Abstract

The prediction of hydraulic coefficients, which varies for different materials, is an essential criterion for designing open channels and related hydraulic structures. The main objective of this research is to compare experimental hydraulic coefficients on fixed and mobile bed materials in open channel flow. The study considered three types of bed roughness; namely steel beds, 2 mm uniform beds, and 25 mm uniform grain-sized beds in a rectangular flume model with a 1:500 and 1:200 adjustable bed slope. Manning's roughness coefficient and Chezy's equations were used to model channel bed roughness concerning the variation of flow characteristics for different bed conditions. The results showed that Manning's roughness coefficient had an inverse relationship with the discharge, flow depth, hydraulic radius, flow conveyance, and Chezy’s coefficient, but it was directly proportional to grain size. The values of mean velocity in the mobile bed along the channel are comparatively less than that of the fixed bed. The percentage change of average n value due to changes in channel slope and bed condition is slightly increased by 144.95% and 150.23% under the channel slope of 1:200, and 143.02% and 148.04% under the channel slope of 1:500. Also, the percentage change of the average C value is slightly decreased by 59.97% and 59.98% under the channel slope of 1:200, and 58.75% and 59.42% under the channel slope of 1:500 when the bed changed from fixed to mobile 2 mm and 25 mm grain sizes, respectively.

1 Introduction

Recently, the number of water resources and hydraulic engineering projects have been quickly rising all over the world; accordingly, the prediction of hydraulic coefficients is an essential criterion to design open channels and related hydraulic structures (Azamathulla et al. 2013; Bahramifar et al. 2013; Bilgil and Altun 2008). The hydraulic coefficient is a fundamental control of flow hydraulics in streams and rivers (Powell 2014), controlling not only the amount of water a channel can transport through its influence on velocity and thus flow depth, but also the distribution of shear stress around the channel boundary, as well as the magnitude and distribution of bed and bank erosion (Ferguson 2010). It is also a measure of the effect that bed material size, elements of the roughness of channel surface (channel configuration), channel geometry (channel shape and size), channel curvature (sinuosity), vegetation type and density, channel alignment and irregularities, channel obstruction (tree stumps, logs, boulders, bedrock outcrops, etc.), stage, and discharge, suspended sediment load and bed sediment loads, and other factors have on flow resistance (Moretto 2014; Vargas-Luna et al. 2016; Yager et al. 2018).

Hydraulic resistance is typically described using a friction factor or roughness coefficient. This coefficient can be described in a variety of ways, and a wide range of equations have been presented to connect the mean channel flow velocity to flow resistance, particularly in the situation of shallow flow over rough surfaces (Nguyen and Niansheng 2014). To find out the general equation, Griffiths (1981) and Limerinos (1970) intended to correlate various hydraulic coefficients to the characteristics of beds and flows (for instance, flow discharge or depth, bed roughness, river width, and slope). Hydraulic coefficients are calculated using flow equations such Manning's roughness equation, Chezy’s equation, and the Darcy-Weisbach equation based on the roughness of the channel bed (Ferguson et al. 2019; Konwar and Sarma 2015). However, when a channel bed is movable, and the sediments are transported as bed loads, the phenomena are more complicated than clear water flowing in a channel with fixed beds (Hajbabaei et al. 2017; Mrokowska and Rowinski 2019).

Hydraulic coefficient (flow resistance) in open channels plays an important part in river engineering and has been studied for many years (Yen 2002). In hydraulic engineering, Manning’s roughness coefficient n is defined as a crucial parameter in designing hydraulic structures, calculating velocity distribution, an accurate determination of energy losses, expression of channel roughness, and flow resistance (Bilgil and Altun 2008). Previous research demonstrated that there are many significant parameters impacting the velocity in a certain channel, such as water area, wetted perimeter, maximum surface velocity, the slope of the water surface, maximum depth, roughness coefficient, and water temperature (Huthoff and Augustijn 2005). This implies that when the channel bed has high roughness material, the bed shear stress will be higher. However, rivers with beds including various materials will have major effects on roughness resistance to flow, which have been examined by numerous researchers (Yochum 2018).

