﻿ 往复流作用下推移质输沙强度的计算公式
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 哈尔滨工程大学学报  2019, Vol. 40 Issue (2): 247-253  DOI: 10.11990/jheu.201708073 0

### 引用本文

LI Ming, YU Meixin, YU Guoliang. Formula for predicting bedload sediment transport rate under oscillatory flows[J]. Journal of Harbin Engineering University, 2019, 40(2), 247-253. DOI: 10.11990/jheu.201708073.

### 文章历史

1. 上海交通大学 船舶与海洋工程系, 上海 200240;
2. 中船第九设计研究院工程有限公司, 上海 200063

Formula for predicting bedload sediment transport rate under oscillatory flows
LI Ming 1, YU Meixin 2, YU Guoliang 1
1. Department of Naval Architecture and Ocean Engineering, Shanghai Jiao Tong University, Shanghai 200240, China;
2. China Shipbuilding NDRI Engineering Co., Ltd., Shanghai 200063, China
Abstract: To examine the influence of unsteady effect of oscillatory flow on bedload sediment transport and accurately predict the bedload sediment transport rate under oscillatory flows, this study presents an analysis of force balancing and energy conservation in sediment particle movement. Calculation formulas for bedload sediment transport rate under oscillatory flows are derived and, in the meantime, Keulegan-Carpenter (KC) number is introduced to correct the phase-lag effect. Finally, experimental data on oscillatory flows are fitted, thereby deriving a semi-theoretical semi-empirical formula used in calculating the half-cycle averaged bedload sediment transport rate. Compared with the existing quasi-steady flow model, the new formula shows an obvious improvement in predicting the measure results, especially when the KC number is small.
Keywords: bedload sediment transport    oscillatory flow    U-shaped tunnel experiment    phase-lag effect    KC number    semi-theoretical semi-empirical formula

1 推移质输沙强度计算式的推导 1.1 相位滞后效应与KC数的关系

 $\frac{1}{2}\frac{{{U^2}}}{{\left( {s-1} \right)g{W_s}T}}\;\;\;或\;\;\;\frac{{2\pi \Delta }}{{{W_s}T}}$

 $\frac{{{\upsilon ^{0.25}}{U^{0.5}}}}{{{W_s}{T^{0.75}}}}\exp \left[{-{{\left( {\frac{{{U_{w.{\text{cesf}}}}}}{U}} \right)}^2}} \right]\;\;\;或\;\;\;\frac{\Delta }{{{W_s}T}}\left( {1 -\frac{{\xi {u_w}}}{{{c_w}}}} \right)$

 ${\text{KC' = }}\frac{{\left( {u-{u_p}} \right)T}}{D}$ (1)

 ${\text{KC' = }}\frac{{{{\left( {u-{u_p}} \right)}^2}/D}}{{\left| {u-{u_p}} \right|/T}} \sim \frac{{{{\left( {u-{u_p}} \right)}^2}}}{{D\left| {\frac{{{\text{d}}u}}{{{\text{d}}t}} - \frac{{{\text{d}}{u_p}}}{{{\text{d}}t}}} \right|}}$

1.2 推移质输沙强度计算公式

 Download: 图 1 散粒体泥沙颗粒受力情况 Fig. 1 Force analysis for sediment granular particles
 $\begin{gathered} {\rho _s}V\frac{{{\text{d}}{u_p}}}{{{\text{d}}t}} = \rho V\frac{{{\text{d}}u}}{{{\text{d}}t}} + \rho {C_a}V\left( {\frac{{{\text{d}}u}}{{{\text{d}}t}}-\frac{{{\text{d}}{u_p}}}{{{\text{d}}t}}} \right) + \hfill \\ \;\;\;\;\;\;\frac{1}{2}\rho {C_d}A{\left( {u-{u_p}} \right)^2}\frac{{u-{u_p}}}{{\left| {u - {u_p}} \right|}} - \hfill \\ \;\;\;\beta \left( {\left( {{\rho _s} - \rho } \right)Vg - \frac{1}{2}\rho {C_l}A{{\left( {u - {u_p}} \right)}^2}} \right) \hfill \\ \end{gathered}$ (2)

 $\begin{gathered} 4\left( {s-1} \right)D\frac{{{\text{d}}u}}{{{\text{d}}t}}/\left( {\frac{{{\text{d}}u}}{{{\text{d}}t}}-\frac{{{\text{d}}{u_p}}}{{{\text{d}}t}}} \right) = 4D\left( {s + {C_a}} \right) + \hfill \\ 3D\left( {{C_d}\frac{{u-{u_p}}}{{\left| {u - {u_p}} \right|}} + \beta {C_l}} \right){\left( {u - {u_p}} \right)^2}/\left( {\frac{{{\text{d}}u}}{{{\text{d}}t}} - \frac{{{\text{d}}{u_p}}}{{{\text{d}}t}}} \right) - \hfill \\ \;\;\;\;\;\;\;\;\;\;\;4D\beta \left( {s - 1} \right)g/\left( {\frac{{{\text{d}}u}}{{{\text{d}}t}} - \frac{{{\text{d}}{u_p}}}{{{\text{d}}t}}} \right) \hfill \\ \end{gathered}$ (3)

${\left( {u-{u_p}} \right)^2}/\left( {\frac{{{\text{d}}u}}{{{\text{d}}t}}-\frac{{{\text{d}}{u_p}}}{{{\text{d}}t}}} \right)$用一个与KC′有关的函数D·f(KC′)代替:

