Abstract

Flow friction is the key to studying water movement and has been one of the most important research topics in hydraulics and river dynamics. The roughness coefficient in the Manning formula represents friction applied to the flow by channel and changes with the river section characteristics, water level, and flow velocity. However, the Manning formula tends to simulate the friction with little variability, which contributes to large errors in the simulation of water level and flow velocity. To solve this problem, we proposed an improved friction formula based on the relationships between roughness coefficient and energy gradient and developed a differential model of one-dimensional flow with the proposed friction formula. The developed model was tested against both the experimental flood data and observed flow data in Qiantang River, China. The results indicated that the proposed friction formula provides a better simulation of target friction than the original Manning friction formula. The parameters in the proposed friction formula are less sensitive to the river section characteristics. Our results also showed that the developed differential model using the proposed friction formula can simulate the water level and flow velocity well in both the calibration and validation period and can improve the simulation of water level in tidal reach.

INTRODUCTION

Flow friction is the key to understanding water movement. In either the Navier–Stokes equations or the Saint–Venant equations, the structure of the friction formula is empirically based, which may cause significant errors in real application. As a result, hydraulic friction or friction loss has been one of the most important research topics in hydraulics and river dynamics (Jones 1976; Bathurst 1985). Research on this topic has focused on: (1) the development of the flow resistance formula under the condition of fixed bed, including the Chezy formula (Herschel 1897), Du Buat formula (Du Buat 1822), Eytelwein formula (Eytelwein 1826), Darcy–Weisbach formula (Chow et al. 1988), Darcy–Bazin formula (Darcy & Bazin 1865), Manning formula (Willcocks & Holt 1899), etc.; (2) the roughness research under the conditions of turbid water-movable bed (Dou 1982; Grant & Madsen 1982; Zhao & Zhang 1997). However, the complex mechanism behind the friction has not been comprehensively investigated.

The Manning formula is the common expression of friction in the channel flow:  
formula
(1)
where u is the mean velocity at a cross-section (m/s), R is hydraulic radius (m), n is roughness coefficient (), S is the gradient. We define fm () as the friction calculated by the Manning formula (m/s2):  
formula
(2)
where g is acceleration of gravity (m/s2). This formula reflects that the friction is proportional to u2, and inversely proportional to . The roughness coefficient factor usually varies with the characteristic of river section, water level, flow velocity, etc. (De Doncker et al. 2009).

As one of the parameters in flood calculation, the roughness of a natural river is determined by the roughness degree of the riverbed and quay wall, shape of river section, state of flow, sediment concentration, etc. (Limerinos 1970; Arcement & Schneider 1989; Xiekang et al. 2000). Some common methods for the estimation of roughness include using empirical formulae or charts, hydraulic methods, roughness inverse techniques (Khatibi et al. 2000; Dong & Yang 2002; Yen 2002), roughness curves (Khatibi et al. 2000), and the comprehensive roughness method (Yang et al. 2007). Bao et al. (2009) examined the impact of roughness on the water level of the tidal reach and separated the roughness into floodplain roughness and main channel roughness in cross-sections with obvious floodplain. Their results indicate that the changes of roughness have a great impact on water level calculation. The floodplain water level is dominated by the floodplain roughness while the low water level is mainly affected by the main channel roughness (Bao et al. 2010). Based on these findings, the relationship between roughness and water level was proposed (Bao et al. 2009).

More and more studies have indicated that flow energy plays an important role in the study of flow law. The energy equation has been widely applied in various research topics including energy dissipation of bend flow (Li & Fang 2010), sediment movement (Yang 1972; McLaren & Bowles 1985), study of river meandering mechanism (Ikeda et al. 1981; Nanson & Knighton 1996; Yao et al. 2010), numerical modeling on 1D river flow (Murillo & García-Navarro 2014), correlation between the flow energy loss and the channel deposition and erosion (Qi et al. 2013), and numerical modeling of flow resistance (Pannone et al. 2013). These studies examined the correlations between flow energy and flow motion law as well as riverbed friction. However, they did not apply the differential energy gradient frictional resistance formula in flow motion equation, or use it in practice.

In this study, we strive to investigate the relationship between roughness and the energy gradient change to: (1) examine the changing characteristics of friction during flood processes; (2) develop a function for roughness and energy gradient and propose an improved friction formula of one-dimensional flow; and (3) evaluate the proposed friction formula with the flow data from both experiments and a case study in Qiantang River.