The influence of flow resistance gives some effect to flow rate and the roughness characteristics (McKeon 2008). Emphasizing the experimental studies of hydraulic roughness, Wang et al. (2011) analyzed the effect of roughness on the flow structure in a gravel bed channel using particle image velocimetry. Christodoulou (2013) experiments with flow in a 16.5% sloping channel across several types of submerged artificial large-scale roughness elements (flow resistance is determined by the shape, size, and density of the object used as a barrier). Ebrahimi et al. (2018), and Sadeque et al. (2009) provide the results of an experimental examination of flow around cylindrical objects on a rough bed in an open channel.

To consider the sediment movement impact of the riverbed on the water flow structure, it is necessary to study the mobile bed resistance of mountainous rivers. The mobile bed resistance for alluvial rivers is the resistance of sand waves and develops at various stages. However, a large amount of coarse grain mountainous pebbles in the river bed makes it difficult to form sand waves (Hou et al. 2019).

In open channels, several experimental studies have been done on the determination of hydraulic coefficients (flow resistance coefficients or roughness coefficients) of different bed materials in recent years around the world (Lau and Afshar 2013). However, many problems remain unsolved concerning how the channel roughness is predicted, in as much as it varies with the bed configuration and the amount of sediment load over fixed and mobile bed materials, and also the effect of hydraulic coefficients on the values of hydraulic characteristics (flow parameters) for open channel flow, over the fixed and mobile bed materials. When a channel has a mobile bed and sediment is being transported, the problem of determining the hydraulic coefficients is much more complicated than in the simple case of clear water flowing in a channel with a fixed bed. This problem has not been studied sufficiently to enable hydraulic engineers to predict hydraulic coefficients when man-made changes in the river cross-section are being planned, such as for flood control, power generation, navigation, and water supply. Therefore, in this laboratory flume study, the research gaps have been addressed by using Manning's roughness and Chezy's equations to select the best hydraulic performing bed from fixed and mobile beds of the non-uniform channel bed surface gradient. Particularly, through the comparison of the effect of experimentally obtained roughness coefficients on discharge, sediment bed load on the velocity (i.e., mean and shear) profile, channel bed slope (hydraulic gradient) on roughness coefficients, selected bed materials on roughness coefficients, bed roughness coefficients on velocity and shear distributions (profiles) of the flow for fixed, as well as mobile bed materials in both clear water flow and sediment-laden flow conditions.

2 Materials and Methodology

2.1 Experimental setup

For this research, experimental tests were conducted at the hydraulics laboratory on a free surface recirculating an artificial straight simple rectangular open channel flume with uniform cross section, under the faculty of Hydraulic and Water Resources Engineering, Water Technology Institute, Arba Minch University, Ethiopia. The length, width, and height of the glass sided tilting rectangular open channel flume, which is uniform throughout the longitudinal direction, are 7.5, 0.3, and 0.45 m, respectively, with adjustable bed slopes of 1:200 (0.005) and 1:500 (0.002) to obtain uniform flow conditions. An adjustable gate was provided at the upstream section of the flume to reduce the turbulence of the incoming water and velocity of approach in the flow. At the downstream end, another adjustable overflow tailgate was provided to control the flow depth and maintain a uniform flow in the channel (flume). Throughout the 30 mm thick bed, the flow rate remained constant. The transported solid concentrations of the sediment samples for the mobile bed were collected by a sediment trap net placed downstream at the end of the flume. This experimental setup is schematically shown in Figure 1.

Figure 1 Schematic of the study flume layout and experimental arrangement site.

2.2 Materials or measuring equipment

The materials used to investigate the experimental analysis of flow over a fixed and mobile bed are fixed and mobile-bed materials (Figure 2), centrifugal pump, digital flow meter, Vernier height point gauge, stopwatch, and digital mass balance.

