The Richtmyer-Meshkov instability on a three-dimensional (3D) single-mode interface accelerated by a planar shock is numerically studied and the emphasis is placed on the effect of principle curvatures of the initial interface. A 3D numerical method is developed by implementing the Weighted Essentially Non-Oscillation scheme and the Level-Set method combined with real Ghost Fluid Method, and is validated by simulating typical shock-interface interactions. Three single-mode interfaces including a two dimensional (2D) single-mode interface, a typical 3D single-mode interface (3D+) and a minimum surface featured single-mode interface (3D-) are considered. Due to different combinations of interfacial principle curvatures, the three interfaces present distinct morphologies. It can be seen in the 3D+ case, shock convergence and divergence at the interface are more severe than that in the 2D case, and in the 3D- case, shock pattern is more complex and shock reflection at the boundary suppresses the interface amplitude growth. Quantitatively, the growth rate at the symmetry plane is measured and it is found that, comparing with the 2D single mode case, identical signed principle curvatures in the 3D+ case enhance the perturbation growth, while opposite signed principle curvatures in the 3D- case weaken the growth. It is also found that the growth rates are in good agreement with the ones predicted by the extended 3D implusive model proposed by Luo. et.al.. It is more interesting to see that the 3D effect existed in the interface evolution is significant, especially the interlayer momentum transition can not be neglected.
