# Estimation of Bed Shear Stress in Shallow Transitional Flows under Condition of Incipient Motion of Sand Particles Using Turbulence Characteristics

^{1}

^{2}

^{3}

^{*}

## Abstract

**:**

^{2}

_{0}, v′

^{2}

_{0}, w′

^{2}

_{0}and TKE

_{0}by 0.17, 0.33, 1.24 and 0.2, respectively. However, it was found that these values were slightly proportional to the shear Reynolds number. Additionally, the one-point measurement approach was assessed. The TKE method which applies all three components of Reynolds normal stresses was preferred to the u′

^{2}, v′

^{2}and w′

^{2}methods. Results showed that, u′

^{2}

_{0}, v′

^{2}

_{0}and w′

^{2}

_{0}have values of 60.5, 31.3 and 8.2 percent of the total, respectively.

## 1. Introduction

_{xz}= −ρu′w′, τ

_{xy}= −ρu′v′, and τ

_{yz}= −ρv′w′) and Reynolds normal stresses (σ

_{x}= −ρu′

^{2}, σ

_{y}= −ρv′

^{2}, and σ

_{z}= −ρw′

^{2}), where u′, v′, and w′ are the velocity fluctuations in the streamwise, spanwise, and vertical directions, respectively [5]. For simplification, the Reynolds stress is often divided by the mass density ρ with the dimension of [L

^{2}/T

^{2}]. These terms are called either the second moments or second-order correlations which are not never zero. They are used to describe characteristics of turbulent flow in the Cartesian coordinate system, which are investigated in this study to estimate bed shear stress, along with TKE.

^{2}+ v′w′

^{2})

^{0.5}, to estimate the bed shear stress [14,20,21,22]. This approach can be especially useful in the field study of rivers and seas, where correct aligning the ADV probe with the flow streamwise direction is prone to error. However, the necessity of applying this approach in laboratory conditions needs to be investigated, especially when the flow is uniform, the secondary current is negligible and the ADV probe is well-aligned with the flow streamwise direction.

^{2}, v′

^{2,}and w′

^{2}divided by 2 multiplied by the mass density of water. Some researchers claimed that TKE is proportional to the bed shear stress with a simple linear relationship which can be used to determine the bed shear stress [7,19]. Obviously, applying all three components of the u′

^{2}, v′

^{2,}and w′

^{2}in three directions [18], makes the TKE approach less sensitive to the misalignment of the ADV probe from the correct flow streamwise direction [19]. To estimate the bed shear stress, TKE should be multiplied by a specified coefficient of 0.19 [19,21,23,24,25,26,27], or 0.20 [28,29] or 0.21 [7], generally related to oceanography studies. Kim et al. [7] claimed that the TKE method is the most reliable approach, but more research work is required in the case of the coefficient that relates TKE to bed shear stress. This approach is widely accepted in oceanography studies, but surprisingly was rarely used in fluvial studies [2,26]. Biron et al. [2] successfully used this method with a coefficient of 0.19 in river studies. Pope et al. [23] claimed that the TKE approach could be used effectively in situations such as the presence of vegetation patches and bedforms in river beds where the application of some other well-known methods is almost impossible.

^{2}more accurately than u′

^{2}and claimed that u′

^{2}is more affected by noise errors. In this way, Kim et al. [7] tried to modify the TKE method and proposed that w′

^{2}

_{0}could be multiplied by a coefficient of 0.9 to estimate the bed shear stress. Results of experimental studies by Zhang et al. [26] showed that the w′

^{2}method for measuring bed shear stress in rivers could be more handful than the TKE method, but the value of the coefficient needs to be modified. However, it seems that this approach needs to be further studied for clarifying to what extent the estimation of the bed shear stress can only rely on w′

^{2}.

^{2}can be applied using the one-point measurement approach, especially in oceanography [2,7,23,31]. Biron et al. [2] and Pope et al. [23] satisfactorily applied the one-point measurement approach to estimate the bed shear stress by using both −u′w′ and TKE, respectively. Kim et al. [7] reported that the one-point measurement approach for determining the bed shear stress by using −u′w′ was a reliable method under a tidal condition. They recommended that the ADV sampling volume should be sufficiently close to the bed, located inside the constant shear stress layer. They also suggested that the ADV sampling volume should be at certain distance from the bed to avoid the influence of the bed materials on the sampling volume. Rashid [31] argued that the depth for the one-point measurement approach should be as close as possible to the bed, and thus recommended a depth of 4 mm above the sand bed for the one-point measurement method to estimate the bed shear stress using −u′w′, TKE, and w′

^{2}.

^{2}are well-known methods used by oceanographers but have not received enough attention in river studies by river engineers. It is necessary to examine the applicability of these approaches in river studies and determine the appropriate coefficients in order to estimate the bed shear stress, especially under the conditions mentioned above. Additionally, the possibility of using other Reynolds normal stresses including u′

^{2}and v′

^{2}to estimate the bed shear stress should be examined. The necessity of the approach of the vector addition of −u′w′ and −v′w′ for different conditions in the laboratory needs to be investigated. By analyzing and simplifying all vertical distribution of Reynolds shear and normal stresses as well as TKE, the main purpose of this study is to investigate all possible ways for estimating bed shear stress under the mentioned specific conditions and answer some questions such as: is it sufficient to estimate bed shear stress by multiplying the values of Reynolds normal shear stresses including u′

^{2}, v′

^{2,}and w′

^{2}as well as TKE at the bed by a specified constant coefficient? Are the determined coefficients dependent on the values of shear Reynolds number within the range of transitional flow conditions?

## 2. Materials and Methods

^{3}were used in this experimental study. The median grain size of all four groups is less than 2 mm in the range of sand grain size [34]. According to the well-known grain size classification [5], group “I” was classified as medium sand, group II as coarse sand, and both groups III and IV as very coarse sand. The geometric standard deviation of particle size distribution is between 1.16 to 1.45, indicating an acceptable uniform distribution of sediment particles. The grain size distribution curves of four sediment groups were presented in Figure 3a together with a picture of a sample of sediment used in this experiment (Figure 3b). Three flow depths H1, H2, and H3 equal to 100, 120, and 140 mm, respectively, were used for experiments using sediment groups “I”, “II”, and “III”. For sediment group “IV”, three flow depths of 91, 104, and 120 mm, respectively, were used. The bed of a flume section which is 4 m long was covered with a sand layer with a thickness of 3 cm. It should be noticed that this homogeneous sandy bed is different from the real situation in natural rivers, but this assumption is necessary for this study to evaluate the flow characteristics under the condition of incipient motion of different sizes of sediment particles. The flume section between this sand bed and flume entrance was 8 m long and can provide suitable conditions for a fully developed flow. To reduce the effect of the end slide-gate, the flume section with the sand bed was located 3 m upstream from the end slide-gate of the flume. The flume bed of both the upstream and downstream sections from the sand bed section was covered with coarse material that was unable to move during the experiments.

