Application of modified Manning formula in the determination of vertical profile velocity in natural rivers

The velocity distribution attracts more attention these days, especially in hydrological process research (Chiu 1989; Marini et al. 2011; Fontana et al. 2013). Empirical models have been developed by earlier investigators for predicting the point velocity distribution within the whole cross section based on experimental and field data. Entropy theory has been widely used in velocity distribution, and its limitations analyzed (Marini et al. 2011; Bonakdari & Moazamnia 2014; Corato et al. 2014). Because the specific assumptions and hydraulic conditions during the experiment or field sampling vary widely, the equation derived from one cross section is not suitable for another cross section or under different water levels. The vertical distribution of the velocity has been systematically studied and distribution functions have been proposed under various hydraulic conditions (Bergstrom et al. 2001; Huai et al. 2009; Bowers et al. 2012). However, widely accepted lateral distribution formula for the flow vertical profile velocity have only rarely been studied (Wiberg & Smith 1991; Chen et al. 1999; Cheng & Gartner 2003; Hu et al. 2008).

In this paper, we present the application of the modified Manning empirical formula in river vertical profile velocity determination, with hydraulic data collected from a Northern Germany lowland catchment. The main aim was to test the practicality of determining the vertical profile velocity based on Manning roughness, vertical depth and river bed slope; to evaluate the influence of the water level and aspect ratio on the velocity determination; and finally to provide a new and simple approach to vertical velocity determination, especially for a network area without gauge data.

Ten river sections were surveyed under a moderate water level period (September 2011), and surveys were then repeated at the same cross sections in the flood season (January 2012). This provided cross-sectional profiles of water surface width, depth, velocity, discharge and river bed elevation for seven cross sections evenly distributed along the 300 m river reach. At every cross section, five to ten vertical profiles evenly distributed along the river width were sampled. Each vertical profile was positioned 0.5–2 m apart depending on the river top width (Figure 1(b) and 1(c)). In addition, the roughness coefficient of each cross section was estimated during the field campaign. In total, 140 cross sections and 1050 vertical profiles were sampled.

River gradient is an essential parameter for the application of the modified Manning formula in a river vertical profile. Nowadays, a total station is normally used to collect large-scale land surface elevation data, but high-precision river gradient data is still unavailable (Huang et al. 2002). Dhondia & Stelling (2002) pointed out that the hydraulic model is a relatively reliable data source for river gradients under current technological constraints. In our study, the river bed elevation was derived from the vertical distance between the highest bank point and the lowest river bed point. The elevation of the highest bank point was extracted from the 1 m Digital Elevation Model (DEM), while the maximum water depth and vertical distance between the highest bank point and water surface were measured from the field (Figure 1(d)). The bed slope between cross sections was then calculated and calibrated in the hydraulic model.

The measurement is carried out in classical vertical profiles across the river width. At each position, the ADQ automatically records the water depth and flow velocity of the cells from top to bottom. For velocity measurement, three ultrasonic transducers emit sound pulses (beams 1, 2, 3), which are then reflected by moving particles in the water column (plankton, branches, leaves, air bubbles etc.). According to the frequency shift between the transmitted and received pulses, the relative velocity between the instrument and the suspended material can be calculated (Figure 2(b) and 2(c)). River depth is calculated from the time difference between the emitted and reflected sound pulse 4, which is directed downward to the riverbed. This methodology provided a reliable database for our study.

Hydrologic Engineering Centers River Analysis System (HEC-RAS, US Army Corps of Engineers) is a one-dimensional (1D) flow routing approach based on the St Venant/Shallow Water Equations (Horritt & Bates 2002; Pappenberger et al. 2005). It has been around in hydraulic design and engineering research for a few decades and is adopted in this study because of its well-performing theoretical basis (Horritt & Bates 2002; Drake et al. 2010; Morche et al. 2010). Based on the data collected in September 2011 and 1 m DEM data, HEC-RAS models for ten selected river sections in the Stör catchment were set up and calibrated, and during this process a combination of roughness and bed gradient calibration was involved. Due to the difficulty in determining a representative Manning's n value, our main attention was focused on roughness calibration to achieve the minimum error between real measured and modeled water surface elevations, maximum depth, hydraulic depth and mean cross-sectional velocity. Finally, the averaged errors between measured and modeled data of all the models were within ±5% (Song et al. 2014). The dataset from January 2012 was used for model validation, and the validation quality is shown in Table 2. The HEC-RAS output and field surveyed data were positively correlated in the validation model. All the correlation coefficients were higher than 0.9. This indicated reliable performance of the steady model, and provided the basis for the application of the modified Manning formula.

