Experimental Study on Hydraulic Characteristics of the Aerated Flow over Spillway in Lianghekou Hydropower Station

doi:10.11988/ckyyb.20240057

Abstract

Abstract:

Based on the mechanism experiments conducted for the spillway of the Lianghekou hydropower station, this study investigates the variation tendencies of the flow regime, dynamic pressure, air vent cavities, wind speed, and aeration concentration under three types of aeration thresholds. The applicability of formulas for calculating cavity length, gas-water ratio, and water content ratio to the studied project are analyzed, and the derived conclusions could be employed for further investigation on Lianghekou hydropower station and other similar hydraulic projects. Research findings indicate that aeration in the water flow becomes more pronounced with the increasing of spillway discharge, and several hydraulic indicators increase simultaneously, including the root mean square of time-averaged pressure and fluctuating pressure, aeration rate, cavity negative pressure, and cavity length. At a given discharge, the air concentration increases along the spillway, and the vertical distribution of aeration concentration displays successively C-shaped, S-shaped, and I-shaped patterns. According to the research findings, we recommend that Shi Qisui’s formula for cavity length, Chen Zhaohe’s formula for gas-water ratio, and Hall’s formula for water content ratio are most suitable for the estimations of the present study.

Key words: deep hole spillway tunnel, cavity length, water content ratio, aeration concentration, gas-water ratio

CLC Number:

TV135.2泄水建筑物水力学

Cite this article

WANG Ji-min, LIU Zong-xian, DU Cheng-bo, MA Bin, ZHANG Zi-liang. Experimental Study on Hydraulic Characteristics of the Aerated Flow over Spillway in Lianghekou Hydropower Station[J]. Journal of Changjiang River Scientific Research Institute, 2025, 42(4): 103-111.

share this article

0
/ / Recommend

Add to citation manager EndNote|Ris|BibTeX

URL: http://ckyyb.crsri.cn/EN/10.11988/ckyyb.20240057

http://ckyyb.crsri.cn/EN/Y2025/V42/I4/103

Figures/Tables 21

Fig.1 Overall model

Fig.2 Layout of aeration steps

Table 1 Test schemes

掺气坎	试验工况	H_u/m	v/(m·s^-1)	Q_w/(m³·s^-1)	Fr
	1	45	33.21	1 173.22	5.92
4^#	2	75	38.58	1 542.93	6.46
	3	108	44.88	1 858.42	7.38
	4	45	36.28	1 173.22	6.75
5^#	5	75	39.57	1 542.93	6.72
	6	108	40.99	1 858.42	6.44
	7	45	36.66	1 173.22	6.86
6^#	8	75	39.73	1 542.93	6.75
	9	108	40.11	1 858.42	6.23

Table 1 Test schemes

掺气坎	试验工况	H_u/m	v/(m·s^-1)	Q_w/(m³·s^-1)	Fr
	1	45	33.21	1 173.22	5.92
4^#	2	75	38.58	1 542.93	6.46
	3	108	44.88	1 858.42	7.38
	4	45	36.28	1 173.22	6.75
5^#	5	75	39.57	1 542.93	6.72
	6	108	40.99	1 858.42	6.44
	7	45	36.66	1 173.22	6.86
6^#	8	75	39.73	1 542.93	6.75
	9	108	40.11	1 858.42	6.23

Fig.3 Measurement parameters

Fig.4 Overflow state under different flow rates of different aerators

Fig.5 Time-averaged pressure and fluctuating pressure along the way for the three aerator types

Fig.6 Schematic diagram of time-averaged pressure and fluctuating pressure along the way

Table 2 Peak values of the root mean squares of time-averaged pressure and fluctuating pressure and their corresponding pile number

