The present study developed an unsteady RANS approach based on the lattice Boltzmann method (LBM), which can perform direct aeroacoustic simulations of low-speed fans at lower computational cost compared with the conventional LBM-LES approach. In this method, the k-? turbulence model is incorporated into the LBM flow solver, where the transport equations of k and ? are also computed by the lattice Boltzmann method, similar to the Navier-Stokes equations. In addition, moving boundaries such as fan rotors are considered by a direct-forcing immersed boundary method. This numerical method was validated in a two-dimensional simulation of a cross-flow fan. As a result, the simulation was able to capture an eccentric vortex structure in the rotor, and the pressure rise by the work of the rotor can be reproduced. Also, the peak sound of the blade passing frequency can be 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.