## 3. Results and Discussion

#### 3.1. Characteristics of Flow over the Sand Bed

#### 3.2. Bed Shear Stress Estimation Using the −u′w′ Method

- Increasing zone: a linear increase in −u′w′ from the water surface to the damping zone.
- Damping zone: a limited portion near the bed with relatively unchanged values of −u′w′.
- Decreasing zone: a linear decrease in −u′w′ from the damping zone to the bed.

_{0}was determined. This method is the most reliable and well-known method for estimating bed shear stress for uniform flows [5,8,9,10], and was applied as the main method for comparing results to those using other methods which considered in this study. Given the approximately constant values of −u′w′ in the damping zone, it is highly recommended to select the depth of one-point measurement within this zone. As clearly shown in Table 2, the ratio of −u′w′

_{0}to the values of −u′w′ in the damping zone −u′w′

_{d}are from 1.09 to 1.33 (with an average of 1.22). Therefore, it is recommended that the measured values using the one-point measurement method in the damping zone −u′w′

_{d}be multiplied by 1.22 to estimate the bed shear stress. This is in contrast to some reported studies [2] which assumed the value obtained by the one-point measurement method could be used to represent the bed shear stress. For confirmation, by multiplying the values in the damping zone by 1.22, the ratio of the results to −u′w′

_{0}was obtained and ranged from 0.92 to 1.12 (standard deviation of 0.07), which indicates a good agreement.

_{0}in Table 2 were 1.32 to 2.00 (with an average of 1.66) times larger than the values obtained by extending a line towards the bed in the decreasing zone −u′w′

_{b}. This finding revealed that this approach for estimating the bed shear stress leads to a considerable underestimation and unreliable results.

#### 3.3. Bed Shear Stress Estimation Using −u′v′ and −v′w′

_{0}were obtained and presented in Table 3. Obviously, it is not possible to create a relationship between −u′w′

_{0}and −u′v′

_{0}and estimate the bed shear stress.

_{0}were determined. As shown in Table 4, for the first nine experiments, the values of −v′w′

_{0}are very close to zero. However, the values for the last three experiments are −0.6, −0.4, and −0.3, respectively. The reason for such large absolute values of −v′w′

_{0}for these three experiments seem to be related to a probable slight misalignment of the ADV probe from the downstream direction. Similar to the −u′v′ distribution, there is no relationships between the values of −u′w′

_{0}and −v′w′

_{0}, indicating that it is not possible to estimate the bed shear stress using −v′w′

_{0}values.

#### 3.4. Bed Shear Stress Estimation Using the Vector Addition of −u′w′ and −v′w′

^{2}+ −v′w′

^{2})

^{0.5}and the vertical distribution of −u′w′ are nearly similar. As shown in Table 5, the values of vector addition of −u′w′ and −v′w′ at the bed [(−u′w′

^{2}+ −v′w′

^{2})

^{0.5}]

_{0}, obtained by extending a line in the increasing zone, are equal to −u′w′

_{0}values, implying that the estimation of the bed shear stress using −u′w′ method is not obviously affected by some not-considerable misalignments of the ADV probe in laboratory, where the experimental conditions are well controlled. However, an obvious misalignment of the ADV probe in a natural river happens very likely. Thus, the application of the vector addition of −u′w′ and −v′w′ can be more recommended.

#### 3.5. Bed Shear Stress Estimation Using u′^{2}

^{2}has the similar distribution pattern to that of −u′w′, including a linear increase from the zero value of u′

^{2}at the water surface to a damping zone and then a decrease towards the bed. Therefore, the vertical distribution of u′

^{2}similar to −u′w′ verticals were simplified into three zones, including the increasing, damping, and decreasing zones. In comparison, the height of the damping zone for the u′

^{2}distribution is less than that of −u′w′ verticals, and the decreasing zone is less recognizable. However, the decreasing zone is clearly visible in verticals for following eight experimental runs: I-H2, I-H3, II-H1, II-H2, II-H3, III-H1, III-H2 and III-H3.

^{2}at the bed u′

^{2}

_{0}were determined by extending a line in the increasing zone to the bed and was presented in Table 6. The ratio of −u′w′

_{0}to u′

^{2}

_{0}is between 0.15 and 0.20 (with an average of 0.17). In this case, by multiplying u′

^{2}

_{0}by 0.17, the ratio of the calculated results (0.17 u′

^{2}

_{0}) to −u′w′

_{0}was obtained between 0.84 and 1.13 (with a standard deviation of 0.09), which indicated that the values of u′

^{2}

_{0}can be used to predict the bed shear stress multiplying by 0.17. Those 12 experiments were divided into two groups, namely, group A includes experiments related to sediment groups I and II with a range of shear Reynolds number between 5 to 14, and the group B includes experiments related to sediment groups III and IV with a range of shear Reynolds number between 34 and 61. The threshold velocity for the incipient motion of bed materials for groups A and B are between 23.6 and 30.6, and between 36.9 and 45.7 cm/s, respectively. One can see from Table 6, the ratio of −u′w′

_{0}to u′

^{2}

_{0}for group A varies from 0.15 to 0.17 (with an average of 0.16), and for group B is between 0.16 and 0.20 (with an average of 0.18). By multiplying u′

^{2}

_{0}values of groups A by 0.16, and u′

^{2}

_{0}values of group B by 0.18, respectively, the ratio of the calculated results (0.16u′

^{2}

_{0}for group A, and 0.18u′

^{2}

_{0}for group B) to −u′w′

_{0}for group A was between 0.97 and 1.07 (with a standard deviation of 0.04), and for group B was between 0.89 and 1.10 (with a standard deviation of 0.07). These values indicate that for a flow under a transitional flow condition, the relationship between u′

^{2}

_{0}and −u′w′

_{0}can be slightly affected by the shear Reynolds number. However, it is feasible for estimating the bed shear stress for the entire range of shear Reynolds number by multiplying the value of u′

^{2}

_{0}by 0.17.