掺气坎	流量/ (m³·s^-1)	时均压强/m				脉动压强均方根/m
掺气坎	流量/ (m³·s^-1)	坎前峰值点桩号/m	坎前峰值	坎后峰值点桩号/m	坎后峰值	峰值点桩号/m	峰值
	1 173.90	4-11.7	9.75	4+20.91	9.00	4+20.91	2.92
4^#	1 540.48	4-11.7	13.02	4+26.88	8.52	4+26.88	2.00
	1 859.73	4-11.7	15.54	4+31.35	7.56	4+29.85	2.08
	1 173.90	5-11.7	11.19	5+22.41	7.80	5+20.91	2.27
5^#	1 540.48	5-11.7	14.79	5+26.88	7.92	5+26.88	2.20
	1 859.73	5-11.7	17.70	5+31.35	8.13	5+32.85	1.89
	1 173.90	6-11.7	11.01	6+25.38	7.65	6+23.88	2.27
6^#	1 540.48	6-11.7	13.89	6+28.38	7.95	6+26.88	1.81
	1 859.73	6-11.7	16.50	6+31.35	8.25	6+32.85	1.94

Table 2 Peak values of the root mean squares of time-averaged pressure and fluctuating pressure and their corresponding pile number

掺气坎	流量/ (m³·s^-1)	时均压强/m				脉动压强均方根/m
掺气坎	流量/ (m³·s^-1)	坎前峰值点桩号/m	坎前峰值	坎后峰值点桩号/m	坎后峰值	峰值点桩号/m	峰值
	1 173.90	4-11.7	9.75	4+20.91	9.00	4+20.91	2.92
4^#	1 540.48	4-11.7	13.02	4+26.88	8.52	4+26.88	2.00
	1 859.73	4-11.7	15.54	4+31.35	7.56	4+29.85	2.08
	1 173.90	5-11.7	11.19	5+22.41	7.80	5+20.91	2.27
5^#	1 540.48	5-11.7	14.79	5+26.88	7.92	5+26.88	2.20
	1 859.73	5-11.7	17.70	5+31.35	8.13	5+32.85	1.89
	1 173.90	6-11.7	11.01	6+25.38	7.65	6+23.88	2.27
6^#	1 540.48	6-11.7	13.89	6+28.38	7.95	6+26.88	1.81
	1 859.73	6-11.7	16.50	6+31.35	8.25	6+32.85	1.94

Fig.7 Relationship between Lj/h and Xmax/h 2.3 通气空腔和风速

Table 3 Hydraulic characteristics of aerator

掺气坎	流量Q_w/ (m³·s^-1)	断面平均流速 v/(m·s^-1)	坎末水深h/m	空腔负压ΔP/m	空腔长度 L_j/m	风速v_a/ (m·s^-1)	掺气量Q_a/ (m³·s^-1)	回水长度 L_h/m	保护长度 L_c/m
	1 173.90	33.19	3.21	-0.15	17.88	14.9	38.74	16.53	54.71
4^#	1 540.48	38.56	3.63	-0.18	22.79	31.99	83.17	21.32	77.91
	1 859.73	44.86	3.77	-0.27	26.13	50.17	130.44	24.15	101.83
	1 173.90	36.26	2.94	-0.18	18.74	19.35	50.31	17.24	53.84
5^#	1 540.48	39.55	3.54	-0.21	22.98	34.94	90.84	21.48	89.39
	1 859.73	40.97	4.13	-0.27	25.82	47.27	122.9	23.93	113.67
	1 173.90	36.64	2.91	-0.21	20.51	23.88	62.09	19.01	61.93
6^#	1 540.48	39.71	3.53	-0.24	23.28	35.16	91.42	21.78	—
	1 859.73	40.09	4.22	-0.27	26.29	46.78	121.63	24.43	—

Table 3 Hydraulic characteristics of aerator

掺气坎	流量Q_w/ (m³·s^-1)	断面平均流速 v/(m·s^-1)	坎末水深h/m	空腔负压ΔP/m	空腔长度 L_j/m	风速v_a/ (m·s^-1)	掺气量Q_a/ (m³·s^-1)	回水长度 L_h/m	保护长度 L_c/m
	1 173.90	33.19	3.21	-0.15	17.88	14.9	38.74	16.53	54.71
4^#	1 540.48	38.56	3.63	-0.18	22.79	31.99	83.17	21.32	77.91
	1 859.73	44.86	3.77	-0.27	26.13	50.17	130.44	24.15	101.83
	1 173.90	36.26	2.94	-0.18	18.74	19.35	50.31	17.24	53.84
5^#	1 540.48	39.55	3.54	-0.21	22.98	34.94	90.84	21.48	89.39
	1 859.73	40.97	4.13	-0.27	25.82	47.27	122.9	23.93	113.67
	1 173.90	36.64	2.91	-0.21	20.51	23.88	62.09	19.01	61.93
6^#	1 540.48	39.71	3.53	-0.24	23.28	35.16	91.42	21.78	—
	1 859.73	40.09	4.22	-0.27	26.29	46.78	121.63	24.43	—

