# Numerical Analysis of Free-Surface Flows over Rubber Dams

^{1}

^{2}

^{*}

## Abstract

**:**

_{2}increases, while the influence of water depth H

_{2}on the lift coefficients is less significant. Furthermore, the discharge coefficients of circular and elliptical dams, computed from the simulated velocity profiles over the crest of the dam, agree with the formulae suggested by previous studies when the downstream depth H

_{2}/H

_{1}< 0.90. In contrast, the discharge coefficient of a tear-shape dam is slightly larger than those of circular dams.

## 1. Introduction

## 2. Numerical Model

_{eff}is the effective viscosity, defined as:

_{eff}= μ + μ

_{SGS}

_{SGS}is the viscosity of the sub-grid scale turbulence. In this study, the sub-grid scale turbulence was modeled using the dynamic model of Lilly–Smagorinsky [21]:

_{s}is the Smagorinsky coefficient, S

_{ij}is the filtered rate of strain tensor [22]:

_{s}is the characteristic length of the spatial filter. For two- and three-dimensional flows, the characteristic length is calculated, respectively, as:

_{s}= 0.30 for two-dimensional flows and C

_{s}= 0.15 for three-dimensional flows. In addition, the projection method [23] was used to decouple the velocity and pressure in the Navier–Stokes equations and to solve the Poisson pressure equation (PPE). For the unsteady flow simulation, the time derivative term used the forward difference scheme.

_{m}occupied by the water in a grid cell can be described by:

_{m}= 1, and the cell is partially occupied by water when 0 < f

_{m}< 1. The wall function is used to calculate the velocity near the channel bed:

_{*}is the shear velocity; and the coefficient A = 17, as suggested by Cabot and Moin [25]. Based on the velocity profile upstream of the dam, the friction velocity can be calculated as u

_{*}= 0.0074 m/s. The grid size near the channel bed is about Δz = 2 mm for Grid 2. Therefore, the distance from the channel bed to the cell center closest to the bed is z = Δz/2 = 1 mm, and the dimensionless distance z

^{+}= zu

_{*}/ν = 7.4.

^{−}

^{6}for the momentum equation. The Courant number was set as Cr = 0.85, and the average time step was Δt = 9.0 × 10

^{−}

^{4}s. The simulation results before the dimensionless time τ = tV

_{1}/D

_{1}= 15 (normalized by the upstream velocity V

_{1}and the diameter D

_{1}of the circular dam) were discarded to exclude the initial transient results from the numerical simulation. The time-averaged velocity and pressure were calculated from the simulation results between the dimensionless time τ = 15–30.

## 3. Model Validation

_{1}= 0.076 m, width W = 0.12 m) was installed in the middle of the flume (see Figure 1 and Figure 2). The discharge rate Q was measured by a 90° V-notch weir. The water depths were measured along the centreline of the flume by a point gauge with a resolution of 0.1 mm.

_{1}= 0.109 m = 1.44 D

_{1}and H

_{2}= 0.0104 m = 0.137 D

_{1}, respectively. The upstream velocity V

_{1}= 0.115 m/s, the upstream Froude no. Fr

_{1}= V

_{1}/(gH

_{1})

^{1/2}= 0.11, and the Reynolds number Re = V

_{1}D

_{1}/ν = 8740.

_{1}. For Zones I, VI, V, and X, non-uniform grids were used, and the stretching ratio was 1.04. The bottom boundary was set as the no-slip boundary condition, the lateral boundaries were set as the free-slip boundary condition, and the downstream boundary was set as zero-gradient for stream-wise velocity [22,28].

