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Approximation Schemes for convective term - structured grids - definitions

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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

Usual using 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 of the control volume 
\boldsymbol{w} 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

NM convectionschemes Stencil 2a.jpg

My wiki