the formulas are contained in the ASME flow meter standards, but some manufacturers, and at least one flow meter handbook also summarize them.
A search on the internet a few years ago indicated that at least one web site list the formulas for free, but they still need to be checked against the source documentation.
The manufacturer typically furnishes a flow calculation and perhaps the software for their integral orifice tramsmitters. I believe that you can get free software from Foxboro.
As per our site wide flow specifications "Integral orifices shall be sized and installed in accordance with manufacturer's recommendations. Manufacturer shall supply meter sizing and flow calculations".