Interface Effects on Harmonics of the Weakly Nonlinear Stage in Rayleigh-Taylor Instability

To better understand the fluid interface effects in different spatial coordinate systems on harmonics of the weakly nonlinear stage in Rayleigh-Taylor instability (RTI), employing the method of the parameter expansion up to the third order, this paper analytically investigated harmonics of the weakly nonlinear stage in classical Rayleigh-Taylor instability on spherical interface, and compared the spherical results with the cylindrical and the planar ones. The results show that the amplitudes of the first four harmonics will recover those in planar RTI as the initial radius of the interface tends to be infinity compared against the initial perturbation wavelength. When the initial radius is small, both the Bell-Plesset effect induced by curvature of the initial interface and the space effect including the planar, cylindrical and spherical geometries make a tremendous impact on the harmonic development. That is to say, the interface effect between two fluids in different coordinate systems plays an important role for the development of the first four harmonics in RTI. For the first four harmonics, the smaller the initial radius is, the faster they grow. This trend is more remarkable for spherical geometry than cylindrical one. Also, the second-order feedback to the zeroth harmonic for the cylindrical and spherical RTI strengthens the contract of the initial unperturbed interface.