DATA

Experiment setup and flood data collection

The flood experiment was conducted in an open flume in the hydraulics laboratory of Hohai University. The slope of the glass rectangular flume could be adjusted upon request. The flume was 7.73 m in length and 0.31 m in width. Three sections were set up to measure the water level and flow velocity (Figure 1). The friction between two sections is related to the flume slope, boundary roughness, flow characteristics, and aquatic plants, etc. During the experiment, the riverbed roughness, sectional shape, and the slope were kept unchanged. The changes of friction and energy were investigated under different discharges (reflecting different flow characteristics) and different densities of aquatic plants (reflecting the aquatic plant factor). The intake pump of the flume was equipped with a flow meter with measured accuracy at 2.8 × 10–6 m3/s. Discharge could be adjusted by the rotary switch in the intake valve. The maximum discharge of the intake pump was 0.0283 m3/s. The aquatic plant factors can be controlled by different quantities and densities of the branch fences in the flume. Eleven densities of fence branches were used in the experiment. Letters a–k represent different densities of each fence branches (a: lowest density; k: highest density). Twelve types of fence were set up with different fence combinations. The fence combination at each position is presented in Table 1. The experiment was conducted in two steps. First, the relationship between water level and discharge under different conditions of water plant was investigated. In this situation, the slope and channel characteristics were the same and there was no backwater effect, the relationship between water level and discharge was only related to the type of fence. For 12 types of fence and three sections (upper, middle and lower), 36 stage-discharge relationships were obtained and used for calculating the flood process in the second step. In the second step, flood data was collected. For each type of fence, four floods were simulated. The peak water level, peak time and period number of 48 floods are shown in Table 2.

Table 1

Fence setup in lab experiment

Type of fence Fence setting position
 
1# 2# 3# 4# 5# 6# 
     
     
    
   
 
c;d  
e;f 
c;d i;j 
d;k e;f 
10 d;k e;f;g 
11 d;k e;f;g i;j 
12 c;d;k e;f;g i;j 
Type of fence Fence setting position
 
1# 2# 3# 4# 5# 6# 
     
     
    
   
 
c;d  
e;f 
c;d i;j 
d;k e;f 
10 d;k e;f;g 
11 d;k e;f;g i;j 
12 c;d;k e;f;g i;j 
Table 2

Sampling statistics of 48 floods

Flood no. Upper section
 
Middle section
 
Lower section
 
Period number 
Peak water level (m) Peak time (s) Peak water level (m) Peak time (s) Peak water level (m) Peak time (s) 
1-1 0.1773 88 0.1725 88 0.1685 96 46 
1-2 0.1523 60 0.1455 56 0.1445 56 30 
1-3 0.1338 56 0.1305 64 0.1270 52 34 
1-4 0.1068 52 0.1025 52 0.1020 52 33 
2-1 0.1833 84 0.1775 84 0.1700 76 43 
2-2 0.1613 64 0.1565 68 0.1510 72 38 
2-3 0.1383 60 0.1325 60 0.1265 64 32 
2-4 0.1168 60 0.1135 56 0.1025 60 30 
3-1 0.1858 68 0.1750 68 0.1690 72 34 
3-2 0.1663 68 0.1575 68 0.1470 68 35 
3-3 0.1426 60 0.1335 56 0.1255 60 25 
3-4 0.1248 56 0.1155 56 0.1050 64 25 
4-1 0.1868 64 0.1695 60 0.1640 64 34 
4-2 0.1653 64 0.1455 60 0.1405 56 31 
4-3 0.1378 64 0.1245 64 0.1155 68 30 
4-4 0.1218 56 0.1095 56 0.0980 60 23 
5-1 0.2003 88 0.1855 88 0.1630 84 46 
5-2 0.1708 72 0.1555 80 0.1440 72 36 
5-3 0.1458 64 0.1325 68 0.1195 64 30 
5-4 0.1213 52 0.1095 52 0.1005 56 25 
6-1 0.2058 64 0.1815 64 0.1645 64 44 
6-2 0.1828 76 0.1575 76 0.1450 76 47 
6-3 0.1518 56 0.1245 56 0.1150 60 34 
6-4 0.1358 72 0.1085 68 0.1020 68 36 
7-1 0.2098 88 0.1915 96 0.1655 76 44 
7-2 0.1838 80 0.1695 80 0.1455 84 41 
7-3 0.1563 60 0.1435 64 0.1230 64 38 
7-4 0.1343 56 0.1225 52 0.1030 60 32 
8-1 0.2233 72 0.1985 76 0.1675 80 48 
8-2 0.1978 68 0.1735 72 0.1455 80 41 
8-3 0.1773 64 0.1555 64 0.1285 68 37 
8-4 0.1438 56 0.1265 60 0.0990 56 34 
9-1 0.2228 80 0.1935 84 0.166 84 52 
9-2 0.1928 76 0.1645 76 0.1405 76 43 
9-3 0.1623 68 0.1395 72 0.1170 80 39 
9-4 0.1403 68 0.1195 68 0.1030 68 37 
10-1 0.2318 80 0.1955 84 0.1695 96 53 
10-2 0.2008 76 0.1715 76 0.1445 84 43 
10-3 0.1733 60 0.1425 68 0.1215 60 38 
10-4 0.1448 52 0.1185 60 0.1000 56 34 
11-1 0.2353 76 0.2015 84 0.1670 76 55 
11-2 0.2043 76 0.1725 80 0.1415 88 50 
11-3 0.1833 64 0.1555 64 0.1265 68 40 
11-4 0.1483 56 0.1255 56 0.0995 60 33 
12-1 0.2353 72 0.1985 72 0.1650 84 55 
12-2 0.2138 72 0.1785 72 0.1485 72 42 
12-3 0.1743 64 0.1445 68 0.1190 68 38 
12-4 0.1523 56 0.1245 56 0.0980 64 34 
Flood no. Upper section
 