Figure 2(a-d) Fixed and mobile bed materials used for bed roughness in the flume.

2.3 Methods used for numerical analysis

Hydraulic coefficients (flow resistance coefficients or roughness coefficients) are mainly expressed (quantified) by Manning’s roughness coefficient n, Chezy's coefficient C, and the Darcy-Weisbach coefficient f. Out of the three resistance coefficients, the most commonly used is Manning’s coefficient (Chow 1959; Ferguson 2010; Konwar and Sarma 2015).

Manning's roughness formula

Manning's formula, developed by Irish engineer Robert Manning in 1885 and later published, serves as an alternative to various resistance coefficients. Engineers find it more convenient to use in practical applications and frictional constant calculations because of its gradual development and modification (Azamathulla et al. 2013; Azamathulla and Jarrett 2013; Sihag et al. 2022). The formula can be expressed mathematically as:

(1)

Where:

A	=	cross-sectional area of flow (m2),
R	=	hydraulic radius (m),
S	=	channel bed slope, and
n	=	Manning’s roughness coefficient (based upon channel material and condition) in s/m1/3

Chezy's formula

Chezy's formula, in terms of velocity, is represented in Equation 2 as follows:

(2)

Where:

C	=	Chezy's coefficient, depending upon the various characteristics of the channel and their comparison with those of another similar channel (m1/2/s), and
V	=	mean flow velocity (m/s).

Shear stress distribution

Average shear stress at the boundary for uniform flow can be expressed as:

(3)

Where:

γ

=

unit weight of water (kN/m3),

Conveyance of the channel section formulae

The conveyance of the channel section is a measure of the carrying capacity of the channel section. For Manning’s formula, conveyance can be expressed as:

(4)

Similarly, for Chezy's formula:

(5)

Where:

K	=	conveyance of the channel section, and
A	=	wetted area (m2).

2.4 Measurement procedures

A series of tests were performed in the flume. In each test, a particular uniform flow was established and became stably adjusted with its bed and remained so for several seconds during which the test was performed. The following procedures were used for each experimental test:

1. Fill the storage tank under the ground of the laboratory building with water.
2. Pump the water from the storage tank into an overhead tank on the roof of the laboratory building by using electrically operated centrifugal pumps.
3. Set up and fix the selected materials in the flume bed.
4. Adjust the bed slope of the flume by using a leveling device mounted on the flume.
5. Open the flow control inlet valves and admit water into the flume until the optimum flow rate is achieved.
6. When the flow is constant throughout the flume, read the flow rate from the digital flow meter.
7. Set up a Vernier height point gauge and measure the flow depth from the bottom of the flume to the water surface for a smooth channel (originally stainless-steel bed), and from the crest of the rough elements to the water surface for 2 mm and 25 mm grain sized fixed and mobile bed materials.
8. Measure the volume in litres, and time in seconds, to manually determine the volumetric flow rate in litres per second.
9. Measure solid mass concentration resulting from sediment entrainment by using a digital mass balance for mobile bed materials.
10. Repeat the experimental steps from number 5 to 8 by increasing flow depth. 

3 Results and Discussion

3.1 Effect of bed material (surface) on the hydraulic coefficient

In Tables 1 and 2, Manning's roughness coefficient and Chezy's constant average values, and ranges for the various bed surfaces with varying discharges under the slopes of 1:200 and 1:500 are displayed. From these tables, the average values of Manning’s roughness coefficient in steel for a fixed bed is the smallest compared to 2 mm and 25 mm grain size surfaces, whereas the average Chezy’s constants show the reverse. However, for the mobile bed surfaces average, Chezy’s constants are unavailable because of the non-erodibility of the flume bed materials. The average values of Manning’s roughness coefficient for 2 mm and 25 mm grain size mobile bed surfaces are higher than that of fixed-bed surfaces under both channel slopes. It was also observed that increasing the slope from 1:500 to 1:200 increased and decreased the average values of Manning’s roughness coefficient and Chezy’s constant from smooth to rough (fixed and mobile) surface, respectively. From Tables 1 and 2, it was concluded that the rougher the bed surface, the larger and smaller values of Manning’s roughness coefficient and Chezy’s constant, while the smoother the bed surface, the lower and higher Manning’s roughness coefficient, and Chezy’s constant, respectively.

