If the pipe is sufficiently thin relative to the wavelength of the vibrations, it makes no difference whether it is straight or bent. For example long organ pipes are often made with sharp 90 or 180 degree bends to fit them into the available space, and this has no practical effect on the sound the produce.
For a pipe open at both ends, the relative phase of the velocity at each end is arbitrary. At resonance, the two velocities can be either in phase or out of phase.
If you bend the pipe and connect the ends, the velocities at the two "ends" much be in phase with each other. If they are not, the pressure at that point will not stay constant at zero.
So if the open pipe modes have frequencies 0, f, 2f, 3f, etc., the donut modes will be 0, 2f, 4f, etc.
However, apart from the zero frequency, the donut modes can have their nodes and antinodes at any position around the ring (the donut doesn't have "ends" like an open pipe). So a resonance with a frequency 2f, 4f, can be considered as an arbitrary linear combination of two modes with the same frequency, but with nodes and antinodes at different positions around the donut.