# Scour, Velocities and Pressures Evaluations Produced by Spillway and Outlets of Dam

^{*}

^{†}

## Abstract

**:**

^{3}/s) and half-height outlets (1760 m

^{3}/s), with three complementary procedures: empirical formulae obtained in models and prototypes, semi-empirical methodology based on pressure fluctuations-erodibility index and computational fluid dynamics simulations. The free surface weir could generate a scour around 21 m, while the intermediate outlet could reach the intact rock, located 34 m below the initial river bed. A pre-excavated basin is proposed and the velocities and pressures are analyzed. The results demonstrated the suitability of combining different methodologies to achieve an adequate resolution of this complex phenomenon.

## 1. Dam Characteristic

_{4}= 700 m

^{3}/s (return period TR = 4 years) and a half-height outlet with two almost symmetrical ducts (5.00 m × 5.80 m). Considering the maximum normal operating level (924 m), the intermediate outlet design flow is Q

_{40}= 1760 m

^{3}/s (return period TR = 40 years). Hence, the total design flow of the weir and half-height outlet is Q

_{100}= 2340 m

^{3}/s (return period TR = 100 years). The bottom outlet consists of four radial gates (5.00 m × 6.80 m). According to Consorcio PCA [1], the total discharge capacity of the dam is Q

_{10000}= 5520 m

^{3}/s (return period TR = 1000 years).

## 2. Empirical Formulae

_{s}is the scour depth below tailwater level, Y

_{0}the tailwater depth, Y

_{s}the scour depth below the original bed, Γ an experimental coefficient, q the specific flow, H

_{n}the net energy head, g the gravitational acceleration, and d the characteristic size of bed material.

_{0}the energy loss in the duct, H

_{n}= H

_{0}= H

_{B}− t

_{0}the net energy head at the exit of the outlet, H the falling height from reservoir level to tailwater level (ski-jump and free surface weir), h

_{0}the vertical distance between the outlet exit and the tailwater level, B

_{i}, U

_{i}, θ

_{i}the thickness, velocity and angle of the jet in initial condition, B

_{j}, U

_{j}, θ

_{j}the total thickness, velocity and angle of the jet in the impingement conditions.

^{3}/s), the scour could reach a depth of 17 m. However, taking into account the mean value + 0.50 standard deviation, then the design flow fully penetrates the alluvial (scour 24 m). Considering the mean value + 1 standard deviation, the design flow could reach the intact rock layer (scour 34 m).

_{22}= 1320 m

^{3}/s. The design flow (Q

_{40}= 1760 m

^{3}/s) would penetrate over 32 m and not reach the intact rock. However, with the mean value + 0.50 standard deviation, the flow of 1550 m

^{3}/s would already completely erode the weathered rock layer (scour 34 m).

## 3. Semi-Empirical Methodology

_{g}is the thickness due to gravity effect, ξ the jet lateral spread distance due to the turbulence effect, q the specific flow, H the fall height, and h is the energy head at the crest weir. φ = K

_{φ}T

_{u}, with T

_{u}being the turbulence intensity and K

_{φ}an experimental parameter (1.14 for circular jets and 1.24 for the three-dimensional nappe flow case).

_{s}is the number of resistance of the mass, K

_{b}the number of the block size, K

_{d}the number of resistance to shear strength on the discontinuity contour, and J

_{s}the number of structure relative of the grain.

^{2}, is calculated and based on the erodibility index K.

_{p}) and the fluctuating dynamic pressure (C

_{p}'). We can use these dynamic pressure coefficients as estimators of the stream power reduction coefficients, by the effect of the jet disintegration in the air and their diffusion in the stilling basin (Annandale [31]). Hence, the dynamic pressures are also a function of the fall height to disintegration height ratio (H/L

_{b}) and water cushion to impingement jet thickness (Y/B

_{j}). Thus, the total dynamic pressure can be expressed as:

_{p}(Y/B

_{j}) is the mean dynamic pressure coefficient, C

_{p}'(Y/B

_{j}) the fluctuating dynamic pressure coefficient, P

_{jet}the stream power per unit of area, and F the reduction factor of the fluctuating dynamic pressure coefficient. In the rectangular jet case (nappe flow), Castillo et al. [27] adjusted the formulae to calculate the disintegration height (L

_{b}), and the mean and fluctuating dynamic pressure coefficients (C

_{p}and C

_{p}', respectively) by using new laboratory data (see Figure 4, Figure 5 and Figure 6). These formulae were used to estimate the P

_{total}value.

_{s}= 24 m (tailwater depth Y

_{0}= 6 m), the flow rate of 500 m

^{3}/s would have the power to erode the weathered rock, although the design flow of 700 m

^{3}/s, would not have enough power to reach the intact rock. This confirms that the maximum scour of the free surface weir could be near to D

_{s}= 26 m (alluvial 20 m, tailwater depth Y

_{0}= 6 m).

_{p}and C

_{p}' are valid for H/L

_{b}≤ 0.50 (Ervine et al. [21]). However, for the design flow (Q

_{40}= 1760 m

^{3}/s) the H/L

_{b}= 1.67. For this reason the calculations in Figure 8 were carried out using the rectangular jet analogy.

