Weakly nonlinear incompressible Rayleigh-Taylor instability in spherical geometry

Abstract

In this research, a weakly nonlinear (WN) model for the incompressible Rayleigh-Taylor instability in cylindrical geometry [Wang et al., Phys. Plasmas 20, 042708 (2013)] is generalized to spherical geometry. The evolution of the interface with an initial small-amplitude single-mode perturbation in the form of Legendre mode (Pn) is analysed with the third-order WN solutions. The transition of the small-amplitude perturbed spherical interface to the bubble-and-spike structure can be observed by our model. For single-mode perturbation Pn, besides the generation of P2n and P3n, which are similar to the second and third harmonics in planar and cylindrical geometries, many other modes in the range of P0–P3n are generated by mode-coupling effects up to the third order. With the same initial amplitude, the bubbles at the pole grow faster than those at the equator in the WN regime. Furthermore, it is found that the behavior of the bubbles at the pole is similar to that of three-dimensional axisymmetric bubbles, while the behavior of the bubbles at the equator is similar to that of two-dimensional bubbles.

Simulation of the Weakly Nonlinear Rayleigh-Taylor Instability in Spherical Geometry*

“…The RTI in ICF or astrophysics concerns nonplanar geometry. To describe the nonplanar WN RTI, Wang et al [11] extended the WN model to cylindrical geometry, and Zhang et al [12] extended the WN model to spherical geometry. The spherical WN model shows that mode coupling processes are more complicated than those in planar cases.…”

“…If we define the wave number to be the square root of the Laplacian eigenvalue of Legendre polynomial 𝑃 𝑛 (cos 𝜃), i.e., 𝑘 𝑛 = 2𝜋/𝜆 𝑛 = √︀ 𝑛(𝑛 + 1)/𝑅 0 , and denote the equivalent Atwood number as 𝐴 ′ 𝑇 = (𝜌 ex − 𝜌 in )/( √︀ (𝑛 + 1)/𝑛𝜌 in + √︀ 𝑛/(𝑛 + 1)𝜌 ex ), we again obtain 𝛾 𝑛 = √︀ 𝐴 ′ 𝑇 𝑔𝑘 𝑛 . If the equations are solved to the third order [12] , mode coupling processes at the second and the third order will be recovered, which generate modes besides 𝑃 𝑛 . From here on, we use 𝑃 𝑛 to denote the initial perturbation mode, and use 𝑃 l to denote the modes involved in the coupling processes.…”

“…However, it shall be noticed that the modes from the (3𝑛 + 1)th to the 4𝑛th are non-negligible, and also for 𝑃 9 , the even modes from the (2𝑛+1)th to the 3𝑛th (i.e., 20th, 22th, 24th, 26th) are non-negligible. These modes can not emerge from the 2nd-and the 3rd-order coupling theory, [12] which indicates that higher order coupling processes can not be neglected.…”

Interface Width Effect on the Weakly Nonlinear Rayleigh–Taylor Instability in Spherical Geometry

“…When the perturbation driven by the external gravity or acceleration grows, several modes emerge as a result of mode coupling, i.e., 𝜂(𝜃, 𝑡) = ∑︀ 𝑙 𝜂 𝑙 (𝑡)𝑃 𝑙 (cos 𝜃). Zhang et al [15] extended the third-order WN model to spherical geometry and studied the mode coupling processes. In this model, the perturbation and the velocity potentials are expanded into the power series of a formal parameter 𝜖 for each mode 𝑃 𝑙 (cos 𝜃),…”

Weakly Nonlinear Rayleigh–Taylor Instability in Cylindrically Convergent Geometry

“…[15] The WN theory can at least help us improve the physical understanding of the onset of nonlinearity (i.e., mode-coupling effect). The RTI growth in planar, [16−18] cylindrical, [19] and spherical [20] geometry have been studied through a third-order WN theory for fixed interface. Recently, the WN effect of BP growth for a uniformly converging cylinder (purely converging geometry) [21,22] is investigated.…”