有限水深自由面格林函数的新表述
New formulation of the finite depth free surface Green function
摘要：
对于三维有限深流体域中的脉动自由面源，本文给出了John [F。漂浮物体的运动2。简谐运动，通信。Pure Appl.Math.3（1950）45-101]叠加为朗肯源势、镜像源势和无限域$(0, \infty)$中的波积分。当源点与场点一起位于自由表面上时，John积分及其梯度不收敛，因为相应被积函数的积分$\int^\infty_κ$不像$κ$趋于$\infty$那样以一致的方式趋于零。因此，格林函数的计算不是基于波积分的直接积分，而是通过早期研究中的近似展开获得的。在本研究中，五个图像的源相对于自由表面镜和水床镜的图像方法被用来重新制定的波动积分。因此，源的自由表面格林函数被分解为Rankine势、五个镜像源势和一个新的波动积分，其中被积函数用一个光滑的快速衰减函数来近似。格林函数的梯度进一步制定，使相同的积分稳定性与波积分证明。本研究的意义在于改进的格林函数及其梯度的波动积分法是收敛的。因此，格林函数的评价是通过积分的被积函数在一个简单的方式。将该方法应用于规则波中浮体或潜体的运动，表明该近似方法在实际计算线性波浪载荷时具有足够的精度。
英文摘要：
For a pulsating free surface source in a three-dimensional finite depth fluid domain, the Green function of the source presented by John [F. John, On the motion of floating bodies II. Simple harmonic motions, Communs. Pure Appl. Math. 3 (1950) 45-101] is superposed as the Rankine source potential, an image source potential and a wave integral in the infinite domain $(0, \infty)$. When the source point together with a field point is on the free surface, John's integral and its gradient are not convergent since the integration $\int^\infty_κ$ of the corresponding integrands does not tend to zero in a uniform manner as $κ$ tends to $\infty$. Thus evaluation of the Green function is not based on direct integration of the wave integral but is obtained by approximation expansions in earlier investigations. In the present study, five images of the source with respect to the free surface mirror and the water bed mirror in relation to the image method are employed to reformulate the wave integral. Therefore the free surface Green function of the source is decomposed into the Rankine potential, the five image source potentials and a new wave integral, of which the integrand is approximated by a smooth and rapidly decaying function. The gradient of the Green function is further formulated so that the same integration stability with the wave integral is demonstrated. The significance of the present research is that the improved wave integration of the Green function and its gradient becomes convergent. Therefore evaluation of the Green function is obtained through the integration of the integrand in a straightforward manner. The application of the scheme to a floating body or a submerged body motion in regular waves shows that the approximation is sufficiently accurate to compute linear wave loads in practice.
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