Apparent Friction Coefficient Used for Flow Calculation in Straight Compound Channels

Water flow in channels with a compound cross-section involves an exchange of water mass and momentum between the slower flowing water in the floodplains and the faster water in the main channel. This process is called the streams interaction. As a result, the water velocity in the main channel decreases, and at the same time the velocity and depth of flow increase in the part of the floodplains adjacent to the main channel. Diversification of the surface roughness of the main channel and floodplains significantly affects the form of interactions. The results of laboratory experiments were used to characterize the influence of interactions on the discharge capacity of the channel with diversified roughness. The reduction in velocity of the main channel caused by the stream interactions is described with the apparent friction coefficients introduced at the boundary between the main channel and the floodplain. The obtained values of resistance coefficients, supplemented with the values from experiments reported in the literature, were used to establish a relationship useful in assessing the discharge capacity of such channels.


Introduction
For the practice, the kinematic structure of the stream in the channel with a compound cross-section can be characterized with sufficient accuracy, as the distribution of the depth averaged velocity in the cross-section. This distribution depends on the channel shape in the plane, the shape of the cross-section, the roughness of the side slopes and the bottom of the main channel, and flow resistance caused by the turbulent exchange of water masses and momentum between slower flowing water in the floodplains, and faster in the main channel. The process of the momentum exchange, along with the formation of eddy structures in the transition region between floodplains and the main channel was called the "kinematic effect" [1], and nowadays is described as the streams interaction. As a result of the interaction, the water velocity in the main channel decreases, while the velocity and depth of flow increase in parts of the floodplains adjacent to the main channel [2]. Diversification of the surface roughness of the main channel and the floodplains intensifies the process of creating eddies and secondary flows in the main channel, and affects the capacity of the channel with a compound cross-section [3][4][5][6]. Despite the spatial nature of the phenomena, in practice, many different methods are used to assess the discharge capacity of channels with compound cross-sections, and many calculation programs for the water profile in the channels are based on one-dimensional models [7]. Among them, the Divided Channel Method (DCM) is most often mentioned. The phenomenon of interaction emerging in the transition region between the main channel and the floodplain is described by separating the two streams, most often with vertical lines, on which the apparent shear stresses were assigned. The concept of the apparent tangential stresses at the division boundaries of the channel compound cross-sections was introduced by Wright and Carstens (1970) [8].
From the eighties of the last century, a number of formulas have been introduced to calculate the flow resistance due to the momentum transfer between the main channel and floodplain, according to concept of the apparent shear stress [9][10][11][12][13]. They were elaborated as a result of hydraulic experiments on the channel, using small and large scale models. A review of formulas derived to describe apparent tangential stresses at the boundary of streams in the main channel and floodplains can be found in Moreta and Martin-Vide (2010) [14].
The knowledge of apparent shear stresses makes it possible to determine the values of the dimensionless resistance coefficients used to calculate the average velocity in the steady uniform flow in the main channel of the compound cross-section, using the Darcy-Weisbach formula: where: v m -average flow velocity in the main channel, g-gravitational acceleration, f m -resistance coefficient for the main channel cross-section, calculated for the wetted perimeter, accounting for the length of the cross-section division plane, side slopes and the bottom of the main channel, R m -hydraulic radius of the main channel cross-section, S o -longitudinal channel slope. The flow resistance coefficients at the division boundary of the compound cross-section, according to Nuding (1998) [15], calculated on the basis of apparent shear stresses, depend on the following parameters of the channel (Figure 1): where: f a -apparent coefficient of resistance at the boundary between the main channel and floodplain area, f mb -resistance coefficient of the main channel bottom, f ms -resistance coefficient of the main channel side slopes, f m -resistance coefficient in the main channel, f fb -resistance factor of the bottom of the floodplain, H-water depth in the main channel, h f -water depth in the floodplain, b m -bottom width of the main channel, b f -floodplain width, 1:m-aspect of the side slope of the main channel and floodplains, k mb -absolute surface roughness of the main channel, k ms -absolute roughness of the main channel side slopes, k fb -absolute surface roughness of the floodplain, k fs -absolute roughness of the floodplain side slopes.
