How does the free surface affect the hydrodynamics of a shallowly submerged submarine?

It has long been recognized that a shallowly submerged submarine traveling beneath the free surface experiences a larger resistance force in conjunction with a lift force and a pitch moment, which all vary periodically with respect to Froude number. As is well known, the periodic behavior of the forces and moment mainly has to do with the interference effects between the dominant wave systems inside the submarine wake, which predominantly originate from the bow, stern and shoulders. In naval architecture, the principal type of interference is typically considered between the bow and stem waves, where the geometry undergoes abrupt changes. However, as the aft shoulder of a shallowly submerged submarine operates in a closer proximity to the free surface compared to the stem, it is surmised that interference between the bow and aft shoulder waves may have a more significant effect on the behavior of the forces and moments. Accordingly, the main purpose of the present study is to investigate whether the interaction between the bow and aft shoulder waves or the interaction between the bow and stem waves has a more dominant effect on the hydrodynamic behavior of a shallowly submerged submarine. In this regard, the straight-ahead simulations of a generic submarine with constant forward velocities are performed in commercial code STARCCM + using URANS equations with a Reynolds stress turbulence model at submergence depths and Fronde numbers ranging from h = 1.1D to h = infinity (D: submarine diameter) and from Fn = 0.205 to Fn = 0.512, respectively. The numerical model is partially validated against the existing experimental resistance force data. The analysis of the obtained results demonstrates that in case of the shallowly submerged submarines, the interaction between the bow and aft shoulder waves has a dominant effect on the behavior of the resistance force, lift force and pitch moment.