I am stuck in the design phase - what is a good software program to assist with determining whether the head at the downstream inlets causes a tailwater upstream?
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I have been trying hec-ras for a time and seems to fit your question.
HY-8 is FHWA Culvert program
HEC-RAS is COE hydraulic modeling program with detailed culvert analysis equations.
if you don't want to download anything then try: