The Effect of the Second-order Difference-frequency Wave Force on the Motions of an Underwater Vehicle Near the Surface
Authors: Qin Rui , Lu Haining , Chen Jiahua 
Published: December 2022

_
When an underwater vehicle performs tasks, such as power recharge, air exchange, deployment, and retrieval near the surface, the low-frequency suction force and pitch moment will interfere with its motion. This paper presents a study on the effect of second-order difference-frequency wave force on the motions of the DARPA SUBOFF model at various submergence depths and environmental conditions. Ignoring the second-order wave force component induced by the second-order difference-frequency force may underestimate the second-order effect in irregular waves in the time domain. Comparisons were made between Newman’s approximation, which sets the off-diagonal difference-frequency quadratic transfer function (QTF) value to the average over the corresponding diagonal values, and the full QTFs based on the Pinkster approximation in the frequency and time domains. The QTFs-based prediction is significantly larger values than Newman’s approximation. Time-domain simulations were conducted, to investigate the effect of the second-order difference-frequency wave loads on motion responses at different nondimensional submergence depths and various sea conditions. The results of the computation indicate that ignoring the second-order difference-frequency wave force may underestimate the motion and dynamic responses of underwater vehicles; in addition, the pitch and heave motion responses are dominated by the low-frequency response.

Introduction
When an underwater vehicle travels close to the water surface, it experiences an upward force and pitch moment, known as surface suction. This can significantly affect the underwater vehicle’s behavior when power recharge, air exchange, deployment, and retrieval are incorporated into the underwater vehicle simulations. Veillon et al. (1996) noted that for a 10,000-ton submarine at a depth of 50 m, the surface suction due to waves will require around 20–30 tons of compensation to stop the submarine from surfacing. In addition, Hirom (1974) indicated that in waves, the fluctuating forces on a submarine can be on the order of 1000 kN, with a steady component of approximately 10 kN. In calm water, suction occurs owing to a higher flow velocity and a lower pressure above an underwater vehicle when sailing near the surface, and the wave pattern generated by the underwater vehicle complicates the surface effect. The longitudinal position of the center of pressure is after the amidships, which induces the pitch moment. In the presence of waves, the first-order wave force, which is the oscillatory force at the wave frequency, acts on the underwater vehicle. In addition, waves and model motions induce low-frequency suction force and pitch moment. Using any internal compensation and control surfaces to offset the induced suction force and pitch moment requires a good and early understanding of the magnitude of these forces, for designing sufficient-capacity compensation tanks (Crossland 2013). Therefore, the surface suction and pitch moment effects are essential components of the hydrodynamic characteristics of underwater vehicles sailing near the water surface.