Water 2019, 11, x FOR PEER REVIEW 2 of 12 From the eighties of the last century, a number of formulas have been introduced to calculate the flow resistance due to the momentum transfer between the main channel and floodplain, according to concept of the apparent shear stress [9][10][11][12][13]. They were elaborated as a result of hydraulic experiments on the channel, using small and large scale models. A review of formulas derived to describe apparent tangential stresses at the boundary of streams in the main channel and floodplains can be found in Moreta and Martin-Vide (2010) [14].
The knowledge of apparent shear stresses makes it possible to determine the values of the dimensionless resistance coefficients used to calculate the average velocity in the steady uniform flow in the main channel of the compound cross-section, using the Darcy-Weisbach formula: where: vm-average flow velocity in the main channel, g-gravitational acceleration, fm-resistance coefficient for the main channel cross-section, calculated for the wetted perimeter, accounting for the length of the cross-section division plane, side slopes and the bottom of the main channel, Rmhydraulic radius of the main channel cross-section, So-longitudinal channel slope.
The flow resistance coefficients at the division boundary of the compound cross-section, according to Nuding (1998) [15], calculated on the basis of apparent shear stresses, depend on the following parameters of the channel ( Figure 1): where: fa-apparent coefficient of resistance at the boundary between the main channel and floodplain area, fmb-resistance coefficient of the main channel bottom, fms-resistance coefficient of the main channel side slopes, fm-resistance coefficient in the main channel, ffb-resistance factor of the bottom of the floodplain, H-water depth in the main channel, hf-water depth in the floodplain, bm-bottom width of the main channel, bf-floodplain width, 1:m-aspect of the side slope of the main channel and floodplains, kmb-absolute surface roughness of the main channel, kms-absolute roughness of the main channel side slopes, kfb-absolute surface roughness of the floodplain, kfsabsolute roughness of the floodplain side slopes. Bretschneider and Özbek T (1997) [16] used measurements of average water velocity in the main channel, and apparent tangential stresses at the division boundary of the cross-section on large-scale hydraulic models, as part of the SERC (Science and Engineering Research Council) program at the Hydraulic Research Laboratory in Wallingford, England, to determine the apparent resistance Bretschneider and Özbek T (1997) [16] used measurements of average water velocity in the main channel, and apparent tangential stresses at the division boundary of the cross-section on large-scale hydraulic models, as part of the SERC (Science and Engineering Research Council) program at the Hydraulic Research Laboratory in Wallingford, England, to determine the apparent resistance coefficients on vertical division lines, discussed in this work. The goal of the present study and performed hydraulic experiments was to explain how the surface roughness of the main channel and floodplains affects the values of these coefficients.

Study on Discharge Capacity of Channel with Compound Cross-Section
Study on the capacity of the channel with the compound cross-section was carried out in the hydraulic laboratory of the Department of Water Engineering of the Warsaw University of Life Sciences on a concrete model of a rectilinear section of the channel with a constant slope of 0.5 .
The channel with a compound trapezoidal cross-section was 16 m long and 2.10 m wide ( Figure 2). The bottom of the main channel and symmetrical floodplains in the cross-section were horizontal. The channel model was supplied by five pumps with a total discharge of 0.50 m 3 /s in a closed water cycle. In the initial section of the channel, a row of 0.30 m long PVC (Polyvinyl Chloride) pipes was laid, calming the flow, and directing water into the model. At the end of the channel, a tilting gate for controlling the water levels was mounted, which was used to force a steady uniform flow into the channel. Measured values were the water depths in the main channel and in the floodplains, the flow velocity at the cross-section points, the water temperature, and the flow rate in the channel. Measurements of the water flow velocity were carried out in a section located halfway along the channel. To measure the velocity components, an 11 × 33 mm ellipsoid electrostatic PEMS was installed on a sliding measuring carriage. Spot velocity measurements were carried out with an accuracy of 1 cm/s in 77 measuring profiles (Figure 2), and in nearly 500 cross-section points. The required length of the measurement time series for the longitudinal velocity was determined in the respect of an error of the calculated mean longitudinal velocity, with an allowable absolute error of the mean velocity of δ = 1 cm/s. It was found that the velocity should be measured 50 times, in order to give a 5 s measurement period with 0.1 s interval, using the electromagnetic probe. The water depths were measured with a pin gauge having an accuracy of 0.1 mm. To measure the flow rate in the channel, a calibrated circular measuring overflow with a diameter of 540 mm was used. The water head on the length of the channel was measured with a differential pressure gauge, based on the difference in water levels in piezometers located in the bottom axis of the main channel, at a distance of 4.0 m and 12.0 m from the beginning of the channel. The electro-probe and differential pressure gauge were connected to a computer measurement logger. Study on the capacity of the channel with the compound cross-section was carried out in the hydraulic laboratory of the Department of Water Engineering of the Warsaw University of Life Sciences on a concrete model of a rectilinear section of the channel with a constant slope of 0.5‰.
