粘弹性海底淤泥对自由表面水波衰减效应的研究
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摘要
大量观测表明，海底淤泥能够有效地耗散波能，是自由表面浅水波衰减的重要因素之一。淤泥与波浪相互作用的研究具有重要的理论意义和广泛的应用价值，是海岸工程领域关心的重要问题。
线性粘弹性模型是常见的一类淤泥模型，在波泥相互作用的理论研究中被广泛使用。然而这类研究多局限于线性简谐波。在有限的针对弱非线性水波和淤泥相互作用的理论研究中，Boussinesq型方程组已被证明是有力的工具。然而已有的Boussinesq型方程组只针对牛顿淤泥模型，不能描述某些真实淤泥的粘弹性效应。将Boussinesq型方程组拓展到线性粘弹性淤泥模型，并利用其描述线性粘弹性淤泥作用下的水波衰减问题，就是本文的主要目的。
本文选取Maxwell模型和Kelvin-Voigt模型分别作为粘弹性流体淤泥模型和粘弹性固体淤泥模型的代表。在粘弹性淤泥和水层组成的双层流体模型的基础上，本论文利用摄动法、傅立叶展开和拉普拉斯变换建立起了相应淤泥模型的Boussinesq型方程组。当Maxwell模型和Kelvin-Voigt模型蜕化为牛顿流体时，其蜕化后的Boussinesq型方程组与文献中牛顿模型的Boussinesq型方程组一致。
本文将Boussinesq型方程组应用于一维水波问题，得到了线性简谐波的衰减率和孤立波波幅随时间的演化方程。对于偏流体的Maxwell模型，其线性简谐波的衰减率受低阶模态的影响，会出现多个共振峰值。当孤立波经过后，Maxwell淤泥层中会出现沿竖直方向传播的剪应力波。而对于偏固体的Kelvin-Voigt模型，其线性简谐波的衰减率在一阶模态的主导下只有一个较为明显的共振峰值。当孤立波经过后，Kelvin-Voigt淤泥层中的水平速度剖面则逐渐趋向于一阶模态的速度剖面。造成上述现象的原因是：Maxwell模型的低阶模态同步衰减，而Kelvin-Voigt模型的一阶模态衰减得最慢，从而主导了淤泥层中的运动。
此外，本文针对传统的整数阶粘弹性模型的局限性，引入了分数阶Maxwell模型。该分数阶Maxwell模型在拟合某些真实淤泥的数据时，相较于传统模型具有优势。本文还给出了分数阶Maxwell淤泥模型存在时线性单色波的衰减率，并初步探讨了模型参数对衰减率的影响。
It has been observed that seabed mud can dissipate the energy of surfacewater waves effectively. As an important mechanism of wave attenuation, thewave-mud interaction has been extensively studied by coastal engineers fordecades.
Among varies models for mud rheology, linear viscoelastic models arecommonly used in analytical studies on wave-mud interaction. However, mostof these studies focus on linear sinusoidal waves. In the very limited studies onthe interaction between the weakly nonlinear waves and seabed mud, theBoussinesq-type equations are powerful tools. The limitation of currentBoussinesq-type equations is that the mud is modeled by Newtonian fluid,which cannot cover the complexity of mud rheology. Hence, it is necessary toextend the Boussinesq-type equations to linear viscoelastic models, which is thegoal of present work.
In the present work, we choose the Maxwell model and the Kelvin-Voigtmodel as mud models, which are typical models for viscoelastic fluid andviscoelastic solid, respectively. Basing on a two-layer system composed ofinviscid water and viscoelastic mud, and combining the perturbation analysis,the Fourier expansion and the Laplace transformation, we set upBoussinesq-type equations for surface waves over the Maxwell model and theKelvin-Voigt model, respectively. When the Maxwell model and theKelvin-Voigt model degenerating to the Newtonian model, our Boussinesq-typeequations can degenerate to the existing Boussinesq-type equations in theliterature.
Applying the Boussinesq-type equations to one-dimensional waves, thedamping rate of linear sinusoidal waves and the evolution equation for theamplitude of a solitary wave are obtained. For the Maxwell model, the dampingrate of linear sinusoidal waves is dominated by low-order modes, and shearwave induced by solitary waves propagates along the vertical direction in themud layer. For the Kelvin-Voigt model, the first mode dominates the mud motion. Hence the damping rate of linear sinusoidal waves has only one obviouspeak and the velocity profile of the mud layer tends to that of the first modeafter the solitary wave crest. Modal analysis shows that for the Maxwell model,all the modes decay simultaneously, while for the Kelvin-Voigt model, the firstmode decays slowest and takes domination.
We also introduce a Fractional-ordered Maxwell model, which can overcome thedefect of traditional viscoelastic models when fitting the experimental data of some realmud. Basing on this fractional-order Maxwell model, we obtain the damping rate of alinear sinusoidal wave. The influence of model parameters on the damping rate is alsodiscussed.
