工程科学与技术   2017, Vol. 49 Issue (2): 62-68

1. 武汉大学 水资源与水电工程科学国家重点实验室, 湖北 武汉 430072;
2. 黄河水利科学研究院 黄河小浪底研究中心, 河南 郑州 450003;
3. 黄河勘测规划设计有限公司, 河南 郑州 450003

Vertical Velocity Distribution and Improved Discriminant Formula of Turbidity Current at the Plunging Point in Reservoir
LI Tao1,2, XIA Junqiang1, ZHANG Junhua2, GAO Guoming1,2, XIA Runliang2, WAN Zhanwei3, WANG Zenghui3
1. State Key Lab. of Water Resources and Hydropower Eng. Sci., Wuhan Univ., Wuhan 430072, China;
2. Research Centre on Xiaolangdi of Yellow River, Yellow River Inst. of Hydraulic Research, Zhengzhou 450003, China;
3. Yellow River Eng. Consulting Co. Ltd.; Zhengzhou 450003, China
Abstract: The discriminant formula is a mathematics formula which can decide whether turbidity current plunges.There are an implicit relation between in the discriminant of plunging point, due to the past researchers ignoring or hard to give the mathematics formula on the vertical velocity distribution at the point which could not directly discrete sediment content from Froude number.With mathematic analysis on surveyed data and experiments, physical figure of vertical velocity distribution was decided by parabola.Based on plunging point data during water and sediment regulation period in Xiaolangdi reservoir in 2001 to 2015, and data from tests on physical model of Xiaolangdi reservoir, empirical coefficient of the vertical velocity distribution formula at the point was acquired with calibration, which revealed the theoretical hypothesis was right.Then the momentum correction factor of turbidity current in reservoir is obtained respectively by 1.2 in theory and 1.13, which were closely.Based on the above research, an improved discriminant formula is validated by the data of flumes by Fan Jiahua[13], Cao Ruxuan[14] and Jiao Enze[15], surveyed data of Xiaolangdi reservoir in 2001 to 2005, and data from tests on physical model of Xiaolangdi reservoir.The prediction results of the formula by data from Xiaolangdi reservoir and Sanmenxia reservoir were pretty well with the surveyed.This could be applied in water and sediment regulation program compilation and numerically simulating in sediment-laden reservoirs.
Key words: plunging discriminant formula    vertical velocity distribution    parabolic distribution    momentum correction factor    turbidity current in reservoir

1 潜入点垂线流速分布形式 1.1 潜入点位置

 图1 水库异重流示意图 Fig. 1 Sketch of turbidity current in reservoir

 图2 2014年7月小浪底水库原型异重流潜入点 (距坝18 km) Fig. 2 Plunging point in Xiaolangdi reservoir on July, 2014(18 km to the dam)

 图3 小浪底水库实体模型试验中异重流潜入点 (距坝53 km) Fig. 3 Plunging point in physical model test of Xiaolangdi reservoir (53 km to the dam)

1.2 流速与含沙量分布

1) 在离潜入点较远的上游，水深较小，流速较大，含沙量较大，流速和含沙量沿水深呈正常分布；

2) 到离潜入点不远处，水深增大，流速和含沙量分布呈不正常状态，最大流速位置向库底移动；

3) 在水深增大到一定程度，浑水开始潜入库底，此处为异重流潜入点，这里流速及含沙量沿垂线分布很不均匀，在水面处流速为0，含沙量也几乎为0，最大流速位置进一步向库底靠近；

4) 潜入点往下，异重流已经形成，异重流的流速和含沙量沿水深分布比较均匀，异重流之上形成横轴环流，含沙量的零点在水面以下。在潜入点处，有漂浮物聚集，这通常是判定异重流发生的一个直观标志。潜入点的水流泥沙条件可以作为判定异重流是否发生的条件。

 图4 2002年7月8日小浪底水库实测主流线流速、含沙量沿程分布 Fig. 4 Vertical velocity and sediment content distribution along the Xiaolangdi reservoir in July 8, 2002

 $\frac{{{u}^{'}}}{{{V}_{\text{m}}}}=a{{\left( \frac{z}{{{h}_{\text{p}}}} \right)}^{2}}+b\left( \frac{z}{{{h}_{\text{p}}}} \right)$ (1)

 图5 潜入点流速沿垂线分布图 Fig. 5 Vertical velocity distribution at plunging point

 $\frac{{{u}^{'}}}{{{V}_{\text{m}}}}=4{{\left( \frac{z}{{{h}_{\text{p}}}} \right)}^{2}}+4\left( \frac{z}{{{h}_{\text{p}}}} \right)$ (2)

 $\frac{{{u}^{'}}}{{{V}_{\text{m}}}}=1.79{{\left( \frac{z}{{{h}_{\text{p}}}} \right)}^{2}}+2.36\left( \frac{z}{{{h}_{\text{p}}}} \right)$ (3)

