工程科学与技术   2019, Vol. 51 Issue (1): 52-59

1. 北京林业大学 水土保持学院 重庆缙云山三峡库区森林生态系统国家定位观测研究站，北京 100083;
2. 北京市水土保持工程技术研究中心，北京 100083

Characteristics of Horseshoe Vortex Upstream of the Cylinder in Shallow Water with Low Cylinder Reynolds Number
YANG Pingping1, ZHANG Huilan1,2, WANG Yunqi1,2, WANG Yujie1,2
1. Jinyun Forest Ecosystem Research Station, School of Soil and Water Conservation, Beijing Forestry Univ., Beijing 100083, China;
2. Beijing Eng. Research Center of Soil and Water Conservation, Beijing Forestry Univ., Beijing 100083, China
Abstract: The horseshoe vortex (HV) is formed at the upstream of a vertical cylinder when flow passes the cylinder and it is responsible for the local scouring at the base of the cylinder. Extensive works had been carried out to investigate the characteristics of HV in open channel flow with high Reynold number and large flow depth. However, it was difficult to measure HV experimentally in low Reynold number and shallow flow depth, in view of limitation of experimental technology. To capture HV accurately in shallow water flow, a high resolution and high frequency particle image velocimetry (HR-PIV) was employed in present study. Subsequently, the flow fields upstream of the cylinder were captured by HR-PIV in 6 experimental groups with shallow flow depth. The separation points of each groups were obtained by analyzing the characteristics of time-averaged flow fields. The HV was calculated by λci criterion where λci represented the swirling strength of vortex. Then the locations of HV were obtained in accordance with the maximal swirling strength point. In addition, the radius of HV was calculated by superposition of Oseen vortex and pure shear model. The results showed that within a low cylinder Reynolds number (ReD) where ReD<5 000, as the increase ofReD, the location of separation point and HV were rapidly approaching the cylinder simultaneously, whereas the HV moved rapidly towards the flume bed, while the radius of HV decreased and the swirling strength increased. Under shallow water flow conditions and the cylinder diameter keeping constant, as the increase of flow depth, the locations of separation point moved towards upstream; HV moved towards upstream and free surface, simultaneously, while the radius of HV increased. Theses HV parameters in the present flow conditions were larger than those in open channel flows. Derived from previous works, it was found that the separation point and HV would display a different manner as ReD became lager. When 5 000<ReD <8 000, the separation point was still rapidly moving downstream while HV remained stable as increasing ReD. While ReD>8 000, the separation point was slowly moving downstream and HV still remained stable. The research results can provide a basis and reference for engineering design for preventing local scouring at the base of cylinder.
Key words: flow around cylinder    horseshoe vortex    particle image velocimetry    shallow water flow    low Reynolds number

1 试验与方法 1.1 试验系统

 图1 试验示意图 Fig. 1 Schematic diagram of experimental set-up

1.2 涡旋识别方法

 ${M}{\text{ = }}\left[\!\!\! {\begin{array}{*{20}{c}} {\displaystyle\frac{{\partial u}}{{\partial x}}}&{\displaystyle\frac{{\partial u}}{{\partial y}}}\\ {\displaystyle\frac{{\partial v}}{{\partial x}}}&{\displaystyle\frac{{\partial v}}{{\partial y}}} \end{array}}\!\!\!\right]$ (1)

${\lambda {\rm _{ci}}}$ 是该矩阵特征值的虚部，在2维平面下计算方法为：

 {\lambda _{{\rm ci}}} = \left\{ {\begin{aligned} & {\sqrt {Q - \frac{{{P^2}}}{4}} }{\text{，}}{Q - \frac{{{P^2}}}{4} < 0}{\text{；}}\\ & {0,}\qquad\qquad\;\,{Q - \frac{{{P^2}}}{4} < 0} \end{aligned}} \right. (2)

