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Approximation Schemes for convective term - structured grids - definitions

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(→definitions for NV diagram)
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we discoverd such definitions as
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we discovered such definitions as
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<math>\boldsymbol{\phi_{i}}</math> is a function of <math>\boldsymbol{\phi_{i+1/2}}</math>
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<math>\boldsymbol{ \hat{\phi_{i}} }</math> is a function of <math>\boldsymbol{ \hat{\phi}_{i+1/2}} </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>

Revision as of 18:50, 29 September 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

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

\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}}

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