Characteristics of the flow in channels with extended boundaries

The Flow in Channels with Extended Boundaries (FCEB) was introduced and defined by Kamal et al. (2020). It is the flow where the channel sides are extended covering parts of the channel flow surface symmetrically or asymmetrically. As the extensions just touch the flow-free surface, the pressure domain is not affected and the pressure at the top flow-free surface is still approximately atmospheric. On the other hand, the stagnant extensions affect the distributions of both velocities and discharges in the flow cross-section.

Available in the literature, one may find many attempts to study the FCEB applications mentioned above and the velocity domain of other types of flow. However, only a few attempts exist in studying the FCEB.

Regarding applications of FCEB, herein some research and engineering works are briefly demonstrated. The UNEPA (1998) divided wetlands into three types, namely natural, constructed, and floating. Wu et al. (2015) reviewed the application of constructed wetlands as green technology and summarized the key design parameters for their sustainable operation. Shahid et al. (2018) appraised the floating wetland as a wastewater treatment technology regarding effectiveness and sustainability. Neese et al. (2002) evaluated the use of floating bridges for trail crossings in streams and very wet areas and introduced a complete study on the effectiveness of using different materials for the floating structure. The company of Goulburn-Murray Water (2012) sets the technical specifications for the floating jetties in Australia. El-Maadawy et al. (2018) focused on the accuracy of load determination in checking the structural safety of a floating dry dock. They modeled an existing floating dock in the Port Said shipyard as a case study and developed a three-dimensional finite element model using full technical data of the floating dock. Clark & Wall (2016) monitored the Dauphin River in central Manitoba, Canada. They stressed the many river ice processes that can be observed on the river. Shishmarev et al. (2016) built a numerical model to study the stress–strain relation of ice layers extended from the sides of a frozen channel. They studied the effects of different parameters, such as channel boundaries, load speed, and channel width and depth on the hydro-elastic response of the ice extensions.

For the investigation of the velocity domain, the following may be demonstrated. Huai et al. (2019) modeled rivers with increasing overall roughness by placing artificial flexible vegetation. They used Acoustic Doppler Velocimeter (ADV) to sample the stream-wise velocity field. Yunwen et al. (2022) constructed a laboratory-generalized meandering channel and measured the instantaneous velocities using ADV. They showed that the positions with large turbulent kinetic energy were in the mainstream area. For the same discharge, the transverse water surface gradients are large at the downstream positions adjacent to the apex sections and small near the middle positions of the crossover areas. Yeganeh & Heidari (2022) used the entropy concept and the principle of maximum entropy to drive one-dimensional velocity distribution in an open channel. Results showed that the Tsallis entropy method is more accurate than other forms of entropy, showing much better simulation.

As mentioned above, only a few efforts were made to investigate the hydraulics of FECB. Most of the research work was conducted as side studies for ice rivers. Tang & Davar (1982) studied five models of flow in rivers with partial ice cover. Each model has a different cover ratio. For the sake of comparison, an open channel case was tested. Their analysis mainly depended on the Manning equation and Streeter velocity distribution logarithm. Mitchel (2015) studied the velocity distribution below layers of ice in the stream. A corresponding Manning roughness coefficient was calculated for the model boundaries and ice cover. Chen et al. (2018) focused on the problems of predicting precise streamflow and the difficulties faced when managing waterways that are covered with ice. They introduced a formula with a K-factor in the form of a general Manning equation. Results showed that the K-factor is affected by the resistance ratio of the boundary. Wazney et al. (2019) studied the change in ice dynamics due to an unsteady flow. They used the Rising Limb Analysis Method (RLAM) to study the changes caused by the surge wave to the hydraulic parameters like velocity, discharge, and shear stress. They noticed a temporary increase in these parameters with high water levels. Beltaos & Peters (2020) showed that the inundation effectiveness of ice jams in the Peace River, Alberta, Canada depends on the magnitude of the prevailing river flow. They demonstrated the adopted ‘lagged-flow’ methodology via basic hydrodynamic principles and compared it with the independently generated numerical-modelling data.