The channel with a compound trapezoidal cross-section was 16 m long and 2.10 m wide ( Figure  2). The bottom of the main channel and symmetrical floodplains in the cross-section were horizontal. The channel model was supplied by five pumps with a total discharge of 0.50 m 3 /s in a closed water cycle. In the initial section of the channel, a row of 0.30 m long PVC (Polyvinyl Chloride) pipes was laid, calming the flow, and directing water into the model. At the end of the channel, a tilting gate for controlling the water levels was mounted, which was used to force a steady uniform flow into the channel. Measured values were the water depths in the main channel and in the floodplains, the flow velocity at the cross-section points, the water temperature, and the flow rate in the channel. Measurements of the water flow velocity were carried out in a section located halfway along the channel. To measure the velocity components, an 11 × 33 mm ellipsoid electrostatic PEMS was installed on a sliding measuring carriage. Spot velocity measurements were carried out with an accuracy of 1 cm/s in 77 measuring profiles ( Figure 2), and in nearly 500 cross-section points. The required length of the measurement time series for the longitudinal velocity was determined in the respect of an error of the calculated mean longitudinal velocity, with an allowable absolute error of the mean velocity of δ = 1 cm/s. It was found that the velocity should be measured 50 times, in order to give a 5 s measurement period with 0.1 s interval, using the electromagnetic probe. The water depths were measured with a pin gauge having an accuracy of 0.1 mm. To measure the flow rate in the channel, a calibrated circular measuring overflow with a diameter of 540 mm was used. The water head on the length of the channel was measured with a differential pressure gauge, based on the difference in water levels in piezometers located in the bottom axis of the main channel, at a distance of 4.0 m and 12.0 m from the beginning of the channel. The electro-probe and differential pressure gauge were connected to a computer measurement logger. Diversification of the surface roughness in the channel was obtained by painting the concrete of a blurred surface with paint (called a smooth surface), or by applying a terrazzo layer with a grain diameter of 6-12 mm (called a rough surface).
Channel capacity experiments were carried out for the following variants ( Figure 3):  Diversification of the surface roughness in the channel was obtained by painting the concrete of a blurred surface with paint (called a smooth surface), or by applying a terrazzo layer with a grain diameter of 6-12 mm (called a rough surface).
Channel capacity experiments were carried out for the following variants (  Values of the absolute roughness of flume surfaces were determined from the distribution of mean velocity in the region where it satisfies the log-law [17].
The list of experiments carried out during the experiments, measured flow rates in the main channel and the adjacent floodplains, is summarized in Table 1.  Values of the absolute roughness of flume surfaces were determined from the distribution of mean velocity in the region where it satisfies the log-law [17].
The list of experiments carried out during the experiments, measured flow rates in the main channel and the adjacent floodplains, is summarized in Table 1.   On the basis of spot velocity measurements it was possible to plot lines of constant velocities (isovels) in the cross-sections of the channel. Examples of isovels for the experiment variant W 1.0, W 2.0 and W 3.0 at similar water depths are shown in Figure 4.

Resistance Coefficients in the Main Channel
The values of dimensionless resistance coefficients in the main channel with different bottom and slope roughness, as well as resistance factors fa in the plane of distribution of the cross-section of the channel, were calculated using the Einstein method (1934) [18]. According to this method, for each surface roughness along the perimeter of the cross-section the flow area can be found, in which this roughness shapes the flow conditions. The area can be determined using the graph of isovels in the cross-section of the main channel. The division of the cross-section into these areas is carried out with lines perpendicular to the isovels, starting from the wetted perimeter points, separating the perimeter into sections with different roughness ( Figure 5). This technique for determining division lines assumes that they are free from the shear stress, and forces are not transferred between the separated areas.

