I am trying to calculate surface integral for the following problem (see figure). I have the vertex coordinates x,y,z as well as face center coordinates xm,ym,zm. If I have all the interpolated values that are needed for this calculation, then how should I approximate the surface integral for the angled surface shown above (v is velocity vector and u, v, w values are saved in xm,ym,zm coordinates)? Could you kindly describe a little? Another question, which interpolation scheme would be best for this kind of problem? If my question is not clear please let me know.
For the sake of simplicity I'd just calculate the area of the grid-cells, multiply the velocity-vectors with those areas and add the products together. That should give you some area-averaged velocity.