In this study, we developed an unsteady RANS approach based on the lattice Boltzmann method (LBM) to perform direct aeroacoustic simulations of low-speed fans at a lower computational cost compared with the conventional LBM-LES approach. In this method, the k-ω turbulence model is incorporated into the LBM flow solver, and the transport equations of k and ω are computed using the LBM, similar to the Navier-Stokes equations. In addition, moving boundaries, such as fan rotors, are considered using a direct-forcing immersed boundary method. This numerical method was validated through a two-dimensional simulation of a cross-flow fan. The simulation was able to capture an eccentric vortex structure in the rotor, and a pressure rise due to the work of the rotor could be reproduced. In addition, the peak sound of the blade-passing frequency was successfully predicted by the present method. Furthermore, the simulation results showed that the peak sound is generated by the interaction between the rotor blade and the flow around the tongue part of the casing.