How to calculate inventory turnover without Beginning inventory?
Are there any other formula to calculate inventory turnover other than COGS/Average inventory?

http://www.investopedia.com/terms/i/inventoryturnover.asp

You can use Sales/Inventory.

If you don't have beginning inventory, just use ending inventory. It won't give you the best picture, but it will be better than nothing.

Or you can calculate beginning inventory if that is all that you are missing.

Beg Inventory + Purchases – COGS = Ending Inventory