NONLINEAR DYNAMIC RESPONSE OF A FLOATING ICE SHEET TO A MOVING LOAD
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摘要: 本文考虑非线性、惯性和阻尼的影响, 研究了任意深度二维理想流体顶部浮冰的振动. 对相关的拟微分算子进行展开并将非线性项保留至三阶后, 完全非线性问题被简化为仅与自由面上的变量相关的三阶截断模型. 为了验证简化模型的准确性, 重点关注了自由孤立波解. 在不考虑阻尼的情况下, 采用多重尺度方法推导了三阶非线性薛定谔方程(NLS), 利用该方程预测了任意水深下原始欧拉方程中自由波包型孤立波解的存在性及三阶截断模型的准确性. 相比于Dinvay等所提出的二阶模型, 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, 能够在最小相速度附近更好地模拟冰层的动力学响应. 进一步地对自由孤立波解进行数值计算, 数值结果表明三阶截断模型在分岔曲线和孤立波波形上均与完全欧拉方程吻合良好, 准确性高于二阶截断模型. 基于三阶截断模型, 探究了匀速局域化载荷作用下的浮冰非线性动力学响应并将时间依赖解与实验测量数据进行比较, 数值计算结果与实验记录吻合良好.Abstract: Vibrations of a floating ice cover on top of a two-dimensional ideal fluid of arbitrary depth are studied when the effects of nonlinearity, inertia, and damping are all considered. We reduce the fully nonlinear problem to a cubic-truncation system involving variables on the free surface by expanding the 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 wavepacket solitary wave solutions. In the absence of damping, the normal form analysis is performed to derive the cubic nonlinear Schrödinger equation, which predicts the existence of free wavepacket solitary waves in the primitive equations and the accuracy of the cubic-truncation model. The main advantage of the cubic-truncation approximation over the quadratic-truncation model is that the resultant NLS equation has correct coefficient of the nonlinear term, which allows a better approximation of dynamic responses of the ice cover near the phase speed minimum. Solitary waves are then numerically computed, and it is shown that the cubic-truncation approximation agrees well with the full Euler equations for bifurcation curves and wave profiles, indicating that the reduced model is more accurate than the quadratic truncation model. The nonlinear dynamic response of a floating ice sheet to a fully localized constant-moving load is investigated based on the 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.
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Key words:
- hydroelastic wave /
- wave-structure interaction /
- free-surface flow
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图 2 (a) 水深
$ h = 250\;\mathrm{m} $ , 惯性$ \alpha = 0.069 $ 时, 孤立波解分岔图. 实线: 完全欧拉方程的分岔曲线, 短划线: 三阶截断模型的分岔曲线, 点线: 二阶截断模型的分岔曲线. (b) 波速$ c = 1.287\;8 $ 时, 欧拉方程、三阶截断模型和二阶截断模型的孤立波解. 实线: 完全欧拉方程, 短划线: 三阶截断模型, 点线: 二阶截断模型Figure 2. The bifurcation curves of wavepacket solitary waves for
$ h = 250\;\text{m} $ and$ \alpha = 0.069 $ . Solid line: full Euler equations, dashed line: third-truncation model, dotted line: quadratic-truncation model. (b) The typical profiles for$ c = 1.287\;8 $ . Solid line: full Euler equations, dashed line: third-truncation model, dotted line: quadratic-truncation model
图 1 (a) 水深
$ h = 6.8\;\text{m} $ , 惯性系数$ \alpha = 0.069 $ 时, 孤立波解分岔图. 实线: 完全欧拉方程的分岔曲线, 短划线: 三阶截断模型的分岔曲线, 点线: 二阶截断模型的分岔曲线. (b) 波速$c = 1.287\;8$ 时, 欧拉方程、三阶截断模型和二阶截断模型的孤立波解. 实线: 完全欧拉方程, 短划线: 三阶截断模型, 点线: 二阶截断模型Figure 1. (a) Bifurcation curves of wavepacket solitary waves for
$ h = 6.8\;\text{m} $ and$ \alpha = 0.069 $ . Solid line: full Euler equations, dashed line: third-truncation model, dotted line: quadratic-truncation model. (b) The typical profiles for$c = 1.287\;8$ . Solid line: full Euler equations, dashed line: third-truncation model, dotted line: quadratic-truncation model
图 3 三阶截断模型的数值结果与文献[1]实验中所记录的冰层形变对比. 其中虚线代表Takizawa的实验结果, 红点代表实验中的运动载荷经过冰层形变测量仪的垂直位置. 实线表示三阶截断模型式(5)和式(7)的数值结果, 蓝点代表数值计算中载荷经过监测点的垂向位置
Figure 3. Comparisons of the numerical results of Eq. (3) the numerical results of cubic-truncation model and the experimental records of Ref. [1]. The dashed lines represent the experimental data, and the red dots indicates the position of the load as it passed the deflectometer. The solid lines shows the numerical results of Eqs. (5) and (7), and the blue dots indicates the z-position of the load as it passes the point where the time series are obtained
图 4 三阶截断模型的数值结果与文献[2]实验中所测量的冰层应变对比, 图中应变值量级为
$ 10^{-6} $ . 虚线代表Squire的实验结果, 实线表示三阶截断模型式(5)和式(7)的数值结果Figure 4. The comparison of microstrain between the numerical results of cubic-truncation model and numerical records of Ref. [2]. The microstrain is of
$ 10^{-6} $ . Dashed lines: experimental records, solid lines: numerical approximation of the cubic-truncation model Eqs. (5) and (7)Table 1. Values and units of the physical parameters in the shallow-water[1] and deep-water[2] experiments
Physical parameter Symbol Lake Saroma McMurdo sound Young's modulus $E/( { {\rm{N} } \cdot { {\rm{m} }^{ - 2} } } )$ $5.1\times{10^8}$ $4.2\times{10^9}$ Poisson ratio $\nu$ $0.3$ $0.3$ ice thickness $d/{\rm{m}}$ $0.17$ $1.6$ water depth $h/{\rm{m}}$ $6.8$ $\infty$ flexural rigidity $D/({ {\rm{N} } } \cdot { {\rm{m} } } )$ $2.35\times{10^5}$ $1.8\times{10^9}$ water density $\rho_{w}/(\mathrm{kg \cdot m^{-3} })$ $1024$ $1026$ ice density $\rho_{i}/(\mathrm{kg \cdot m^{-3} })$ $917$ $917$ 
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