^{2}

_{0}to the values of in the damping zone u′

^{2}

_{d}is between 1.08 and 1.17 (with an average of 1.12). By multiplying u′

^{2}

_{d}values by 1.12, the ratio of the calculated results (1.12u′

^{2}

_{d}) to −u′

^{2}

_{0}was obtained between 0.96 and 1.04 (a standard deviation of 0.07), which indicates that the value of u′

^{2}

_{0}could be effectively estimated with the one-point measurement method in the appropriate depth of damping zone (with an average of z/h = 0.08).

#### 3.6. Bed Shear Stress Estimation Using v′^{2}

^{2}(note: v′ is the velocity fluctuations in the spanwise direction), which may be related to some unexpected doppler noises. Despite this problem, the values of v′

^{2}decrease linearly towards the bed without any detectable turning point or decreasing zone. By extending a line from zero value at the water surface towards the bed, the values of v′

^{2}at the bed (v′

^{2}

_{0}) were determined, as shown in Table 7. The ratio of −u′w′

_{0}to v′

^{2}

_{0}is between 0.28 and 0.36 (with an average of 0.33). We multiplied, the values of v′

^{2}

_{0}by 0.33, and the ratio of the results (0.33v′

^{2}

_{0}) to −u′w′

_{0}was obtained with a range from 0.92 to 1.19 (with a standard deviation of 0.09). This result indicates that it is possible to estimate the bed shear stress by means of the values of v′

^{2}

_{0}. Results showed that the coefficient for group A is between 0.28 and 0.34 (with an average of 0.30), and for group B is between 0.34 and 0.36 (with an average of 0.35). For sand group A, the v′

^{2}

_{0}values are multiplied by 0.30, and for sand group B, the v′

^{2}

_{0}values are multiplied by 0.35. Then, the ratio of results (0.30v′

^{2}

_{0}) to −u′w′

_{0}for group A was between 0.89 and 1.08 (with a standard deviation of 0.06), and the ratio of results (0.35v′

^{2}

_{0}) to −u′w′

_{0}for group B was between 0.97 and 1.02 (with a standard deviation of 0.02). These values indicate that the range of the shear Reynolds number could somehow influence the relationship between v′

^{2}

_{0}and −u′w′

_{0}. In the case of the estimation of v′

^{2}

_{0}using the one-point measurement method, due to the presence of scattered points in the verticals, there is a risk that the measured single data may be invalid. Anyway, given the linear distribution from the zero value of v′

^{2}at the water surface to the bed, the value of v′

^{2}

_{0}could be estimated to be equal to the one-point measured value divided by the difference between one and the relative depth of the point.

#### 3.7. Estimation of Bed Shear Stress Using w′^{2}

^{2}for all experiments are shown in Figure 12 (note: w′ is the velocity fluctuations in the vertical direction). Obviously, the values of w′

^{2}in the main flow are almost constant and starts to decrease at a specific depth near the bed without any recognizable damping zone that is comparable to the finding of Grass [38]. Grass showed that the values of w′

^{2}in the main flow are almost constant above a depth of about z/h = 0.1 near the bed, albeit with a slightly increase, and then decrease to the bed for all smooth, transitional and rough flows. However, in this study, the vertical distribution of w′

^{2}was simplified into two distinct zones, as below:

- Unchanged zone: unchanged w′
^{2}values from the water surface to a specified depth near the bed where w′^{2}starts to decrease. - Decreasing zone: a linear decrease in w′
^{2}values near the bed towards the bed.

^{2}at the bed w′

^{2}

_{0}were determined. According to Table 8, the ratio of −u′w′

_{0}to w′

^{2}

_{0}is varied from 1.06 to 1.43 (with an average of 1.24). This average ratio differs significantly from the ratio of 0.9 proposed by Kim et al. [7] for the tidal deep-water condition. This result of the present study confirms the recommendation of Zhang et al. [26] regarding the modification the coefficient proposed by Kim et al. [7], especially in river studies. The values of w′

^{2}

_{0}were multiplied by 1.24 and the ratios of 1.24w′

^{2}

_{0}to −u′w′

_{0}were obtained and ranged from 0.87 to 1.17 (with a standard deviation of 0.10), which indicates that the bed shear stress can be estimated using the values of w′

^{2}

_{0}. According to Table 8, the coefficient for group A is between 1.06 and 1.16 (with an average of 1.13), and for group B is between 1.28 and 1.43 (with an average of 1.36). For sand group A, the w′

^{2}

_{0}values are multiplied by 1.13, and for sand group B, the w′

^{2}values are multiplied by 1.36. Then, the ratio of results (1.13 w′

^{2}) to −u′w′

_{0}for group A was between 0.97 and 1.07 (with a standard deviation of 0.04), and the ratio of results (1.36 w′

^{2}) to −u′w′

_{0}for group B was between 0.95 and 1.06 (with a standard deviation of 0.04). This finding indicates that the coefficient is proportional to the shear Reynolds number. These values show that although one unique coefficient of 1.24 applied for the entire range of the shear Reynolds number was sufficient, one can obtain better results (or closer values to −u′w′

_{0}) by applying two different coefficients for groups A and B, respectively.

^{2}

_{0}using the one-point measurement method, the measuring depth should be selected at the higher depth of the decreasing zone. In this case, it is important to determine the depth of the boundary between the unchanged and decreasing zones. According to Table 8, this depth is between z/h = 0.08 to 0.14 (with an average of 0.11), which is in good agreement with the result of Grass [38]. Therefore, the depth of the one-point measurement method should be averagely higher than z/h = 0.11.

#### 3.8. Estimation of Bed Shear Stress Using TKE

^{2}, v′

^{2}and w′

^{2}divided by 2, the pattern of the vertical distribution of TKE be developed based on all velocity fluctuation components of turbulent flow. As shown in Figure 13, the verticals of TKE are affected by the scattering of data observed in the vertical distribution of v′

^{2}related to the particle size of bed material, namely, sediment groups I and II. Given that the u′

^{2}possesses the largest portion of TKE values compared to v′

^{2}and w′

^{2}, the vertical distribution of TKE is expected to have the similar pattern to that of u′

^{2}. However, due to the effects of v′

^{2}and w′

^{2}, an obvious damping zone is not observed, and only a small decreasing zone is located near the bed. Thus, the TKE verticals include a decreasing and increasing zones with a turning point near the bed.