Fig.8 Vertical aeration concentration distribution for the three aerator types

Fig.9 Aeration concentration distribution along the spillway for the three aerator types

Table 4 Calculation formulae of cavity length

公式名称	计算公式
潘水波公式^[13]	$\begin{array}{c}L=B C h_{0}\left\{F r^{2} \frac{\cos (\alpha-A \theta)}{\cos ^{2} \alpha}\left[\sin (A \theta)+\sqrt{\sin ^{2}(A \theta)+\frac{2 g\left(\Delta+h_{0} / 2\right)}{v_{0}^{2}} \cos \alpha}\right]+\right. \\\left.\frac{\Delta+h_{0} / 2}{h_{0}} \tan \alpha\right\}\end{array}$	L为空腔长度;A为出射角修正系数;B为能量损失系数;C为距离矫正系数;Δ为坎高;θ为挑角;α为坡角; h₀为水深; v₀为流速;Fr为弗劳德数
时启燧公式^[14]	$\left\{\begin{array}{l}\frac{L_{\mathrm{c}}}{h_{1}}=\frac{L_{\mathrm{p}}}{h_{1}}=0.155+2.961 X_{1}-1.674 X_{1}^{-1} \\X_{1}=\left(U_{1} / \sqrt{g h_{1}}\right)\left(\sqrt{\Delta / h_{1}} /(\cos \alpha \cos \theta)\right)\end{array}\right.$	L_c、L_p分别为根据水流的掺气浓度和底板压力分布划定的空腔长度;X₁为综合水力参数;U₁为流速;h₁为水深;θ为挑角;坎高Δ=0.3~2.5 cm;U₁=6.58~15.4 m/s
徐一民公式^[15]	$\left\{\begin{array}{l}\frac{v^{2}}{g} y^{\prime \prime}=\left(1+y^{\prime 3}\right) \cos \alpha-y^{\prime}\left(1+y^{\prime 2}\right) \sin \alpha+\frac{\Delta p}{\gamma h}\left(1+y^{\prime 2}\right)^{\frac{3}{2}} \\y^{\prime}(0)=\tan \theta \\y(0)=\Delta+y_{0}\end{array}\right.$	v为断面平均流速;α为底坡坡角;Δp为射流上下表面压差;θ为挑角;Δ为坎高;y₀为跌坎高度;γ为射流冲击角;g为重力加速度;t为时间
罗铭公式^[16]	$\left\{\begin{array}{l}L_{\mathrm{jet}}=V_{1} T \cos \varphi_{1}+0.5 g\left(\sin \alpha-0.00625 F r^{2}\right) T^{2} \\T=\frac{V_{1} \sin \varphi_{1}}{g\left(\cos \alpha+P_{\mathrm{N}}\right)}\left[1+\sqrt{1+\frac{\left(2 