In the present study, the FCEB is simulated experimentally for the case of asymmetrical extensions. The ADV is used in collecting the velocity data to investigate the flow domain. It is hoped that this research work will give new insight into researchers and engineers in the field of hydraulic engineering.

Since the early attempts of researchers in the field of fluid flow, the velocity distribution has always been considered the cornerstone in investigating the flow characteristics. The discharge, the momentum force and the kinetic energy driving the flow, and the shear stress are all functions of the velocity distribution. In this study, laboratory experiments were conducted to collect the velocity data of the FCEB. Details of the experimental work are introduced herein.

Eight experiments were conducted (experiments 3–10), and another two more experiments were used from Kamal et al. (2020) for the sake of comparison (experiments 1 and 2). Table 1 shows the boundary conditions for these experiments. The following are defined in Table 1:

• Column (1) gives the experiment number.

• Columns (2–5) give the values of the boundary extension ratio (⁠⁠) for every experiment. The values of BER used in these experiments are 0, 25, 50, and 67%. As the flume width b = 1,200 mm, then the extension widths corresponding to these BERs are 0, 150, 300, and 402 mm, respectively.

• Columns (6–9) indicate if the boundary is rough or smooth in the experiment. The abbreviations RB, RE, SB, and SE mean rough bed, rough extension, smooth bed, and smooth extension, respectively.

• Columns (10 and 11) show whether the value of the aspect ratio (a = b/D) used in the experiment equals 10 or 6. As the flume width (b) was kept constant as 1,200 mm, then a = 10 means that D = 120 mm, and a = 6 means that D = 200 mm.

• For experiments 1 and 2, BER= 0 (no extensions exist) and the bed and sides are smooth. That is, experiments 1 and 2 are open channel flow (OCF) in rectangular smooth channels. The values of a were set to a = 10 (D = 120 mm) and 6 (D = 200 mm), respectively.

Table 1

The boundary conditions of the experiments

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The ADV data were collected in a complete mesh in the y–z plane. The mesh is 50 mm (y-direction) × 10 mm (z-direction). Near boundaries, the mesh distances are decreased to 15 mm × 7 mm. This gives approximately 400 sampling locations in case of a = 10, and 600 sampling positions in case of a = 6. Due to the measuring limitations of the ADV, no data were collected in the two-channel corners.

Prior to conducting the experiments, preliminary experiments were conducted to set the following:

• The range of different flow parameters such as the discharge and flow depth for the flume.

• The ADV optimum number of readings/min. Starting from 12,000 readings/min, the optimum number of readings/min was found = 3,000 to give acceptable and accurate readings. The sampling time/location = 3 min.

Table 2 gives the flow parameters and the number of readings for every experiment. In Table 2, Q is the discharge, and u and u_max are the average and maximum flow velocities, respectively.

The following may be noticed:

• A deformation in the symmetric shape of the velocity distribution takes place due to the asymmetricity of the two extensions. The maximum normalized cross-sectional velocity distribution is not in the middle (at y = 0) but is shifted toward the extension with less BER.

• The grey zone where most of the discharge passes is shifted toward the extension with smaller BER.

• The above-mentioned criteria are not so clear in Figure 10. This is due to the small difference between BER= 25 and 50%.

• The existence of the stagnant extensions added new boundaries to the flume cross-section forming low-velocity zones (LVZs) bounded by the extensions, the side boundaries, and the flume bed. In LVZ, the flow resembles that of a closed conduit with a maximum velocity value near the middle, but with a major difference that the iso-lines are not closed due to the absence of the fourth boundary.

• In the field, the consequences of the formation of LVZ are the accumulation of dirt and weeds, existence of bad smells due to unexpected non-desirable chemical reactions, and high probability of sedimentation.

• That is why the engineer should take measures to deal with water in the LVZ as part of his management for any engineering application where FCEB exists, especially if the extension is constructed for touristic and recreational purposes.

• • In Figures 13 and 14 representing the iso-lines of u/U of experiments 8 and 9, new LVZs are noticed to exist near the flow surface.

• • During conducting these two experiments, high turbulence was noticed, and the x-component of velocity (u) at the surface was nearly zero. The main velocity component was v (y-direction) in the form of severe cross-currents.

