Approximation Schemes for convective term - structured grids - definitions

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