_{1}= 0.115 m/s is the velocity outside the boundary layer; δ = 0.1H

_{1}is the thickness of the boundary layer, and the exponent α = 0.2. Since the boundary layer thickness δ was much smaller than the upstream water depth, H

_{1}, the approaching flow was close to a uniform flow. Figure 4 demonstrates that the predicted water depths h

_{p}by different computational grids are all very close to the measured water depth h

_{m}. Grids 1–3 are two-dimensional grids used by the 2D LES model, while Grid 4 is a three-dimensional grid computed by the 3D LES model. The average error between the predicted and measured water depths are defined as:

_{m}. Table 1 summarizes the relative errors Δ

_{h}of different computational grids. As can be seen in Table 1, the result of the three-dimensional model was not outstandingly better than that of the two-dimensional models; this demonstrates that the non-uniformity of the wake flow in the y-direction (span-wise direction) did not affect the integral features on the mid-plane of the cylinder. Therefore, Grid 2 (120 grid points around the dam surface) was used for the rest of the simulation to save computing time.

_{1}is the undisturbed upstream velocity. The total pressure can be separated into the hydrostatic pressure P

_{stat}and the dynamic pressure P

_{d}.

_{p}, on the dam surface of the validation case (Case A1). The maximum pressure occurred at the location θ = −90° due to the hydrostatic pressure was the largest near the upstream channel bed. The pressure coefficient was negative between θ = 90°–180°, owing to the large velocity on the top and leeward side of the circular dam. In addition, the difference between the measured and predicted time-averaged pressure coefficients by Grid 2 was 3.2–6.3%. Note that the predicted pressures by the two- and three-dimensional (Grid 4) LES model were very close. The good agreement between the measured and predicted pressures validates the accuracy of the present LES model.

_{i}is the angle; n (= 120) is the number of angular locations (grid points) on the dam surface; R is the radius of the dam; and W is the dam width. The dimensionless drag coefficient, C

_{D}, is defined as:

_{D}is the drag acting on the dam, and A = WD

_{1}is the projected area of the circular dam. The lift coefficient, C

_{L}, is defined as:

## 4. Results and Discussion

#### 4.1. Hydrodynamic Loading

_{1}= 0.076 m, upstream water depth H

_{1}= 0.109 m, and velocity V

_{1}= 0.115 m/s were identical to that of the flume experiment. However, the downstream water depth varied in the range of H

_{2}= 0.010 m–0.107 m (depth ratio H

_{2}/H

_{1}= 0.10–0.98). Therefore, the downstream velocity V

_{2}= 0.117–1.21 m/s, and the downstream Froude number was Fr

_{2}= 0.12–3.53.

_{2}/H

_{1}). Note that it only shows the simulated water depths in the range of x/D

_{1}= −2.0–4.0, while the downstream boundary is set at x/D

_{1}= 15. When the depth ratio H

_{2}/H

_{1}≤ 0.2, the upstream flow conditions were sub-critical flows, while the downstream conditions were super-critical flows. The water surface dropped rapidly right behind the dam. As the downstream water depth H

_{2}increased, the downstream conditions became sub-critical flows, and the water surface was disturbed by the strong turbulence behind the dam. The disturbance lessened, and the water surface gradually became smoother as it goes downstream.

_{2}/H

_{1}= 0.2, 0.6, and 0.98. The flow conditions distinctly differed when the water depth H

_{2}changed. When the depth ratio H

_{2}/H

_{1}= 0.2, there was a downward jet flow on the leeward side of the dam, and the downstream was a super-critical flow. In contrast, a submerged hydraulic jump (a high-speed flow near the channel bed and a reversed flow near the water surface) occurred in the wake of the dam when H

_{2}/H

_{1}= 0.6. When the depth ratio H

_{2}/H

_{1}= 0.98, the water flowed horizontally over the dam, and the high-speed flow stayed close to the free surface.

_{p total}, and dynamic pressure coefficients, C

_{pd}, on the centerline of the dam surface for different depth ratios. The total and dynamic pressures on the frontal side (θ = −90°–30°) of the dam were identical since the upstream water depths; subsequently, the hydrostatic pressures were the same for different depth ratios. The leeward total pressures increased as the downstream depths increased owing to the hydrostatic pressure. The leeward dynamic pressures for the cases of H

_{2}/H

_{1}= 0.1, 0.2, and 0.4 (see Figure 9b) were identical due to the similar downstream flow conditions. The negative dynamic pressure (caused by the nappe flow) on the leeward side of the dam increased as the depth ratio H

_{2}/H

_{1}decreased. Notice that the location of the maximum suction shifted from θ = 90° to θ = 150° as the depth H

_{2}decreased due to the change of the flow attachment. The simulated pressure distribution can be used for the structural design of the rubber dams.

