## 1 Introduction  
More and more people are traveling by air, so the demand for economical and environmentally friendly aircraft is also increasing. Meeting these needs will require improvements in aerodynamic efficiency and in reduced drag. In addition, these goals must be met while addressing environmental concerns, such as reducing carbon dioxide emissions to help combat global warming. Since skin friction drag constitutes over half of the total drag on an aircraft [1], reducing it is key to both increasing aerodynamic performance and reducing fuel consumption. Riblets are one of the methods used to reduce aircraft surface drag [1; 2; 3; 4]. They are a passive means of turbulent flow control near the wall and were first used by the NASA Langley Research Center in the 1970s [3; 5]. They imitate the fine structure of the skin of a shark, which is known for being able to swim long distances at high speed. Riblets are fine grooves aligned in the direction of a flow across a surface that induce a streamwise vortex, which reduces the turbulent flow in a boundary layer that causes surface drag, thus reducing skin friction drag. To date, much research has studied the mechanism [3; 4; 5; 6] and effectiveness [7] of riblets in reducing skin friction drag, with many industrial applications to wind turbine blades, railways, and aircraft [8]. These studies have clarified a number of design criteria, such as the height, width, and skewness of the peak shape of the riblets in reducing skin friction drag for flowfields of interest. The 3M company has demonstrated riblets that reduced skin friction drag by 4% to 6% over the flowfield with no pressure gradients or zero incidence [8]. With the aid of computer simulation and high-resolution measurements, riblets can reduce skin friction drag by 10% by optimizing the geometry of the riblets for the flowfield [9; 10; 11; 12]. The Airbus company has conducted wind tunnel tests with riblets attached to an entire airframe surface and has demonstrated a reduction in skin friction drag of 2% [1; 13]. According to an estimate by the Airbus company, even a 1% drag reduction can reduce annual operating costs by more than a million dollars per aircraft [1; 13]. Therefore, reducing skin friction drag has a significant economic impact [4]. To improve the drag-reducing performance further, multi-dimensional riblet shapes, such as two-dimensional 'blade riblets', which aligned a row of thin fences along the streamwise direction [12], and three-dimensional zigzag riblets [14], and sinusoidal riblets [9; 14; 15; 16; 17], have emerged. Among those intensive past studies, three-dimensional sinusoidal riblets [9; 14; 15; 16; 17] are found to be more effective compared to conventional straight-shaped riblets. [PERSON] et al. [14] investigated the influence of wavelengths and amplitudes of sinusoidal riblets on the effectiveness of drag-reducing performance and concluded that certain combinations of a height/spacing ratio can improve drag reduction. However, the drag-reducing performance by the straight-shaped riblets could become comparable once the geometry of the riblet grooves aligned to the streamwise direction is optimized for the flowfield of interest. Additionally, three-dimensional riblet shapes may cause adverse pressure gradients, which results in the benefit of drag-reducing ability being suppressed by inducing additional turbulence in the boundary layer. [PERSON] et al. [17] addressed the design criteria for three-dimensional sinusoidal riblets with respect to their amplitude and wavelength and showed that approximately a 2% drag-reduction improvement can be expected over a conventional straight-shaped riblet. Thus, three-dimensional riblets with well-designed geometry for a flowfield of interest are considered to be effective for the reduction of skin friction drag.