关键词:圆柱绕流;方柱绕流;改进的延迟分离涡模拟方法;高雷诺数 Abstract Flow past circular cylinders and square cylinders at high Reynolds numbers are simulated by improved delayed detached-eddy simulation (IDDES), including a circular cylinder, a square cylinder, two tandem circular cylinders and two tandem square cylinders. The mean drag coefficient, the RMS values of lift coefficient and the Strouhal number are computed for various Reynolds numbers, which show a good agreement with previous experimental and numerical simulation data. It is found that the effect of Reynolds number on the global quantities for square cylinders is not much in this range of Reynolds numbers, which is different for circular cylinders. There is no drag crisis phenomenon for flow past a square cylinder at . The Strouhal number is Reynolds-independent for , and the Reynolds-independent is also observed for the mean drag coefficient and the RMS lift coefficient. Simulation for two tandem circular cylinders is performed at Reynolds numbers of and for five different spacing to diameter ratios: , 2.5, 3.0, 3.5 and 4.0. At the critical spacing ( there is found a distinct step-like jump of mean drag coefficient and RMS lift coefficient of the subcritical Reynolds number of , and the mean drag coefficient of the downstream circular cylinder is negative for . However, the mean drag coefficient and the RMS lift coefficient are seen to be slightly affected by spacing for , and the mean drag coefficient of the downstream circular cylinder is always positive. Flow past two tandem square cylinders is considered at Reynolds numbers of and . The abrupt change in mean drag coefficient and RMS lift coefficient at the critical spacing is clearly seen on both upstream and downstream square cylinders for both Reynolds numbers. When , the mean drag coefficient of the downstream cylinder is negative for both Reynolds numbers.
单圆柱计算域及边界条件的设定如图1所示,入口距离圆柱中心为 ( 为圆柱直径),出口距离圆柱中心为 ,左右两边均距离圆柱中心 ,圆柱的展向长度为 ,将入口边界设为速度入口,出口边界为压力出口,左右两边也设为速度入口,上下面为对称面边界条件,圆柱表面为无滑移壁面边界条件. 单方柱计算域及边界条件与单圆柱类似,计算域也为 ,此时 为方柱边长. 显示原图|下载原图ZIP|生成PPT 图1单圆柱计算域及边界条件 -->Fig. 1Computational domain and boundary conditions for a circular cylinder -->