Middle section
 
Lower section
 
Period number 
Peak water level (m) Peak time (s) Peak water level (m) Peak time (s) Peak water level (m) Peak time (s) 
1-1 0.1773 88 0.1725 88 0.1685 96 46 
1-2 0.1523 60 0.1455 56 0.1445 56 30 
1-3 0.1338 56 0.1305 64 0.1270 52 34 
1-4 0.1068 52 0.1025 52 0.1020 52 33 
2-1 0.1833 84 0.1775 84 0.1700 76 43 
2-2 0.1613 64 0.1565 68 0.1510 72 38 
2-3 0.1383 60 0.1325 60 0.1265 64 32 
2-4 0.1168 60 0.1135 56 0.1025 60 30 
3-1 0.1858 68 0.1750 68 0.1690 72 34 
3-2 0.1663 68 0.1575 68 0.1470 68 35 
3-3 0.1426 60 0.1335 56 0.1255 60 25 
3-4 0.1248 56 0.1155 56 0.1050 64 25 
4-1 0.1868 64 0.1695 60 0.1640 64 34 
4-2 0.1653 64 0.1455 60 0.1405 56 31 
4-3 0.1378 64 0.1245 64 0.1155 68 30 
4-4 0.1218 56 0.1095 56 0.0980 60 23 
5-1 0.2003 88 0.1855 88 0.1630 84 46 
5-2 0.1708 72 0.1555 80 0.1440 72 36 
5-3 0.1458 64 0.1325 68 0.1195 64 30 
5-4 0.1213 52 0.1095 52 0.1005 56 25 
6-1 0.2058 64 0.1815 64 0.1645 64 44 
6-2 0.1828 76 0.1575 76 0.1450 76 47 
6-3 0.1518 56 0.1245 56 0.1150 60 34 
6-4 0.1358 72 0.1085 68 0.1020 68 36 
7-1 0.2098 88 0.1915 96 0.1655 76 44 
7-2 0.1838 80 0.1695 80 0.1455 84 41 
7-3 0.1563 60 0.1435 64 0.1230 64 38 
7-4 0.1343 56 0.1225 52 0.1030 60 32 
8-1 0.2233 72 0.1985 76 0.1675 80 48 
8-2 0.1978 68 0.1735 72 0.1455 80 41 
8-3 0.1773 64 0.1555 64 0.1285 68 37 
8-4 0.1438 56 0.1265 60 0.0990 56 34 
9-1 0.2228 80 0.1935 84 0.166 84 52 
9-2 0.1928 76 0.1645 76 0.1405 76 43 
9-3 0.1623 68 0.1395 72 0.1170 80 39 
9-4 0.1403 68 0.1195 68 0.1030 68 37 
10-1 0.2318 80 0.1955 84 0.1695 96 53 
10-2 0.2008 76 0.1715 76 0.1445 84 43 
10-3 0.1733 60 0.1425 68 0.1215 60 38 
10-4 0.1448 52 0.1185 60 0.1000 56 34 
11-1 0.2353 76 0.2015 84 0.1670 76 55 
11-2 0.2043 76 0.1725 80 0.1415 88 50 
11-3 0.1833 64 0.1555 64 0.1265 68 40 
11-4 0.1483 56 0.1255 56 0.0995 60 33 
12-1 0.2353 72 0.1985 72 0.1650 84 55 
12-2 0.2138 72 0.1785 72 0.1485 72 42 
12-3 0.1743 64 0.1445 68 0.1190 68 38 
12-4 0.1523 56 0.1245 56 0.0980 64 34 
Figure 1

Flood simulator. (a) picture of experimental installation; (b) diagram of experimental installation.

Figure 1

Flood simulator. (a) picture of experimental installation; (b) diagram of experimental installation.

River section and data

The observed data in Zhijiang hydrologic station (water level and discharge) and Wenjiayan water level station (water level) were collected for the validation and assessment of the proposed friction formula. Doppler ultrasonic wave equipment was set up in Zhijiang hydrologic station for accurate measurements. The distance between the two stations was 616 m. There was no tributary in this river reach so that no lateral inflow would contribute to the streamflow during non-rainy period.

