array} \right)
			</math>
			,
			<math>
			\left( \begin{array}{c}\rho_R\\P_R\\v_R\end{array}\right)
			=
			\left( \begin{array}{c}0.125\\0.1\\0.0\end{array}\right)
			</math>
			</center>
			where
			::*<math>\rho</math> is the density
			::*P is the pressure
			::*v is the velocity

			The time evolution of this problem can be described by solving the [[Euler equations]].
			Which leads to three characteristics, describing the propagation speed of the
			various regions of the system. Namely the rarefaction wave, the contact discontinuity and
			the shock discontinuity.
			If this is solved numerically, one can test against the analytical solution,
			and get information how well a code captures and resolves shocks and contact discontinuities
			and reproduce the correct density profile of the rarefaction wave.

			==Analytic derivation==
			The different states of the solution are separated by the time evolution of the
			three [[Method of characteristics\|characteristics]] of the system, which is due to the finite speed
			of information propagation. Two of them are equal to the speed
			of sound of the both states
			::<math>cs_1 = \sqrt{\gamma \frac{P_L}{\rho_L}}</math>
			::<math>cs_5 = \sqrt{\gamma \frac{P_R}{\rho_R}}</math>
			The first one is the position of the beginning of the rarefaction wave while
			the other is the velocity of the propagation of the shock.

			Defining:
			::<math>\Gamma = \frac{\gamma - 1}{\gamma + 1}</math>, <math>\beta = \frac{\gamma - 1}{2 \gamma}</math>
			The states after the shock are connected by the [[Rankine-Hugoniot equation\|Rankine Hugoniot]]
			shock jump conditions.
			::<math>\rho_4 = \rho_5 \frac{P_4 + \Gamma P_5}{P_5 + \Gamma P_4}</math>
			But to calculate the density in Region 4 we need to know the pressure in that region.
			This is related by the contact discontinuity with the pressure in region 3 by
			::<math>P_4 = P_3</math>
			Unfortunately the pressure in region 3 can only be calculated iteratively, the right
			solution is found when <math>u_2</math> equals <math>u_4</math>
			::<math>u_4 = \left(P_3' - P_5\right)\sqrt{\frac{1-\Gamma}{\rho_R(P_3'+\Gamma P_5)}}</math>
			::<math>u_2 =\left(P_1^\beta-P_3'^\beta\right) \sqrt{\frac{(1-\Gamma^2)P_1^{1/\gamma}}{\Gamma^2 \rho_L}}</math>
			::<math>u_2 - u_4 = 0</math>
			This function can be evaluated to an arbitrary precision thus giving the pressure in the
			region 3
			::<math>P_3 = \operatorname{calculate}(P_3,s,s,,)</math>
			finally we can calculate
			::<math>u_3 = u_5 + \frac{(P_3 - P_5)}{\sqrt{\frac{\rho_5}{2}((\gamma+1)P_3 +(\gamma-1)P_5)}}</math>
			::<math>u_4 = u_3</math>
			and <math>\rho_3</math> follows from the adiabatic gas law
			::<math>\rho_3 = \rho_1 \left(\frac{P_3}{P_1}\right)^{1/\gamma}</math>

			==References==
			*{{cite journal \| title=A Survey of Several Finite Difference Methods for Systems of Nonlinear Hyperbolic Conservation Laws \| first=G. A. \| last=Sod \| year=1978 \| journal=[[Journal of Computational Physics\|J. Comput. Phys.]] \| volume=27\| pages=1–31 \| url=http://www.osti.gov/energycitations/product.biblio.jsp?osti_id=6812922 \| doi=10.1016/0021-9991(78)90023-2 \|bibcode = 1978JCoPh..27....1S }}
			*{{cite book \| first=Eleuterio F.\| last=Toro\| year=1999 \| title=Riemann Solvers and Numerical Methods for Fluid Dynamics\| publisher=Springer Verlag\|location=Berlin \| id=ISBN 3-540-65966-8}}

			==See also==
			*[[Shock tube]]
			*[[Computational fluid dynamics]]

			[[Category:Fluid dynamics]]
			[[Category:Computational fluid dynamics]]

en>Rjwilmsi: typos fixed: the 1970’s → the 1970s using AWB (8323)

2012-09-01T12:31:36Z

typos fixed: the 1970’s → the 1970s using AWB (8323)

New page

The writer's title is Andera and she believes it seems quite good psychic [[http://www.taehyuna.net/xe/?document_srl=78721 http://www.taehyuna.net]]. For a whilst I've been in Mississippi but now I'm contemplating other choices. Credit authorising is how he makes money. To play lacross is the factor I adore most of all.

Helium atom scattering - Revision history

en>Hhhippo: removed Category:Materials testing; added Category:Materials science using HotCat

en>Rjwilmsi: typos fixed: the 1970’s → the 1970s using AWB (8323)