Resistance Coefficients in the Main Channel
The values of dimensionless resistance coefficients in the main channel with different bottom and slope roughness, as well as resistance factors f a in the plane of distribution of the cross-section of the channel, were calculated using the Einstein method (1934) [18]. According to this method, for each surface roughness along the perimeter of the cross-section the flow area can be found, in which this roughness shapes the flow conditions. The area can be determined using the graph of isovels in the cross-section of the main channel. The division of the cross-section into these areas is carried out with lines perpendicular to the isovels, starting from the wetted perimeter points, separating the perimeter into sections with different roughness ( Figure 5). This technique for determining division lines assumes that they are free from the shear stress, and forces are not transferred between the separated areas.  According to Einstein, the average flow velocity in each of these sections is equal to the average velocity across the entire main channel cross-section, i.e., v i = v m . Expressing the velocity with the Darcy-Weisbach equation with this condition, the following dependence can be obtained: then: where: f m denotes the average Darcy's friction factor in a main channel-being the substitutionary coefficient of resistance for the cross-section of the main channel calculated for the wetted perimeter P m that includes lengths of the section dividing lines (P m = P l + P lsb + P b + P rsb + P r ), R m symbolizes the hydraulic radius of the entire cross-section of the main channel (R m =A m /P m ), and R i is the hydraulic radius of the cross-sectional area per given roughness (R = A i /P i ).
The coefficient of resistance f m in the cross-section of the main channel is calculated on the basis of the average velocity (v m = Q m /A m ) and the calculated hydraulic radius of the main channel R m .
The determined areas of the cross-sectional area A i , in which the flow conditions are shaped under the influence of a constant roughness over the length of the wetted P i perimeter, were used to calculate the hydraulic radius R i and the coefficients of resistance f i .
The values of apparent resistance coefficients calculated in this way, as well as the resistance coefficients of the bottom and left slope of the main channel, are summarized in Table 2. Table 2. Cross-section areas A i for sections of the wetted perimeter with a constant roughness, hydraulic radius R i and flow resistance coefficients in the cross-section of the main channel determined on the basis of the isovels.  The calculated values of apparent resistance coefficients f a and resistance coefficients of the main channel f m as well as of the bottom f mb , the left side slope of the main channel, f ms , the bottom of the left floodplain f fb , of the compound cross section in experiments made in variants W 1.0, W 2.0 and W 3.0, are presented as a function of the depth ratio H/h f in Figure 6.    The variability of the apparent resistance coefficients is expressed in relation to the resistance coefficients of the bottom of the main channel fa/fmb. Pasche (1984) [19], and later German DVWK guidelines (1991) [20] recommended in the calculations of the capacity of channels with compound cross-sections using DCM for the ratio of depths in the main channel and floodplains fulfilling the condition H/hf > 3, the value of the apparent resistance coefficient equal to fa = 3fmb. However, at depths meeting the condition H/h < 3, it is recommended to take the value of the apparent resistance coefficient on the cross-section lines equal to the resistance coefficient of the bottom fa = fmb. Figure 9 presents the variability of ratios of the apparent resistance coefficients to the coefficients of the bottom of the main channel fa/fmb in respect of the relative depth H/hf obtained from the conducted experiments.     The variability of the apparent resistance coefficients is expressed in relation to the resistance coefficients of the bottom of the main channel fa/fmb. Pasche (1984) [19], and later German DVWK guidelines (1991) [20] recommended in the calculations of the capacity of channels with compound cross-sections using DCM for the ratio of depths in the main channel and floodplains fulfilling the condition H/hf > 3, the value of the apparent resistance coefficient equal to fa = 3fmb. However, at depths meeting the condition H/h < 3, it is recommended to take the value of the apparent resistance coefficient on the cross-section lines equal to the resistance coefficient of the bottom fa = fmb. Figure 9 presents the variability of ratios of the apparent resistance coefficients to the coefficients of the bottom of the main channel fa/fmb in respect of the relative depth H/hf obtained from the conducted experiments.   The variability of the apparent resistance coefficients is expressed in relation to the resistance coefficients of the bottom of the main channel f a /f mb . Pasche (1984) [19], and later German DVWK guidelines (1991) [20] recommended in the calculations of the capacity of channels with compound cross-sections using DCM for the ratio of depths in the main channel and floodplains fulfilling the condition H/h f > 3, the value of the apparent resistance coefficient equal to f a = 3f mb . However, at depths meeting the condition H/h < 3, it is recommended to take the value of the apparent resistance coefficient on the cross-section lines equal to the resistance coefficient of the bottom f a = f mb . As shown in Figure 9, with the ratio of the depth in the main channel and the floodplain 3 < H/hf < 7 in the variant W 1.0, the apparent resistance coefficients took values = 3 ÷ 8 . The introduction of rough floodplain in the variant W 2.0, and the side slopes of the main channel in the variant W 3.0, caused that at 3 < H/hf < 7, the values of the apparent resistance coefficients become equal.