METHODS

Ideal friction calculation

The ideal friction calculation is derived from the Saint–Venant equation. As a result, the basic assumption of the ideal friction calculation is that the Saint–Venant equation could describe the unsteady flow of an open channel. The one-dimensional flow motion equation expressed with section average flow velocity u (m/s) and water level z (m) can be written as Equation (3). The effect of the bed slope is considered in item , which could be expressed as :  
formula
(3)
 
formula
(4)
where f is friction, z is the waterhead, h is the flow depth, i is the bed slope, and g is gravitational acceleration. The most ideal friction formula can be expressed as Equation (4). Hence, the differential expression of the ideal friction with the Pressiman differential scheme weighted by θ can be written as:  
formula
(5)
where , i is time t, j is position, i+ 1 is the next time step(), and j+ 1 is the next position step().

Any proposed friction formula should be evaluated based on the goodness of fit between simulated friction and ideal friction (Equations (4) and (5)). In this paper, the ideal friction was used as the target friction for evaluation.

Manning friction calculation

The differential form of the Manning friction formula (Equation (2)) with the Pressiman differential scheme weighted by can be written as:  
formula
(6)
represents the mean value between the reach section [j, j+ 1] and period [i, i+ 1]. For the experimental floods, the flow velocity, water level (or water depth) of the upper and lower sections [j, j+1] are all available for the calculation of both ideal friction and Manning friction values with Equations (5) and (6).

Changes of friction during flood processes

The structure of the friction formula could be evaluated with simulated friction changes during flood processes. Figure 2 shows the changes of ideal friction and Manning friction with changes of water level. The target friction increases rapidly at the beginning of a flood and then shows a sharp decrease before the flood peak appears. It reaches its minimum value around the flood peak. Changes of target friction are determined by spatiotemporal variation of flow velocity and surface slope. There are multiple peaks during the water rising and recession period. Manning friction is relatively larger when the water level is low (water rising and recession period) as the hydraulic radius is smaller during low water level periods than that during flood peak periods. Manning friction is proportional to flow velocity (u2) and is inversely proportional to hydraulic radius (R4/3). However, the hydraulic radius changes in the same direction with flow velocity, thus the synchronization between flow velocity and hydraulic radius often has an offset effect on Manning friction, which explains the smaller variation of Manning friction compared to target friction. The small correlation coefficient (0.2055) indicates the inconsistency between changes of target friction and Manning friction.

Figure 2

Changes of target friction and Manning friction with water level during a flood (Flood 1-2).

Figure 2

Changes of target friction and Manning friction with water level during a flood (Flood 1-2).

Changes of Manning roughness coefficient during flood processes

Based on the analyses, the constant Manning roughness coefficient may introduce large errors to Manning friction estimation. As a result, understanding how the roughness coefficient changes during a flood is the key to improving the calculation of Manning friction. For every time step we can change the roughness coefficient to make the Manning friction and target friction equal. In this situation, the time-varying roughness coefficient can be expressed as:  
formula
(7)
The total energy (S) of the flow in the channel section is expressed as:  
formula
(8)
To investigate the relation between roughness coefficient and energy gradient, we calculated the energy gradient using a Pressiman differential scheme weighted by θ as below:  
formula
(9)
where Se is the energy gradient and α is the proportional coefficient between the kinetic energy of the mean velocity in the section and point velocity. Figure 3 shows the changes of Manning roughness coefficient, water level, flow velocity, and energy gradient during a flood (Flood 1-2). Manning roughness coefficient exhibits a poor correlation with both water level and flow velocity. However, it shows a high negative correlation with energy gradient (–0.7674). To confirm the correlation between the Manning roughness coefficient and energy gradient, we calculated the correlation coefficient between them for 48 floods as listed in Table 3. Among the three examined flood factors, the energy gradient exhibits the highest average correlation coefficient (–0.871) with Manning roughness coefficient.
Table 3

Correlation coefficients between roughness and water level, flow velocity and energy gradient