_{D}is the drag acting upon the water flow by the dam; V

_{1}(z) and V

_{2}(z) are the stream-wise velocities up- and downstream of the dam, respectively. Assuming that the up- and downstream are both uniform flows, and the bed friction is negligible, the drag is equal to:

_{1}H

_{1}W = V

_{2}H

_{2}W. The first term on the right-hand side of Equation (20) represents the difference between the up- and downstream hydrostatic pressures, and the second term on the right-hand side is the net momentum fluxes.

_{2}/H

_{1}in Figure 10a. The average relative error between the drag coefficients computed by the momentum integration method and from the simulated total pressures was 6.5%. This validated the present LES model to compute the drag of the dam. Notice that the drag coefficients (from the LES model) were almost the same (C

_{D}= 89.5) when the depth ratio H

_{2}/H

_{1}≤ 0.40. This large drag was caused by the difference between the up- and downstream hydrostatic pressures. When H

_{2}/H

_{1}> 0.40, the drag coefficient C

_{D}(and the difference of hydrostatic pressure) steadily decreased with the increasing downstream water depth H

_{2}.

_{L}= 97–110. The dimensionless buoyancy C

_{B}= 88.6 for different downstream depths; therefore, the lift coefficients without the buoyancy (without the hydrostatic pressure) C

_{L}– C

_{B}= 8.5–21.4. The reason that the lift coefficients of large depth ratios (H

_{2}/H

_{1}> 0.60) were slightly larger than those of small depth ratios (H

_{2}/H

_{1}≤ 0.40) is due to the maximum negative pressures occurred on the top side, rather than on the leeward side of the dam when downstream depth H

_{2}was large.

_{1}D

_{1}/ν on the hydrodynamic loading of the circular dam, the surface pressure on a circular cross-section, with the diameter D

_{1}= 3.0 m, was simulated by the present LES model. The scale ratio of the full-size dam to the model dam is 39.5. The up- and downstream water depths were set as: H

_{1}= 4.30 m and H

_{2}= 0.41 m; the up- and downstream velocities V

_{1}= 0.45 m/s and V

_{2}= 4.72 m/s. Hence, the depth ratios, H

_{2}/H

_{1}= 0.095, H

_{1}/D

_{1}= 1.43 and the Froude number Fr

_{1}= 0.07, were close to those of the validation case (Case A1). The upstream was a sub-critical flow, and the downstream was a super-critical flow.

^{6}). As can be seen in Figure 11a, the total pressure in front of the full-size prototype dam was much larger than that of the scaled-down model owing to the difference in hydrostatic pressure. Figure 11b illustrates that the maximum suction pressure occurred at the angular location of 145° (on the leeward side of the dam). In contrast, the suction pressure of the prototype dam was much larger than that of the scaled-down model, resulting from the velocity of the nappe flow over the prototype dam was larger than that of the scaled-down model. Furthermore, the drag and lift coefficients (C

_{D}= 249 and C

_{L}= 272) of the prototype dam were different from that of the scaled-down model (C

_{D}= 89.5 and C

_{L}= 97). The lift coefficients without the dimensionless buoyancy were C

_{L}–C

_{B}= 8.5 and 43.9 for scaled-down and prototype dam, respectively. In other words, the dimensionless force coefficients obtained from the scaled-down model experiments are not applicable to the full-size circular dams due to the Reynolds number effect. The present numerical model can be used to compute the dynamic loading of the full-size dams for the purpose of structural design.

