Approximation Schemes for convective term - structured grids - definitions

(Difference between revisions)
 Revision as of 14:01, 29 September 2005 (view source)Michail (Talk | contribs)← Older edit Latest revision as of 01:14, 9 November 2005 (view source)Michail (Talk | contribs) (→node stencil) (25 intermediate revisions not shown) Line 1: Line 1: - Here we will develop a commone definitions and regulations because of + == Goals of this section == - * in different articles was used defferent definitions and notations + 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 * we are searching for common approach and generalisation - == Usual using definition for convected variable == + '''Please note:''' ''as we still developing this section, you can find the rest of non-unificated definitions'' + + == Usual used definition for convected variable == $[itex] Line 12: Line 16: [itex] [itex] \boldsymbol{\phi} \boldsymbol{\phi} -$ + [/itex] + + we shall use here $\boldsymbol{\phi}$ == definition of considered face, upon wich approximation is applied == == definition of considered face, upon wich approximation is applied == - usually considered the face + usually (in the most articles) west face $+ \boldsymbol{w}$ of the control volume  is considered ''without loss of generality'' - $+ for which '''flux is directed from the left to the right''' i.e. \boldsymbol{U_{f} \triangleright 0 } - \boldsymbol{w} + -$ + + 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 + + + + ---- + Return to [[Numerical methods | Numerical Methods]] - for which flux is directed from left to the right + Return to [[Approximation Schemes for convective term - structured grids]]

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