Figure 6(b) and 6(c) reveal a similar phenomenon to Figure 3(b) and 3(c). The regression R² and Nash–Sutcliffe coefficient were higher for the data from deeper profiles. This strengthens the conclusion mentioned before – the improved Manning formula (2) works better under higher water depth.

The velocity determination was affected by different hydraulic conditions. According to Table 3, the catchment size, discharge, water depth and width are correlated with each other, while the aspect ratio is independent from other parameters. Besides, the previous results showed higher determination quality in the flood season and in the middle part of the cross section. Therefore, we mainly focus on the effects of water depth and aspect ratio on the determination quality of the modified Manning formula.

A similar trend existed between the CV of relative error and water depth, and the inverse proportional trend was even more regular and distinct (Figure 8(b)). This seems to suggest that when the water depth was greater, the determination errors of the profile velocities from the middle part of the cross sections were closer to each other and the determination quality was relatively higher. The distribution histogram exposed that nearly 50% of the CV were less than 0.1.

The averaged cross-sectional relative error and its CV under similar aspect ratio bands were calculated, and the results are shown in Figure 8(c) and 8(d). The plot revealed the positive proportion between aspect ratio and the other two parameters when the aspect ratio was under 15. When the aspect ratio was higher than 15, the trend becomes uncertain. There were around ten cross sections with an aspect ratio higher than 15 and valued from 15 to 30 in our study. Due to the insufficient number and wide range of the samples, the analysis of the cross-sectional parameters with an aspect ratio higher than 15 failed to represent the real situation.

The method proposed by Shiono and Knight (SKM) provides an analytical solution, and has been most widely adopted for the vertical profile velocity and its lateral distribution determination (Tang & Knight 2008; Liu et al. 2013; Choi & Lee 2015). Along slightly different lines to the SKM, many vertical profile velocity models have been developed. Ervine et al. (2000) improved geometry and roughness boundaries (Ervine et al. 2000). Castanedo et al. (2005) introduced the λ-method, including three different forms of expression for the lateral turbulent shear stress (Castanedo et al. 2005). All these models involve parameters such as water density, gravitational acceleration, flow depth, bed slope in the stream wise direction, bed slope in the lateral direction, lateral eddy viscosity, bed shear stress, geometric factors and secondary flow effects, etc. The accuracy of these methods is considered to be high, but the applicability was limited due to the inaccessibility of most parameters, especially the bed slope in the lateral direction, lateral eddy viscosity, bed shear stress and the secondary flow.

The modified formula proposed in this paper adopted three parameters: the Manning coefficient, water depth and river bed slope. All three parameters are easier to measure compared to the parameters in SKM, which simplified the calculation process significantly. Therefore, the main advantage of this formula is the distinct reduction of data collection and computation effort with acceptable accuracy. Although the synthetic results near the bank showed higher error, this method is still quite significant for basic hydrology research due to the low velocity and discharge portion near the bank.

Results analysis indicated that the modified Manning formula is applicable for the determination of vertical profile velocity, especially in the middle part of the river. Data from the flood season show a better regression quality. Further analysis for vertical data from the middle region of each cross section revealed that the removal of data near the river bank will largely improve the conformity of V_S and V_M. Five factors may be responsible for this phenomenon:

1. The relatively low accuracy of measured velocity near the river bank sampled by ADQ. Former measurements proved that the distances from the bank are negatively correlated with the mean vertical profile velocity error, and the repeatability of measurement is lower near the river bank (Song et al. 2012). The relative error of the ADQ velocity measurement near the river bank was around 0.2, while the relative error of the synthetic river bank velocity was 0.5 or even higher. Due to the uncertainty of the measurement for the verticals near the bank, the determination error of the modified formula were with higher uncertainty.

