When simulating a jet flame and due to lack of computational capacity only simulate a quarter of the 3-dimensional model, how is the hydraulic diameter defined. If the edge boundry conditions are defined as periodic are the hydraulic diameter calculated from the quarter of the circle or the whole circle?

Help would be highly appreciated!

I believe it should be calculated from the dimensions of the quarter itself. Please note that you should be specifying the flow rate through inlet only for the quarter. i.e. (Total flow rate)/4. In case you're specifying velocity it should be specified as the same.