Flood number Ch Cu CSe Flood number Ch Cu CSe 
1-1 −0.0368 0.0007 −0.796 7-1 0.6252 0.0068 −0.9221 
1-2 −0.0466 0.0049 −0.7674 7-2 0.6406 0.0055 −0.8988 
1-3 −0.0228 0.0049 −0.5728 7-3 0.5876 0.0039 −0.8911 
1-4 0.1321 0.0107 −0.7439 7-4 0.6849 0.0036 −0.8104 
2-1 −0.2156 −0.0136 −0.8388 8-1 0.5546 0.0048 −0.9472 
2-2 −0.0859 −0.0024 −0.7775 8-2 0.6864 0.0052 −0.9132 
2-3 0.0375 0.0022 −0.7829 8-3 0.7248 0.0045 −0.9044 
2-4 0.0755 0.0024 −0.7584 8-4 0.7438 0.0033 −0.8305 
3-1 −0.0325 0.0003 −0.8263 9-1 0.7088 0.0056 −0.9280 
3-2 0.0480 0.0025 −0.8566 9-2 0.7420 0.0048 −0.9128 
3-3 0.0450 0.0019 −0.8705 9-3 0.6583 0.0037 −0.9347 
3-4 0.2143 0.0041 −0.7938 9-4 0.7690 0.0030 −0.8420 
4-1 0.1691 0.0049 −0.8913 10-1 0.6457 0.0047 −0.9494 
4-2 0.1039 0.0027 −0.8714 10-2 0.6694 0.0041 −0.9321 
4-3 0.2358 0.0037 −0.7980 10-3 0.7318 0.0035 −0.9095 
4-4 0.3390 0.0035 −0.7129 10-4 0.7932 0.0029 −0.8775 
5-1 0.4386 0.0068 −0.7658 11-1 0.5721 0.0042 −0.8973 
5-2 0.2924 0.0041 −0.9271 11-2 0.4487 0.0025 −0.957 
5-3 0.4044 0.0044 −0.8139 11-3 0.5589 0.0029 −0.8877 
5-4 0.3993 0.0032 −0.7588 11-4 0.6697 0.0026 −0.8259 
6-1 0.3366 0.0047 −0.8743 12-1 0.4470 0.0028 −0.8965 
6-2 0.2929 0.0032 −0.9372 12-2 0.5012 0.0027 −0.8729 
6-3 0.5359 0.005 −0.8361 12-3 0.5047 0.0021 −0.8452 
6-4 0.5731 0.0042 −0.7349 12-4 0.5174 0.0019 −0.7882 
Flood number Ch Cu CSe Flood number Ch Cu CSe 
1-1 −0.0368 0.0007 −0.796 7-1 0.6252 0.0068 −0.9221 
1-2 −0.0466 0.0049 −0.7674 7-2 0.6406 0.0055 −0.8988 
1-3 −0.0228 0.0049 −0.5728 7-3 0.5876 0.0039 −0.8911 
1-4 0.1321 0.0107 −0.7439 7-4 0.6849 0.0036 −0.8104 
2-1 −0.2156 −0.0136 −0.8388 8-1 0.5546 0.0048 −0.9472 
2-2 −0.0859 −0.0024 −0.7775 8-2 0.6864 0.0052 −0.9132 
2-3 0.0375 0.0022 −0.7829 8-3 0.7248 0.0045 −0.9044 
2-4 0.0755 0.0024 −0.7584 8-4 0.7438 0.0033 −0.8305 
3-1 −0.0325 0.0003 −0.8263 9-1 0.7088 0.0056 −0.9280 
3-2 0.0480 0.0025 −0.8566 9-2 0.7420 0.0048 −0.9128 
3-3 0.0450 0.0019 −0.8705 9-3 0.6583 0.0037 −0.9347 
3-4 0.2143 0.0041 −0.7938 9-4 0.7690 0.0030 −0.8420 
4-1 0.1691 0.0049 −0.8913 10-1 0.6457 0.0047 −0.9494 
4-2 0.1039 0.0027 −0.8714 10-2 0.6694 0.0041 −0.9321 
4-3 0.2358 0.0037 −0.7980 10-3 0.7318 0.0035 −0.9095 
4-4 0.3390 0.0035 −0.7129 10-4 0.7932 0.0029 −0.8775 
5-1 0.4386 0.0068 −0.7658 11-1 0.5721 0.0042 −0.8973 
5-2 0.2924 0.0041 −0.9271 11-2 0.4487 0.0025 −0.957 
5-3 0.4044 0.0044 −0.8139 11-3 0.5589 0.0029 −0.8877 
5-4 0.3993 0.0032 −0.7588 11-4 0.6697 0.0026 −0.8259 
6-1 0.3366 0.0047 −0.8743 12-1 0.4470 0.0028 −0.8965 
6-2 0.2929 0.0032 −0.9372 12-2 0.5012 0.0027 −0.8729 
6-3 0.5359 0.005 −0.8361 12-3 0.5047 0.0021 −0.8452 
6-4 0.5731 0.0042 −0.7349 12-4 0.5174 0.0019 −0.7882 

Note: Ch, Cu, and CSe represent the correlation coefficient between the roughness coefficient and water depth, flow velocity and energy gradient, respectively.

Figure 3

Change process of n, U, H and Se during Flood 1–2.

Figure 3

Change process of n, U, H and Se during Flood 1–2.