_{0}) were determined. As shown in Table 9, in accordance with Soulsby [28,29], the ratio of −u′w′

_{0}to TKE

_{0}is varied from 0.18 to 0.23 (with an average of 0.20). The ratio of 0.2TKE

_{0}to −u′w′

_{0}was obtained, which varied from 0.85 to 1.11 (with a standard deviation of 0.08), indicating that TKE

_{0}can be effectively used to estimate the bed shear stress. Additionally, since the TKE method applies all three components of Reynolds normal stresses in three directions [18], it is less sensitive to the probable misalignment of the ADV probe [19], it is more preferred compared to the u′

^{2}, v′

^{2}and w′

^{2}approach. According to Table 9, the ratio of −u′w′

_{0}to TKE

_{0}for sand group A is between 0.18 and 0.20 (with an average of 0.19) which is consistent with results of other researchers [19,21], and the ratio of −u′w′

_{0}to TKE

_{0}for sand group B is between 0.20 and 0.23 (with an average of 0.22) which is comparable with 0.21 proposed by Kim et al. [7]. For sand group A, the TKE values are multiplied by 0.19, and for sand group B, the TKE values are multiplied by 0.22, respectively. The ratio of results (0.19TKE) to −u′w′

_{0}for group A was calculated between 0.93 and 1.05 (with a standard deviation of 0.04), and the ratio of results (0.22TKE) to −u′w′

_{0}for group B was calculated between 0.94 and 1.07 (with a standard deviation of 0.05), indicating an improvement in the results. However, applying the unique coefficient of 0.2 is acceptable and can be recommended. According to Table 9, the depth of turning point between the increasing zone and the decreasing zone is varied from z/h = 0.04 to z/h = 0.06 (with an average of 0.05). For the one-point measurement approach, considering the risk of the influence from the invalid data points, the value of TKE

_{0}could be estimated to equal to the one-point measured value divided by the difference between one and the relative depth of the point.

^{2}

_{0}/TKE

_{0}, v′

^{2}

_{0}/TKE

_{0}, and w′

^{2}

_{0}/TKE

_{0}are varied from 1.16 to 1.25 (with an average of 1.21), from 0.60 to 0.66 (with an average of 0.63) and from 0.15 to 0.18 (with an average of 0.16), respectively. These values indicate that the variation of the ratios in all experiments is considerably small. Additionally, the values of u′

^{2}

_{0}, v′

^{2}

_{0}, and w′

^{2}

_{0}are about 60.5%, 31.3%, and 8.2% of the values of (u′

^{2}

_{0}+ v′

^{2}

_{0}+ w′

^{2}

_{0}), respectively. In this case, by knowing one of the values of u′

^{2}

_{0}, v′

^{2}

_{0,}w′

^{2}

_{0}, or TKE

_{0}, the other terms can be estimated with acceptable accuracy.

## 4. Conclusions

_{0}was determined, which was considered as the bed shear stress and used to evaluate results using other methods. Results showed that, under such a laboratory condition with uniform flow and well-aligned ADV probe, the vector addition of the −u′w′ and −v′w′ is not necessary. The vertical distribution of u′

^{2}has the same distribution profile as that of −u′w′, and can be simplified into three zones, but with a less height of the damping zone, and less recognizable the decreasing zone. The values of v′

^{2}decreased linearly towards the bed without any detectable turning points. With respect of w′

^{2}, the values in the main flow were almost constant and started to decrease at the depth of z/h = 0.11 without any obvious damping zone. Along TKE verticals, a dominated increasing zone and a small decreasing zone were observed with a turning point near the bed at a depth of z/h = 0.05. The bed shear stress can be effectively estimated by multiplying the values of u′

^{2}

_{0}, v′

^{2}

_{0}, w′

^{2}

_{0}, and TKE

_{0}by 0.17, 0.33, 1.24, and 0.20, respectively. The estimated coefficient for TKE is in agreement with those in the literature. Since the TKE method applies all three components of Reynolds normal stresses in three directions, the TKE method seems to be preferred to comparing to the u′

^{2}, v′

^{2}, and w′

^{2}methods. By classifying the laboratory experiments into two groups of A and B with a range of shear Reynolds number, respectively, from 5 to 14 and from 34 to 61, the bed shear stress for group A was estimated by multiplying the values of u′

^{2}

_{0}, v′

^{2}

_{0}, w′

^{2}

_{0}, and TKE

_{0}by 0.16, 0.30, 1.13, and 0.19, respectively, and for group B 0.18, 0.35, 1.36 and 0.22, respectively. This means under a transitional flow condition, to estimate −u′w′

_{0}, the coefficients for multiplying by u′

^{2}

_{0}, v′

^{2}

_{0}, w′

^{2}

_{0}, and TKE

_{0}are slightly increased from the hydraulically smooth to hydraulically rough flow conditions. For the one-point measurement approach, the middle point of the damping zone (at a depth of z/h = 0.13 for −u′w′ and z/h = 0.08 for u′

^{2}) was recommended to be used for estimating −u′w′

_{0}and u′

^{2}

_{0}by multiplying the related values by 1.22 for −u′w′ and 1.12 for u′

^{2}, respectively. Given the linear distribution of v′

^{2}and TKE from the zero value at the water surface to the bed, the value at the bed could be estimated which is equal to the one-point measured value divided by the difference between one and its relative depth at that point, considering z/h > 0.05 for TKE. In the case of w′

^{2}, it is recommended the depth for estimating w′

^{2}

_{0}using the one-point measurement approach should be selected in the unchanged zone with the depth of z/h > 0.11. It was estimated that the values of u′

^{2}

_{0}, v′

^{2}

_{0}, and w′

^{2}

_{0}are, respectively, about 60.5%, 31.3%, and 8.2% of their summation. The results of this study could be applied in such conditions similar to the current experiments. However, further studies should be carried out for other situations such as different particle sizes of bed material, bed load conditions, different water depths, and flow regimes.

## Author Contributions

## Funding

## Data Availability Statement

## Conflicts of Interest

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**Figure 1.**(

**a**) A schematic view of the flume used for experiments and related equipment, and (

**b**) a photographic view of the flume.

**Figure 2.**Some experimental equipment used for this experimental study: (

**a**) electromotor pump; (

**b**) slide-gate at the end of the flume; (

**c**) variable frequency drive; (

**d**) electromagnetic flowmeter.

**Figure 3.**(

**a**) Grain size distribution curves of four sediment groups (from the left to the right, groups I, II, III, and IV, respectively), (

**b**) a picture of a sample of the natural quartz sand used as sediment in this study.

**Figure 5.**Flow velocity verticals at three successive locations within the sand bed for verification of fully development flows along the sand bed reach. Note: “III-H2-P3” implies a velocity vertical over sediment group of “III” and the water depth of “H2” at the location of “P3”.