2 潜入点判别条件 2.1 动量修正系数理论分析

 \begin{align} &{{J}_{\text{b}}}-\frac{{{f}_{\text{m}}}}{8}\frac{u_{\text{m}}^{2}}{{{\eta }_{\text{g}}}^{'}g{{h}_{\text{m}}}}-\frac{{{\tau }_{\text{c}}}}{{{h}_{\text{m}}}\left( {{k}_{\text{m}}}{{\gamma }_{\text{m}}}-{{\gamma }_{\text{c}}} \right)}-\frac{\partial {{h}_{\text{m}}}}{\partial x}= \\ &\ \ \ \ \ \frac{1}{{{\eta }_{\text{g}}}^{'}g}\left( \frac{\partial {{u}_{\text{m}}}}{\partial t}+{{\alpha }_{\text{m}}}{{u}_{\text{m}}}\frac{\partial {{u}_{\text{m}}}}{\partial x} \right) \\ \end{align} (4)

 ${{\alpha }_{\text{m}}}=\frac{\int_{0}^{{{h}_{\text{p}}}}{{{u}^{'}}{{u}^{'}}\text{d}z}}{u_{\text{m}}^{2}{{h}_{\text{m}}}}$ (5)

 ${{k}_{\text{m}}}=\frac{\int_{0}^{{{h}_{\text{m}}}}{\left( \int_{z}^{{{h}_{\text{m}}}}{{{\gamma }_{\text{m}}}^{'}\text{d}z} \right)\text{d}z}}{0.5{{\gamma }_{\text{m}}}h_{\text{m}}^{2}}$ (6)

km的大小取决于含沙量S沿水深的变化趋势，目前可利用常见的含沙量沿水深分布公式近似计算。

 \begin{align} &\frac{\text{d}}{\text{d}x}\left( {{q}_{\text{m}}} \right)=\frac{\text{d}}{\text{d}x}\left( {{h}_{\text{m}}}{{u}_{\text{m}}} \right)={{h}_{\text{m}}}\frac{\text{d}}{\text{d}x}\left( {{u}_{\text{m}}} \right)+{{u}_{\text{m}}}\frac{\text{d}}{\text{d}x}\left( {{h}_{\text{m}}} \right)=0 \\ &或\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \frac{\text{d}{{u}_{\text{m}}}}{\text{d}x}=-\frac{{{u}_{\text{m}}}}{{{h}_{\text{m}}}}\frac{\text{d}{{h}_{\text{m}}}}{\text{d}x} \\ \end{align} (7)

 $\frac{\partial {{h}_{\text{m}}}}{\partial x}=\frac{{{J}_{\text{b}}}-\frac{{{f}_{\text{m}}}}{8}\frac{u_{\text{m}}^{2}}{{{\eta }_{\text{g}}}^{'}g{{h}_{\text{m}}}}-\frac{{{\tau }_{\text{c}}}}{{{h}_{\text{m}}}\left( {{k}_{\text{m}}}{{\gamma }_{\text{m}}}-{{\gamma }_{\text{c}}} \right){{\eta }_{\text{g}}}^{'}g}}{1-\frac{{{\alpha }_{\text{m}}}}{{{\eta }_{g}}^{'}g}{{u}_{\text{m}}}\frac{{{u}_{\text{m}}}}{{{h}_{\text{m}}}}}$ (8)

 $\frac{{{\alpha }_{\text{m}}}u_{\text{p}}^{2}}{{{\eta }_{g}}^{'}g{{h}_{\text{p}}}} < 1$ (9)

 ${{u}_{\text{m}}}=\frac{\int_{0}^{{{h}_{\text{p}}}}{{{V}_{\text{m}}}\left( \frac{bz}{{{h}_{\text{p}}}}+\frac{a{{z}^{2}}}{h_{\text{p}}^{2}} \right)\text{d}z}}{{{h}_{\text{p}}}}={{V}_{\text{m}}}\left( \frac{b}{2}+\frac{a}{3} \right)$ (10)
 \begin{align} &\int_{0}^{{{h}_{\text{p}}}}{{{u}^{'}}{{u}^{'}}\text{d}z}=V_{m}^{2}\int_{0}^{{{h}_{\text{p}}}}{\left( \frac{bz}{{{h}_{\text{p}}}}+\frac{a{{z}^{2}}}{h_{\text{p}}^{2}} \right)\text{d}z}= \\ &\ \ \ \ \ \ V_{\text{m}}^{2}{{h}_{\text{p}}}\left( \frac{{{b}^{2}}}{3}+\frac{{{a}^{2}}}{5}+\frac{ab}{2} \right) \\ \end{align} (11)

