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

## 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)
 $\gamma = \left( \frac{\partial x}{ \partial \xi } \right)^2 + \left( \frac{\partial y}{ \partial \xi } \right)^2$ (6)
 $\beta = \frac{\partial x}{ \partial \xi} \frac{\partial x}{ \partial \eta} + \frac{\partial y}{ \partial \xi} \frac{\partial y}{ \partial \eta}$ (7)

 $J = \frac{\partial x}{ \partial \xi} \frac{\partial y}{ \partial \eta} - \frac{\partial y}{ \partial \xi} \frac{\partial x}{ \partial \eta}$ (8)