**Figure 6.**Vertical distribution of −u′w′ which divided into the increasing, damping, and decreasing zones.

**Figure 7.**Vertical distribution of −u′v′ and the extended line from the zero value at the water surface towards the bed.

**Figure 8.**Vertical distribution of −v′w′ and the extended line from the zero value at the water surface towards the bed.

**Figure 9.**Vertical distribution of the vector addition of −u′w′ and −v′w′ that is the same as −u′w′ verticals.

**Figure 10.**Vertical distribution of u′

^{2}which divided into the increasing, damping, and decreasing zones.

Experiment Name: | I-H1 | I-H2 | I-H3 | II-H1 | II-H2 | II-H3 | III-H1 | III-H2 | III-H3 | IV-H1 | IV-H2 | IV-H3 |
---|---|---|---|---|---|---|---|---|---|---|---|---|

d—mm ^{1} | 0.43 | 0.43 | 0.43 | 0.83 | 0.83 | 0.83 | 1.38 | 1.38 | 1.38 | 1.94 | 1.94 | 1.94 |

h—mm ^{2} | 100 | 120 | 140 | 100 | 120 | 140 | 100 | 120 | 140 | 91 | 104 | 120 |

B/h ^{3} | 9.0 | 7.5 | 6.4 | 9.0 | 7.5 | 6.4 | 9.0 | 7.5 | 6.4 | 9.9 | 8.7 | 7.5 |

d/h ^{4} | 0.004 | 0.004 | 0.003 | 0.008 | 0.007 | 0.006 | 0.014 | 0.012 | 0.010 | 0.021 | 0.019 | 0.016 |

T—°C ^{5} | 17 | 15 | 19 | 18 | 20 | 19 | 21 | 22 | 17 | 24 | 24 | 21 |

ν—cm^{2}/s ^{6} | 0.011 | 0.011 | 0.010 | 0.011 | 0.010 | 0.010 | 0.010 | 0.010 | 0.011 | 0.009 | 0.009 | 0.010 |

Q—lit/s ^{7} | 21.25 | 27.31 | 33.42 | 27.10 | 33.03 | 38.50 | 33.20 | 41.60 | 49.28 | 35.70 | 41.60 | 49.35 |

U = Q/(Bh)—cm/s ^{8} | 23.6 | 25.3 | 26.5 | 30.1 | 30.6 | 30.6 | 36.9 | 38.5 | 39.1 | 43.6 | 44.4 | 45.7 |

Re = 4 Uh/ν ^{9} | 87,390 | 106,472 | 144,645 | 114,369 | 146,507 | 166,632 | 150,801 | 193,334 | 202,663 | 173,202 | 201,826 | 224,157 |

Fr = U/(gh)^{0.5}, ^{10} | 0.24 | 0.23 | 0.23 | 0.30 | 0.28 | 0.26 | 0.37 | 0.36 | 0.33 | 0.46 | 0.44 | 0.42 |

Re* = u*Ks/ν ^{11} | 5 | 5 | 6 | 13 | 14 | 14 | 34 | 37 | 34 | 61 | 61 | 59 |

Zv- mm ^{12} | 0.4 | 0.4 | 0.4 | 0.3 | 0.3 | 0.3 | 0.2 | 0.2 | 0.2 | 0.2 | 0.2 | 0.2 |

Zb- mm ^{13} | 2.6 | 2.5 | 2.1 | 1.9 | 1.8 | 1.8 | 1.2 | 1.1 | 1.2 | 1.0 | 0.9 | 1.0 |

nvb ^{14} | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |

nw ^{15} | 12 | 13 | 14 | 12 | 14 | 14 | 12 | 12 | 14 | 10 | 11 | 12 |

ni ^{16} | 8 | 9 | 9 | 8 | 9 | 9 | 8 | 9 | 9 | 8 | 8 | 9 |

nf ^{17} | 0 | 0 | 1 | 0 | 0 | 1 | 0 | 0 | 1 | 0 | 0 | 0 |

^{1}d: Median size of sediment particles;

^{2}h: water depth;

^{3}B/h: aspect ratio; B: flume width;

^{4}d/h: relative roughness;

^{5}T: water temperature;

^{6}ν: kinematic viscosity;

^{7}Q: discharge;

^{8}U: velocity;

^{9}Re: Reynolds number;

^{10}Fr: Froude number;

^{11}Re*: shear Reynolds number; u*: shear velocity obtained by the −u′w′ method; Ks: Nikuradse’s equivalent roughness assumed equal to median size of sediment particles;

^{12}Zv: upper limitation of the viscous sublayer depth (z = 5 ν/u*);

^{13}Zb: upper limitation of the transition or buffer layer depth (z = 30 ν/u*);

^{14}nvb: number of ADV data points in both viscous sublayer and buffer layer (z < 30 ν/u*);

^{15}nw: number of ADV data points in the turbulent wall shear layer or logarithmic layer (30 ν/u* ≤ z < 0.2 h);

^{16}ni: number of ADV data points in the intermediate layer (0.2 h ≤ z ≤ 0.6 h);

^{17}nf: number of ADV data points in the free surface layer (0.6 h < z ≤ h).

Experiment Name: | I-H1 | I-H2 | I-H3 | II-H1 | II-H2 | II-H3 | III-H1 | III-H2 | III-H3 | IV-H1 | IV-H2 | IV-H3 |
---|---|---|---|---|---|---|---|---|---|---|---|---|

−u′w′_{0} | 1.5 | 1.8 | 2.2 | 2.9 | 2.9 | 3.0 | 5.7 | 6.4 | 6.9 | 8.2 | 8.4 | 9.0 |

−u′w′_{d} | 1.2 | 1.4 | 1.7 | 2.5 | 2.5 | 2.6 | 4.4 | 4.8 | 5.2 | 7.5 | 7.5 | 7.6 |

−u′w′_{0}/−u′w′_{d} | 1.25 | 1.29 | 1.29 | 1.16 | 1.16 | 1.15 | 1.30 | 1.33 | 1.33 | 1.09 | 1.12 | 1.18 |

α(−u′w′_{d}) | 1.5 | 1.7 | 2.1 | 3.1 | 3.1 | 3.2 | 5.4 | 5.9 | 6.3 | 9.2 | 9.2 | 9.3 |