Friction formula and dynamic differential model based on energy-gradient-dependent roughness coefficient

Manning roughness coefficient and energy gradient for each time step in Flood 1-2 are plotted in Figure 4, which shows an approximately linear relationship. We assume that the linear relationship between them can be expressed as:  
formula
(10)
Then the differential form of the relationship between the roughness coefficient and energy gradient can be written as:  
formula
(11)
The improved Manning formula fmr can be expressed as:  
formula
(12)
where n0 is a constant coefficient (), αf is the proportional coefficient of frictional resistance variation caused by energy variation (), fmr is the friction calculated by Manning formula (m/s2), and nr is the roughness coefficient
Figure 4

Relation between the roughness coefficient and energy gradient.

Figure 4

Relation between the roughness coefficient and energy gradient.

Without lateral inflow, the continuity equation expressed with water depth h (m) and flow velocity u (m/s) can be written as:  
formula
(13)
Therefore, Equations (3), (12) and (13) constitute the improved hydrodynamic model, and the differential form with Pressiman difference scheme (Bao et al. 2009) can be expressed as:  
formula
(14)
where:  
formula
(15)
when αf = 0 and n2 = n0, Equation (15) is converted into the dynamic differential model with the Manning formula as friction.

RESULTS AND DISCUSSION

Experimental flood simulation with original Manning friction

Twelve types of fence represent 12 types of channel segment characteristics. To test the Manning friction formula for the same reach, the parameter in the Manning formula under each type of fence was calibrated. For each fence module, three floods were used to calibrate parameter and one flood was used for validation. The simulation effect was evaluated by absolute relative error value (RE) using Equation (16):  
formula
(16)
where OB and OBC are the observed and calculated variable values, respectively.

Table 4 shows the calibrated parameter (n2) and the absolute relative error for both calibrated and validated floods. The low relative errors indicate poor simulation effects in both the calibration and the validation stage. The poor simulation was mainly due to the error introduced by the Manning friction. Figure 5 shows the comparison between the measured and calculated water level and flow velocity in the lower section. The differences between measured and calculated water level and flow velocity can be divided into three periods. (1) At the water rising period, the Manning friction was smaller than the target friction (Figure 2). Hence, the flow spread from the upper section to the lower section was much faster than the actual, which led to larger calculated water depth and smaller calculated flow velocity than the observed ones in the lower section. (2) Around the flood peak, the Manning friction is larger than the target friction (Figure 2) which leads to smaller calculated water depth and larger calculated flow velocity. (3) At the recession period, the calculated water level and flow velocity showed fluctuations around the measured water level and flow velocity because of the accumulated error during the calculation processes (Figure 5). The results of other floods showed similar characteristics, which indicates that the Manning friction calculation error may be the main reason for the calculated error of water depth and flow velocity.

Table 4

Simulation results of water level (H) and flow velocity (U) by the Manning formula

Type of fence n2 RE(HRE(UVer.RE(HVer.RE(U
0.0264 64.5 49.9 63.4 44.1 
0.0314 66.8 38.7 67.5 53.6 
0.0446 70.1 40.0 75.1 68.6 
0.0411 72.3 63.8 68.6 54.3 
0.0501 71.8 57.2 73.2 53.4 
0.0595 74.2 63.5 74.1 63.9 
0.0653 73.6 63.2 68.4 51.2 
0.0669 76.1 69.2 73.3 64.4 
0.0703 76.2 70.3 71.8 65.2 
10 0.0716 75.5 69.9 72.9 65.6 
11 0.0548 73.5 65.8 70.9 62.7 
12 0.0630 73.5 49.3 69.9 16.8 
Average 0.645 72.3 58.4 70.8 55.3 
Type of fence n2 RE(HRE(UVer.RE(HVer.RE(U
0.0264 64.5 49.9 63.4 44.1 
0.0314 66.8 38.7 67.5 53.6 
0.0446 70.1 40.0 75.1 68.6 
0.0411 72.3 63.8 68.6 54.3 
0.0501 71.8 57.2 73.2 53.4 
0.0595 74.2 63.5 74.1 63.9 
0.0653 73.6 63.2 68.4 51.2 
0.0669 76.1 69.2 73.3 64.4 
0.0703 76.2 70.3 71.8 65.2 
10 0.0716 75.5 69.9 72.9 65.6 
11 0.0548 73.5 65.8 70.9 62.7 
12 0.0630 73.5 49.3 69.9 16.8 
Average 0.645 72.3 58.4 70.8 55.3 
Figure 5

Comparison of the measured and simulated water level and flow velocity by the Manning friction formula and proposed friction formula for Flood 1-2.

Figure 5

Comparison of the measured and simulated water level and flow velocity by the Manning friction formula and proposed friction formula for Flood 1-2.

