# Performance Improvement in a Wavy Twisted Rudder by Alignment of the Wave Peak
## 1 Introduction  
As environmental conditions have become highly unpredictable, the safety and reliability of ship operations have acquired unprecedented significance. For safer operation, research on the development of rudders with superior maneuverability is essential, especially in rough seas. Special rudders such as flaps and schilling rudders have been developed to improve performance by delaying the stall point and producing a high lift. Numerous studies have been conducted on their performance. [PERSON] [1] reported a 12-30% flap NACA 0018 profile using a 1.5 flap-to-linkage ratio that could generate a 50% greater lift than an all-moveable rudder of equal area. [PERSON] et al. [2] and [PERSON] [3] demonstrated that a 20% flap NACA 66 profile could achieve a 59% greater lift than the original NACA 66 profile. [PERSON] et al. [4] studied the distribution of velocity and energy behind a flap rudder based on variations in the flap angle. [PERSON] et al. [5] studied the maneuvering performance and turning ability using a numerical method in shallow and deep-sea conditions based on the increase in the performance of the rudder. [PERSON] [6] proposed that flapped rudders could provide a considerably greater lift curve slope and a 60-70% greater maximum lift compared to a conventional rudder in the same area. [PERSON] et al. [7] investigated the increase in turning ability with an increase in rudder force. A comparative study was conducted with a conventional (semi-spade) and special rudder (flap) for the same full-scale ships, as displayed in Figure 1 [8]. Maneuvering performances such as the zig-zag and stops were compared to determine the effectiveness of the special rudder. [PERSON] et al. [9] analyzed the influences of the three major properties of aflapped rudder on its hydrodynamic coefficients using Reynolds-averaged Navier-Stokes simulations. Although these special rudders offer superior maneuverability by delaying stall and generating high lift, the additional drag and risk of operational failure due to their complicated structure are a serious concern in actual operating conditions, which have not been addressed thus far. The wavy twisted rudder was newly developed and introduced in a previous study to improve the performance of the existing twisted rudder [10; 11]. Full-spade twisted rudders are extensively used in commercial ships instead of semi-spade and plain full-spade rudders because the flow-adapted twisting section of the rudder provides a more efficient hydrodynamic performance and has a lower cavitation risk under more favorable flow conditions. The wavy twisted rudder is a biomimetic technology that was modeled after a hump-back whale fin. The wavy twisted rudder was developed to remove the cavitation risk around the non-smoothing twisting leading-edge region, and to delay the stall point by making the flow more turbulent with a counter-rotating vortex pair from the wavy leading edge [12]. Numerous studies on humpback whale fin technology have been conducted. [PERSON] et al. [13] presented a morphological study and suggested that the undulations act as vortex generators, re-energizing the boundary layer. They also suggested that the undulations could increase the effective span of a finite wing by diminishing the spanwise flow component, thus reducing the strength of the wing-tip vortex. [PERSON] et al. [14] performed numerical studies on the aerodynamic characteristics of two modified NACA 0012 airfoils with different wavy surfaces using large-eddy simulations. [PERSON] and [PERSON] [15] numerically investigated the flow around the wings with spanwise waviness for Reynolds numbers Re = 1000, where several combinations of wavelength and amplitude were considered. [PERSON] et al. [16] presented wall-resolved large-eddy simulations for flow over an NACA 0021 wing with leading-edge undulations. When an undulation amplitude of 1.5% of the chord length possessing a wavelength of 21% of the chord length was applied, a 36% increase in mean lift and a 25% decrease in mean drag were observed.