α(−u′w′_{d})/−u′w′_{0} | 0.98 | 0.95 | 0.94 | 1.05 | 1.05 | 1.06 | 0.94 | 0.92 | 0.92 | 1.12 | 1.09 | 1.03 |

Z1/h | 0.07 | 0.08 | 0.07 | 0.07 | 0.08 | 0.07 | 0.08 | 0.08 | 0.07 | 0.06 | 0.06 | 0.07 |

Z2/h | 0.21 | 0.22 | 0.22 | 0.14 | 0.16 | 0.14 | 0.23 | 0.25 | 0.25 | 0.10 | 0.11 | 0.16 |

avg(Z1/h, Z2/h) | 0.14 | 0.15 | 0.15 | 0.11 | 0.12 | 0.11 | 0.15 | 0.17 | 0.16 | 0.08 | 0.09 | 0.11 |

Z1, mm | 7.0 | 9.6 | 9.8 | 7.0 | 9.6 | 9.8 | 8.0 | 9.6 | 9.8 | 5.5 | 6.2 | 8.4 |

Z2, mm | 21.0 | 26.4 | 30.8 | 14.0 | 19.2 | 19.6 | 22.8 | 30.0 | 34.5 | 9.1 | 11.4 | 18.7 |

avg(Z1, Z2), mm | 14.0 | 18.0 | 20.3 | 10.5 | 14.4 | 14.7 | 15.4 | 19.8 | 22.1 | 7.3 | 8.8 | 13.5 |

−u′w′_{b} | 0.9 | 1.0 | 1.1 | 1.8 | 1.8 | 1.8 | 3.2 | 3.6 | 3.7 | 6.2 | 6.2 | 6.2 |

−u′w′_{0}/−u′w′_{b} | 1.67 | 1.80 | 2.00 | 1.61 | 1.61 | 1.67 | 1.78 | 1.78 | 1.86 | 1.32 | 1.35 | 1.45 |

_{0}, −u′w′

_{d}and −u′w′

_{b}indicate the values at the bed by extending a linear line in the increasing, damping and decreasing zones, respectively, α is the average of −u′w′

_{0}/−u′w′

_{d}equal to 1.22, Z1 and Z2 are the lower and upper limitations of the damping zone, respectively.

Experiment Name: | I-H1 | I-H2 | I-H3 | II-H1 | II-H2 | II-H3 | III-H1 | III-H2 | III-H3 | IV-H1 | IV-H2 | IV-H3 |
---|---|---|---|---|---|---|---|---|---|---|---|---|

−u′v′_{0} (cm/s)^{2} | 0.5 | 0.2 | 0.1 | 0.9 | 0.5 | 0.2 | 1.2 | 0.7 | 0.4 | 0.3 | −0.3 | 0.4 |

Experiment Name: | I-H1 | I-H2 | I-H3 | II-H1 | II-H2 | II-H3 | III-H1 | III-H2 | III-H3 | IV-H1 | IV-H2 | IV-H3 |
---|---|---|---|---|---|---|---|---|---|---|---|---|

−v′w′_{0}, (cm/s)^{2} | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.1 | 0.0 | −0.6 | −0.4 | −0.3 |

Experiment Name: | I-H1 | I-H2 | I-H3 | II-H1 | II-H2 | II-H3 | III-H1 | III-H2 | III-H3 | IV-H1 | IV-H2 | IV-H3 |
---|---|---|---|---|---|---|---|---|---|---|---|---|

[(−u′w′^{2} + −v′w′^{2})^{0.5}]_{0} | 1.5 | 1.8 | 2.2 | 2.9 | 2.9 | 3.0 | 5.7 | 6.4 | 6.9 | 8.2 | 8.4 | 9.0 |

$\frac{-{\mathrm{u}\prime \mathrm{w}\prime}_{0}}{{\left[{\left(-{\mathrm{u}\prime \mathrm{w}\prime}^{2}+-{\mathrm{v}\prime \mathrm{w}\prime}^{2}\right)}^{0.5}\right]}_{0}}$ | 1.00 | 1.00 | 1.00 | 1.00 | 1.00 | 1.00 | 1.00 | 1.00 | 1.00 | 1.00 | 1.00 | 1.00 |

^{2}+ −v′w′

^{2})

^{0.5}]

_{0}indicate the vector addition of −u′w′ and −v′w′ at the bed.

Experiment Name: | I-H1 | I-H2 | I-H3 | II-H1 | II-H2 | II-H3 | III-H1 | III-H2 | III-H3 | IV-H1 | IV-H2 | IV-H3 |
---|---|---|---|---|---|---|---|---|---|---|---|---|

u′^{2}_{0} (cm/s)^{2} | 10.0 | 12.0 | 13.3 | 18.0 | 18.0 | 19.2 | 31.4 | 33.6 | 34.2 | 49.0 | 51.4 | 51.7 |

−u′w′_{0}/u′^{2}_{0} | 0.15 | 0.15 | 0.17 | 0.16 | 0.16 | 0.16 | 0.18 | 0.19 | 0.20 | 0.17 | 0.16 | 0.17 |

β′u′^{2}_{0} (cm/s)^{2} | 1.7 | 2.0 | 2.3 | 3.1 | 3.1 | 3.3 | 5.3 | 5.7 | 5.8 | 8.3 | 8.7 | 8.8 |

β′u′^{2}_{0}/−u′w′_{0} | 1.13 | 1.13 | 1.03 | 1.06 | 1.06 | 1.09 | 0.94 | 0.89 | 0.84 | 1.02 | 1.04 | 0.98 |

Β″u′^{2}_{0} (cm/s)^{2} | 1.6 | 1.9 | 2.1 | 2.9 | 2.9 | 3.1 | - | - | - | - | - | - |

β″u′^{2}_{0}/−u′w′_{0} | 1.07 | 1.07 | 0.97 | 0.99 | 0.99 | 1.02 | - | - | - | - | - | - |

Β‴u′^{2}_{0} (cm/s)^{2} | - | - | - | - | - | - | 5.7 | 6.0 | 6.2 | 8.8 | 9.3 | 9.3 |

β‴u′^{2}_{0}/−u′w′_{0} | - | - | - | - | - | - | 0.99 | 0.95 | 0.89 | 1.08 | 1.10 | 1.03 |

Z1/h | 0.05 | 0.05 | 0.06 | 0.05 | 0.05 | 0.05 | 0.06 | 0.04 | 0.04 | 0.04 | 0.05 | 0.04 |

