The nonlinear saturation amplitude of two-mode Rayleigh–Taylor instability in plane geometry

A weakly nonlinear theoretical model is established for the two-dimensional incompressible two-mode Rayleigh–Taylor instability (RTI). An exact third-order analytical solution of the small perturbed interface is derived. The root mean square of all perturbation amplitudes is employed to define the nonlinear saturation amplitude (NSA) of the two-mode RTI. It is found that the two-mode feedback becomes significant in affecting the nonlinear saturation of both modes and even their rms amplitudes within specific ranges of the wavenumber ratio and initial amplitude ratio. The dependence of the NSA of rms amplitude on the Atwood number, the wavenumber ratio, and the initial amplitude ratio is obtained. When two-mode feedback becomes noticeable, an approximate analytical solution for the NSA of rms amplitude is derived and compared with the Haan model at different wavenumber ratios. In addition, a fitting model for the NSA of the rms amplitude over a wider range is obtained using the least squares method. Finally, the theoretical model is compared with the numerical simulation using high-order and low-dissipation methods and exhibits good agreement in the linear and weakly nonlinear stages.