Experimental flood simulation with proposed friction formula

Roughness coefficient is considered to change with energy gradient in the proposed friction formula. It can be evaluated in two steps: (1) examine how well the proposed friction formula can simulate the target friction; (2) evaluate the ability of differential models with proposed friction formula to simulate the water depth and flow velocity. Table 5 shows the simulation results using the proposed friction formula. Table 5 indicates that the proposed friction formula can simulate the target friction well and the differential model driven by the proposed friction formula can simulate the water level and flow velocity well in both calibration and validation phases. The relative range of the parameters in both Manning () and proposed friction formula (, ) was also investigated for different fence types which represent different river section characteristics. The range of the parameters could be used to indicate the variability of the parameters for different river section characteristics.

Table 5

Simulation results of target friction (f), water level (H), and flow velocity (U) using the proposed friction formula

Type of fence n0 × 100 αf RE(fRE(HRE(UVer.RE(fVer.RE(HVer.RE(U
–0.249 0.973 0.898 89.4 94.2 0.875 90.3 96.1 
–0.318 0.973 0.820 92.6 95.0 0.779 92.6 96.2 
–0.288 0.973 0.899 92.0 95.0 0.895 90.9 95.9 
–0.339 0.973 0.922 91.9 96.3 0.907 89.9 96.7 
–0.339 0.953 0.936 89.6 95.8 0.925 87.7 97.1 
–0.336 0.937 0.917 89.1 94.7 0.908 86.2 94.6 
–0.258 0.920 0.828 89.9 93.4 0.838 85.3 91.0 
–0.225 0.913 0.856 89.6 92.3 0.863 86.5 90.9 
–0.263 0.918 0.838 89.8 91.4 0.826 86.2 92.0 
10 –0.173 0.909 0.859 89.2 92.5 0.872 84.7 91.0 
11 –0.333 0.926 0.914 88.3 91.8 0.910 83.8 93.0 
12 –0.339 0.931 0.765 90.8 93.0 0.763 86.6 95.0 
Average 3.46 0.942 0.871 90.2 93.8 0.863 87.6 94.1 
Type of fence n0 × 100 αf RE(fRE(HRE(UVer.RE(fVer.RE(HVer.RE(U
–0.249 0.973 0.898 89.4 94.2 0.875 90.3 96.1 
–0.318 0.973 0.820 92.6 95.0 0.779 92.6 96.2 
–0.288 0.973 0.899 92.0 95.0 0.895 90.9 95.9 
–0.339 0.973 0.922 91.9 96.3 0.907 89.9 96.7 
–0.339 0.953 0.936 89.6 95.8 0.925 87.7 97.1 
–0.336 0.937 0.917 89.1 94.7 0.908 86.2 94.6 
–0.258 0.920 0.828 89.9 93.4 0.838 85.3 91.0 
–0.225 0.913 0.856 89.6 92.3 0.863 86.5 90.9 
–0.263 0.918 0.838 89.8 91.4 0.826 86.2 92.0 
10 –0.173 0.909 0.859 89.2 92.5 0.872 84.7 91.0 
11 –0.333 0.926 0.914 88.3 91.8 0.910 83.8 93.0 
12 –0.339 0.931 0.765 90.8 93.0 0.763 86.6 95.0 
Average 3.46 0.942 0.871 90.2 93.8 0.863 87.6 94.1 
The range for roughness factor n2 in the Manning formula is:  
formula
The range of the parameters n0 and αf proposed friction formula are:  
formula

Our results show that the parameters in the proposed friction formula are more stable than the parameters in the Manning friction formula. The range of the parameter αf was only 6.8%, which means that this parameter is not sensitive to the changes of river section characteristics. Table 6 lists the simulation effect of the floods with the different types of fence using the proposed friction formula with the same parameter (, ). The good match between simulated and measured water level and flow velocity further confirms the generality of parameters in the proposed friction formula. To illustrate the simulation effect for a single flood, we selected Flood 1-2 for explanatory purposes. Figure 5 shows the comparison of the measured and simulated water level and flow velocity by the Manning friction formula and proposed friction formula for Flood 1-2. Figure 6 shows the comparisons among the target friction (f), the friction (fmr) calculated by the proposed friction formula, and friction (fm) calculated by the Manning formula. Figures 5 and 6 illustrate the improvement in simulating target friction, water level, and flow velocity with the proposed friction formula.

Table 6

Simulation results of water level (H) and flow velocity (U) by the proposed friction formula with the same parameter for 12 types of fencesa

Type of fence 10 11 12 
RE(H84.5 87.7 86.2 86.4 85.8 85.5 85.9 85.3 85.5 85.1 83.4 86.0 
RE(U91.1 93.5 93.2 93.2 92.9 92.9 91.0 90.4 90.9 90.4 91.4 93.0 
Type of fence 10 11 12 
RE(H84.5 87.7 86.2 86.4 85.8 85.5 85.9 85.3 85.5 85.1 83.4 86.0 
RE(U91.1 93.5 93.2 93.2 92.9 92.9 91.0 90.4 90.9 90.4 91.4 93.0 

aParameters used in this simulation: n0 = –0.002146625; af = 0.915392.