Z2/h | 0.13 | 0.14 | 0.13 | 0.10 | 0.09 | 0.10 | 0.13 | 0.08 | 0.09 | 0.08 | 0.09 | 0.08 |

avg(Z1/h, Z2/h) | 0.09 | 0.10 | 0.10 | 0.08 | 0.07 | 0.08 | 0.10 | 0.06 | 0.07 | 0.06 | 0.07 | 0.06 |

Z1, mm | 5.0 | 6.0 | 8.4 | 5.0 | 6.0 | 7.0 | 6.0 | 4.8 | 5.6 | 3.6 | 5.2 | 4.8 |

Z2, mm | 13.0 | 16.8 | 18.2 | 10.0 | 10.8 | 14.0 | 13.0 | 9.6 | 12.6 | 7.3 | 9.4 | 9.6 |

avg(Z1, Z2), mm | 9.0 | 11.4 | 13.3 | 7.5 | 8.4 | 10.5 | 9.5 | 7.2 | 9.1 | 5.5 | 7.3 | 7.2 |

u′^{2}_{d} (cm/s)^{2} | 8.6 | 10.3 | 11.6 | 16.3 | 16.3 | 17.3 | 27.5 | 31.0 | 31.1 | 45.3 | 47.1 | 47.3 |

−u′^{2}_{0}/u′^{2}_{d} (cm/s)^{2} | 1.16 | 1.17 | 1.15 | 1.10 | 1.10 | 1.11 | 1.14 | 1.08 | 1.10 | 1.08 | 1.09 | 1.09 |

γu′^{2}_{d} (cm/s)^{2} | 9.6 | 11.5 | 13.0 | 18.3 | 18.3 | 19.4 | 30.8 | 34.7 | 34.8 | 50.7 | 52.8 | 53.0 |

γu′^{2}_{d}/−u′^{2}_{0} | 0.96 | 0.96 | 0.98 | 1.01 | 1.01 | 1.01 | 0.98 | 1.03 | 1.02 | 1.04 | 1.03 | 1.02 |

^{2}

_{0}and u′

^{2}

_{d}indicate the values obtained by extending a line in the increasing and damping zones, respectively, β′, β″, and β‴ are the average of −u′w′

_{0}/u′

^{2}

_{0}for all experiments, experiments group A and experiments group B equal to 0.17, 0.16 and 0.18, respectively. Z1 and Z2 are lower and upper limitation depth of the damping zone, respectively. γ is the average of −u′

^{2}

_{0}/u′

^{2}

_{d}equal to 1.12.

Experiment Name: | I-H1 | I-H2 | I-H3 | II-H1 | II-H2 | II-H3 | III-H1 | III-H2 | III-H3 | IV-H1 | IV-H2 | IV-H3 |
---|---|---|---|---|---|---|---|---|---|---|---|---|

v′^{2}_{0} (cm/s)^{2} | 5.4 | 6.2 | 6.5 | 9.8 | 9.8 | 9.8 | 16.4 | 17.8 | 19.5 | 23.7 | 24.4 | 25.3 |

−u′w′_{0}/v′^{2}_{0} | 0.28 | 0.29 | 0.34 | 0.30 | 0.30 | 0.31 | 0.35 | 0.36 | 0.35 | 0.35 | 0.34 | 0.36 |

δ′v′^{2}_{0} (cm/s)^{2} | 1.8 | 2.0 | 2.1 | 3.2 | 3.2 | 3.2 | 5.4 | 5.9 | 6.4 | 7.8 | 8.1 | 8.3 |

δ′v′^{2}_{0}/−u′w′_{0} | 1.19 | 1.14 | 0.98 | 1.12 | 1.12 | 1.08 | 0.95 | 0.92 | 0.93 | 0.95 | 0.96 | 0.93 |

δ″v′^{2}_{0} (cm/s)^{2} | 1.6 | 1.9 | 2.0 | 2.9 | 2.9 | 2.9 | - | - | - | - | - | - |

δ″v′^{2}_{0}/−u′w′_{0} | 1.08 | 1.03 | 0.89 | 1.01 | 1.01 | 0.98 | - | - | - | - | - | - |

δ‴v′^{2}_{0} (cm/s)^{2} | - | - | - | - | - | - | 5.7 | 6.2 | 6.8 | 8.3 | 8.5 | 8.9 |

δ‴v′^{2}_{0}/−u′w′_{0} | - | - | - | - | - | - | 1.01 | 0.97 | 0.99 | 1.01 | 1.02 | 0.98 |

^{2}

_{0}represents the value obtained by extending a line from the zero value at the water surface towards the bed, δ′, δ″ and δ‴ are the average of −u′w′

_{0}/v′

^{2}

_{0}for all experiments, experiments group A and experiments group B equal to 0.33, 0.30 and 0.35, respectively.

Experiment Name: | I-H1 | I-H2 | I-H3 | II-H1 | II-H2 | II-H3 | III-H1 | III-H2 | III-H3 | IV-H1 | IV-H2 | IV-H3 |
---|---|---|---|---|---|---|---|---|---|---|---|---|

w′^{2}_{0} (cm/s)^{2} | 1.3 | 1.7 | 1.9 | 2.5 | 2.5 | 2.8 | 4.0 | 5.0 | 5.1 | 5.8 | 6.2 | 6.7 |

−u′w′_{0}/w′^{2}_{0} | 1.15 | 1.06 | 1.16 | 1.16 | 1.16 | 1.07 | 1.43 | 1.28 | 1.35 | 1.41 | 1.35 | 1.34 |

ζ′w′^{2}_{0} (cm/s)^{2} | 1.6 | 2.1 | 2.4 | 3.1 | 3.1 | 3.5 | 5.0 | 6.2 | 6.3 | 7.2 | 7.7 | 8.3 |

ζ′w′^{2}_{0}/−u′w′_{0} | 1.07 | 1.17 | 1.07 | 1.07 | 1.07 | 1.16 | 0.87 | 0.97 | 0.92 | 0.88 | 0.92 | 0.92 |

ζ″w′^{2}_{0} (cm/s)^{2} | 1.5 | 1.9 | 2.1 | 2.8 | 2.8 | 3.2 | - | - | - | - | - | - |

ζ″w′^{2}_{0}/−u′w′_{0} | 0.98 | 1.07 | 0.98 | 0.97 | 0.97 | 1.05 | - | - | - | - | - | - |

ζ‴w′^{2}_{0} (cm/s)^{2} | - | - | - | - | - | - | 5.4 | 6.8 | 6.9 | 7.9 | 8.4 | 9.1 |

