Difference between revisions of "User:Tohline/PGE/Hybrid Scheme"

From VistrailsWiki
Jump to navigation Jump to search
(Begin page that discusses hybrid scheme)
(→‎Component Forms: Begin example #2)
Line 91: Line 91:




===CartNon & CartNon==
===Example #2===
Consider the familiar case of transport of
This component set has been spelled out in, for example, equations (5) - (7) of [http://adsabs.harvard.edu/abs/1978ApJ...224..497N Norman & Wilson (1978)] and equations (11), (12), & (3) of  [http://adsabs.harvard.edu/abs/1997ApJ...490..311N New & Tohline (1997)].
<div align="center">
<table border="0" cellpadding="3">
 
<tr>
  <td align="right">
<math>\boldsymbol{\hat{e}}_R: ~~~~\frac{\partial (\rho v_R)}{\partial t} + \nabla\cdot[(\rho v_R) \vec{v}~]</math>
  </td>
  <td align="center">
<math>~=~</math>
  </td>
  <td align="left">
<math>
-~\frac{\partial P}{\partial R} - \rho \frac{\partial |\Phi}{\partial R} + \frac{A^2}{\rho R^3} + \rho\Omega_0^2 R + \frac{2\Omega_0 A}{R} \, ,
</math>
  </td>
</tr>
 
<tr>
  <td align="right">
<math>\boldsymbol{\hat{e}}_\phi: ~~~~\frac{\partial (A)}{\partial t} + \nabla\cdot[A \vec{v}~]</math>
  </td>
  <td align="center">
<math>~=~</math>
  </td>
  <td align="left">
<math>
-~\frac{\partial P}{\partial \phi} - \rho \frac{\partial |\Phi}{\partial \phi} - 2\Omega_0 RS \, ,
</math>
  </td>
</tr>
 
<tr>
  <td align="right">
<math>\boldsymbol{\hat{e}}_z: ~~~~\frac{\partial (\rho v_z)}{\partial t} + \nabla\cdot[(\rho v_z) \vec{v}~]</math>
  </td>
  <td align="center">
<math>~=~</math>
  </td>
  <td align="left">
<math>
-~\frac{\partial P}{\partial z} - \rho \frac{\partial |\Phi}{\partial z} \, .
</math>
  </td>
</tr>
</table>
</div>


=Related Discussions=
=Related Discussions=

Revision as of 02:08, 26 February 2014

Hybrid Scheme

Whitworth's (1981) Isothermal Free-Energy Surface
|   Tiled Menu   |   Tables of Content   |  Banner Video   |  Tohline Home Page   |

Traditional Eulerian Representation (Review)

Here we review the traditional Eulerian representation of the Euler Equation, as has been discussed in detail earlier.

in terms of velocity:

The so-called "Eulerian form" of the Euler equation can be straightforwardly derived from the standard Lagrangian representation to obtain,

Eulerian Representation
of the Euler Equation,

<math>~\frac{\partial\vec{v}}{\partial t} + (\vec{v}\cdot \nabla) \vec{v}= - \frac{1}{\rho} \nabla P - \nabla \Phi</math>

in terms of momentum density:

Also, we can multiply this expression through by <math>~\rho</math> and combine it with the continuity equation to derive what is commonly referred to as the,

Conservative Form
of the Euler Equation,

<math>~\frac{\partial(\rho\vec{v})}{\partial t} + \nabla\cdot [(\rho\vec{v})\vec{v}]= - \nabla P - \rho \nabla \Phi</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>~\rho</math><math>~\vec{v}</math>) and the velocity vector <math>~\vec{v}</math> and is sometimes written as, <math>\nabla\cdot [(\rho \vec{v}) \otimes \vec{v}]</math>.

Component Forms

Let's split the vector Euler equation into its three scalar components; various examples are identified in Table 1.

Example #

Grid Basis

Grid Rotation

Momentum Basis

Momentum Frame

1

Cartesian

Nonrotating

Cartesian

Inertial

2

Cylindrical

Nonrotating

Cylindrical

Inertial


Example #2

This component set has been spelled out in, for example, equations (5) - (7) of Norman & Wilson (1978) and equations (11), (12), & (3) of New & Tohline (1997).

<math>\boldsymbol{\hat{e}}_R: ~~~~\frac{\partial (\rho v_R)}{\partial t} + \nabla\cdot[(\rho v_R) \vec{v}~]</math>

<math>~=~</math>

<math> -~\frac{\partial P}{\partial R} - \rho \frac{\partial |\Phi}{\partial R} + \frac{A^2}{\rho R^3} + \rho\Omega_0^2 R + \frac{2\Omega_0 A}{R} \, , </math>

<math>\boldsymbol{\hat{e}}_\phi: ~~~~\frac{\partial (A)}{\partial t} + \nabla\cdot[A \vec{v}~]</math>

<math>~=~</math>

<math> -~\frac{\partial P}{\partial \phi} - \rho \frac{\partial |\Phi}{\partial \phi} - 2\Omega_0 RS \, , </math>

<math>\boldsymbol{\hat{e}}_z: ~~~~\frac{\partial (\rho v_z)}{\partial t} + \nabla\cdot[(\rho v_z) \vec{v}~]</math>

<math>~=~</math>

<math> -~\frac{\partial P}{\partial z} - \rho \frac{\partial |\Phi}{\partial z} \, . </math>

Related Discussions


Whitworth's (1981) Isothermal Free-Energy Surface

© 2014 - 2021 by Joel E. Tohline
|   H_Book Home   |   YouTube   |
Appendices: | Equations | Variables | References | Ramblings | Images | myphys.lsu | ADS |
Recommended citation:   Tohline, Joel E. (2021), The Structure, Stability, & Dynamics of Self-Gravitating Fluids, a (MediaWiki-based) Vistrails.org publication, https://www.vistrails.org/index.php/User:Tohline/citation