Viscous effects on fully coupled resonant-triad interactions: an analytical approach

This paper is concerned with viscous effects on the development of a fully coupled resonant triad consisting of Rayleigh waves. Complementary to the numerical study of Lee (1995), we attack this problem analytically. The fully coupled amplitude equations are derived with all the kernels involved being expressed in closed forms. The amplitude equations are then solved numerically. It is found that viscosity reduces the growth of the disturbance in the parametric-resonance stage and delays the final occurrence of the finite-time singularity. But viscosity does not appear to be able to eliminate the singularity. While the analysis is performed for the temporally evolving instability waves, we demonstrate its broad application by showing that it can be slightly modified to obtain the amplitude equations for the spatially growing Rayleigh waves, and the equations which describe the development of the resonant-triad of Tollmien–Schlichting waves in the fully interactive stage.