ζ‴w′^{2}_{0}/−u′w′_{0} | - | - | - | - | - | - | 0.95 | 1.06 | 1.01 | 0.96 | 1.00 | 1.01 |

Zt/h | 0.10 | 0.10 | 0.10 | 0.10 | 0.10 | 0.08 | 0.11 | 0.13 | 0.10 | 0.10 | 0.12 | 0.14 |

Zt-mm | 10.0 | 12.0 | 14.0 | 10.0 | 12.0 | 11.2 | 11.0 | 15.6 | 14.0 | 9.1 | 12.5 | 16.8 |

^{2}

_{0}represents the value obtained by extending a vertical line in the unchanged zone towards the bed, ζ′, ζ″ and ζ‴ are the average of −u′w′

_{0}/w′

^{2}

_{0}for all experiments, experiments group A and experiments group B equal to 1.24, 1.13 and 1.36, respectively, Zt is the depth where w′

^{2}begins to decrease (the depth of the boundary between the unchanged and decreasing zones).

Experiment Name: | I-H1 | I-H2 | I-H3 | II-H1 | II-H2 | II-H3 | III-H1 | III-H2 | III-H3 | IV-H1 | IV-H2 | IV-H3 |
---|---|---|---|---|---|---|---|---|---|---|---|---|

TKE_{0} (cm/s)^{2} | 8.3 | 9.9 | 10.8 | 15.2 | 15.2 | 15.9 | 25.9 | 28.2 | 29.4 | 39.3 | 41.0 | 41.9 |

−u′w′_{0}/TKE_{0} | 0.18 | 0.18 | 0.19 | 0.19 | 0.19 | 0.19 | 0.22 | 0.23 | 0.23 | 0.21 | 0.20 | 0.21 |

η′TKE_{0} (cm/s)^{2} | 1.7 | 2.0 | 2.2 | 3.0 | 3.0 | 3.2 | 5.2 | 5.6 | 5.9 | 7.9 | 8.2 | 8.4 |

η′TKE_{0}/−u′w′_{0} | 1.11 | 1.10 | 1.03 | 1.05 | 1.05 | 1.06 | 0.91 | 0.88 | 0.85 | 0.96 | 0.98 | 0.93 |

η″TKE_{0} (cm/s)^{2} | 1.6 | 1.9 | 2.1 | 2.9 | 2.9 | 3.0 | - | - | - | - | - | - |

η″TKE_{0}/−u′w′_{0} | 1.05 | 1.05 | 0.98 | 1.00 | 1.00 | 1.01 | - | - | - | - | - | - |

η‴TKE_{0} (cm/s)^{2} | - | - | - | - | - | - | 5.7 | 6.2 | 6.5 | 8.6 | 9.0 | 9.2 |

η‴TKE_{0}/−u′w′_{0} | - | - | - | - | - | - | 1.00 | 0.97 | 0.94 | 1.05 | 1.07 | 1.02 |

Zt/h | 0.05 | 0.05 | 0.06 | 0.05 | 0.05 | 0.05 | 0.06 | 0.04 | 0.04 | 0.04 | 0.05 | 0.04 |

Zt-mm | 5.0 | 6.0 | 8.4 | 5.0 | 6.0 | 7.0 | 6.0 | 4.8 | 5.6 | 3.6 | 5.2 | 4.8 |

_{0}indicates the value obtained by extending a line towards the bed, Zt/h is the depth of the turning point, η′, η″, and η‴ are the average of −u′w′

_{0}/TKE

_{0}for all experiments, experiments group A and experiments group B equal to 0.20, 0.19 and 0.22, respectively.

Experiment Name: | I-H1 | I-H2 | I-H3 | II-H1 | II-H2 | II-H3 | III-H1 | III-H2 | III-H3 | IV-H1 | IV-H2 | IV-H3 |
---|---|---|---|---|---|---|---|---|---|---|---|---|

u′^{2}_{0}/TKE_{0} | 1.20 | 1.21 | 1.23 | 1.19 | 1.19 | 1.21 | 1.21 | 1.19 | 1.16 | 1.25 | 1.25 | 1.24 |

v′^{2}_{0}/TKE_{0} | 0.65 | 0.62 | 0.60 | 0.65 | 0.65 | 0.62 | 0.63 | 0.63 | 0.66 | 0.60 | 0.60 | 0.60 |

w′^{2}_{0}/TKE_{0} | 0.16 | 0.17 | 0.18 | 0.17 | 0.17 | 0.18 | 0.15 | 0.18 | 0.17 | 0.15 | 0.15 | 0.16 |

u′^{2}_{0}/(u′^{2}_{0} + v′^{2}_{0} + w′^{2}_{0}) | 0.60 | 0.60 | 0.61 | 0.59 | 0.59 | 0.60 | 0.61 | 0.60 | 0.58 | 0.62 | 0.63 | 0.62 |

v′^{2}_{0}/(u′^{2}_{0} + v′^{2}_{0} + w′^{2}_{0}) | 0.32 | 0.31 | 0.30 | 0.32 | 0.32 | 0.31 | 0.32 | 0.32 | 0.33 | 0.30 | 0.30 | 0.30 |

w′^{2}_{0}/(u′^{2}_{0} + v′^{2}_{0} + w′^{2}_{0}) | 0.08 | 0.09 | 0.09 | 0.08 | 0.08 | 0.09 | 0.08 | 0.09 | 0.09 | 0.07 | 0.08 | 0.08 |

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## Share and Cite

**MDPI and ACS Style**

Shahmohammadi, R.; Afzalimehr, H.; Sui, J.
Estimation of Bed Shear Stress in Shallow Transitional Flows under Condition of Incipient Motion of Sand Particles Using Turbulence Characteristics. *Water* **2022**, *14*, 2515.
https://doi.org/10.3390/w14162515

**AMA Style**

Shahmohammadi R, Afzalimehr H, Sui J.
Estimation of Bed Shear Stress in Shallow Transitional Flows under Condition of Incipient Motion of Sand Particles Using Turbulence Characteristics. *Water*. 2022; 14(16):2515.
https://doi.org/10.3390/w14162515

**Chicago/Turabian Style**

Shahmohammadi, Reza, Hossein Afzalimehr, and Jueyi Sui.
2022. "Estimation of Bed Shear Stress in Shallow Transitional Flows under Condition of Incipient Motion of Sand Particles Using Turbulence Characteristics" *Water* 14, no. 16: 2515.
https://doi.org/10.3390/w14162515