Nonlinear deformations of a two-dimensional gas bubble are investigated in the framework of a Hamiltonian formulation involving surface variables alone. The Dirichlet–Neumann operator is introduced to accomplish this dimensional reduction and is expressed via a Taylor series expansion. A recursion formula is derived to determine explicitly each term in this Taylor series up to an arbitrary order of nonlinearity. Both analytical and numerical strategies are proposed to deal with this nonlinear free-boundary problem under forced or freely oscillating conditions. Simplified models are established in various approximate regimes, including a Rayleigh–Plesset equation for the time evolution of a purely circular pulsating bubble, and a second-order Stokes wave solution for weakly nonlinear shape oscillations that rotate steadily on the bubble surface. In addition, a numerical scheme is developed to simulate the full governing equations, by exploiting the efficient and accurate treatment of the Dirichlet–Neumann operator via the fast Fourier transform. Extensive tests are conducted to assess the numerical convergence of this operator as a function of various parameters. The performance of this direct solver is illustrated by applying it to the simulation of cycles of compression-dilatation for a purely circular bubble under uniform forcing, and to the computation of freely evolving shape distortions represented by steadily rotating waves and time-periodic standing waves. The former solutions are validated against predictions by the Rayleigh–Plesset model, while the latter solutions are compared to laboratory measurements in the case of mode-2 standing waves.

中文翻译：

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模拟二维​​气泡非线性变形的边界摄动方法

在仅涉及表面变量的哈密顿公式的框架中研究了二维气泡的非线性变形。引入狄利克雷-诺依曼算子来完成这种降维，并通过泰勒级数展开来表达。推导递归公式以明确确定该泰勒级数中的每一项直至任意阶的非线性。提出了解析和数值策略来处理强制或自由振荡条件下的非线性自由边界问题。在各种近似状态下建立了简化模型，包括用于纯圆形脉动气泡时间演化的瑞利-普莱塞方程，以及用于在气泡表面稳定旋转的弱非线性形状振荡的二阶斯托克斯波解。此外，还开发了一种数值方案来模拟完整的控制方程，通过快速傅里叶变换利用狄利克雷-诺依曼算子的高效和准确处理。进行了大量的测试来评估该算子作为各种参数的函数的数值收敛性。该直接求解器的性能通过将其应用于均匀受力下纯圆形气泡的压缩-膨胀循环的模拟以及以稳定旋转波和时间周期驻波为代表的自由演化形状畸变的计算来说明。前一种解决方案根据 Rayleigh-Plesset 模型的预测进行验证，而后一种解决方案则与 2 型驻波情况下的实验室测量结果进行比较。