• • This is due to the integrated effects of the small depth of flow (a = 10), the roughness of the extensions, and the large discharge.

• • ADV data sampled in these experiments contain the three velocity components and the three turbulence components. In most of the experiments, the values of the y and z velocity components (v, w) and the turbulence components were too small to be considered.

• • This is not the case in experiments 8 and 9 where the values of v were too large to be ignored.

• • To shed more light on these bizarre results of experiments 8 and 9, Figures 16 and 17 were drawn. They present the iso-lines of the y-component of velocity normalized by average flow velocity (v/U) using color coding. The following can be shown:

• • o The dark blue color means that the values of v/U are as small as zero. In the yellow zones, the values of v/U are as high as 2.6 in experiment 8 and 0.9 in experiment 9.

  • o In most of the cross-section, values of v/U are nearly zero and the flow is one-dimensional.

  • o In the new LVZ near the surface, the values of u/U are nearly zero and the values of v/U are very high. This explains the severity of the cross-currents visualized during the experiment.

  • o That is the kinetic energies driving the flow in these new LVZs are consumed by these cross-currents and the y-component of velocity (v).

• • In Figures 12 and 15 of experiments (7 and 10), the iso-lines of maximum u/U are closed, and the maximum u/U exists at z/D ≈ 0.75 (not 1.0 as was expected).

• • This may be explained by the rather weak turbulence and cross-currents noticed at the flow surface with less power than those of experiments (8 and 9).

• • The main difference between the two cases is the smaller aspect ratio in experiments 7 and 10 (a = 6) that means larger flow depth (D = 200 mm). The relatively large depth of flow decreased the effects of the turbulence, bringing about relatively weak cross-currents on the flow surface in experiments 7 and 10. This denotes the effect of aspect ratio of the FCEB characteristics.

To show the effects of bed roughness, a comparison between the two cases of rough and smooth beds is carried out. In experiments 3–8, the depths of the bed boundary layers are very small due to the smoothness of the HDOP material. In experiments 9 and 10, the depths of the bed boundary layers are slightly larger due to the bed roughness.

To study the effects of the extension roughness, a comparison is carried out between the iso-lines of the ratio u/U in cases of experiments 3 and 7 through Figures 8 and 12:

• The boundary conditions in these two experiments are the same except that the extensions in experiment 3 are smooth and those in experiment 7 are rough.

• The depth of the small u/U zone under the 67% extension is much larger in experiment 7.

• In Figure 8, the maximum values of u/U are situated near the middle (z/D ≈ 0.5), while in Figure 12, the maximum values are shifted by the effects of the extension roughness downward to a section of z/D ≈ 0.375.

• The flow in the LVZ in experiment 7 no longer resembles the case of the pipeline with one side opened.

Here, more details will be demonstrated by analyzing two groups of relations:

The discharge distribution denotes the share of every part of the cross-section in passing the flow. In FCEB, the extension forms a new static boundary to the flow forming the LVZ with less flow velocity and less passing discharge. Consequently, more discharges should pass through the rest of the section (the uncovered part of the cross-section). This matches well with the studies of Shen & Ackermann (1980), and Mitchel (2015) for the case of the flow partially covered with ice layers.

Using the experimental data, the discharges in any part of the flow section may be calculated based on the discharge definition equation: ⁠, where A is the area perpendicular to the direction of the velocity vector and n is the number of readings.

The SURFER software was used to calculate the discharge passing through every zone of the cross-section. Table 3 presents the results of experiments 3–6, and Table 4 introduces the results of experiments 7–10.

For Table 3, the following can be shown:

• • Column (2):

• • o The first two rows give the total discharges in experiments 1 and 2 (OCF) that are used as references.

  • o The following four rows give the total discharge passing through the flume in the four different experiments (3–6).

• • Columns (3), (5), and (7) give the shares of the discharges passing in the first 25, 50, and 67% parts of the sections. In cases where the experiment did not have one of the three extensions, the corresponding cells are left blank.

• • Columns (4), (6), and (8) give the outcomes of dividing the contents of columns (3), (5), and (7) on the corresponding cells of column (2).