 $P = - \frac{{\partial u}}{{\partial x}} - \frac{{\partial v}}{{\partial y}}$ (3)
 $Q = \frac{{\partial u}}{{\partial x}}\frac{{\partial v}}{{\partial y}} - \frac{{\partial u}}{{\partial y}}\frac{{\partial v}}{{\partial x}}$ (4)

1.3 马蹄涡的尺度提取

 \begin{aligned}[b] & {u = \frac{\varGamma }{{2{\text{π}} }} \cdot \left[ {1 - \exp \left( - \frac{{{x^2} + {y^2}}}{{{R^2}}}\right)} \right] \cdot \left( - \frac{y}{{{x^2} + {y^2}}}\right)}+\\ & {\;\;\; k{{\cos }^2}\theta y - k\cos \theta \sin x} \end{aligned} (5)
 \begin{aligned}[b] & {v = \frac{\varGamma }{{2{\text{π}} }} \cdot \left[ {1 - \exp \left( - \frac{{{x^2} + {y^2}}}{{{R^2}}}\right)} \right] \cdot \frac{x}{{{x^2} + {y^2}}}}+\\ & {\;\;\; k\sin \theta \cos \theta y - k{{\sin }^2}\theta x} \end{aligned} (6)

 图2 实测流场与拟合流场对比 Fig. 2 Compared of simulated and measured fields

2 试验结果与讨论 2.1 流动分离点

 图3 流动分离点位置随柱体雷诺数变化关系 Fig. 3 Relationship between location of separation point and cylinder Reynolds number

 图4 流动分离点位置随水深的变化关系 Fig. 4 Relationship between location of separation point and depth

2.2 马蹄涡的特征

 图5 马蹄涡识别 Fig. 5 Horseshoe vortex system extraction

 图6 马蹄涡纵向及垂向位置随柱体雷诺数变化关系 Fig. 6 Relationship between location of horseshoe voretex and cylinder Reynolds number

 图7 马蹄涡半径随柱体雷诺数的变化关系 Fig. 7 Relationship between radius of horseshoe vortex and cylinder Reynolds number

 图8 旋转强度随柱体雷诺数的变化关系 Fig. 8 Relationship between swirling rate of horseshoe voretex and cylinder Reynolds

 图9 主马蹄涡位置及半径随水深变化关系 Fig. 9 Relationship between both location and radius of horseshoe vortex and cylinder Reynolds number

2.3 流动过程

 图10 柱体前端马蹄涡模型 Fig. 10 Model of horseshoe vortex

3 结　论

1）在低柱体雷诺数条件下（1 600< ${{\mathop{Re}\nolimits} _{\rm{D}}}$ <4 400），随着柱体雷诺数的增加流动分离点和马蹄涡的纵向位置皆急剧向下游运动，马蹄涡的垂向位置减小逐渐靠向床面，马蹄涡半径变小但其旋转强度变强。

2）在浅薄层水流条件下（0.48< $h/D$ <0.58），当柱体直径不变时，随着水深增加，流动分离点和马蹄涡的纵向位置向上游运动，马蹄涡的垂向位置和半径增加，且显著大于明渠水流条件下。

3）与前人的研究数据对比，发现马蹄涡的运动状态呈3个阶段：500< ${{\mathop{Re}\nolimits} _{\rm{D}}}$ <5 000，流动分离点、马蹄涡纵垂向位置、半径与 ${{\mathop{Re}\nolimits} _{\rm{D}}}$ 呈反比，与马蹄涡旋转强度呈正比；5 000< ${{\mathop{Re}\nolimits} _{\rm{D}}}$ <8 000，流动分离点仍与 ${{\mathop{Re}\nolimits} _{\rm{D}}}$ 呈反比，马蹄涡各项参数稳定； ${{\mathop{Re}\nolimits} _{\rm{D}}}$ >8 000，流动分离点缓慢向下游移动，马蹄涡参数稳定，主马蹄涡的纵向位置稳定在0.17 $D$ 、垂向位置在0.06 $D$ ，半径大小在0.04 $D$ 左右。

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