Since the values of the apparent resistance coefficients also depend on the width of the floodplain and the main channel (2), it was decided to identify such a dependence. This was done on the basis of results reported in the literature. Bretschneider and Özbek (1997) [16] calculated the apparent, and those for the bottom of the main channel, coefficients of resistance, on the basis of measured apparent shear stresses, in experiments performed on large scale models of a compound channel as part of the SERC program at the Hydraulic Research Laboratory in Wallingford, England. The ratios of apparent and bottom resistance coefficients from their own experiments in the variant W 1.0 were compared with the results of the Bretschneider and Özbek calculations (1997) [16] for a series of tests performed in a smooth two-section channel with 1:1 side slopes, and different ratios of floodplain and the main channel widths bf/bm, and these are shown in Figure 10. The dependencies presented there show that the increase in the width of the floodplain in relation to the width of the main channel significantly affects the discussed ratio of resistance coefficients. This effect decreases with the increase of the flow depth. The character of changes in the resistance coefficients of fa/fmb as a function of ratios of depth H/hf and the widths of the floodplain and the main channel bf/bm from our own research, presented in Figure 10, is explained by the regression equation: Figure 10. The values of the resistance coefficients of fa/fmb in the function of H/hf from our own research in the variant W 1.0, together with the results of Bretschneider and Özbek (1997) [16] for a series of experiments performed in a two-section channel with side slopes of 1:1.
As it results from the Equation (5) for very narrow floodplains, with bf/bm = 1.0 and H/hf = 3, the ratio of resistance coefficients takes the values fa/fmb = 2.4. However, for wide floodplains, with bf/bm = 4.5 and H/hf = 3, the ratio of these coefficients reaches the value of fa/fmb = 5.0.

Conclusions
The analysis of the values of resistance coefficients determined for the main channel with the compound cross-section showed that: • In a smooth channel with a compound cross-section, the values of the resistance coefficients of the bottom and side slopes of the main channel do not change significantly with an increase in depth; an increase in the surface roughness of the floodplain area causes the increase of resistance coefficients in the smooth main channel, • the values of apparent resistance coefficients are several times greater than the resistance coefficients for side slopes and bottoms of the main channel and floodplains, • apparent resistance coefficients decrease with increasing depth, • the ratios of apparent, and the bottom of the main channel resistance coefficients, increase along with the increase in the width of the floodplain in relation to the width of the main channel.

Conclusions
The analysis of the values of resistance coefficients determined for the main channel with the compound cross-section showed that:

•
In a smooth channel with a compound cross-section, the values of the resistance coefficients of the bottom and side slopes of the main channel do not change significantly with an increase in depth; an increase in the surface roughness of the floodplain area causes the increase of resistance coefficients in the smooth main channel, • the values of apparent resistance coefficients are several times greater than the resistance coefficients for side slopes and bottoms of the main channel and floodplains, • apparent resistance coefficients decrease with increasing depth, • the ratios of apparent, and the bottom of the main channel resistance coefficients, increase along with the increase in the width of the floodplain in relation to the width of the main channel.