t_{\mathrm{r}}+t_{\mathrm{s}}\right) g\left(P_{\mathrm{N}}+\cos \alpha\right)}{\left(V_{1} \sin \varphi_{1}\right)^{2}}}\right] \\\varphi_{1}=\xi_{1} \varphi \\\xi_{1}=\sqrt{\frac{\mathrm{e}^{X}-\mathrm{e}^{-X}}{\mathrm{e}^{X}+\mathrm{e}^{-X}}} \\X=\frac{t_{\mathrm{r}}}{h_{1} \varphi} \mathrm{e}^{\frac{2.7725}{F r_{1}-1}} \\V_{1}=\xi_{2} V_{0}\end{array}\right.$	L_jet为射流的空腔长度;V₀、φ、α、P_N分别为断面平均流速、挑角、坡角、空腔负压指数; t_r、t_s分别为坎高及跌坎高度;T为水舌上缘上升高度;φ₁为实际射流出射角;V₁为射流底缘实际流速;Fr为弗劳德数;ξ₁、ξ₂为由试验研究确定的修正系数;φ<15°;0.908<ξ₂<1.0;g为重力加速度
夏毓常公式^[17]	$\frac{L}{h}=\frac{1}{\cos \alpha}\left[\frac{\Delta}{h} \sin \alpha+F r \frac{\cos (\alpha-\theta)}{\cos \alpha}\left(F r \sin \theta+\sqrt{F r^{2} \sin ^{2} \theta+2 \frac{\Delta}{h} \cos \alpha}\right)\right]$	Fr为弗劳德数;L为空腔长度; Δ为挑高; α为坡角; θ为挑角;h为水深
苗宝广公式^[17]	$\left\{\begin{array}{l}X=\frac{F r \cos \alpha_{1}}{\cos \alpha_{2} \cos \theta} \sqrt{\frac{t_{\mathrm{s}}+t_{\mathrm{r}}}{h}} \\\frac{L}{h}=2.4995 X+0.9049\end{array}\right.$	X为综合水力参数;Fr为弗劳德数;h为水深; t_s、t_r分别为跌坎高度及坎高; α₁、α₂、θ分别为上底坡坡角、下底坡坡角及挑角
Rutschmann公式^[15]	$\left\{\begin{array}{l}L=L_{\max }(1-0.4 \sqrt{\Delta \bar{P}}) \\\Delta \bar{P}=\frac{\Delta P}{\gamma h_{0}} \\L_{\max }=\frac{F r^{2}-\theta h_{0}}{\cos \alpha}\left(1+\sqrt{2+\frac{2 T \cos \alpha}{\theta F r^{2}}}\right) \\T=\left(\Delta+y_{0}\right) / h_{0} \\\bar{\theta}=\theta \sqrt{\operatorname{th}\left(\frac{\Delta}{\theta h_{0}}\right)} \\F r=\frac{v_{0}}{\sqrt{g h_{0}}}\end{array}\right.$	ΔP为空腔压力; Δ、y₀分别为坎高及跌坎高度;γ为射流冲击角;h₀为来流水深;v₀为来流流速;L为空腔长度;L_max为最大空腔长度;θ为挑角;α为底坡坡角;t为时间

