A seal assembly is provided that extends along an axial centerline, and includes a rotor seal element that engages with a stator seal element. The rotor seal element extends radially between an inner element side and an outer element side, and includes a channel, a plurality of first passages and a plurality of second passages. The channel extends radially into the rotor seal element from the inner element side. The first and the second passages are fluidly coupled with the channel. Each of the first passages extends radially through the rotor seal element to a respective first passage outlet. Each of the second passages extends axially through the rotor seal element to a respective second passage outlet.