Table 1 Average n and C values of bed materials for different discharges under the slope of 1:200.

Bed material	Average of n for fixed bed	Range of n for fixed bed	Average of n for mobile bed	Range of n for mobile bed
Steel	0.0113	0.0103 – 0.0121	NA	NA
2 mm grain size	0.0198	0.0164 – 0.0222	0.0485	0.0410 – 0.0555
25 mm grain size	0.0219	0.0201 – 0.0240	0.0548	0.0501 – 0.0601
Bed material	Average of C for fixed bed	Range of C for fixed bed	Average of C for mobile bed	Range of C for mobile bed
Steel	50.783	37.975 – 60.606	NA	NA
2 mm grain size	29.807	20.643 – 38.142	11.933	8.277 – 15.257
25 mm grain size	26.255	19.134 – 31.198	10.506	7.644 – 12.478

NOTE: NA denotes ‘not available’.

Table 2 Average n and C values of bed materials for different discharges under the slope of 1:500.

Bed material	Average of n for fixed bed	Range of n for fixed bed	Average of n for mobile bed	Range of n for mobile bed
Steel	0.0112	0.0102 – 0.0119	NA	NA
2 mm grain size	0.0172	0.0147 – 0.0190	0.0429	0.0368 – 0.0473
25 mm grain size	0.0204	0.0163 – 0.0235	0.0506	0.0386 – 0.0585
Bed material	Average of C for fixed bed	Range of C for fixed bed	Average of C for mobile bed	Range of C for mobile bed
Steel	51.445	38.490 – 61.531	NA	NA
2 mm grain size	33.697	24.172 – 42.503	13.482	9.699 – 17.001
25 mm grain size	28.589	19.553 – 38.313	11.602	7.852– 16.200

NOTE: NA denotes ‘not available’.

Table 3 shows the percentage change (%) of average n and C values due to changes in channel slope and bed condition. It is observed that the values of Manning’s roughness coefficient n under the slope of 1:200 increased by 144.95% and 150.23% when the bed changed from fixed to mobile of 2 mm and 25 mm grain size. Under the slope of 1:500, the values increased by 143.02% and 148.04% when the bed changed from fixed to mobile of 2 mm and 25 mm grain size. While the Chezy’s constants C, under the slope of 1:200, are decreased by 59.97% and 59.98% when the bed changed from fixed to mobile of 2 mm and 25 mm grain size, and under the slope of 1:500 are decreased by 58.75% and 59.42%, which is indicated by the negative sign when the bed changed from fixed to mobile of 2 mm and 25 mm grain size, respectively.

Table 3 Percentage change (%) of average n and C values due to changes in channel slope and bed condition.

Bed slope	Percentage change (%) of average n values		Percentage change (%) of average C values
	When the bed changed from fixed to mobile of 2 mm grain size	When the bed changed from fixed to mobile of 25 mm grain size	When the bed changed from fixed to mobile of 2 mm grain size	When the bed changed from fixed to mobile of 25 mm grain size
1:500	143.02	148.04	-58.75	-59.42
1:200	144.95	150.23	-59.97	-59.98

NOTE: Negative sign (-) indicates that the % change of average C value is decreased as the channel bed changed from fixed to mobile under both slopes.

