The wavefronts are still circular.
Work Step by Step
Try it and see! The waves are formed in the reference frame of the water. The water (and the circular waves) simply travel downstream together. Here's another way to look at it. Picture a moving cruise ship, with a child tossing a toy into the onboard pool, which makes circular waves. From the viewpoint of an observer on shore, this is analogous to the situation described in this question.