串列双圆柱计算域及边界条件的设定如图2所示,入口距离上游圆柱中心为10 ,两圆柱相距 ,出口距离下游圆柱中心为20 ,左右两边均距离圆柱中心10 ,圆柱的展向长度为2 . 间距比 表示两圆柱中心距离 与圆柱直径 的比值,本文计算的 包括2.0, 2.5, 3.0, 3.5和4.0五种工况. 同理,串列双方柱计算域和边界条件与双圆柱类似,计算域也为 ,此时 为方柱边长, 为两方柱中心距离. 显示原图|下载原图ZIP|生成PPT 图2双圆柱计算域及边界条件 -->Fig. 2Computational domain and boundary conditions for two tandem circular cylinders -->
双圆柱及双方柱网格划分如图3和图4所示(以 为例),为了保证计算精度, 值取小于1.0,边界层网格增长率为1.15,单圆柱网格以及单方柱网格同理. 显示原图|下载原图ZIP|生成PPT 图3双圆柱网格划分示意图 -->Fig. 3Grid distribution of two tandem circular cylinders -->
显示原图|下载原图ZIP|生成PPT 图4双方柱网格划分示意图 -->Fig. 4Grid distribution of two tandem square cylinders -->
2 计算结果与分析
2.1 网格收敛性分析及数值模型验证
网格的疏密程度不仅会影响计算精度,还会影响计算效率. 网格收敛性分析以 的圆柱绕流工况为例,选取3套由疏到密的网格进行计算,无量纲时间步长( , 为时间步长, 为来流速度, 为圆柱直径)取0.02,计算结果如表1所示. 表1给出了3套网格下计算的平均阻力系数 、升力系数均方根 和斯特劳哈尔数 与实验结果和其他文献CFD计算结果的对比情况. 3套网格计算的 和 与实验结果吻合较好, 要比实验值大,但与其他文献的CFD计算结果较吻合. 对比3套网格的结果可以看出,标准网格G2与精细网格G3的结果很接近,而粗糙网格G1的结果则与其他两套网格有较大差距,综合计算成本和效率,本文选取标准网格来进行数值模拟. Table 1 表1 表1网格收敛性分析 Table 1Sensitivity of the results to different grids
单圆柱绕流的平均阻力系数 、升力系数均方根 及斯特劳哈尔数 随雷诺数变化的情况如图5所示. 从图5(a)中可以看出本文的计算结果与实验结果吻合良好,随着雷诺数的增大,单圆柱绕流的 先维持平稳而后出现迅速减小(即阻力危机现象),最后又缓慢增大. 从图5(b)可以看出本文计算的 有的点与实验值吻合良好,部分计算值则相对 显示原图|下载原图ZIP|生成PPT 图5不同雷诺数下单圆柱绕流的平均阻力系数、升力系数均方根和斯特劳哈尔数 -->Fig. 5Mean drag coefficient, RMS lift coefficient and Strouhal number of a circular cylinder versus -->
显示原图|下载原图ZIP|生成PPT 图5(续)不同雷诺数下单圆柱绕流的平均阻力系数、升力系数均方根和斯特劳哈尔数(续) -->Fig. 5(continued) Mean drag coefficient, RMS lift coefficient and Strouhal number of a circular cylinder versus -->
实验值偏大,但与文献[28]的计算结果接近, 随雷诺数的增大呈现先增大后减小最后维持平稳的变化趋势. 从图5(c)中可以看出,本文计算的 在 区间内与实验值相差较大. 其原因是,在此雷诺数区域内,圆柱绕流的边界层从层流分离转化为湍流分离,流动较为复杂,涡脱表现为不规则,因此在此区域内计算结果不太理想,但与其他CFD结果较为吻合. 图6为单方柱绕流的平均阻力系数 、升力系数均方根 及斯特劳哈尔数 随雷诺数的变化情况. 从图6(a)可以看出,本文计算的 与Sohankar用LES计算的结果相符,但都比实验值略大,误差在10%左右. 从图6(b)看出本文计算的 均略小于文献[11]的计算结果,更接近于实验数据,随雷诺数的增大, 呈现出先保持平稳而后下降最后再增大的变化趋势,但总体变化幅度不大. 从图6(c)可以看出本文计算的 与实验数据吻合良好,随雷诺数的增大, 数无明显变化,保持在0.12 0.13范围内. 显示原图|下载原图ZIP|生成PPT 图6不同雷诺数下单放柱绕流的平均阻力系数、升力系数均方根和斯特劳哈尔数 -->Fig. 6Mean drag coefficient, RMS lift coefficient and Strouhal number of a square cylinder versus -->
在 区间内,单方柱绕流的系数 , 及 不随雷诺数变化而发生大幅度变化,这与单圆柱绕流情况区别明显,未出现类似于单圆柱绕流的阻力危机现象,Sohankar [11]用LES方法数值模拟了较高雷诺数(最高达106量级)下的方柱绕流,其研究同样表明方柱绕流未出现阻力危机现象. 图7为 和 时,单圆柱绕流的涡量图. 从图7可以看出, 时圆柱绕流的分离点后移,尾涡区变窄,即压差阻力减小,从而导致总阻力减小. 图8为 和 时,单方柱绕流的涡量图. 从图8可以看出,方柱绕流的分离点固定,压差阻力无明显变化,因此单方柱绕流未出现阻力危机现象. 显示原图|下载原图ZIP|生成PPT 图7不同雷诺数下,单圆柱绕流的瞬时涡量图 -->Fig. 7Instantaneous vorticity distributions for flow past a circular cylinder at different Reynolds numbers -->
