Calculating surface integral on Cartesian grid

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.

[Bjorn Gustavsson]

Is that integral:

or

it will make a bit of a difference...

[Asif Shahriar Nafi]

It is the second one

Thanks a lot

Answer 1

[Bjorn Gustavsson]

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.

HTH

[Asif Shahriar Nafi]

Thanks