The net current (streaming) in a turbulent bottom boundary layer under waves above a flat bed, identified as potentially relevant for sediment transport, is mainly determined by two competing mechanisms: an onshore streaming resulting from the horizontal non-uniformity of the velocity field under progressive free surface waves, and an offshore streaming related to the nonlinearity of the waveshape. The latter actually contains two contributions: oscillatory velocities under nonlinear waves are characterized in terms of velocity-skewness and acceleration-skewness (with pure velocity-skewness under Stokes waves and acceleration-skewness under steep sawtooth waves), and both separately induce offshore streaming. This paper describes a 1DV Reynolds-averaged boundary layer model withk-εturbulence closure that includes all these streaming processes. The model is validated against measured period-averaged and time-dependent velocities, from 4 different well-documented laboratory experiments with these processes in isolation and in combination. Subsequently, the model is applied in a numerical study on the waveshape and free surface effects on streaming. The results show how the dimensionless parameterskh (relative water depth) and A/kN (relative bed roughness) influence the (dimensionless) streaming velocity and shear stress and the balance between the mechanisms. For decreasing kh, the relative importance of waveshape streaming over progressive wave streaming increases, qualitatively consistent with earlier analytical modeling. Unlike earlier results, simulations for increased roughness (smaller A/kN) show a shift of the streaming profile in onshore direction for all kh. Finally, the results are parameterized and the possible implications of the streaming processes on sediment transport are shortly discussed.