显示原图|下载原图ZIP|生成PPT 图8不同雷诺数下,单方柱绕流的瞬时涡量图 -->Fig. 8Instantaneous vorticity distributions for flow past a square circular cylinder at different Reynolds numbers -->
2.3 串列双圆柱绕流及串列双方柱绕流
图9为雷诺数 时,不同间距比( )下串列双圆柱绕流的平均阻力系数 和升力系数均方根 ,其中 表示上游圆柱, 表示下游圆柱. 本文计算的上游圆柱的 和实验结果及其他文献的CFD计算结果吻合较好,下游圆柱在间距比小时本文的计算结果比实验值要大,但与Kitagawa的计算结果接近,在较大间距比下本文的计算结果则与实验值及Kitagawa的计算结果吻合良好. 从图9(b)中可以看出,本文计算的 除下游圆柱在间距比较大时比实验值偏大,其他工况均与实验值较为吻合. 时,串列双圆柱绕流在 区间内存在一临界间距比( )使得串列双圆柱绕流的 和 在临界间距比前后均发生跳跃性变化. 如图10所示,当 (即 )时,前柱后方还没有单独涡脱落,而当 (即 )时,前柱后方已经有单独的涡脱落现象,正是由于这种流态的变化才使得 和 发生跳跃性变化. 且当 时,下游圆柱的阻力系数为负数,当 时,下游圆柱的阻力系数才为正数. 显示原图|下载原图ZIP|生成PPT 图9Re=2.2×104时,不同间距比下串列双圆绕流的${\rm{\bar C}}$d和C'l -->Fig. 9Mean drag coefficient and RMS lift coefficient of upstream and downstream circular cylinders for different spacing to diameter ratios (L/D) at Re=2.2×104 -->
显示原图|下载原图ZIP|生成PPT 图9(续)Re=2.2×104时,不同间距比下串列双圆绕流的${\rm{\bar C}}$d和C'l(续) -->Fig. 9(continued) Mean drag coefficient and RMS lift coefficient of upstream and downstream circular cylinders for different spacing to diameter ratios (L/D) at Re=2.2×104(continued) -->
显示原图|下载原图ZIP|生成PPT 图10Re=2.2×104时,圆柱中心高度处不同间距比下的瞬时涡量图 -->Fig. 10Instantaneous vorticity distributions of circular cylinders at middle of cylinder span in the case of different spacing to diameter ratios at Re=2.2×104 -->
图11为 和 时,串列双圆柱绕流的 和 随间距比的变化情况,其中 为湍流分离工况, 为层流分离工况. 从图中可以看出,当雷诺数处于超临界雷诺数区域时( ),串列双圆柱绕流不再存在临界间距比,随着间距比的增大,下游圆柱的阻力系数和升力系数均缓慢增大,而上游圆柱的阻力系数和升力系数则趋于稳定,均无出现明显的跳跃性变化,而且下游圆柱的阻力系数始终为正数. 对比图10(b)与图12(b)可以看出,对于 的工况,当 时,虽然上游圆柱后方有单独涡脱落现象,但是漩涡形成的位置离上游圆柱较远,对上游圆柱的影响较小,因此其水动力系数没有出现明显跳跃性变化现象. 显示原图|下载原图ZIP|生成PPT 图11不同雷诺数时,不同间距比下串列双圆柱绕流的${\rm{\bar C}}$d和C'l -->Fig.11Mean drag coefficient and RMS lift coefficient of upstream and downstream circular cylinders for different spacing to diameter ratios (L/D) at different Re -->
显示原图|下载原图ZIP|生成PPT 图12时,圆柱中心高度处不同间距比下的瞬时涡量图 -->Fig. 12Instantaneous vorticity distributions of circular cylinders at middle of cylinder span in the case of different spacing to diameter ratios at -->
