Numerical Simulation of 3D Nonlinear Flows around a Vertical Cylinder

本研究之主要目的是以無網格方法中的MFS方法探討均勻流經過直立圓柱之非線性水面波動的數值模擬。本方法是利用三維線性基本解(Fundamental solution)求解Laplace equation而得到速度勢及速度勢梯度，配合Mixed Eulerian-Lagrangian自由液面邊界條件，更新計算點位置求出自由液面高程，並以二維二次多變數函數(multiquadric function)內插出圓柱與自由液面交界處之計算點高程；而時間域則以三層中央二階差分法(蛙跳法)對時間離散之顯示法，求出下一時間步的計算值。 本研究以移動座標之轉換，並假設在無黏性與不可壓縮條件下之直立移動圓柱與底床從靜止狀態加速到恆定均勻流狀態進行模擬計算。本數值方法之自由液面動力邊界條件在忽略非線性項後，可輕易的計算出線性結果，並與前人Sadathosseini et al. (2008)與Kawamura (2002)的研究互相比較。非線性波數值結果與線性波比較後可顯示出非線性效應的影響。本模式在圓柱水波的模擬上，趨勢與Sadathosseini et al. (2008)計算的水波趨勢相當接近。而在於圓柱前方停滯點的水位與能量公式推導出之停滯點水位則以本模式較為符合物理意義且合理。但與Kawamura (2002)比較上有相當大之差異，推測本模式與Kawamura (2002)之間的差異是由於水波相位差的不同所造成。The objective of this study is aimed to numerically simulate nonlinear waves for uniform flows passing through a vertical cylinder by using the MFS method, one of the meshless methods. The method uses the fundamental solution of three-dimensional Laplace equation to solve the velocity potential and its gradient. With the Mixed Eulerian-Lagragian free surface conditions, the elevation can be interpolated by updating calculated positions by two-dimensional multi-quadratic function, particularly the interface between the free surface and the cylinder. A three-level central second-order difference method (leap-frog method) has been used for time discretization to develop an explicit marching computation scheme. Assuming the fluid is inviscid and incompressible, this study also employs moving coordinate transformation to accelerate a vertical cylinder with its bed in a still water from the rest to a terminal velocity in a short time. Present numerical model can easily reduced into a linear case by ignoring the nonlinear terms in the dynamic free surface boundary condition. Present numerical results are compared with results of previous study of Sadathosseini et al. (2008) and Kawamura (2002). Numerical results of nonlinear waves generated are also compared with calculated linear cases in order to see the nonlinear effects. It is found that present results show a variation trend of free surface displacement around the cylinder similar to those of Sadathosseini et al. (2008). A new nondimensional parameter of the stagnation elevation at the front point of the cylinder encountered the uniform flows to justify present results are physically more reasonable. But in Kawamura (2002) cases, the difference between present calculations and previous results are quite different. It is believed that the difference may possibly due to the phase difference between present computation and previous ones