Figure 6

Target friction, Manning friction, and simulated friction by proposed friction formula for Flood 1-2.

Figure 6

Target friction, Manning friction, and simulated friction by proposed friction formula for Flood 1-2.

Validation of the proposed friction formula with Qiantang River flood data

The water level in Wenjianyan section can be calculated using the differential equation (Equation (14)) with measured water level and flow velocity in the upper section (Zhijiang section) as input. The differential model using the proposed friction formula can be tested by the comparisons between calculated and measured water level. Table 7 shows the simulation results of water level during the selected periods. The model was tested against the water level during both the wet and dry seasons. Event numbers 1–4 in Table 7 were selected for parameter calibration. The calibrated parameters and are –0.0013008 and 0.8140032 respectively. For explanatory purposes, we plotted the observed and simulated water level graph of Wenjiayan section in Figure 7 during the period 2010-5-9 8:00 to 2010-5-13 7:00. The simulation results indicate that the differential model using the proposed friction formula can simulate the water level in the study location accurately. The water level is determined by complex interactions between tides and floods since Wenjiayan is located in the tidal reach. Traditional hydrodynamic models always present a poor performance in simulating the water level within a tidal reach (Bao et al. 2010). However, the proposed friction determination could improve the simulation of water levels in a tidal reach.

Table 7

Simulation results of the water level (H) at Wenjiayan location

Number Period RE(H
2009.11.1.8:00–2009.11.5.8:00 0.913 
2009.11.5. 8:00–2009.12.1.8:00 0.988 
2010.4.9.9:00–2010.4.16.9:00 0.977 
2010.6.8.9:00–2010.7.1.8:00 0.787 
2009.12.1.8:00–2009.12.30.8:00 0.974 
2010.5.8.10:00–2010.6.7.7:00 0.872 
Number Period RE(H
2009.11.1.8:00–2009.11.5.8:00 0.913 
2009.11.5. 8:00–2009.12.1.8:00 0.988 
2010.4.9.9:00–2010.4.16.9:00 0.977 
2010.6.8.9:00–2010.7.1.8:00 0.787 
2009.12.1.8:00–2009.12.30.8:00 0.974 
2010.5.8.10:00–2010.6.7.7:00 0.872 
Figure 7

Measured water level and simulated water level using the proposed friction formula in Wenjiayan location.

Figure 7

Measured water level and simulated water level using the proposed friction formula in Wenjiayan location.

CONCLUSION

In this study, the mechanism of friction was thoroughly analyzed. It was found that the Manning formula tends to simulate the friction with little variability, which contributes to large errors in the simulation of water level and flow velocity. An improved friction formula is proposed based on the relationships between roughness coefficient and energy gradient in open-channel sections. Then, the differential model of one-dimensional flow was constructed based on the proposed friction formula. The improved friction formula was tested with both experimental flood data and the data from Qiantang River. The testing and simulation results can be summarized as below:

  • (1)

    The proposed friction formula provides a good simulation of target and the differential model using the proposed friction formula can simulate the water level and flow velocity well in both calibration and validation phases.

  • (2)

    The parameters in the proposed friction formula are less sensitive to the river section characteristics represented by different types of fences in this study than that of the Manning friction formula. In other words, the parameters in the proposed formula remain steadier than those in the Manning formula regarding different river section characteristics. In practice, the characteristics of the river section are extremely complex and may change along the river. The determination of parameters could be very difficult. If the parameters are relatively stable, we could use the same set of parameters for all river sections with different characteristics and still provide reliable simulations. As a result, the proposed friction formula has strong adaptability.

  • (3)

    The differential model using the proposed friction formula can improve the simulation of water level in tidal reaches where the water level is hard to simulate due to the complex interactions between tides and floods. The water level forecasting in a tidal reach is among the most critical problems for flood management and flood control because the water level is impacted by the interaction of upper reach flood wave and downstream tide wave (Qu et al. 2009). The calibrated differential model using the proposed friction formula could be used as a tool for water level forecasting in tidal reaches.

ACKNOWLEDGEMENTS

This study was supported by the National Natural Science Foundation of China (Grant Nos. 51279057 and 41371048), the Fundamental Research Funds for the Central Universities (Grant No. 2014B35314), Ministry of water resources public welfare industry special scientific research Project (Grant No. 201401034), Major Program of National Natural Science Foundation of China (Grant Nos. 51190090 and 51190091), IWHR Research & Development Support Program (JZ0145B012019), and the Open Research Fund Program of State Key Laboratory of Hydrology-Water Resources and Hydraulic Engineering (2015490611).

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