Table 4 Calculation formulae of cavity length

公式名称	计算公式
潘水波公式^[13]	$\begin{array}{c}L=B C h_{0}\left\{F r^{2} \frac{\cos (\alpha-A \theta)}{\cos ^{2} \alpha}\left[\sin (A \theta)+\sqrt{\sin ^{2}(A \theta)+\frac{2 g\left(\Delta+h_{0} / 2\right)}{v_{0}^{2}} \cos \alpha}\right]+\right. \\\left.\frac{\Delta+h_{0} / 2}{h_{0}} \tan \alpha\right\}\end{array}$	L为空腔长度;A为出射角修正系数;B为能量损失系数;C为距离矫正系数;Δ为坎高;θ为挑角;α为坡角; h₀为水深; v₀为流速;Fr为弗劳德数
时启燧公式^[14]	$\left\{\begin{array}{l}\frac{L_{\mathrm{c}}}{h_{1}}=\frac{L_{\mathrm{p}}}{h_{1}}=0.155+2.961 X_{1}-1.674 X_{1}^{-1} \\X_{1}=\left(U_{1} / \sqrt{g h_{1}}\right)\left(\sqrt{\Delta / h_{1}} /(\cos \alpha \cos \theta)\right)\end{array}\right.$	L_c、L_p分别为根据水流的掺气浓度和底板压力分布划定的空腔长度;X₁为综合水力参数;U₁为流速;h₁为水深;θ为挑角;坎高Δ=0.3~2.5 cm;U₁=6.58~15.4 m/s
徐一民公式^[15]	$\left\{\begin{array}{l}\frac{v^{2}}{g} y^{\prime \prime}=\left(1+y^{\prime 3}\right) \cos \alpha-y^{\prime}\left(1+y^{\prime 2}\right) \sin \alpha+\frac{\Delta p}{\gamma h}\left(1+y^{\prime 2}\right)^{\frac{3}{2}} \\y^{\prime}(0)=\tan \theta \\y(0)=\Delta+y_{0}\end{array}\right.$	v为断面平均流速;α为底坡坡角;Δp为射流上下表面压差;θ为挑角;Δ为坎高;y₀为跌坎高度;γ为射流冲击角;g为重力加速度;t为时间
罗铭公式^[16]	$\left\{\begin{array}{l}L_{\mathrm{jet}}=V_{1} T \cos \varphi_{1}+0.5 g\left(\sin \alpha-0.00625 F r^{2}\right) T^{2} \\T=\frac{V_{1} \sin \varphi_{1}}{g\left(\cos \alpha+P_{\mathrm{N}}\right)}\left[1+\sqrt{1+\frac{\left(2 t_{\mathrm{r}}+t_{\mathrm{s}}\right) g\left(P_{\mathrm{N}}+\cos \alpha\right)}{\left(V_{1} \sin \varphi_{1}\right)^{2}}}\right] \\\varphi_{1}=\xi_{1} \varphi \\\xi_{1}=\sqrt{\frac{\mathrm{e}^{X}-\mathrm{e}^{-X}}{\mathrm{e}^{X}+\mathrm{e}^{-X}}} \\X=\frac{t_{\mathrm{r}}}{h_{1} \varphi} \mathrm{e}^{\frac{2.7725}{F r_{1}-1}} \\V_{1}=\xi_{2} V_{0}\end{array}\right.$	L_jet为射流的空腔长度;V₀、φ、α、P_N分别为断面平均流速、挑角、坡角、空腔负压指数; t_r、t_s分别为坎高及跌坎高度;T为水舌上缘上升高度;φ₁为实际射流出射角;V₁为射流底缘实际流速;Fr为弗劳德数;ξ₁、ξ₂为由试验研究确定的修正系数;φ<15°;0.908<ξ₂<1.0;g为重力加速度
夏毓常公式^[17]	$\frac{L}{h}=\frac{1}{\cos \alpha}\left[\frac{\Delta}{h} \sin \alpha+F r \frac{\cos (\alpha-\theta)}{\cos \alpha}\left(F r \sin \theta+\sqrt{F r^{2} \sin ^{2} \theta+2 \frac{\Delta}{h} \cos \alpha}\right)\right]$	Fr为弗劳德数;L为空腔长度; Δ为挑高; α为坡角; θ为挑角;h为水深
苗宝广公式^[17]	$\left\{\begin{array}{l}X=\frac{F r \cos \alpha_{1}}{\cos \alpha_{2} \cos \theta} \sqrt{\frac{t_{\mathrm{s}}+t_{\mathrm{r}}}{h}} \\\frac{L}{h}=2.4995 X+0.9049\end{array}\right.$	X为综合水力参数;Fr为弗劳德数;h为水深; t_s、t_r分别为跌坎高度及坎高; α₁、α₂、θ分别为上底坡坡角、下底坡坡角及挑角
Rutschmann公式^[15]	$\left\{\begin{array}{l}L=L_{\max }(1-0.4 \sqrt{\Delta \bar{P}}) \\\Delta \bar{P}=\frac{\Delta P}{\gamma h_{0}} \\L_{\max }=\frac{F r^{2}-\theta h_{0}}{\cos \alpha}\left(1+\sqrt{2+\frac{2 T \cos \alpha}{\theta F r^{2}}}\right) \\T=\left(\Delta+y_{0}\right) / h_{0} \\\bar{\theta}=\theta \sqrt{\operatorname{th}\left(\frac{\Delta}{\theta h_{0}}\right)} \\F r=\frac{v_{0}}{\sqrt{g h_{0}}}\end{array}\right.$	ΔP为空腔压力; Δ、y₀分别为坎高及跌坎高度;γ为射流冲击角;h₀为来流水深;v₀为来流流速;L为空腔长度;L_max为最大空腔长度;θ为挑角;α为底坡坡角;t为时间

