  We study the necessary condition under which a resonantly driven exciton
polariton superfluid flowing against an obstacle can generate turbulence. The
value of the critical velocity is well estimated by the transition from
elliptic to hyperbolic of an operator following ideas developed by Frisch,
Pomeau, Rica for a superfluid flow around an obstacle, though the nature of
equations governing the polariton superfluid is quite different. We find
analytical estimates depending on the pump amplitude and on the pump energy
detuning, quite consistent with our numerical computations.