 ${{\alpha }_{\text{m}}}=\frac{\int_{0}^{{{h}_{\text{p}}}}{{{u}^{'}}{{u}^{'}}\text{d}z}}{u_{\text{m}}^{2}{{h}_{\text{m}}}}=\frac{\left( \frac{{{b}^{2}}}{3}+\frac{{{a}^{2}}}{5}+\frac{ab}{2} \right)}{{{\left( \frac{b}{2}+\frac{a}{3} \right)}^{2}}}$ (12)

z=hp时，理论值${{u}_{\text{m}}}=\frac{2}{3}{{V}_{\text{m}}}$, 代入式 (12) 可得：

 ${{\alpha }_{\text{m}}}=1.20$ (13)

z=hp时，实测值um=0.34Vm, 代入式 (12) 可得：

 ${{\alpha }_{\text{m}}}=1.13$ (14)

2.2 潜入点判别条件推导

 ${{\eta }_{\text{g}}}'=\frac{{{\gamma }^{'}}-{{\gamma }_{0}}}{{{\gamma }^{'}}}=\frac{\left( {{\gamma }_{\text{s}}}-{{\gamma }_{0}} \right)S}{\left( {{\gamma }_{\text{s}}}-{{\gamma }_{0}} \right)S+{{\gamma }_{0}}{{\gamma }_{\text{s}}}}$ (15)

γs=2 650 kg/m3时，式 (15) 可变为:

 ${{\eta }_{\text{g}}}'=\frac{1.65S}{1.65S+2\ 650}=1-\frac{1}{1+\frac{S}{1\ 606}}$ (16)

 ${{\left( {{\eta }_{\text{g}}}' \right)}^{'}}=\frac{\frac{1}{1\ 606}}{{{\left( 1+\frac{S}{1\ 606} \right)}^{2}}}>0$ (17)
 ${{\left( {{\eta }_{\text{g}}}' \right)}^{''}}=-\frac{\frac{1}{1\ {{606}^{2}}}}{{{\left( 1+\frac{S}{1\ 606} \right)}^{3}}} < 0$ (18)

 ${{\eta }_{\text{g}}}'=-1.747\ 4{{\left( {{S}_{\text{V}}} \right)}^{2}}+1.611\ 5{{S}_{\text{V}}}+0.001\ 1$ (19)

 $Fr_{\text{p}}^{2}=u_{\text{p}}^{2}/\left( g{{h}_{\text{p}}} \right)=f\left( {{S}_{\text{V}}}/{{\alpha }_{\text{m}}} \right)$ (20)

 图6 潜入点处Frp2与SV/αm关系图 Fig. 6 Relation between Frp2 and SV/αm

 \begin{align} &Fr_{\text{p}}^{2}=u_{\text{p}}^{2}/\left( g{{h}_{\text{p}}} \right)=0.263{{\left( {{S}_{\text{V}}} \right)}^{0.76}}, \\ &或\ \ \ \ F{{r}_{\text{p}}}={{u}_{\text{p}}}/\sqrt{g{{h}_{\text{p}}}}=0.51{{\left( {{S}_{\text{V}}} \right)}^{0.38}} \\ \end{align} (21)

 \begin{align} &F{{r}^{'}}_{\text{p}}^{2}=u_{\text{m}}^{2}/\left( {{\eta }_{\text{g}}}g{{h}_{\text{m}}} \right)=0.24{{\left( {{S}_{\text{V}}} \right)}^{-0.137}} \\ &或\ \ \ \ F{{r}_{\text{p}}}^{'}={{u}_{\text{m}}}/\sqrt{{{\eta }_{\text{g}}}g{{h}_{\text{m}}}}=0.49{{\left( {{S}_{\text{V}}} \right)}^{-0.069}} \\ \end{align} (22)

qm=uphp代入式 (22)，则可得潜入点水深与单宽流量及体积比含沙量之间的关系表达式，即

 ${{h}_{\text{p}}}=0.738\ 6q_{\text{m}}^{2/3}/S_{\text{V}}^{0.764/3}$ (23)

 图7 潜入点计算与实测水深对比 Fig. 7 Plunging depth of calculated and surveyed

3 结论

1) 在对异重流潜入点流速垂线分布进行文献分析的基础上，提出其数学表达式近似接近截距等零的抛物线，抛物线的系数理论值分别为-4、4。将小浪底水库实测的异重流潜入点流速沿垂线分布资料进行率定，其系数分别为-1.79、2.76。两者数值接近，正负及数量级相同，其中的差别在于实际测验的误差、浑水阻力变化与水流、泥沙运动的复杂性。

2) 根据动量修正系数的定义，将潜入点流速垂线分布公式进行计算后，相应得到动量修正系数的理论值为1.2，小浪底水库实测及物理模型实测值为1.13，二者甚为接近。

3) 根据2) 的结果，收集2006—2015年小浪底库区潜入点、1961—1962年三门峡库区资料、小浪底库区模型试验资料，结合前人研究，对潜入点判别条件进行了修正，得到了新的潜入点修正公式，其计算结果与实测值接近。

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