I have the following equation for the net mass flow out of a control volume through a surface $S$ - $$\int \int_S p \overrightarrow{V} \cdot \overrightarrow{d}S$$
(Actually there should be an ellipse lying across both of those integral signs but the latex code doesn't seem to work for it).
Why do we need a double integral? Surely one integral sign should suffice as we are summing up all the infinitesimal surfaces $dS$ over the surface $S$?
You are integrating over a closed surface $S$, hence the oval. It's an extension of the loop integral $\oint$ notation to closed surface integrals.
In higher physics we generally drop all but one integral sign and don't bother with circles or ovals. The appearance of more than one integral either points to the level of the material or the author wanting to make a point.