• • The contents of the last four rows give the percentage reductions between the case at any of the four experiments compared to its corresponding case in experiments 1 and 2 with the same value of the aspect ratio.

The following may be cleared in the first two rows:

• The total discharge (Q) is not equally distributed across the flow section because of the boundaries and boundary layers.

• The value in the cell under the 25% extension side gives the discharge passing in the first part of the flow area 150 mm wide. This equals 150/1,200 = 12.5% of the flow area close to the vertical side boundary. This area passes only 9.319 and 11.431% of the section full discharge for the cases of the aspect ratios of 10 and 6, respectively. These values are less than 12.5% of the total discharge as it would be expected.

• Going far from the boundary (closer to the centerline), the value in the cell under the 50% extension side gives the discharge passing in the first part of the flow area 300 mm wide. The table shows that 300/1,200 = 25% of the flow area passes only 21.191 and 23.975% for the cases of the aspect values of 10 and 6, respectively. Any of the two values in the two cases does not equal 25% of the total discharge as it was expected.

• Again, the value in the cell under the 67% extension side means the first 402/1,200 = 33.5% of the flow area. The table shows that 33.5% of the flow area passes 30.321 and 32.607% for the cases of the aspect values of 10 and 6, respectively. Any of the two values in the two cases does not equal 33.5% of the total discharge as it was expected.

• The percentage of the flow that passes through any part of the flow cross-section in case of a = 10 is less than the corresponding part in case of a = 6. This is since the increase in the flow depth decreases the effects of the boundaries on the flow.

• The reductions in the percentages of discharges in the four experimental cases due to the existence of the 25% extension, 50% extension, and 67% extension with respect to the reference case of OCF are shown in the last four rows of Table 3.

• In experiments 3, 4, and 6, the reductions in the flow under the 67% extension side are more than that under the 25% extension side (experiments 3 and 4) or that under the 50% extension side (experiment 6). That is the effect of the wide extension is much clearer.

• Again, the case of 25 and 50% extensions in experiment 5 does not lead to firm conclusion.

• The values of these reductions range between 0.378 and 6.055%.

Referring to Table 4 and bearing in mind what was cleared in Table 3, the following can be shown:

• The values of the reductions range between 0.423 and 12.697%.

• The reductions in the flow under the 67% extension side are more than that under the 25% extension side in the same experiment.

• This means that the cases of wider extensions are much worse from the practical point of view. The existence of wide LVZs beneath the wide floating application results in less velocities, less discharges, and less energy of flow.

• This gives more chance for the accumulation of dirt and weeds, existence of bad smells due to unexpected non-desirable chemical reactions, and more probabilities of sedimentation.

An experimental study was conducted to investigate the characteristics of FCEB. The study stressed the case of asymmetrical extensions. Three-boundary extension ratios were adopted as follows: 25, 50, and 67%. Within the limits of the parameters and conditions studied, the following are concluded:

• Due to asymmetricity in the extensions, the axis of maximum relative flow velocity component is shifted toward the extension of less BER.

• LVZs were formed bounded by the extensions, the side boundaries, and the flume bed where the flow resembles that of a closed conduit with maximum velocity value near the middle, but with a major difference that the iso-lines are not closed due to the absence of the fourth boundary.

• In experiments with small depth of flow, rough extensions, and large discharge, there exist high turbulence and severe cross-currents creating a new zone near the flow surface with small x-component of velocity and considerably large y-component.

• The roughness of the extension increases the depth of its boundary layer, pushing the maximum velocity component in the LVZ downward.

• The variation of u/u_max with z/D matches well the Prandtl power law distribution in cases of smooth bed and extensions.

• In cases of rough bed and extensions, the values of u/u_max lie outside the Prandtl power law envelope.

• The existence of LVZ under the extensions decreases the discharge passing through this zone. In field, the consequences of the formation of LVZ are the accumulation of dirt and weeds, existence of bad smells due to unexpected non-desirable chemical reactions, and more probabilities of sedimentation.

• The engineer should take engineering measures to clean the water in LVZ as part of his management for any engineering application where FCEB exists, especially if the extension is constructed for touristic and recreational purposes.

All relevant data are included in the paper or its Supplementary Information.

The authors declare there is no conflict.