#### 4.2. Discharge Coefficient

_{2}/H

_{1}< 0.90. Notice that the velocities, due to the boundary layer at the dam surface, decreased as it approached the dam crest. In contrast, the horizontal velocities became smaller for the depth ratio H

_{2}/H

_{1}= 0.90 and 0.98, under the same upstream flow condition. The depth-averaged velocity V

_{c}and water depth h

_{c}above the dam crest are used to compute the Froude number at the dam crest, Fr

_{c}= V

_{c}/(gh

_{c})

^{0.5}= 0.96 and 0.81 when H

_{2}/H

_{1}= 0.90 and 0.98, respectively. In other words, they did not reach the critical flow condition at the dam crest.

_{u}is the upstream (x/D

_{1}= −5) head above the dam crest. The discharge coefficients of the circular dam are plotted against the upstream head in Figure 13. The formulae suggested by Heidarpour and Chamani [11] and Matthew [31] for the discharge coefficients are also plotted in the figure for comparison. As can be seen, the discharge coefficient C

_{d}= 1.18 for H

_{2}/H

_{1}≤ 0.80 is very close to the curves suggested by Heidarpour and Chamani [11] and Matthew [31]. However, the discharge coefficients (C

_{d}= 1.10 and 1.04) for the cases of H

_{2}/H

_{1}= 0.90 and 0.98 are lower than the curves. The reduced discharge coefficient is resulting from the flow condition above the dam crest not being critical flow when H

_{2}/H

_{1}≥ 0.90. In other words, the formulae for discharge coefficients will over-estimate the discharge rate when the downstream water depth close to the upstream depth (the Froude number at the dam crest, Fr

_{c}< 1.0).

_{v}of the dam body:

_{v}= 2.19 × 10

^{9}N/m

^{2}. When the pressure variation on the dam surface is ΔP = 10,000 Pa (about 1 m depth of water over the dam), the change rate of dam volume is about Δ

^{−6}. If the rubber dam is air-inflatable, the bulk modulus of the dam body will be around E

_{v}= 1 × 10

^{5}N/m

^{2}(close to air). Under the same pressure change ΔP = 10,000 Pa, the change rate of the dam volume is about Δ

_{1}= 0.076 m, and the horizontal diameter D

_{2}= 1.2D

_{1}= 0.091 m. The second one is a tear-shape dam, with height D

_{1}= 0.076 m and length L = 0.114 m, with the frontal part a concave curve and the leeward part a semi-circular shape. The upstream water depth is 0.109 m, velocity is 0.115 m/s, and the downstream water depth is 0.0104 m, identical to the validation case (Case A1).

_{2}/D

_{1}= 1.2), and tear-shape dam are compared in Figure 14. Due to the leeward curves of the circular and tear-shape dams are the same, the nappe flows over these two dams are quite similar. Nonetheless, the water surface on the leeward part of the elliptical dam is different from that of the circular dam. The time-averaged velocity vectors over elliptical and tear-shape dams are shown in Figure 15. As can be seen, the downstream flow fields of the elliptical and tear-shape dams are very similar. In contrast, the tear-shape dam did not have the re-circulating flows in front of the dam, such as those in the circular and elliptical dams. In other words, the tear-shape dam could prevent sediment accumulation upstream of the dam.

## 5. Concluding Remarks

_{2}decreased, owing to the difference in hydrostatic pressures of the up- and down-stream flows. In addition, the drag coefficients calculated from the simulated surface pressures were very close to the drag coefficients computed by the momentum integration method. On the other hand, the lift coefficient increased slightly when the downstream depth H

_{2}increased, resulting from the negative uplift pressure on the dam crest. Furthermore, the dimensionless force coefficients of full-size dams are larger than those of scaled-down dam models due to the difference in the Reynolds number.

_{2}/H

_{1}< 0.9. Nevertheless, their formulae over-estimate the discharge coefficient when the downstream depth H

_{2}/H

_{1}≥ 0.90, owing to the flow condition above the dam crest not being critical flow. In addition, the discharge coefficient of the tear-shape dam is slightly larger than those of circular dams. In brief, the present numerical model can simulate the hydrodynamic loading and discharge rate of full-size rubber dams of any shape to avoid the scale effect of flume experiments.

