Calculation on non-orthogonal curvelinear structured grids, finite-volume method
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</math> | </math> | ||
</td><td width="5%">(4)</td></tr></table> | </td><td width="5%">(4)</td></tr></table> | ||
+ | |||
+ | <table width="70%"><tr><td> | ||
+ | :<math> | ||
+ | \alpha = \left( \frac{\partial x}{\partial \eta } \right)^2 + \left( \frac{\partial y}{\partial \eta } \right)^2 | ||
+ | </math> | ||
+ | </td><td width="5%">(5)</td></tr></table> | ||
+ | |||
+ | <table width="70%"><tr><td> | ||
+ | :<math> | ||
+ | \gamma = \left( \frac{\partial x}{ \partial \xi } \right)^2 + \left( \frac{\partial y}{ \partial \xi } \right)^2 | ||
+ | </math> | ||
+ | </td><td width="5%">(6)</td></tr></table> | ||
+ | |||
+ | <table width="70%"><tr><td> | ||
+ | :<math> | ||
+ | \beta = \frac{\partial x}{ \partial \xi} \frac{\partial x}{ \partial \eta} + \frac{\partial y}{ \partial \xi} \frac{\partial y}{ \partial \eta} | ||
+ | </math> | ||
+ | </td><td width="5%">(7)</td></tr></table> | ||
+ | |||
+ | |||
+ | <table width="70%"><tr><td> | ||
+ | :<math> | ||
+ | J = \frac{\partial x}{ \partial \xi} \frac{\partial y}{ \partial \eta} - \frac{\partial y}{ \partial \xi} \frac{\partial x}{ \partial \eta} | ||
+ | </math> | ||
+ | </td><td width="5%">(8)</td></tr></table> | ||
+ | |||
+ | Using the finite volume method the trnsformed equations can be integrated as follows: | ||
+ | |||
+ | <table width="70%"><tr><td> | ||
+ | :<math> | ||
+ | \left[ \left( \rho U \Delta \eta \right) \right]^{e}_{w} + \left[ \left( \rho V \Delta \xi \right) \right]^{n}_{s} = \left[ \frac{\Gamma \Delta \eta}{J} \left( \alpha \frac{\partial \phi}{\partial \xi} - \beta \frac{\partial \phi}{ \partial \eta} \right) \right]^{e}_{w} + \left[ \frac{\Gamma \Delta \xi}{J} \left( \gamma \frac{\partial \phi}{\partial \eta } - \beta \frac{\partial \phi}{\partial \xi} \right) \right]^{n}_{s} + \left( J \Delta \xi \Delta \eta \right) \overline{S}^{\phi}_{P} | ||
+ | </math> | ||
+ | </td><td width="5%">(9)</td></tr></table> | ||
+ | |||
+ | The convection terms are approximated as described in section http://www.cfd-online.com/Wiki/Discretization_of_the_convection_term . | ||
+ | |||
+ | Diffusion terms are approximated by the second-oder central differencing scheme. | ||
+ | |||
+ | The standard form of the finite volume equation can be obtained as | ||
+ | |||
+ | <table width="70%"><tr><td> | ||
+ | :<math> | ||
+ | A^{\phi}_{P} \phi_{P} = A^{\phi}_{E} \phi_{E} + A^{\phi}_{W} \phi_{W} + A^{\phi}_{N} \phi_{N} + A^{\phi}_{S} \phi_{S} + b^{\phi} | ||
+ | </math> | ||
+ | </td><td width="5%">(10)</td></tr></table> | ||
+ | |||
+ | where | ||
+ | |||
+ | <table width="70%"><tr><td> | ||
+ | :<math> | ||
+ | A^{\phi}_{E} = \left(\frac{\Gamma}{J} \alpha \frac{\Delta \eta}{\Delta \xi} \right)_{e} + max \left[ 0, - \left( \rho U \Delta \eta \right)_{e} \right] | ||
+ | </math> | ||
+ | </td><td width="5%">(11)</td></tr></table> | ||
+ | |||
+ | <table width="70%"><tr><td> | ||
+ | :<math> | ||
+ | A^{\phi}_{W} = \left(\frac{\Gamma}{J} \alpha \frac{\Delta \eta}{\Delta \xi} \right)_{w} + max \left[ 0, \left( \rho U \Delta \eta \right)_{w} \right] | ||
+ | </math> | ||
+ | </td><td width="5%">(12)</td></tr></table> | ||
+ | |||
+ | <table width="70%"><tr><td> | ||
+ | :<math> | ||
+ | A^{\phi}_{N} = \left( \frac{ \Gamma }{J} \gamma \frac{\Delta \xi}{\Delta \eta} \right)_{n} + max \left[ 0, - \left( \rho V \Delta \xi \right)_{n} \right] | ||
+ | </math> | ||
+ | </td><td width="5%">(13)</td></tr></table> | ||
+ | |||
+ | <table width="70%"><tr><td> | ||
+ | :<math> | ||
+ | A^{\phi}_{S} = \left( \frac{ \Gamma }{J} \gamma \frac{\Delta \xi}{\Delta \eta} \right)_{s} + max \left[ 0, \left( \rho V \Delta \xi \right)_{s} \right] | ||
+ | </math> | ||
+ | </td><td width="5%">(14)</td></tr></table> | ||
+ | |||
+ | <table width="70%"><tr><td> | ||
+ | :<math> | ||
+ | b^{\phi} = \left( J \Delta \xi \Delta \eta \right) \overline{S}^{\phi}_{P} - \left[ \frac{\Gamma \Delta \eta}{J} \left( \beta \frac{\partial \phi}{\partial \eta} \right) \right]^{e}_{w} - \left[ \frac{\Gamma \Delta \xi}{J} \left( \beta \frac{\partial \phi}{\partial \xi} \right) \right]^{n}_{s} | ||
+ | </math> | ||
+ | </td><td width="5%">(15)</td></tr></table> |
Latest revision as of 10:24, 20 August 2010
2D case
For calculations in complex geometries boundary-fitted non-orthogonal curvlinear grids is usually used.
General transport equation is transformed from the physical domain into the computational domain as the following equation
| (2) |
where
| (3) |
| (4) |
| (5) |
| (6) |
| (7) |
| (8) |
Using the finite volume method the trnsformed equations can be integrated as follows:
| (9) |
The convection terms are approximated as described in section http://www.cfd-online.com/Wiki/Discretization_of_the_convection_term .
Diffusion terms are approximated by the second-oder central differencing scheme.
The standard form of the finite volume equation can be obtained as
| (10) |
where
| (11) |
| (12) |
| (13) |
| (14) |
| (15) |