Computation of symmetric, time-periodic solutions of the vortex sheet with surface tension
Computation of symmetric, time-periodic solutions of the vortex sheet with surface tension
复制标题
计算具有表面张力的涡流片的对称、时间周期解
DOI：
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发表时间：
2010
影响因子：
11.1
通讯作者：
J. Wilkening
中科院分区：
文献类型：
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作者：
D. Ambrose;J. Wilkening
介绍了一种计算具有表面张力的时间周期涡旋片的数值方法，该涡旋片可分离两种不混溶、不旋转、等密度的二维理想流体。该方法是基于最小化初始条件和假设周期的非线性泛函，除非解是周期的，在这种情况下它是零。采用了一种基于伴随的最优控制技术来有效地计算该泛函的梯度。在处理伴随公式中的奇异积分时需要特别注意。从平面静止态线性化问题的解出发，发现了一类光滑对称的呼吸体，它们以四分之一周期的时间间隔交替地通过最大动能的平面状态和所有能量以势能形式存储在界面中的静止状态。在某些情况下，在返回到初始的平面配置之前，接口会翻转。结果表明，描述这些解的分岔图包含若干条由近分岔事件分隔的不相交曲线。
A numerical method is introduced for the computation of time-periodic vortex sheets with surface tension separating two immiscible, irrotational, two-dimensional ideal fluids of equal density. The approach is based on minimizing a nonlinear functional of the initial conditions and supposed period that is positive unless the solution is periodic, in which case it is zero. An adjoint-based optimal control technique is used to efficiently compute the gradient of this functional. Special care is required to handle singular integrals in the adjoint formulation. Starting with a solution of the linearized problem about the flat rest state, a family of smooth, symmetric breathers is found that, at quarter-period time intervals, alternately pass through a flat state of maximal kinetic energy, and a rest state in which all the energy is stored as potential energy in the interface. In some cases, the interface overturns before returning to the initial, flat configuration. It is found that the bifurcation diagram describing these solutions contains several disjoint curves separated by near-bifurcation events.