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Calculation on non-orthogonal curvelinear structured grids, finite-volume method

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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 (x,y) into the computational domain \left( \xi , \eta \right) as the following equation


 
\frac{\partial}{\partial \xi} \left( \rho U \phi \right) + \frac{\partial }{ \partial \eta } \left( \rho V \phi \right) = \frac{\partial}{\partial \xi} \left[ \frac{\Gamma }{J} \left( \alpha \frac{\partial \phi}{\partial \xi} - \beta \frac{\partial \phi}{ \partial \eta} \right) \right] + \frac{\partial}{\partial \eta} \left[ \frac{\Gamma}{J} \left( \gamma \frac{\partial \phi}{\partial \eta} - \beta \frac{\partial \phi}{ \partial \xi} \right) \right] + J S^{\phi}
(2)

where

 
U = \overline{u} \frac{\partial y}{\partial \eta} - \overline{v} \frac{\partial x}{\partial \eta}
(3)
 
V = \overline{v} \frac{\partial x}{ \partial \xi} - \overline{u} \frac{\partial y}{ \partial \xi}
(4)
 
\alpha = \left( \frac{\partial x}{\partial \eta } \right)^2 + \left( \frac{\partial y}{\partial \eta } \right)^2
(5)
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