MECHANISM ANALYSIS OF PRESSURE OSCILLATION IN PARTICLE METHOD
ZhuYue*,, JiangShengyao, YangXingtuan, DuanRiqiang*, Key Laboratory of Advanced Reactor Engineering and Safety of Ministry of Education, Collaborative Innovation Center of Advanced Nuclear Energy Technologyrm, Institute of Nuclear and New Energy Technology, Tsinghua University, Beijing 100084 China 文献标识码:A
关键词:移动粒子半隐式法;压力振荡机理;碰撞模型;核函数 Abstract The moving particle semi-implicit method was developed to simulate incompressible fluid using a meshless method. There was a big problem that the space distribution and the time variation of pressure oscillate drastically in the MPS method. In order to investigate the pressure oscillation in the MPS method, the simplified one-dimensional model was developed. The mechanism of pressure oscillation in the MPS method was illustrated by the movement and the relative position between the center particle and its neighbor particles. The collision model was employed in the simulation and the relative position between particles was controlled by choosing different collision parameters. The classical dam break problem was simulated. With the increase of collision parameter, the fluctuation of the deviation of particle number density decreased. According, the amplitude of pressure oscillation was suppressed. Two different kernel functions were also employed to investigate the pressure oscillations. The results showed that the gauss kernel function improved the stabilization of pressure calculation. It was the reason that the same movement of particles lead to less deviation of particle number density when the gauss kernel function was used. And the randomness of particles motion led to random fluctuation in the deviation of particle number density. As a result, the pressure fluctuation in MPS method occurred and it was difficult to be eliminated.
Table 1 表1 表1采用MPS核函数时, 在不同碰撞系数下的粒子数密度偏差的最大波动值 Table 1The maximum fluctuation of the deviation in particle number density of different by MPS kernel function
图8是采用高斯核函数时, 压力?时间、粒子数从图8(a) 中可以看出, 当碰撞系数=5时, 溃坝模型在计算了一段时间后, 计算发生了发散, 这与之前采用MPS 核函数时进行计算的结果一致, 都是由于碰撞系数取值过小, 从而造成在计算过程中出现粒子聚集现象而造成计算发散.对比图8(a)~ 图8(c) 随着碰撞系数 的增加, 压力振荡幅度逐渐下降在MPS方法中, 核函数又称为权函数, 在计算粒子之间相互关系式, 与核函数在支持域, 并且当碰撞系数 > 0.9 时, 压力振荡的幅度无法进一步的降低. 这与之前采用MPS 方法观察得到的现象一致, 即MPS 方法中压力波动粒子数密度偏差的波动具有一定的正相关性, 说明增加碰撞系数 能够有效的抑制住压力振荡现象. 显示原图|下载原图ZIP|生成PPT 图8高斯核函数下压力和粒子数密度差?时间变化示意图 -->Fig.8Time histories of pressure and deviation in particle number density at point A by Gauss kernel function -->