图13为雷诺数 时,不同间距比(L/D)下串列双方柱绕流的 和 ,同理 表示上游方柱, 表示下游方柱. 间距比较小时,本文计算的平均阻力系数 与实验结果及他人计算结果吻合良好,在间距比较大时,本文计算的 比实验值略大. 从图13(b)中可以看出,本文计算的上游方柱 与实验值很接近,下游方柱的 要比实验值偏大,但总体误差不大. 时,串列双方柱绕流在 区间内与层流分离下的串列双圆柱绕流类似,存在一临界间距比( )使得串列双方柱绕流的 和 均发生跳跃性变化. 同理,如图14所示,上下游方柱水动力系数发生跳跃性变化也是由于流态发生变化而导致的. 只有在 时,上游方柱后方才有单独的涡脱落. 且当 时,下游方柱的阻力系数为负数,而当 时,下游方柱的阻力系数变为正数. 显示原图|下载原图ZIP|生成PPT 图13Re=1.6×104时,不同间距比下串列双方柱绕流的${\rm{\bar C}}$d和C'l -->Fig. 13Mean drag coefficient and RMS lift coefficient of upstream and downstream square cylinders for different spacing to diameter ratios at Re=1.6×104 -->
显示原图|下载原图ZIP|生成PPT 图14时,方柱中心高度处不同间距比下的瞬时涡量图 -->Fig. 14Instantaneous vorticity distributions of square cylinders at middle of cylinder span in the case of different spacing to diameter ratios at -->
显示原图|下载原图ZIP|生成PPT 图14(续)Re=1.6×104时,方柱中心高度处不同间距比下的瞬时涡量图(续) -->Fig. 14(continued) Instantaneous vorticity distributions of square cylinders at middle of cylinder span in the case of different spacing to diameter ratios at Re=1.6×104(continued) -->
图15为 和 时,串列双方柱绕流的${\rm{\bar C}}$d和C'l随间距比的变化情况. 从图中可以看出,与串列双圆柱绕流情况不同,串列双方柱绕流在两种雷诺数下均存在一临界间距比Lc/D,当 时,上下游方柱的阻力系数和升力系数都有跳跃性变大的现象, 时 处于 区间内,而 时, 则处于 区间内. 如图16(b)所示,上游方柱后方形成的漩涡的位置与图14(b)类似,均很靠近上游方柱,因此 时也有临界间距比存在. 且当 时,两种雷诺数下的下游方柱阻力系数均为负数. 显示原图|下载原图ZIP|生成PPT 图15不同雷诺数时,不同间距比下串列双方柱绕流的${\rm{\bar C}}$d和C'l -->Fig. 15Mean drag coefficient and RMS lift coefficient of upstream and downstream square cylinders for different spacing to diameter ratios at different Re -->
显示原图|下载原图ZIP|生成PPT 图16时,方柱中心高度处不同间距比下的瞬时涡量图 -->Fig. 16Instantaneous vorticity distributions of square cylinders at middle of cylinder span in the case of different spacing to diameter ratios at -->
3 结论
本文采用改进的延迟分离涡方法数值模拟了较高下的圆柱绕流及方柱绕流,研究了不同雷诺数下圆柱绕流与方柱绕流的水动力特性,本文的计算结果与已有的实验数据以及其他文献的CFD计算结果吻合良好,验证了本文所采用的IDDES方法在数值模拟柱体绕流时的准确性和有效性. 通过结果的对比分析,可以得到以下结论: (1)在 区间内,单方柱绕流未出现类似于单圆柱绕流的阻力危机现象,单方柱绕流的 , 及 均趋于稳定,维持在一定范围内波动,这与分离点的位置有关,方柱绕流分离点固定,而圆柱绕流的分离点会随雷诺数的变化而变化. (2)串列双圆柱绕流中,当 时,存在一临界间距比( )使得 前后上下游圆柱的升阻力系数发生跳跃性变化. 而当 时,串列双圆柱绕流不再存在临界间距比,随着间距比的增大,下游圆柱的阻力系数和升力系数均缓慢增大,且下游圆柱的阻力系数始终为正数,上游圆柱的阻力系数和升力系数则趋于稳定,均无明显的跳跃性变化. (3)串列双方柱绕流中,在 和 下均存在一临界间距比,当 时,上下游方柱的升阻力系数都有跳跃性变大的现象,但是两种雷诺数下的 值不同. The authors have declared that no competing interests exist.
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