Fig.10 Verification of cavity length between calculation and test results

Table 5 Relative error of cavity length between calculation and test results

公式名称	相对误差的平均值/%		相对误差的均方差/%
公式名称	L_j/h	X_max/h	L_j/h	X_max/h
时启燧公式^[14]	2.82	3.49	2.63	3.33
Rutschmann公式^[15]	38.16	15.61	11.79	9.18
徐一民公式^[15]	25.26	7.07	10.28	6.12
罗铭公式^[16]	24.68	6.73	9.78	5.66
夏毓常公式^[17]	27.78	7.80	10.41	7.33
苗宝广公式^[17]	3.36	4.15	3.52	4.44

Table 5 Relative error of cavity length between calculation and test results

公式名称	相对误差的平均值/%		相对误差的均方差/%
公式名称	L_j/h	X_max/h	L_j/h	X_max/h
时启燧公式^[14]	2.82	3.49	2.63	3.33
Rutschmann公式^[15]	38.16	15.61	11.79	9.18
徐一民公式^[15]	25.26	7.07	10.28	6.12
罗铭公式^[16]	24.68	6.73	9.78	5.66
夏毓常公式^[17]	27.78	7.80	10.41	7.33
苗宝广公式^[17]	3.36	4.15	3.52	4.44

Table 6 Calculation formulae of water-gas ratio and parameter significance

公式名称	计算公式
Kalinske 公式^[18]	β=A(Fr_C-1)^B	β为气水比,β=Q_a/Q_w; Q_w为闸门一定开启高度下的流量;A取0.004 8;B取1.42;Fr_C为闸门处水流弗劳德数
陈肇和公式^[19]	β=K(Fr-1)^a^ln(^Fr^-1)+^b-1	β为气水比,β=Q_a/Q_w; Q_w为闸门一定开启高度下的流量(m³/s); Fr为闸门处水流弗劳德数; K、a、b为各区间的系数,K取0.497 5,a取-0.209,b取0.798
潘水波公式^[13]	q_a=kLV	q_a为单宽射流挟气量; k为掺气系数,取0.008 75; L为空腔长度; V为射流流速

Table 6 Calculation formulae of water-gas ratio and parameter significance

公式名称	计算公式
Kalinske 公式^[18]	β=A(Fr_C-1)^B	β为气水比,β=Q_a/Q_w; Q_w为闸门一定开启高度下的流量;A取0.004 8;B取1.42;Fr_C为闸门处水流弗劳德数
陈肇和公式^[19]	β=K(Fr-1)^a^ln(^Fr^-1)+^b-1	β为气水比,β=Q_a/Q_w; Q_w为闸门一定开启高度下的流量(m³/s); Fr为闸门处水流弗劳德数; K、a、b为各区间的系数,K取0.497 5,a取-0.209,b取0.798
潘水波公式^[13]	q_a=kLV	q_a为单宽射流挟气量; k为掺气系数,取0.008 75; L为空腔长度; V为射流流速

Fig.11 Verification of water-gas ratio between calculation and test results

Table 7 Relative error of water-gas ratio between calculation and test results

公式名称	相对误差的平均值/%	相对误差的均方差/%
Kalinske公式^[18]	15.60	14.36
潘水波公式^[13]	35.95	26.58
陈肇和公式^[19]	15.34	13.91

Table 7 Relative error of water-gas ratio between calculation and test results

公式名称	相对误差的平均值/%	相对误差的均方差/%
Kalinske公式^[18]	15.60	14.36
潘水波公式^[13]	35.95	26.58
陈肇和公式^[19]	15.34	13.91

