Based on a quasioptical approach and direct particle-in-cell simulations, we study dynamics of oversized relativistic surface-wave oscillators (SWOs) of the Cherenkov type with 2D periodical corrugated structures of cylindrical geometry. Such corrugation allows significant rarefication of the spectrum of modes with different azimuthal indices. As a result, selective excitation of a mode with a given azimuthal index is possible. Azimuthal index of the generated mode depends on the voltage rise time. For short (nanosecond scale) rise time, generation of an azimuthally symmetric mode can be realized. For longer (hundreds nanoseconds to microseconds) rise time, the modes with high azimuthal indexes would be excited. These conclusions are supported by the experiments where Ka-band SWOs with 2D corrugated structures were realized based on the $300\text{ }\text{ }\mathrm{keV}/100\text{ }\text{ }\mathrm{A}/4\text{ }\text{ }\ensuremath{\mu}\mathrm{s}$ thermionic accelerator SATURN. For an oversize factor of 16, stable narrow-band generation with output power of 1.5--2 MW was obtained at the frequency of 32.5 GHz corresponding to the mode with an azimuthal index of $m=3$. The project of Ka-band subgigawatt power SWOs operating at the azimuthally symmetric mode based on $500\text{ }\text{ }\mathrm{keV}/4\text{ }\text{ }\mathrm{kA}/20\text{ }\text{ }\mathrm{ns}$ high current explosive-emission accelerator SINUS-6 is under development.