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production term on staggered grid

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Old   April 24, 2011, 17:30
Cool production term on staggered grid
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Hi everybody,
I'm writing a code for simulation of a turbulent flow using finite volume, fully staggered grid and the standard k-epsilon equations. Staggered grids have the advantage of being conservative of kinetic energy , which becomes important for turbulent flows. But I have a problem in calculating the production term in the k and epsilon equations. As you know the production term involves calculating u and v gradients with respect to x and y at the center of the volumes. Now, \frac{\partial u}{\partial x} and \frac{\partial v}{\partial y} can be easily found from central differencing:
\frac{\partial u}{\partial x}=\frac{u_e-u_w}{\Delta x}
\frac{\partial v}{\partial y}=\frac{v_n-v_s}{\Delta y},
but i can't find an easy way for finding \frac{\partial u}{\partial y} and \frac{\partial v}{\partial x}, since we don't have the u velocities at north and south faces and v velocities at east and west faces.
Sure I can find these gradients on the corners of the volumes and then average them. But I don't think this is a systematic way and whether this will ruin the whole conservativeness property of the scheme???
I have been thinking about this problem for several weeks now, but I think I lack the knowledge and experience for finding a goodway out of it. Any help is appreciated.
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finite volume, k-epsilon, production term, staggered grid, turbulence

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