显示原图|下载原图ZIP|生成PPT 图8高斯核函数下压力和粒子数密度差—时间变化示意图(续) -->Fig.8Time histories of pressure and deviation in particle number density at point A by Gauss kernel function(continued) -->
3.6 MPS核函数和高斯核函数的对比分析
内不同区域的权重有关, 而具体的核函数的数值小大无关.图9 是两个核函数曲线, 对比两个图形可以看到, MPS 核函数在靠近0 时, 核函数的数值急剧增大, 这意味着两个粒子之间的相互影响也急剧增大, 而高斯核函数在靠近0时, 核函数数值也在增大, 但变化的幅度相对于MPS核函数要小很多. 显示原图|下载原图ZIP|生成PPT 图9两种核函数曲线 -->Fig.9Curves of two kernel functions -->
图10是MPS核函数和高斯核函数下粒子1, 2移动距离和粒子数密度偏差 之间的关系, 从图10(a)可以看到, 在粒子1移动相同距离的情况下, 采用不同的核函数造成的粒子数密度偏差的变化是不同的, 采用MPS 核函数时的粒子数密度偏差大于采用高斯核函数时的粒子数密度偏差.同样的, 在图10(b)可以看到, 对于粒子2 的情况是相同的, 粒子2 移动相同距离时, 采用MPS核函数时产生的粒子数密度偏差大于采用高斯核函数产生的粒子数密度偏差. 这是由于两种不同核函数在支持域内不同分布形式所决定的, MPS核函数在支持域内的变化更为剧烈, 因此移动相同距离时, 其产生的粒子数密度偏差更大.这意味着产生更大的压力振荡幅度. 显示原图|下载原图ZIP|生成PPT 图10两种核函数的粒子1, 2移动距离和粒子数密度偏差$(n_i-n_0)/n_0$之间的关系 -->Fig.10The deviation in particle number density of different particle displacement by two kernel functions -->
图11是MPS核函数、高斯核函数的压力?时间曲线与文献[31]的实验结果对比图, 从图中可以看到.高斯核函数的压力振荡幅度是小于MPS核函数的压力振荡幅度.这是因为在粒子运动相同距离时, 高斯核函数引起的粒子数密度偏差是小于MPS核函数因为的粒子数密度偏差的, 从而其压力振荡的幅度也相对较小.并且从图中可以看出, MPS核函数的容易出现负压力的情况, 这是因为流体处于拉伸状态时, MPS核函数的粒子数密度偏差波动是大于高斯核函数的粒子数密度偏差的波动.而采用高斯核函数时, 能够较好地缓减出现负压力的情况.虽然采用以上两种核函数并不能完全地消除计算过程中的压力振荡现象, 但是两种核函数的压力值与文献[31]的实验值在均值上非常接近的. 显示原图|下载原图ZIP|生成PPT 图11MPS核函数和高斯核函数下 压力?时间变化示意图 -->Fig.11Comparison of time histories of pressure at point A by two kernel functions and Hu’s experimental result -->
图12是MPS核函数和高斯核函数在不同时刻时溃坝压力分布对比图, 对比不同时刻的压力场图可以看到, 高斯核函数的压力分布相对比MPS核函数更为光滑, 采用高斯核函数进行计算时, 流体最底部的压力最大, 而自由表面处压力最小, 这与物理实际相一致.这是因为高斯核函数造成的粒子数密度波动小于MPS 核函数造成的粒子数密度波动, 所以压力振荡相对比与MPS核函数也更小. 显示原图|下载原图ZIP|生成PPT 图12MPS核函数和高斯核函数在不同时刻时溃坝压力分布对比图 -->Fig.12Comparison of dam break pressure field at different instants by two kernel functions -->
4 结论
本文针对在MPS方法中的压力振荡现象进行了研究.首先通过对粒子模型的一维简化, 阐述了粒子运动过程中, 粒子之间的相互位置关系对粒子数密度偏差造成的影响.并且采用了经典的溃坝模型作为本研究的算例, 通过在MPS方法中加入了碰撞模型来控制计算过程中粒子之间的相互位置关系, 从而控制溃坝流体在计算过程的压力振荡.结果表面, 随着碰撞系数 的增加, 溃坝流体在模拟过程中的压力振荡得到了明显的抑制, 但是当碰撞系数{Invalid MML} 时, 压力振荡无法进一步抑制.然后又通过对比分别采用MPS核函数和高斯核函数时, 溃坝流体的压力振荡大小.结果表明, 采用高斯核函数时, 溃坝的压力振荡幅度更小.这是由于在粒子相同幅度的波动情况下, 采用高斯核函数时的粒子数密度偏差波动更小, 从而压力振荡的幅度更小.所以核函数的选取也是影响MPS方法中压力振荡的一个重要因素.并且, 由于在模拟过程中粒子运动具有一定的随机性, 这就不可避免会造成粒子数密度偏差的波动, 从而会产生压力振荡现象.在实际的模拟计算过程中, 可以通过上述手段来抑制压力振荡的幅度, 但无法彻底消除压力的波动. The authors have declared that no competing interests exist.
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