 $\begin{gathered} \left( {4\left( {s-1} \right)D\frac{{{\text{d}}u}}{{{\text{d}}t}} + 4D\beta \left( {s-1} \right)g} \right)/\left( {\frac{{{\text{d}}u}}{{{\text{d}}t}}-\frac{{{\text{d}}{u_p}}}{{{\text{d}}t}}} \right) = \hfill \\ 4D\left( {s + {C_a}} \right) + 3D\left( {{C_d}\frac{{u - {u_p}}}{{\left| {u - {u_p}} \right|}} + \beta {C_l}} \right)f\left( {{\text{KC'}}} \right) \hfill \\ \end{gathered}$ (4)

 $\frac{{{\text{d}}{u_p}}}{{{\text{d}}t}} = \left( {1-\frac{{4\left( {s-1} \right)}}{m}} \right)\frac{{{\text{d}}u}}{{{\text{d}}t}}-\frac{{4\beta \left( {s - 1} \right)g}}{m}$ (5)

 $m = 4\left( {s + {C_a}} \right) + 3\left( {{C_d}\frac{{u-{u_p}}}{{\left| {u-{u_p}} \right|}} + \beta {C_l}} \right)f\left( {{\text{KC'}}} \right)$

 $\begin{gathered} {u_p}\left( t \right) = \left( {1-\frac{{4\left( {s-1} \right)}}{m}} \right)\left( {u\left( t \right)-{u_{cr}}} \right) - \hfill \\ \;\;2\beta \left( {s - 1} \right)g\int_{{t_{cr}}}^t {\frac{1}{m}{\text{d}}t - 12\left( {s - 1} \right)} \cdot \hfill \\ \left( {{C_d}\frac{{u - {u_p}}}{{\left| {u - {u_p}} \right|}} + \beta {C_l}} \right)\int_{{t_{cr}}}^t {\frac{u}{{{m^2}}}f'\left( {{\text{KC'}}} \right){\text{d}}t} \hfill \\ \end{gathered}$ (6)

 ${E_{in}} = \tau {\text{d}}A{\text{d}}x$ (7)

 ${E_k} = {\tau _{cr}}{\text{d}}A{\text{d}}x$ (8)

 ${E_f} = N{\text{d}}A \cdot \frac{\pi }{6}\beta \left( {{\rho _s}-\rho } \right){D^3}g{\text{d}}x$ (9)

 $N = \frac{6}{{\beta \pi }}\left( {\theta-{\theta _{cr}}} \right)\frac{1}{{{D^2}}}$ (10)

 $\varphi \left( t \right) = \frac{{\pi {D^3}N{u_p}}}{{6\sqrt {\left( {s-1} \right)g{D^3}} }}$ (11)

 $\varphi \left( t \right) = \sqrt {\frac{2}{{{f_w}}}} \frac{{\theta-{\theta _{cr}}}}{\beta }\sqrt \theta \frac{{{u_p}}}{{\left| u \right|}}, \;\;\;\;\theta > {\theta _{cr}}$ (12)

 ${\phi _{1/2}} = \frac{r}{{\beta \sqrt {{f_w}} }}\left( {{\theta _m}-{\theta _{cr}}} \right)\sqrt {{\theta _m}} \left( {1-F\left( {{\text{KC''}}} \right)} \right)$ (13)

2 往复流水槽的输沙实验 2.1 实验安排

 Download: 图 3 两种非均匀沙的颗粒特征 Fig. 3 Grain size distributions of the two non-uniform sands

2.2 实验结果

 Download: 图 4 u、up、N在一个周期内的变化(工况Q2) Fig. 4 Variation of u, up, N versus time in a period(test Q2)

3 半理论半经验公式

 ${\varPhi _{1/2}} = \frac{r}{{\beta \sqrt {{f_w}} }}\left( {{\theta _m}-{\theta _{cr}}} \right)\sqrt {{\theta _m}}$ (14)

 Download: 图 7 Φ1/2实验值与式(14)计算值的比较 Fig. 7 Measured half-cycle averaged bedload transport intensity vs. calculated results with Eq. (14)

 ${\varPhi _{1/2}} = \frac{r}{{\beta \sqrt {{f_w}} }}\left( {{\theta _m}-{\theta _{cr}}} \right)\sqrt {{\theta _m}} \left( {1-\frac{{2.17}}{{\ln \left( {{\text{KC''}}} \right)}}} \right)$ (15)

 Download: 图 9 Φ1/2实验值与式(15)计算值的比较 Fig. 9 Measured half-cycle averaged bedload transport intensity vs. calculated results with Eq. (15)

 Download: 图 10 KC″较小时各公式对实验数据的预测情况比较 Fig. 10 Comparison of the performances of Eq. (15) and three quasi-steady models when KC″ is small
4 结论

1) KC数是影响往复流输沙的一个重要指标。当KC数较小，即泥沙颗粒尺寸相比于水质点运动幅度不可忽略时，惯性力对泥沙运动的影响不可忽略，相位滞后效应导致的输沙强度减弱现象明显。

2) 本文提出的半理论半经验输沙强度计算公式由准恒定流项和KC数修正项两部分组成，前者描述了水动力条件的强弱，后者描述了相位滞后相应对输沙强度的削减作用。公式对半周期平均输沙强度的预测效果相比准恒定流模型有很大提高。

3) 当KC″＜8.76时，修正项不具有物理意义；在KC″＞10 000的情况下，修正公式的计算值偏低，这说明该公式有一定的适用范围。