Membrane-less electrolysis is a vital technique for gaseous chemical production, particularly the Hydrogen Evolution Reaction (HER). However, conventional planar electrodes often suffer from bubble adhesion, which masks active sites and creates ``dead areas,'' thereby increasing ohmic resistance and reducing overall efficiency. Consequently, the timely removal of gas bubbles is critical for optimizing performance. Building on experimental evidence suggesting that conical morphologies enable the directional transport of hydrogen bubbles toward the electrode base, this study provides a high-fidelity numerical investigation into the underlying mechanisms. The simulation setup allows for the precise study of bubble growth and motion on the conical electrode. We address this complex two-phase problem using 3D direct numerical simulation (DNS) via the free code repository Basilisk (http://basilisk.fr/). Specifically, we employ embedded boundary conditions to accurately model the conical geometry within a Cartesian coordinate system, integrated with a verified phase-change model to resolve hydrogen diffusion and bubble growth. Our investigations reveal how conical geometry dictates bubble behavior, specifically focusing on growth, self-driven transport, and coalescence. While the unidirectional transport of bubbles on cones is theoretically attributed to the interplay between Laplace pressure and buoyancy, our DNS approach provides a more rigorous, quantitative assessment of these driving forces. These findings offer insights into optimizing electrode morphology to minimize ``dead areas'' and maximize efficiency in gaseous electrolysis.