“…, u, v, w, p are density, velocities in axial, radial, and azimuthal directions, and pressure, respectively. It is worth noting that the velocity equation in azimuthal direction in previous paper 36,37 is cast in terms of w 0 t ¼ @w 0 =@t. Compared to that, the velocity equation for w 0 is expressed directly in MMLEE in this paper. From this formulation, we can know that it not only can account for multi-frequency contribution, but also allows a constant swirl background flow (w 0 6 ¼ 0); therefore, it has wider applications than the single-frequency formulation.…”