Table 8 Calculation formulae of water content ratio and parameter significance

公式名称	计算公式
王俊勇公式^[20]	$\left\{\begin{array}{l}\frac{L_{\mathrm{j}}}{h_{0}}=0.95 X+3.394 ; \\\frac{X_{\max }}{h_{0}}=1.396 X+3.255 \\R^{2}=0.47\end{array}\right.$	β为含水比;B为渠道宽度;h₁为清水水深;h₂为掺气水深;F为弗劳德数的平方;V₁为清水流速;ψ为反映糙率n及水力半径R₁的无因次数
Hall 公式^[20]	$\eta=\frac{1}{k F r^{2}}$	η为含水比;Fr为断面弗劳德数; k为与壁面粗糙度有关的系数,光滑壁面取0.003 5,粗糙壁面取0.01
依伦伯格公式^[21]	$\beta=\left\{\begin{array}{c}0.42 R_{1}^{-0.05} \sin ^{0.26} \theta, \\\sin \theta<0.476 \\0.42 R_{1}^{-0.05} \sin ^{0.74} \theta, \\\sin \theta>0.476\end{array}\right.$	β为含水比;θ为渠槽底坡;R₁为水力半径,R₁≤0.3 m;θ≤45°
斯蒂文斯公式^[21]	$h_{2}=\left[\frac{(q / 8)^{2}}{h_{1}}\right]^{1 / 3}$	q为单宽流量;h₁为清水水深;h₂为掺气水深
柴恭纯公式^[22]	$\left\{\begin{array}{l}\lg \sigma=1.77+0.0081 \frac{V_{1}^{2}}{g R_{1}}, \\\sigma=\frac{h_{2}-h_{1}}{\Delta}\end{array}\right.$	σ为平坡掺气比;Δ为槽壁糙度;V₁为清水流速;h₁为清水水深;h₂为掺气水深;R₁为水力半径,0.6 m<R₁<1.4 m;15 m/s<V₁<30 m/s

Table 8 Calculation formulae of water content ratio and parameter significance

公式名称	计算公式
王俊勇公式^[20]	$\left\{\begin{array}{l}\frac{L_{\mathrm{j}}}{h_{0}}=0.95 X+3.394 ; \\\frac{X_{\max }}{h_{0}}=1.396 X+3.255 \\R^{2}=0.47\end{array}\right.$	β为含水比;B为渠道宽度;h₁为清水水深;h₂为掺气水深;F为弗劳德数的平方;V₁为清水流速;ψ为反映糙率n及水力半径R₁的无因次数
Hall 公式^[20]	$\eta=\frac{1}{k F r^{2}}$	η为含水比;Fr为断面弗劳德数; k为与壁面粗糙度有关的系数,光滑壁面取0.003 5,粗糙壁面取0.01
依伦伯格公式^[21]	$\beta=\left\{\begin{array}{c}0.42 R_{1}^{-0.05} \sin ^{0.26} \theta, \\\sin \theta<0.476 \\0.42 R_{1}^{-0.05} \sin ^{0.74} \theta, \\\sin \theta>0.476\end{array}\right.$	β为含水比;θ为渠槽底坡;R₁为水力半径,R₁≤0.3 m;θ≤45°
斯蒂文斯公式^[21]	$h_{2}=\left[\frac{(q / 8)^{2}}{h_{1}}\right]^{1 / 3}$	q为单宽流量;h₁为清水水深;h₂为掺气水深
柴恭纯公式^[22]	$\left\{\begin{array}{l}\lg \sigma=1.77+0.0081 \frac{V_{1}^{2}}{g R_{1}}, \\\sigma=\frac{h_{2}-h_{1}}{\Delta}\end{array}\right.$	σ为平坡掺气比;Δ为槽壁糙度;V₁为清水流速;h₁为清水水深;h₂为掺气水深;R₁为水力半径,0.6 m<R₁<1.4 m;15 m/s<V₁<30 m/s

Fig.12 Verification of water content ratio between calculation and test results

Table 9 Relative error of water content ratio between calculation and test results

公式名称	相对误差的平均值/%	相对误差的均方差/%
王俊勇公式^[20]	19.79	12.03
Hall公式^[20]	9.09	6.13
斯蒂文斯公式^[21]	13.16	7.63
依伦伯格公式^[21]	26.21	14.97
柴恭纯公式^[22]	11.75	7.45

Table 9 Relative error of water content ratio between calculation and test results

公式名称	相对误差的平均值/%	相对误差的均方差/%
王俊勇公式^[20]	19.79	12.03
Hall公式^[20]	9.09	6.13
斯蒂文斯公式^[21]	13.16	7.63
依伦伯格公式^[21]	26.21	14.97
柴恭纯公式^[22]	11.75	7.45

Metrics

Viewed

Full text

Abstract

Comments

Abstract
Figures/Tables
Related Articles
Recommended Articles
Metrics
Comments

TOP