摘要
本文采用时域内满足自由水面条件格林函数方法,对无限和有限水深上波浪与三维任意形状的物体相互作用所产生的绕射和辐射问题进行了理论研究和数值计算。
本文首先以三维Laplace方程作为基本控制方程,对势函数和格林函数的一阶时间导数应用第二格林定理,建立了时域内有限水深上的积分方程。并利用各种边界和初始条件,进行了一系列的数学推导,使这个积分方程仅建立在物体表面上,大大减少了计算量,降低了对计算机内存的要求。物体在波浪中运动的时候,物体的运动决定了他所受流体力的大小;反过来,受力又影响了物体的运动。因此,物体的积分方程和运动方程的是互相耦合的,需要同时进行求解。本文中采用高阶边界元方法对积分方程进行了求解,运动方程采用四阶Runge-Kutta方法求解。
由于在时域积分方程中含有势函数与格林函数的卷积的时间积分过程。而格林函数的解析表达式是一个从0到∞的积分过程,且呈缓慢衰减的振荡形式,每次计算格林函数都十分耗时。在对积分方程的求解过程中,需要上亿次的调用格林函数,当需要模拟的计算时间较长时,这个计算的计算量和存储量都非常庞大。因此,精确而又快速的求取时域格林函数及其对时间和空间的导数是应用满足时域自由水面条件格林函数方法求解水动力学问题的关键。本文分析了无限和有限深水域中时域格林函数及其对时间和空间导数的特性,分离出快速的振荡元素,针对参数的变化范围推导出了格林函数的级数式、积分式和渐进式用于计算提高计算效率。根据等高线的变化情况,将自变量化分为不同范围并使用Chebyshev多维近似多项式方法来进行分区拟合,建立系数表用于插值,避免了缓慢的直接数值积分计算。为了进一步加快计算速度,又将Chebyshev多项式系数转化为普通多项式系数的形式。最后将格林函数值及其导数值与由此计算出来的近似多项式值进行了比较,研究发现当取合适的截断项时,既大大加快了计算速度同时也可获得所需要的精度。
本文计算了各种形状三维物体在无限水深和有限水深情况下所受到的绕射和辐射波浪力,并与解析解以及频域理论进行了对比,结果符合良好,验证了本文所提出理论的正确性。
A three-dimensional time-domain approach is used to study the wave loads and motion of bodies. In this approach, the exact body boundary condition is satisfied on the wetted body surface while the free-surface boundary conditions are linearized. The problem is solved by using a transient free-surface Green function source distribution. The velocity potential is obtained numerically from a discretized boundary integral equation on the body surface, using a high-order boundary element method.The method is based on using the transient free-surface Green function. Accurate and fast computation of the Green function and its derivations is a hard job. This study is concerned with the Green function and its derivations. Asymptotic expansions and convergent ascending-series expansions for the Green function and its derivations are obtained to replace the numerical evaluation of the relevant integrals. Analysis is required to develop suitable forms for representing singular features when both the source and field points lie on the free surface. A computational approach based on the use of multidimensional Chebyshev polynomial approximations, which greatly decreases the computing cost in numerical evaluation of the Green function, is used. To accelerate the computation further, Chebyshev polynomials can be converted into simpler equivalent ordinary polynomials. In the whole domain the Green function is rapidly oscillatory and includes singularity. It is not very effective to approximate the Green function in the whole domain directly. The approach used in the paper is to divide the physical domain into several zones, and use different approximations in each zone. The flexibility in truncation can be exploited to preserve the form of the polynomial approximations in different sub domain. The polynomial approximation is compared with the directly computing approach. It is found that the algorithm of Chebyshev polynomial approximation with not too many terms can achieve a desired accuracy.Extensive results are presented which validate and demonstrate the efficacy of the method. These results include linear motion and forces without forward speed. Results of diffraction and radiation of a hemi-sphere, a sphere, a cylinder, a box and a taper are presented. The present computations agree with analysis solutions and frequency domain results very well.
引文
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