3.2 Comparison of the effect of slope on roughness coefficient

In Figure 3, a graph between Manning’s roughness coefficients, n of 25 mm uniform bed material with different discharges was used. The effect of channel slope on the Manning’s roughness coefficient was compared with both fixed and mobile beds of 25 mm grain size in Figure 3. The Manning’s roughness coefficient under the slope of 1:500 is lower, compared to a coefficient under the slope of 1:200. This is similar to the results presented by Konwar and Sarma 2015, Lau and Afshar 2013, and Merry 2017 that show how the values of Manning’s n are decreasing from steeper to flatter, which acts in accordance with Manning’s theory. However, due to the entrainment of bed sediment, the values of Manning's roughness coefficient for mobile beds are larger than those for fixed beds, even under the same slope (1:500 or 1:200), as illustrated in Figure 3. In summary, the effect of slope on Manning’s roughness coefficient is increasing gradually for steeper slopes (i.e., n increases with an increase in flume bed slope) and vice versa. Variation of longitudinal velocity distribution with flow depth of 2 mm and 25 mm grain sizes of fixed and mobile beds under the slope of 1:200 and 1:500.

Figure 3 Manning’s roughness n versus discharge Q for 25 mm grain size.

Figures 4 and 5 illustrate the mean velocity distribution of 2 mm and 25 mm grain size for fixed and mobile beds with respect to the depth of flow under the slopes of 1:200 and 1:500. The slope of the channel bed has also affected the velocity of flow in the open channel. At a steeper gradient (1:200), velocity increases, while at a mild gradient (1:500), velocity decreases in both grain sizes and cases (fixed as well as a mobile bed). However, as shown in the figures, the values of mean velocity in the mobile bed are comparatively less than that of the fixed bed because sediment motion requires high water energy. It can be concluded that increasing flow depth along with the channel corresponding to its bed slope increases the mean velocity of flow. The same result has also been obtained in the study of (Khatua et al. 2011) that the increase in longitudinal velocity of flow is nearly in accordance with the increase in flow depth.

Figure 4 Velocity of 2mm grain size fixed bed w.r.t flow depth under the slope of 1:200 and 1:500.

Figure 5 Velocity of 25mm grain size mobile bed w.r.t flow depth under the slope of 1:200 and 1:500.

3.3 Effect of hydraulic radius on Manning’s roughness coefficient

Figure 6 plots the fixed and mobile bed materials to define the effect of the hydraulic radius on the Manning’s roughness coefficient at different discharges under the channel bed slope of 1:200. Figure 6 shows that for the different discharges, the maximum and minimum Manning’s roughness coefficients were located at the hydraulic radius of 0.060 m and 0.009 m, respectively. Consequently, Manning’s roughness coefficient was inversely proportional to the hydraulic radius.

Figure 6 Effect of hydraulic radius on Manning’s roughness coefficient under 1:200.

3.4 Relationship between shear stress and Manning’s roughness coefficient

Figure 7 illustrates the relationship between the shear stress and flow depth, and Manning’s roughness coefficient for steel, fixed, and mobile bed materials under the slope of 1:500 and 1:200. From Figure 7, the shear stress in a rough bed that is 25 mm gain size was significantly the highest compared to a 2 mm gain size and a smooth (steel) bed. This implies that when the channel bed has high roughness material, the bed shear stress will be higher.

Figure 7 Shear stress versus Manning’s roughness coefficient, n.

3.5 Y-Q relationship for fixed and mobile beds

Figure 8 defines the link between the flow depth and flow discharge for fixed and mobile bed materials under the slope of 1:500. The figure shows that the values of flow discharge are lower in mobile beds for both grain sizes, even in the same depth of flow. Figure 8 also showed that the steel bed, when under the different studied heads, gave the highest discharge values for all cases. As a result, the flow discharge was directly proportional to the flow depth for both fixed and mobile beds with varying channel slopes.

Figure 8 Y-Q relationship for fixed and mobile beds under 1:500.

