Appendix/Ramblings/HybridSchemeOld - Revision history

Jet53man: /* Angular Momentum Conservation */

2022-04-11T22:30:24Z

Angular Momentum Conservation

← Older revision		Revision as of 18:30, 11 April 2022
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	As foreshadowed above — see the [[PGE/~~Hybrid_Scheme~~#Recognizing_Statements_of_Conservatio\|subsection titled, ''Recognizing Statements of Conservation'']] — the angular momentum of a Lagrangian fluid element will be conserved if the "source" term, <math>~S = 0</math>. This situation will arise if, at the fluid element's location, the azimuthal pressure variation, <math>~\partial P/\partial\phi</math>, and the azimuthal variation in the gravitational potential, <math>~\partial \Phi/\partial\phi</math>, are both zero, or if the two balance one another (''i.e.,''<math>~\partial P/\partial\phi=-\rho\partial\Phi/\partial\phi</math>).		As foreshadowed above — see the [[PGE/HybridScheme#Recognizing_Statements_of_Conservatio\|subsection titled, ''Recognizing Statements of Conservation'']] — the angular momentum of a Lagrangian fluid element will be conserved if the "source" term, <math>~S = 0</math>. This situation will arise if, at the fluid element's location, the azimuthal pressure variation, <math>~\partial P/\partial\phi</math>, and the azimuthal variation in the gravitational potential, <math>~\partial \Phi/\partial\phi</math>, are both zero, or if the two balance one another (''i.e.,''<math>~\partial P/\partial\phi=-\rho\partial\Phi/\partial\phi</math>).

	Based on the above discussion, we can equally well view the flow from a frame of reference that is rotating with a constant angular velocity, <math>~\Omega_0</math>, and write,		Based on the above discussion, we can equally well view the flow from a frame of reference that is rotating with a constant angular velocity, <math>~\Omega_0</math>, and write,

Jet53man at 22:29, 11 April 2022

2022-04-11T22:29:24Z

← Older revision		Revision as of 18:29, 11 April 2022
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	<tr>		<tr>
	<td align="right">		<td align="right">
	{{ ~~User:Tohline/~~Math/EQ_Continuity01}}		{{ Math/EQ_Continuity01}}
	</td>		</td>
	<td align="center">		<td align="center">
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	As foreshadowed above — see the [[~~User:Tohline/~~PGE/Hybrid_Scheme#Recognizing_Statements_of_Conservatio\|subsection titled, ''Recognizing Statements of Conservation'']] — the angular momentum of a Lagrangian fluid element will be conserved if the "source" term, <math>~S = 0</math>. This situation will arise if, at the fluid element's location, the azimuthal pressure variation, <math>~\partial P/\partial\phi</math>, and the azimuthal variation in the gravitational potential, <math>~\partial \Phi/\partial\phi</math>, are both zero, or if the two balance one another (''i.e.,''<math>~\partial P/\partial\phi=-\rho\partial\Phi/\partial\phi</math>).		As foreshadowed above — see the [[PGE/Hybrid_Scheme#Recognizing_Statements_of_Conservatio\|subsection titled, ''Recognizing Statements of Conservation'']] — the angular momentum of a Lagrangian fluid element will be conserved if the "source" term, <math>~S = 0</math>. This situation will arise if, at the fluid element's location, the azimuthal pressure variation, <math>~\partial P/\partial\phi</math>, and the azimuthal variation in the gravitational potential, <math>~\partial \Phi/\partial\phi</math>, are both zero, or if the two balance one another (''i.e.,''<math>~\partial P/\partial\phi=-\rho\partial\Phi/\partial\phi</math>).

	Based on the above discussion, we can equally well view the flow from a frame of reference that is rotating with a constant angular velocity, <math>~\Omega_0</math>, and write,		Based on the above discussion, we can equally well view the flow from a frame of reference that is rotating with a constant angular velocity, <math>~\Omega_0</math>, and write,
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	=Zach's Dissertation Paper=		=Zach's Dissertation Paper=
	== Section 2.3==		== Section 2.3==
	Building on our introductory discussion of the [[~~User:Tohline/~~PGE/RotatingFrame#Euler_Equation_.28rotating_frame.29\|Euler equation]] (see also Appendix 1.D, §3 of [[~~User:Tohline/~~Appendix/References\|BT87]]), we begin with the,		Building on our introductory discussion of the [[PGE/RotatingFrame#Euler_Equation_.28rotating_frame.29\|Euler equation]] (see also Appendix 1.D, §3 of [[Appendix/References\|BT87]]), we begin with the,

	<div align="center">		<div align="center">
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	=TRASH (do not read below this line)=		=TRASH (do not read below this line)=
	==Traditional Eulerian Representation (Review)==		==Traditional Eulerian Representation (Review)==
	Here we review the traditional Eulerian representation of the Euler Equation, as has been [[~~User:Tohline/~~PGE/Euler#Eulerian_Representation\|discussed in detail earlier]].		Here we review the traditional Eulerian representation of the Euler Equation, as has been [[PGE/Euler#Eulerian_Representation\|discussed in detail earlier]].