## Author Contributions

## Funding

## Data Availability Statement

## Conflicts of Interest

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**Figure 3.**Computational domain and grid arrangement for the circular dam (D is the diameter of the dam).

**Figure 4.**Comparison of measured and simulated water surfaces by different computational grids for the validation case (Case A1).

**Figure 5.**Comparison of measured and predicted pressure coefficients on the surface of the circular dam for the validation case (Case A1).

**Figure 6.**Distribution of pressure coefficients on the centerline of the circular dam for the validation case (Case A1). (

**a**) Total pressure; (

**b**) dynamic pressure.

**Figure 7.**Variation of the water depths over the circular dam of different depth ratios for Cases A.

**Figure 8.**Time-averaged velocity vectors on the mid-plane the circular dam under different downstream depths. (

**a**) H

_{2}/H

_{1}= 0.2; (

**b**) H

_{2}/H

_{1}= 0.6; (

**c**) H

_{2}/H

_{1}= 0.98.

**Figure 9.**Pressure coefficients on the surface of the circular dam under different depth ratios. (

**a**) Total pressure; (

**b**) dynamic pressure.

**Figure 10.**Relationship between the depth ratio and force coefficients (

**a**) drag coefficient; (

**b**) lift coefficient.

**Figure 11.**Comparison of simulated pressure coefficients on the surfaces of scaled-down model and prototype circular dams. (

**a**) Total pressure; (

**b**) dynamic pressure.

**Figure 12.**Profiles of time-averaged horizontal velocity above the crest (x = 0) of the circular dam of different depth ratios.

**Figure 13.**Relationship between the discharge coefficient and upstream water head of different dam shapes, R = D

_{1}/2 is the radius of the dam.

**Figure 14.**Comparison of the simulated water depths for the circular, elliptical, and tear-shape dams.

**Figure 15.**Time-averaged velocity vectors on the central plane of the elliptical and tear-shape dams.

Title | Grid 1 (2D) | Grid 2 (2D) | Grid 3 (2D) | Grid 4 (3D) | ||||

Total grid no. | 104,048 | 159,212 | 233,859 | 955,272 | ||||

Grid no. on the cylinder surface | 90 | 120 | 150 | 120 | ||||

Grid size near the cylinder | 2.65 mm | 2.0 mm | 1.60 mm | 2.0 mm | ||||

Smallest grid size | Δx = 3 mm Δz = 3 mm | Δx = 2 mm Δz = 2 mm | Δx = 1.5 mm Δz = 1.5 mm | Δx = 2 mm Δy = 5 mm Δz = 2 mm | ||||

Water surface Δ _{h} | 8.53% | 5.59% | 5.06% | 6.97% | ||||

Force coeff. | C_{D} | C_{L} | C_{D} | C_{L} | C_{D} | C_{L} | C_{D} | C_{L} |

89.7 | 95.5 | 89.6 | 88.5 | 93.1 | 93.1 | 90.3 | 97.8 | |

CPU time | 18 h | 26 h | 40 h | 768 h |

_{1}= 0.109 m, V

_{1}= 0.115 m/s, H

_{2}= 0.0104 m, V

_{2}= 1.21 m/s.

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Chu, C.-R.; Tran, T.T.T.; Wu, T.-R. Numerical Analysis of Free-Surface Flows over Rubber Dams. *Water* **2021**, *13*, 1271.
https://doi.org/10.3390/w13091271

**AMA Style**

Chu C-R, Tran TTT, Wu T-R. Numerical Analysis of Free-Surface Flows over Rubber Dams. *Water*. 2021; 13(9):1271.
https://doi.org/10.3390/w13091271

**Chicago/Turabian Style**

Chu, Chia-Ren, Truc Thi Thu Tran, and Tso-Ren Wu. 2021. "Numerical Analysis of Free-Surface Flows over Rubber Dams" *Water* 13, no. 9: 1271.
https://doi.org/10.3390/w13091271