在三維系統中加速DLM/FD法並模擬球與橢球在剪力流中相對運動與位移

Abstract

在自然環境與工程應用中，粒子在流體中運動的行為是非常重要的特性，如血液內各細胞的流動情形，或藥物溶液混和的效果等，都與之有關，本文模擬兩粒子在邊界剪力流時使用分佈式拉格朗日乘數/虛擬域(DLM/FD)結合史托克方程式(Stocks equation)與尤拉-牛頓方程式(Euler-Newton’s equations)來近似三維流場中球形與橢球型粒子的移動分布。 本研究主要成果有三，首先將計算兩粒子間之距離方法以球體參數位置導入牛頓疊代法計算，以加速計算速度，再以DLM/FD法模擬球型與橢球型粒子在剪力流中之相對運動，並比較兩種形狀之粒子運動情形之差異，最後模擬多顆球在剪力流中之運動情形，比較相同半徑與不同半徑之差異。 兩粒子間距離計算方法改善前後之結果誤差約為-3.015%左右，差距並不大，時間減少50-75%，速度上得到有效的提升。在模擬結果中粒子形狀以球型與橢球為主，以改變初始位置、半徑及牆距來模擬各情況下兩粒子流動時之軌跡，以達到控制粒子流動型態之效果。 在兩球模擬結果中，可發現在控制流動軌跡時，改變初始位置及牆距之效果較改變半徑明顯，牆距越小與兩球間距離越進近較易出現交換型軌跡，改變半徑是否與軌跡改變關係較小，在兩橢球結果中，會出現另一種滾轉型軌跡，但較無法控制出現，交換型與非交換型軌跡出現時機與兩球之結果相似，在多球模擬結果中，若將半徑隨機改變，交換型軌跡較易出現。

In natural environment and engineering applications, the behavior of particles moving in a fluid is a very important characteristic, such as the flow of cells in the blood, or the effect of mixing drug solutions, etc. This paper simulates two particles in the boundary shear flow using a distributed Lagrangian multiplier/virtual domain (DLM/FD) combined with the Stocks equation and Euler-Newton's equations. Movement distribution of spherical and ellipsoidal particles in three-dimensional flow field. The main results of this research are three. Firstly, the distance between two particles is calculated by introducing the position of the spherical parameter into the Newton iteration method to accelerate the calculation speed. Then the DLM/FD method is used to simulate the relative motion of spherical and ellipsoidal particles in the shear flow. And finally compare the difference between the two particles in the shape of the movement. The error between the two-particle distance calculation methods is about -3.015%, the difference is not large, the time is reduced by 50-75%, and the speed is effectively improved. In the simulation results, the particle shape is mainly spherical and ellipsoidal, and the initial position, radius and wall distance are changed to simulate the trajectory of the two particles flowing under each condition, so as to achieve the effect of controlling the particle flow pattern. In the simulation results of the two spheres, it can be found that when the flow trajectory is controlled, the effect of changing the initial position and the wall distance is obviously changed. The smaller the wall distance and the closer the distance between the two spheres is easier to exchange trajectory than the radius is changed. In the results of the two ellipsoids, another roll transition trajectory will appear, but it is less controllable. The timing of the exchanged and non-exchanged trajectories is similar to the result of the two spheres. In the multi-sphere simulation results, if the radius is randomly changed, Exchange trajectories are more likely to occur.