线性粘弹性模型是常见的一类淤泥模型，在波泥相互作用的理论研究中被广泛使用。然而这类研究多局限于线性简谐波。在有限的针对弱非线性水波和淤泥相互作用的理论研究中，Boussinesq型方程组已被证明是有力的工具。然而已有的Boussinesq型方程组只针对牛顿淤泥模型，不能描述某些真实淤泥的粘弹性效应。将Boussinesq型方程组拓展到线性粘弹性淤泥模型，并利用其描述线性粘弹性淤泥作用下的水波衰减问题，就是本文的主要目的。
本文选取Maxwell模型和Kelvin-Voigt模型分别作为粘弹性流体淤泥模型和粘弹性固体淤泥模型的代表。在粘弹性淤泥和水层组成的双层流体模型的基础上，本论文利用摄动法、傅立叶展开和拉普拉斯变换建立起了相应淤泥模型的Boussinesq型方程组。当Maxwell模型和Kelvin-Voigt模型蜕化为牛顿流体时，其蜕化后的Boussinesq型方程组与文献中牛顿模型的Boussinesq型方程组一致。
本文将Boussinesq型方程组应用于一维水波问题，得到了线性简谐波的衰减率和孤立波波幅随时间的演化方程。对于偏流体的Maxwell模型，其线性简谐波的衰减率受低阶模态的影响，会出现多个共振峰值。当孤立波经过后，Maxwell淤泥层中会出现沿竖直方向传播的剪应力波。而对于偏固体的Kelvin-Voigt模型，其线性简谐波的衰减率在一阶模态的主导下只有一个较为明显的共振峰值。当孤立波经过后，Kelvin-Voigt淤泥层中的水平速度剖面则逐渐趋向于一阶模态的速度剖面。造成上述现象的原因是：Maxwell模型的低阶模态同步衰减，而Kelvin-Voigt模型的一阶模态衰减得最慢，从而主导了淤泥层中的运动。
此外，本文针对传统的整数阶粘弹性模型的局限性，引入了分数阶Maxwell模型。该分数阶Maxwell模型在拟合某些真实淤泥的数据时，相较于传统模型具有优势。本文还给出了分数阶Maxwell淤泥模型存在时线性单色波的衰减率，并初步探讨了模型参数对衰减率的影响。
It has been observed that seabed mud can dissipate the energy of surfacewater waves effectively. As an important mechanism of wave attenuation, thewave-mud interaction has been extensively studied by coastal engineers fordecades.
Among varies models for mud rheology, linear viscoelastic models arecommonly used in analytical studies on wave-mud interaction. However, mostof these studies focus on linear sinusoidal waves. In the very limited studies onthe interaction between the weakly nonlinear waves and seabed mud, theBoussinesq-type equations are powerful tools. The limitation of currentBoussinesq-type equations is that the mud is modeled by Newtonian fluid,which cannot cover the complexity of mud rheology. Hence, it is necessary toextend the Boussinesq-type equations to linear viscoelastic models, which is thegoal of present work.
In the present work, we choose the Maxwell model and the Kelvin-Voigtmodel as mud models, which are typical models for viscoelastic fluid andviscoelastic solid, respectively. Basing on a two-layer system composed ofinviscid water and viscoelastic mud, and combining the perturbation analysis,the Fourier expansion and the Laplace transformation, we set upBoussinesq-type equations for surface waves over the Maxwell model and theKelvin-Voigt model, respectively. When the Maxwell model and theKelvin-Voigt model degenerating to the Newtonian model, our Boussinesq-typeequations can degenerate to the existing Boussinesq-type equations in theliterature.
Applying the Boussinesq-type equations to one-dimensional waves, thedamping rate of linear sinusoidal waves and the evolution equation for theamplitude of a solitary wave are obtained. For the Maxwell model, the dampingrate of linear sinusoidal waves is dominated by low-order modes, and shearwave induced by solitary waves propagates along the vertical direction in themud layer. For the Kelvin-Voigt model, the first mode dominates the mud motion. Hence the damping rate of linear sinusoidal waves has only one obviouspeak and the velocity profile of the mud layer tends to that of the first modeafter the solitary wave crest. Modal analysis shows that for the Maxwell model,all the modes decay simultaneously, while for the Kelvin-Voigt model, the firstmode decays slowest and takes domination.
We also introduce a Fractional-ordered Maxwell model, which can overcome thedefect of traditional viscoelastic models when fitting the experimental data of some realmud. Basing on this fractional-order Maxwell model, we obtain the damping rate of alinear sinusoidal wave. The influence of model parameters on the damping rate is alsodiscussed.
引文
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[3]一苇.浅海石油开采.中国石化,2010(6):27-27.
[4]李家春.海洋工程中极端环境事件的研究进展.第十三届中国海洋（岸）工程学术讨论会论文集,2007.
[5] Smith A, Gordon A D. Large breakwater toe failures. Journal of Waterway, Port, Coastaland Ocean Engineering,1983.109(2):253-255.
[6] Christian J, Taylor P, Yen J, et al. Large diameter underwater pipe line for nuclear powerplant designed against soil liquefaction. Offshore Technology Conference,1974.
[7] van Kessel T, Kranenburg C. Wave-induced liquefaction and flow of subaqueous mudlayers. Coastal Engineering,1998.34(1-2):109-127.
[8] Whiteside P, Ng K, Lee W. Management of contaminated mud in Hong Kong. Terra etAqua,1996,65:10-17.
[9] Gade H G. Effects of a non-rigid, impermeable bottom on plane surface waves in shallowwater. Journal of Marine Research,1958,16(2):61-82.
[10]王以谋,范顺庭.黄河口“烂泥”波浪特性的分析.海洋科学,1995,6:42-46.
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