Can anyone help me find an equation for volumetric flow rate as a function pressure differential across an orifice in a curved channel?

If the duct is large in proportion to the orifice then just use a std orifice equation. The flow will stall and recover around the orifice assuming that the diameter is sufficient to restrict volumetric flow (choked flow).

well i believe that if the radius of the curved channel is small, try going with eccentric orifice. The eccentric point again depends on the radius of curved channel which needs to be calculated. Never thought of such scenario though, good question. If you are looking to have a concentric orifice then you will have different calculation. Exactly what parameters you have to find flow rate? You can find equations(calculations) in API 5167-1 and you can refer part 2 for its design.