Nonlinear free vibration of a pipe conveying fluid around the curved equilibrium configuration resulting from the supercritical flow speed is investigated with the emphasis on dynamics of internal resonances.The governing equation for the pipe system, a nonlinear integro-partial-differential model with variable coefficients, is truncated into a discrete perturbed gyroscopic system via the Galerkin method.A condition for the three-to-one internal resonance is established, and the condition implies that internal resonance is possible in the supercritical regime.