The sediment transport generated by some environmental flow is intense enough to have a deep influence on the flow itself, due to the modifications occurring to the erodible interface. This process is challenging to reproduce in a numerical environment, due to the simultaneous presence of moving boundaries and turbulence structures. Motivated by the need for a three-dimensional model able to deal with the erosion generated by gravity currents, the coupling of a turbulent flow with a mobile bed is developed. The flow is numerically simulated by means of the large eddy simulation, and the evolution of the interface between the bed and the fluid flow is tracked using the level-set method. The implementation of the level-set method using both the upwind and the essentially non-oscillatory schemes is presented. The treatment of the moving boundary is handled using an immersed boundary technique over a fixed Cartesian grid. The level-set method provides a simple and efficient way to manage the time evolution of the geometry and is used to handle the complexity of the moving immersed interface. A channel flow over a sinusoidal bed is simulated for validation purposes, and the numerical results concerning the bed deformation due to the flow shear are discussed. Furthermore, the influence of the motion of the sinusoidal bed on the flow is studied as a separate test case.