2. The difficulty of roughness coefficient determination. Roughness was estimated during the field campaign according to the empirical table and then calibrated with the HEC-RAS model. Due to the vegetation and debris on the intertidal area and its inconsistency during the moderate level season and flood season, it is a challenge to find the proper Manning coefficient for each cross section in different seasons. In addition, the Manning roughness varied for the vertical profiles near the river bank and the vertical profiles in the middle part of the cross section. One constant roughness for the whole river section and the calibration of the roughness value together with the calibration of the river gradient could cause more uncertainty in the results.

3. The inaccuracy of using the unique gradient through the whole cross section. The gradient changed from profile to profile even at the same cross section, especially near the river bank, because of the irregular river geometry. However, the 1D HEC-RAS model treats the river bed along river sections as a single curve and gives an overall gradient for every cross section.

4. Model error. The output of calibrated models for the ten river sections under research had errors within ±10% for all parameters, including velocity, maximum depth and hydraulic depth. These are all factors that can affect gradients and Manning roughness calibration.

5. The influence of the secondary flow. One hypothesis made in this study was the laminar flow of the research flow section. However, this was only a simplified ideal model for the field conditions. The longitudinal secondary flow developed due to the anisotropic turbulence in corner regions or near the water surface (Knight et al. 2007). This indicated the higher disturbance from secondary flow near the river bank region and in the dry season, and resulted in better applicability of the modified Manning formula in the middle part of the cross section and in the flood season. The number, strength and position of the secondary flow cells was considered to be one of the main factors disturbing the lateral vertical profile velocity distribution (Tang & Knight 2015). Leaving out the effects of the secondary flow would lead to some uncertainty.

The hydraulic radius is defined as the ratio of the cross section area to the wet perimeter (R = W*D/P). In the rectangular or trapezoid-shaped cross sections where the width is much larger than the depth, the wet perimeter is closer to the river width, while the hydraulic radius is closer to the depth. This suggests better interchangeability between the hydraulic radius and water depth in rivers with a large aspect ratio. When researchers estimated mean river velocity with river depth instead of hydraulic radius with the Manning formula in the laboratory, the error of determination was inversely proportional to the aspect ratio (Yang & Wen 2007; Jia et al. 2010). Further research on the rectangular river cross section clearly demonstrated that when the aspect ratio is higher than 100, the error caused by substitution of the hydraulic radius with depth was no more than 2%, but when the aspect ratio is around 20, the error was as high as 7%, and it was even higher under a lower aspect ratio. In a triangular or U-shaped cross section, the determination accuracy is slightly higher than the rectangular cross section under the same aspect ratio conditions (Yang & Wen 2007).

However, in our study, a positively proportional relationship between determination uncertainty and the aspect ratio was found. One possible reason might be the lateral shear stress, which is related to the aspect ratio. According to research involving 14 rivers with an aspect ratio ranging from 5 to 25, the influence of lateral shear stress decreased with the increase of aspect ratio (McGahey et al. 2006). In wide, shallow channels, the side walls influenced the channel centre less and the flow was mainly dominated by bed-generated turbulence, while in a narrow deep river the side walls had a stronger influence with greater lateral shearing. The higher disturbance from the bed-generated turbulence in wide shallow rivers would require more detailed hydraulic information near the measured vertical, in addition to depth and Manning roughness. This might corroborate the better applicability of the modified Manning formula in narrow, deep rivers.

Based on data sampled from field campaigns during the moderate water level season and flood season in ten sub-catchments, the application of the modified Manning formula (2) in determining the river vertical profile velocity was analyzed in this paper. Determination accuracy was studied using the linear regression method. The influence of the water depth and aspect ratio on the relative error and coefficient of variation (CV) of the error were explored. The observations in this paper led to the following conclusions:

1. Regression analysis between V_S and V_M showed that the modified Manning formula provides a valid and effective method for vertical profile velocity determination, especially in the middle part of the cross section or during the flood season.

2. Determination error and CV were negatively proportional to the river depth, and were positively proportional to the aspect ratio.

3. The modified formula tends to be more applicable in relatively narrow and deep rivers. More measurements need to be collected to verify the accuracy in rivers with an aspect ratio higher than 15.

The applicability and uncertainty study of the replacement of the hydraulic radius with real water depth in determining the vertical profile velocity with a modified Manning formula is worth further research to reveal the effect of hydraulic conditions on lateral flow dynamics. More application of the modified Manning formula in catchments with various hydrological conditions, especially under high aspect ratio conditions, would be a good resource for future research and would provide more knowledge related to concrete situations.