	===in terms of velocity:===		===in terms of velocity:===

	The so-called "Eulerian form" of the Euler equation can be straightforwardly derived from the [[~~User:Tohline/~~PGE/Euler#Lagrangian_Representation\|standard Lagrangian representation]] to obtain,		The so-called "Eulerian form" of the Euler equation can be straightforwardly derived from the [[PGE/Euler#Lagrangian_Representation\|standard Lagrangian representation]] to obtain,

	<div align="center">		<div align="center">
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	of the Euler Equation,		of the Euler Equation,

	{{~~User:Tohline/~~Math/EQ_Euler02}}		{{Math/EQ_Euler02}}
	</div>		</div>

	===in terms of momentum density:===		===in terms of momentum density:===

	Also, we can multiply this expression through by {{~~User:Tohline/~~Math/VAR_Density01}} and combine it with the continuity equation to derive what is commonly referred to as the,		Also, we can multiply this expression through by {{Math/VAR_Density01}} and combine it with the continuity equation to derive what is commonly referred to as the,

	<div align="center">		<div align="center">
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	of the Euler Equation,		of the Euler Equation,

	{{~~User:Tohline/~~Math/EQ_Euler03}}		{{Math/EQ_Euler03}}
	</div>		</div>

	The second term on the left-hand-side of this last expression represents the divergence of the "dyadic product" of the vector momentum density ({{~~User:Tohline/~~Math/VAR_Density01}}{{~~User:Tohline/~~Math/VAR_VelocityVector01}}) and the velocity vector {{~~User:Tohline/~~Math/VAR_VelocityVector01}} and is sometimes written as, <math>\nabla\cdot [(\rho \vec{v}) \otimes \vec{v}]</math>.		The second term on the left-hand-side of this last expression represents the divergence of the "dyadic product" of the vector momentum density ({{Math/VAR_Density01}}{{Math/VAR_VelocityVector01}}) and the velocity vector {{Math/VAR_VelocityVector01}} and is sometimes written as, <math>\nabla\cdot [(\rho \vec{v}) \otimes \vec{v}]</math>.

	=Component Forms=		=Component Forms=
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	=Related Discussions=		=Related Discussions=
	* Euler equation viewed from a [~~http://www.vistrails.org/index.php?title=User:Tohline/~~PGE/RotatingFrame rotating frame of reference] or [[Main Page]].		* Euler equation viewed from a [[PGE/RotatingFrame\|rotating frame of reference]] or [[Main Page]].
	* An [~~http://www.vistrails.org/index.php/User:Tohline/~~PGE/ConservingMomentum earlier draft of this "Euler equation" presentation].		* An [[PGE/ConservingMomentum\|earlier draft of this "Euler equation" presentation]].


	{{ SGFfooter }}		{{ SGFfooter }}

Jet53man at 22:09, 11 April 2022

2022-04-11T22:09:24Z

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	''March 1, 2014'' by Joel E. Tohline		''March 1, 2014'' by Joel E. Tohline

	<[[~~User:Tohline/~~PGE/~~Hybrid_Scheme_Preface~~\|full preface]]> … This seemed too good to be true. The discovered code modification would allow us to conserve angular momentum very accurately and, at the same time, allow us to use a rotating grid and thereby minimize numerical diffusion … <[[~~User:Tohline/~~PGE/Hybrid_Scheme_Preface\|read more]]>		<[[PGE/HybridSchemePreface\|full preface]]> … This seemed too good to be true. The discovered code modification would allow us to conserve angular momentum very accurately and, at the same time, allow us to use a rotating grid and thereby minimize numerical diffusion … <[[PGE/Hybrid_Scheme_Preface\|read more]]>

	In his dissertation research, Jay Call (see [http://adsabs.harvard.edu/abs/2010CQGra..27q5002C Call, Tohline, & Lehner 2010]) derived a complete description of this hybrid advection scheme in a fully relativistic and generalized coordinate framework. He showed that one can write the system of fluid equations in a manner that facilitates advection of inertial-frame quantities across a rotating grid. In addition — and more importantly — he showed how to write the system of fluid equations to allow advection of inertial-frame angular momentum (generally associated with a cylindrical coordinate mesh) across a rotating ''Cartesian'' grid. In the discussion that follows here, we derive the Newtonian version of Jay's hybrid scheme.		In his dissertation research, Jay Call (see [http://adsabs.harvard.edu/abs/2010CQGra..27q5002C Call, Tohline, & Lehner 2010]) derived a complete description of this hybrid advection scheme in a fully relativistic and generalized coordinate framework. He showed that one can write the system of fluid equations in a manner that facilitates advection of inertial-frame quantities across a rotating grid. In addition — and more importantly — he showed how to write the system of fluid equations to allow advection of inertial-frame angular momentum (generally associated with a cylindrical coordinate mesh) across a rotating ''Cartesian'' grid. In the discussion that follows here, we derive the Newtonian version of Jay's hybrid scheme.

Jet53man: Created page with "FORCETOC =Hybrid Advection Scheme= ==Preface== ''March 1, 2014'' by Joel E. Tohline …..."

2021-07-23T16:47:45Z

Created page with "__FORCETOC__  =Hybrid Advection Scheme= ==Preface== ''March 1, 2014'' by Joel E. Tohline <full preface> …..."

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Appendix/Ramblings/HybridSchemeOld - Revision history

Jet53man: /* Angular Momentum Conservation */

Jet53man at 22:29, 11 April 2022

Jet53man at 22:09, 11 April 2022

Jet53man: Created page with "__FORCETOC__ =Hybrid Advection Scheme= ==Preface== ''March 1, 2014'' by Joel E. Tohline …..."

Jet53man: Created page with "FORCETOC =Hybrid Advection Scheme= ==Preface== ''March 1, 2014'' by Joel E. Tohline …..."