3.6 Effect of Manning’s roughness coefficient on channel conveyance under 1:500

A graph of the effect of Manning’s roughness coefficient on channel conveyance is constructed and plotted in Figure 9. It is observed that the channel conveyance, or carrying capacity of the channel, in fixed steel surfaces is the highest compared to the other two fixed and mobile bed surfaces under slopes. In addition, channel conveyance is the smallest in mobile 25 mm grain size compared with that of steel, fixed 2 mm and 25 mm, and mobile 2 mm grain-sized beds. From the experiment, it is concluded that an increase in resistance, or roughness of the channel, will decrease the flow conveyance, and vice versa.

Figure 9 Manning’s roughness coefficient versus channel conveyance under 1:500.

3.7 Effect of bed material particle size on the Manning’s roughness coefficient

The effect of the particle size of the bed material on resistance to flow (average Manning’s roughness coefficient) is shown in Figure 10. From this figure, it can be seen that the Manning’s roughness coefficient for flow over a 2 mm grain-sized bed without motion (fixed) was slightly smaller, whereas on the same grain-sized bed with motion, it was slightly larger under two-bed slopes (1:500 and 1:200) comparatively. Similarly, results for a 25 mm grain size were analyzed and plotted into the graph as shown in this figure. Sediments require a part of the energy to be consumed from water flow to move when they enter into the channel, increasing the channel resistance (Hou et al. 2019). Figure 10 demonstrates that in both cases, an increase in grain size from 2 mm to 25 mm results in an increase in flow resistance in a channel. Hence, the Manning’s roughness coefficient in a channel is proportional to grain size and decreases as discharge and depth of flow (water) increases.

Figure 10 Graph of average Manning’s roughness coefficient and particle size of the bed material under 1:500 and 1:200.

3.8 Selection of the best hydraulic performance of channel

Due to the increase in the roughness coefficient, the canal could not carry the design discharge, which leads to the tail-end deprivation in the command area. The channel bed should be kept smooth and even the deposits of debris and gravel should be regularly cleaned, and the roughness coefficient should be decreased to maximize the carrying capacity of the channel. Because of the reduced roughness, lined channels can transfer more flow than unlined channels. According to the experimentally acquired results, steel and fixed beds provide the best hydraulic performance of the flume (channel). Because the Manning's roughness coefficient was lower in fixed and steel beds than it was in mobile beds, the value of discharge in these beds was larger. That means the selection was based on the channel carrying capacity concerning the bed resistance (refer to Figure 9).

4 Conclusions

Predicting the flow resistance of roughness elements is significant in hydraulics, because of their importance in practical application. In this experimental study, the prediction of flow resistance coefficients n and C is investigated for water flows, within the rectangular open channel (flume). This work employed a combination of theoretical analysis and practical research to thoroughly investigate and analyze the fixed and mobile bed resistance of an open channel. The value of n over the mobile channel bed is significantly higher than that over the fixed channel bed. The value of mean velocity in the mobile bed along the channel is comparatively lower than that of the fixed bed. Increasing flow depth along with the channel corresponding to its bed slope increases the mean velocity of flow.

Chezy’s constant was directly proportional to actual discharge, while Manning’s constant was inversely proportional to actual discharge. Chezy’s constant has a higher value compared to Manning’s constant. Also, Manning’s standard deviations and variances in 2 mm and 25 mm grain sizes for both fixed and mobile beds are lower compared to Chezy’s. This indicates the accuracy of the resistance coefficient because the lower the value of standard deviation and variance, the higher the level of accuracy. Therefore, the coefficient of resistance is more adaptable, simple, and accurate in Manning’s constant.

Further investigation on the effect of aspect ratio on secondary currents and turbulence intensity distribution in the cross-section of the channel, and the effect of different roughness elements’ spacing (pattern) on flow resistance in open channels needs to be completed. The culture of applying experimental outcomes backed by laboratory equipment and demo structures for the practical world is quite unusual in many engineering studies. It is recommended that laboratory experimental outcomes be developed in practice, based on scientific evidence, distinct fundamental equations, and empirical formulas from the investigation.

Acknowledgement

The authors would like to acknowledge Arba Minch University, School of Post Graduate Studies for financially supporting this research.

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