Difference between revisions of "User:Tohline/Appendix/Ramblings/Hybrid Scheme Implications"
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===Principal Governing Equations=== | ===Principal Governing Equations=== | ||
Quoting from [Ref01] … Among the [[User:Tohline/PGE#Principal_Governing_Equations|principal governing equations]] we have included the | Quoting from [Ref01] … Among the [[User:Tohline/PGE#Principal_Governing_Equations|principal governing equations]] we have included the ''inertial-frame'', | ||
<div align="center"> | <div align="center"> | ||
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</div> | </div> | ||
Shifting into a rotating frame characterized by the angular velocity vector, | |||
<div align="center"> | |||
<math>~\vec{\Omega}_f \equiv \hat\mathbf{k} \Omega_f \, ,</math> | |||
</div> | |||
and applying the operations that are specified at the first few subsections of [Ref02], we recognize the following relationships … | |||
<table border="0" cellpadding="5" align="center"> | |||
<tr> | |||
<td align="right"> | |||
<math>~\vec{v}_\mathrm{inertial}</math> | |||
</td> | |||
<td align="center"> | |||
<math>~=</math> | |||
</td> | |||
<td align="left"> | |||
<math>~\vec{v}_\mathrm{rot} + {\vec\Omega}_f \times \vec{x} \, ,</math> | |||
</td> | |||
</tr> | |||
<tr> | |||
<td align="right"> | |||
<math>~\biggl[ \frac{d \vec{v}}{dt} \biggr]_\mathrm{inertial}</math> | |||
</td> | |||
<td align="center"> | |||
<math>~=</math> | |||
</td> | |||
<td align="left"> | |||
<math>~ | |||
\biggl[ \frac{d \vec{v}}{dt} \biggr]_\mathrm{rot} + 2{\vec\Omega}_f \times {\vec{v}}_\mathrm{rot} | |||
+ {\vec\Omega}_f \times ({\vec\Omega}_f \times \vec{x}) | |||
</math> | |||
</td> | |||
</tr> | |||
<tr> | |||
<td align="right"> | |||
| |||
</td> | |||
<td align="center"> | |||
<math>~=</math> | |||
</td> | |||
<td align="left"> | |||
<math>~ | |||
\biggl[ \frac{d \vec{v}}{dt} \biggr]_\mathrm{rot} + 2{\vec\Omega}_f \times {\vec{v}}_\mathrm{rot} | |||
- \frac{1}{2} \nabla | {\vec\Omega}_f \times \vec{x}|^2 | |||
</math> | |||
</td> | |||
</tr> | |||
<tr> | |||
<td align="right"> | |||
| |||
</td> | |||
<td align="center"> | |||
<math>~=</math> | |||
</td> | |||
<td align="left"> | |||
<math>~ | |||
\biggl[ \frac{\partial \vec{v}}{\partial t} \biggr]_\mathrm{rot} | |||
+ ({\vec{v}}_\mathrm{rot} \cdot \nabla){\vec{v}}_\mathrm{rot} | |||
+ 2{\vec\Omega}_f \times {\vec{v}}_\mathrm{rot} | |||
- \frac{1}{2} \nabla | {\vec\Omega}_f \times \vec{x}|^2 | |||
\, .</math> | |||
</td> | |||
</tr> | |||
</table> | |||
{{LSU_HBook_footer}} | {{LSU_HBook_footer}} | ||
Revision as of 21:45, 25 August 2020
Implications of Hybrid Scheme
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Background
Key H_Book Chapters
[Ref01] Inertial-Frame Euler Equation
[Ref02] Traditional Description of Rotating Reference Frame
[Ref03] Hybrid Advection Scheme
[Ref04] Riemann S-type Ellipsoids
[Ref05] Korycansky and Papaloizou (1996)
Principal Governing Equations
Quoting from [Ref01] … Among the principal governing equations we have included the inertial-frame,
Lagrangian Representation
of the Euler Equation,
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<math>\frac{d\vec{v}}{dt} = - \frac{1}{\rho} \nabla P - \nabla \Phi</math> |
[EFE], Chap. 2, §11, p. 20, Eq. (38)
[BLRY07], p. 13, Eq. (1.55)
Shifting into a rotating frame characterized by the angular velocity vector,
<math>~\vec{\Omega}_f \equiv \hat\mathbf{k} \Omega_f \, ,</math>
and applying the operations that are specified at the first few subsections of [Ref02], we recognize the following relationships …
|
<math>~\vec{v}_\mathrm{inertial}</math> |
<math>~=</math> |
<math>~\vec{v}_\mathrm{rot} + {\vec\Omega}_f \times \vec{x} \, ,</math> |
|
<math>~\biggl[ \frac{d \vec{v}}{dt} \biggr]_\mathrm{inertial}</math> |
<math>~=</math> |
<math>~ \biggl[ \frac{d \vec{v}}{dt} \biggr]_\mathrm{rot} + 2{\vec\Omega}_f \times {\vec{v}}_\mathrm{rot} + {\vec\Omega}_f \times ({\vec\Omega}_f \times \vec{x}) </math> |
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<math>~=</math> |
<math>~ \biggl[ \frac{d \vec{v}}{dt} \biggr]_\mathrm{rot} + 2{\vec\Omega}_f \times {\vec{v}}_\mathrm{rot} - \frac{1}{2} \nabla | {\vec\Omega}_f \times \vec{x}|^2 </math> |
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<math>~=</math> |
<math>~ \biggl[ \frac{\partial \vec{v}}{\partial t} \biggr]_\mathrm{rot} + ({\vec{v}}_\mathrm{rot} \cdot \nabla){\vec{v}}_\mathrm{rot} + 2{\vec\Omega}_f \times {\vec{v}}_\mathrm{rot} - \frac{1}{2} \nabla | {\vec\Omega}_f \times \vec{x}|^2 \, .</math> |
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© 2014 - 2021 by Joel E. Tohline |