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Approximation Schemes for convective term - structured grids - definitions

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(indicators of the local velocity direction)
(indicators of the local velocity direction)
Line 64: Line 64:
<math>U^{+}_{f}</math> and <math>U^{-}_{f}</math>
<math>U^{+}_{f}</math> and <math>U^{-}_{f}</math>
 +
 +
therefore unnormalised form of approximation scheme can be written
 +
 +
<table width="100%"><tr><td>
 +
<math>
 +
\phi_{f}=U^{+}_{f}\phi_{W} + U^{-}_{f}\phi_{P}
 +
</math>
 +
</td><td width="5%">(1)</td></tr></table>
 +
 +
or in more general form
 +
 +
<table width="100%"><tr><td>
 +
<math>
 +
\phi_{f}=U^{+}_{f}\phi_{C} + U^{-}_{f}\phi_{D}
 +
</math>
 +
</td><td width="5%">(1)</td></tr></table>

Revision as of 17:49, 29 September 2005

Here we shall develop a commone definitions and regulations because of

  • in different articles was used defferent definitions and notations
  • we are searching for common approach and generalisation

Usual using definition for convected variable


\boldsymbol{f}


\boldsymbol{\phi}

definition of considered face, upon wich approximation is applied

usually (in the most articles) west face of the control volume 
\boldsymbol{w} is considered (without loss of generality)

for which flux is directed from the left to the right

we shall define it as  \boldsymbol{f}

and convected variable at face as  \boldsymbol{\phi_{f}}

indicators of the local velocity direction

approximation scheme can be written in the next form

 
\phi_{w}=\sigma^{+}_{w}\phi_{W}	+ \sigma^{-}_{w}\phi_{P}
(1)


where \sigma^{+}_{w} and \sigma^{-}_{w} are the indicators of the local velocity direction such that


 
\sigma^{+}_{w} = 0.5 \left( 1 + \frac{\left|U_{w} \right|}{U_{w}} \right)
(1)
 
\sigma^{-}_{w} = 1 - \sigma^{+}_{w}
(1)

and of course

 
\left( U_{w} \neq 0  \right)

(1)

also used such definitions as U^{+}_{w} and U^{-}_{w}

we offer to use

U^{+}_{f} and U^{-}_{f}

therefore unnormalised form of approximation scheme can be written

 
\phi_{f}=U^{+}_{f}\phi_{W} + U^{-}_{f}\phi_{P}

(1)

or in more general form

 
\phi_{f}=U^{+}_{f}\phi_{C} + U^{-}_{f}\phi_{D}

(1)
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