We present a numerical model for simulating boundary-layer turbulence bounded by a dynamic ocean surface and driven by wind stresses. The nonlinear boundary conditions are satisfied on the freely moving surface exactly without any approximation. The time-dependent physical space is mapped onto a rectangular computational domain where the momentum equations are integrated in time for the velocity field and the solenoidal condition is satisfied by solving the transformed pressure Poisson equation. The developed model is validated with a progressive gravity wave and with a gravity-capillary wave. The model capabilities for resolving surface waves of various length scales, ranging from the gravity waves to the capillary ripples, and the nonlinear interactions among the waves are highlighted by simulating the evolution of two-dimensional short-crested gravity-capillary waves and the development of a gravity wave train into crescent patterns. The potential applicability of the model for studying the wind-driven aqueous near-surface layer is then demonstrated. The simulation results reveal fine surface structures, including microscale breaking waves and streamwise velocity streaks, which have been observed in the laboratory and field experiments.