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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}


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

J = \frac{\partial x}{ \partial \xi} \frac{\partial y}{ \partial \eta} - \frac{\partial y}{ \partial \xi} \frac{\partial x}{ \partial \eta}

Using the finite volume method the trnsformed equations can be integrated as follows:

\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}

The convection terms are approximated as described in section .

Diffusion terms are approximated by the second-oder central differencing scheme.

The standard form of the finite volume equation can be obtained as

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}


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]
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]
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]
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]
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}
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