IMECH-IR  > 流固耦合系统力学重点实验室
非线性水弹性波的稳定性及动力学研究
Alternative TitleStudy on Stability and Dynamics of Nonlinear Hydroelastic Waves
孟洋涵
Thesis Advisor王展
2022-05-20
Degree Grantor中国科学院大学
Place of Conferral北京
Subtype硕士
Degree Discipline流体力学
Keyword非线性水弹性波,孤立波,动力学响应,渐近分析,直接数值模拟
Abstract

本文利用渐近方法和数值计算对非线性水弹性波的稳定性及动力学进行了 研究,研究的重点包括两个方面:一是在三维情形下发展长波模型对卫星雷达所 观测到的 lump 及其产生机理进行探究;二是发展截断模型模拟浮冰在移动载荷作用下的非线性动力学响应。针对第一个问题,考虑到三维完全非线性欧拉方程 的复杂性,我们在长波假设下采用经典水波的非局部方法推导了Benney-Luke 型方程,并从该方程出发利用多重尺度方法得到了描述波包和波致均匀流耦合演化的Benney-Roskes-Davey-Stewartson (BRDS) 系统。数值计算结果表明Benney-Luke 型方程中存在与BRDS系统中单参数孤立子族相对应的三类孤立波解,分别为平面孤立波、lump 以及横向周期孤立波。这三类孤立波通过维数破裂分岔相连接,平面孤立波与 lump 被视为两种极限情形而横向周期孤立波为中间态。 同时通过对 Benney-Luke 型方程进行时间积分,我们细致地探究了孤立波的稳定性及相互作用。当局域化载荷以恒定速度在冰面上运动时该简化系统存在一种跨临界机制,在此速度范围内不存在稳态解,而会出现 lump 的周期性脱落现象。 其次,在研究任意水深二维理想流体顶部浮冰的动力学响应时,本文考虑了非线性、惯性以及粘性的影响,通过对相关的拟微分算子进行展开并将非线性项保留至三阶把完全非线性问题转化为与自由面上变量相关的三阶截断模型。为了验证该简化模型的准确性,我们重点关注了自由孤立波解。在不考虑阻尼的情况下建立三阶非线性 Schrödinger (NLS) 方程以预测原始方程中自由孤立波解的存在性和三阶截断模型的准确性。相比于二阶模型,三阶截断模型的优势在于其对应的三阶NLS 具有准确的非线性项系数,在最小相速度附近能够更好地模拟冰层响应。同时,对自由孤立波解的数值计算表明三阶截断模型在任意水深下的准确性均高于二阶模型。最后基于三阶截断模型对不同载荷速度下冰层的挠度及应 变进行数值计算并将数值结果与实验记录进行对比,实现了两者之间的良好一 致性。

Other Abstract

The stability and dynamics of the nonlinear hydroelastic waves are studied via asymptotic analysis and numerical computation in this thesis. The focus of the research includes two aspects: developing a long-wave model in three dimensions to investigate the lumps observed by satellite radar and their generation mechanisms, and exploring the nonlinear dynamic response of a floating ice sheet by a moving load based on the cubic-truncation model. For three-dimensional problem, considering the complexity of the original equations a Benney-Luke-type equation is derived via an explicit non-local formulation of the classic water wave problem under the assumption of shallow water. The normal form analysis is carried out for the newly developed equation, which results in the Benney-Roskes-Davey-Stewartson (BRDS) system governing the coupled evolution of the envelope of a carrier wave and wave-induced mean flow. Numerical results show there are three types of solitary waves in the Benney-Luke-type equation all of which are predicted by the one-parameter solution of the BRDS system: plane solitary wave, lump, and transversally periodic solitary wave. They are linked together by a dimension-breaking bifurcation where the plane solitary waves and lumps are viewed as two limiting cases, and the transversally periodic solitary waves serve as intermediate states. The stability and interaction of solitary waves are investigated via a numerical time integration of the Benney-Luke-type equation. For a localized load moving on the ice sheet with a constant speed, it is found that there exists a transcritical regime of the forcing speed for which there are no steady solutions. Instead, the periodic shedding of lumps are observed if the load moves at a speed in this range. In the second part, vibration of a floating ice sheet on top of two-dimensional ideal fluid of arbitrary depth are studied when the effects of nonlinearity, inertia and damping are all considered. The fully nonlinear problem is reduced to a cubic-truncation model involving variables on the free surface by expanding relevant pseudo-differential operators and retaining nonlinear terms up to the third order. To validate the accuracy of the reduced model, we focus on the free solitary waves. In absence of damping, the nonlinear Schrödinger (NLS) equation is derived to predict the existence of free solitary waves in the primitive equations and the accuracy of the cubic-runcation model. The main advantage of cubic-truncation model over the quadratic-truncation model is that the resultant NLS equation has correct coefficient of the nonlinear term, which allows a better approximation of the dynamic response of an ice sheet near the phase speed minimum. Solitary waves are then numerically computed, and it is shown that the cubic-truncation model is more accurate than the quadratic-truncation model in arbitrary water depth. The dynamic response of a floating ice sheet to a fully localized constant-moving load is investigated based on this cubic-truncation model. The time-dependent solutions are compared with the data from the field measurements, and good agreement is achieved between the numerical results and experimental records.

Language中文
Document Type学位论文
Identifierhttp://dspace.imech.ac.cn/handle/311007/89143
Collection流固耦合系统力学重点实验室
Recommended Citation
GB/T 7714
孟洋涵. 非线性水弹性波的稳定性及动力学研究[D]. 北京. 中国科学院大学,2022.
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