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Approximation Schemes for convective term - structured grids - definitions

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(indicators of the local velocity direction)
(node stencil)
 
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== Goals of this section ==
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Here we shall develop a commone definitions and regulations because of  
Here we shall develop a commone definitions and regulations because of  
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* in different articles was used defferent definitions and notations
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* in different issues was used different definitions and notations
* we are searching for common approach and generalisation
* we are searching for common approach and generalisation
-
== Usual using definition for convected variable ==
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'''Please note:''' ''as we still developing this section, you can find the rest of non-unificated definitions''
 +
 
 +
== Usual used definition for convected variable ==
<math>
<math>
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<math>
<math>
\boldsymbol{\phi}
\boldsymbol{\phi}
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</math>    
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</math>
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we shall use here <math>\boldsymbol{\phi}</math>
== definition of considered face, upon wich approximation is applied ==
== definition of considered face, upon wich approximation is applied ==
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usually (in the most articles) west face of the control volume <math>
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usually (in the most articles) west face <math>
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\boldsymbol{w} </math> is considered (''without loss of generality'')
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\boldsymbol{w} </math> of the control volume  is considered ''without loss of generality''
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for which '''flux is directed from the left to the right''' i.e. <math> \boldsymbol{U_{f} \triangleright 0 } </math>
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for which '''flux is directed from the left to the right'''
 
we shall define it as <math> \boldsymbol{f} </math>
we shall define it as <math> \boldsymbol{f} </math>
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and convected variable at face as <math> \boldsymbol{\phi_{f}} </math>
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and convected variable at face of CV as <math> \boldsymbol{\phi_{f}} </math>
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 +
 
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also you can find in literature such definition as <math> \boldsymbol{i+1/2} </math> , but we suggested it non suitable, because of complication
== indicators of the local velocity direction ==
== indicators of the local velocity direction ==
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</math>
</math>
</td><td width="5%">(1)</td></tr></table>
</td><td width="5%">(1)</td></tr></table>
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== definitions for NV diagram ==
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 +
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we discovered such definitions as
 +
 +
<math>\boldsymbol{ \hat{\phi}_{i+1/2} }</math> is a function of <math>\boldsymbol{ \hat{\phi}_{i}} </math>
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<math>\boldsymbol{ \hat{\phi_{w}} }</math> is a function of <math>\boldsymbol{ \hat{\phi}_{W}} </math>
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we shall use here
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<math>\boldsymbol{ \hat{\phi_{f}} }</math> is a function of <math>\boldsymbol{ \hat{\phi}_{C}} </math>
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== node stencil ==
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Bear in mind this stencil
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 +
 +
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----
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<i> Return to [[Numerical methods | Numerical Methods]] </i>
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<i> Return to [[Approximation Schemes for convective term - structured grids]] </i>

Latest revision as of 01:14, 9 November 2005

Contents

Goals of this section

Here we shall develop a commone definitions and regulations because of

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

Please note: as we still developing this section, you can find the rest of non-unificated definitions

Usual used definition for convected variable


\boldsymbol{f}


\boldsymbol{\phi}

we shall use here \boldsymbol{\phi}

definition of considered face, upon wich approximation is applied

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

for which flux is directed from the left to the right i.e.  \boldsymbol{U_{f} \triangleright 0 }


we shall define it as  \boldsymbol{f}

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


also you can find in literature such definition as  \boldsymbol{i+1/2} , but we suggested it non suitable, because of complication

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)

definitions for NV diagram

we discovered such definitions as

\boldsymbol{ \hat{\phi}_{i+1/2} } is a function of \boldsymbol{ \hat{\phi}_{i}}

\boldsymbol{ \hat{\phi_{w}} } is a function of \boldsymbol{ \hat{\phi}_{W}}

we shall use here

\boldsymbol{ \hat{\phi_{f}} } is a function of \boldsymbol{ \hat{\phi}_{C}}

node stencil

Bear in mind this stencil



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