# Approximation Schemes for convective term - structured grids - definitions

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- | + | == Goals of this section == | |

- | * in different | + | 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 | ||

+ | |||

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

+ | |||

+ | == Usual used definition for convected variable == | ||

+ | |||

+ | <math> | ||

+ | \boldsymbol{f} | ||

+ | </math> | ||

+ | |||

+ | <math> | ||

+ | \boldsymbol{\phi} | ||

+ | </math> | ||

+ | |||

+ | we shall use here <math>\boldsymbol{\phi}</math> | ||

+ | |||

+ | == definition of considered face, upon wich approximation is applied == | ||

+ | |||

+ | usually (in the most articles) west face <math> | ||

+ | \boldsymbol{w} </math> of the control volume is considered ''without loss of generality'' | ||

+ | |||

+ | for which '''flux is directed from the left to the right''' i.e. <math> \boldsymbol{U_{f} \triangleright 0 } </math> | ||

+ | |||

+ | |||

+ | we shall define it as <math> \boldsymbol{f} </math> | ||

+ | |||

+ | and convected variable at face of CV as <math> \boldsymbol{\phi_{f}} </math> | ||

+ | |||

+ | |||

+ | 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 == | ||

+ | |||

+ | approximation scheme can be written in the next form | ||

+ | |||

+ | <table width="100%"><tr><td> | ||

+ | :<math> | ||

+ | \phi_{w}=\sigma^{+}_{w}\phi_{W} + \sigma^{-}_{w}\phi_{P} | ||

+ | </math> | ||

+ | </td><td width="5%">(1)</td></tr></table> | ||

+ | |||

+ | |||

+ | where <math>\sigma^{+}_{w}</math> and <math>\sigma^{-}_{w}</math> are the indicators of the local velocity direction such that | ||

+ | |||

+ | |||

+ | <table width="100%"><tr><td> | ||

+ | :<math> | ||

+ | \sigma^{+}_{w} = 0.5 \left( 1 + \frac{\left|U_{w} \right|}{U_{w}} \right) | ||

+ | </math> | ||

+ | </td><td width="5%">(1)</td></tr></table> | ||

+ | |||

+ | <table width="100%"><tr><td> | ||

+ | :<math> | ||

+ | \sigma^{-}_{w} = 1 - \sigma^{+}_{w} | ||

+ | </math> | ||

+ | </td><td width="5%">(1)</td></tr></table> | ||

+ | |||

+ | and of course | ||

+ | |||

+ | <table width="100%"><tr><td> | ||

+ | <math> | ||

+ | \left( U_{w} \neq 0 \right) | ||

+ | </math> | ||

+ | </td><td width="5%">(1)</td></tr></table> | ||

+ | |||

+ | also used such definitions as <math>U^{+}_{w}</math> and <math>U^{-}_{w}</math> | ||

+ | |||

+ | we offer to use | ||

+ | |||

+ | <math>U^{+}_{f}</math> and <math>U^{-}_{f}</math> | ||

+ | |||

+ | therefore unnormalised form of approximation scheme can be written | ||

+ | |||

+ | <table width="100%"><tr><td> | ||

+ | <math> | ||

+ | \phi_{f}=U^{+}_{f}\phi_{W} + U^{-}_{f}\phi_{P} | ||

+ | </math> | ||

+ | </td><td width="5%">(1)</td></tr></table> | ||

+ | |||

+ | or in more general form | ||

+ | |||

+ | <table width="100%"><tr><td> | ||

+ | <math> | ||

+ | \phi_{f}=U^{+}_{f}\phi_{C} + U^{-}_{f}\phi_{D} | ||

+ | </math> | ||

+ | </td><td width="5%">(1)</td></tr></table> | ||

+ | |||

+ | == definitions for NV diagram == | ||

+ | |||

+ | |||

+ | we discovered such definitions as | ||

+ | |||

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

+ | |||

+ | <math>\boldsymbol{ \hat{\phi_{w}} }</math> is a function of <math>\boldsymbol{ \hat{\phi}_{W}} </math> | ||

+ | |||

+ | we shall use here | ||

+ | |||

+ | <math>\boldsymbol{ \hat{\phi_{f}} }</math> is a function of <math>\boldsymbol{ \hat{\phi}_{C}} </math> | ||

+ | |||

+ | == node stencil == | ||

+ | |||

+ | Bear in mind this stencil | ||

+ | |||

+ | |||

+ | |||

+ | ---- | ||

+ | <i> Return to [[Numerical methods | Numerical Methods]] </i> | ||

+ | |||

+ | <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

we shall use here

## definition of considered face, upon wich approximation is applied

usually (in the most articles) west face of the control volume is considered *without loss of generality*

for which **flux is directed from the left to the right** i.e.

we shall define it as

and convected variable at face of CV as

also you can find in literature such definition as , but we suggested it non suitable, because of complication

## indicators of the local velocity direction

approximation scheme can be written in the next form

| (1) |

where and are the indicators of the local velocity direction such that

| (1) |

| (1) |

and of course

| (1) |

also used such definitions as and

we offer to use

and

therefore unnormalised form of approximation scheme can be written

| (1) |

or in more general form

| (1) |

## definitions for NV diagram

we discovered such definitions as

is a function of

is a function of

we shall use here

is a function of

## node stencil

Bear in mind this stencil

* Return to Numerical Methods *

* Return to Approximation Schemes for convective term - structured grids *