_{weathered_rock}= 16 kW/m

^{2}) does not resist the flow of the annual return period (Q

_{ma}= 136 m

^{3}/s). The intact rock stream power (P

_{rock}= 408 kW/m

^{2}) could resist up to a flow return period of 5 years (Q

_{5}= 820 m

^{3}/s). The Q

_{10}= 1180 m

^{3}/s would exceed the intact rock strength, while the design flow reaches a significant scour in the intact rock. As a solution to the scour, a concrete slab of 20 MPa characteristic strength and thickness of 2 m (P

_{conc}= 788 kW/m

^{2}) is placed directly on the alluvial level (796 MASL). The geometry of the pre-excavated basin should be similar to the geometry of the basin that would be formed with the flow Q

_{40}= 1766 m

^{3}/s. Figure 8 indicates that the concrete slab would resist the power stream of the design flow (P

_{jet}= 666 kW/m

^{2}).

## 4. Numerical Simulation

^{2}/s and H = 120.00 m. With these data, the calculated impingement jet thickness was B

_{j}= 1.46 m.

_{s,i}is the concentration of the suspended sediment, in units of mass per unit volume and $\overline{\mathbf{u}}$ is the mean velocity of the fluid-sediment mixture. The momentum balances for each sediment species and the fluid-sediment mixture are:

**u**

_{s,i}is the velocity of sediment species i; ρ

_{s,i}the density of the sediment species i; f

_{s,i}the volume fraction of sediment species i; P

_{r}the pressure; K

_{i}the coefficient of quadratic drag for species i;

**F**includes body and viscous forces;

**u**

_{r,i}is the relative velocity between the velocity (

**u**

_{s,i}) of sediment species i and the fluid velocity (

**u**

_{f}), and $\overline{\rho}$ is density of fluid-sediment mixture.

**u**

_{drift,i}=

**u**

_{s,i}− $\overline{\mathbf{u}}$ is the velocity needed to compute the transport of sediment due to drift. Assuming that the motion of the sediment is nearly steady at the scale of the computational time and that the advection term is small (i.e., for small drift velocity

**u**

_{drift}), the result of Equation (9) is

_{i}is the drag function and combines shape drag and Stokes drag (Flow Science [37]). The correction to account for particle/particle interactions is an experimentally determined relation referred to as the Richardson-Zaki [38] correlation. Table 8 shows the principal relations to calculate the sediment scour model in FLOW-3D.

_{s}= c

_{rough}d

_{50,packed}. We used the default value of the proportional constant c

_{rough}= 1.0. The entrainment lift velocity of sediment is then computed with Mastbergen and Von den Berg expression and as a function of the dimensionless particle diameter, ${d}_{*}$ (see Table 8). The entrainment velocity

**u**

_{lift,i}is then used to compute the amount of packed sediment that is converted into suspended sediment, effectively acting as a mass source of suspended sediment at the packed bed interface. Once converted to suspended sediment, the sediment subsequently advects and drifts.

_{i}is related to the volumetric bed-load transport rate per unit width, q

_{bi}. The bed-load thickness of the saltating sediment is estimated with the Van Rijn [40] relation (see Table 8). To compute the motion of the sediment in each computational cell, the value of q

_{b,i}is converted into a velocity by ${u}_{bedload,i}={q}_{b,i}/{\delta}_{i}{f}_{b,i}$. The direction of the motion is determined from the motion of the liquid adjacent to the packed bed interface. Therefore, the volumetric flux is ${\mathbf{u}}_{bedload,i}={u}_{bedload,i}(\overline{\mathbf{u}}/\Vert \overline{\mathbf{u}}\Vert )$, where $\overline{\mathbf{u}}/\Vert \underset{\_}{\overline{\mathbf{u}}}\Vert $ is the direction of the fluid-sediment mixture adjacent to the packed interface, and the resulting bed-load velocity, u

_{bedload,i}, is used to transport the packed sediment.

**n**

_{s}is the outward pointing normal to the packed bed interface.

^{TM}technique. At each time step, area and volume fractions describing the packed sediments are calculated throughout the domain. However, the maximum size of the particle that this version can manage is over 35 mm (Flow Science [37]). For this reason, simulations were carried out in a Froude similitude scale 1:50 and the results transformed to prototype scale. The model had 4,159,233 cells and the scour reached the steady state after 60 s of simulation. With this setting, the time required to solve the problem was approximately 26.00 days in an Intel(R) Core(TM) i7-2600 CPU @3.40 GHz processor and 24.0 GB RAM.

^{6}s of CPU, 7.2 days of elapsed time), the maximum scour depth reached the intact rock (34 m). This value is a bit bigger than that obtained with the mean value adjustment of the empirical formulae (32 m), is equal to the mean value + 0.50 standard deviation (34 m), and a bit smaller than the semi-empirical methodology whose value was greater than 34 m (alluvial 24 m, weathered rock 10 m and intact rock > 2 m).

_{s}), the mean value + 0.50 standard deviation (Y

_{s}+ 0.50 SD), and the mean value + 1 standard deviation (Y

_{s}+ SD). The scour obtained with the different methodologies are in agreement.

_{b}= 1.31 according to lab measurements). The velocities reach 43 m/s and are reduced to 6 m/s by diffusion in the water cushion. The instantaneous pressures on the bottom reach 28 m and correspond to hydrostatic pressure of the water cushion.

^{3}/s) and when only one duct only is working (Q = 880 m

^{3}/s), respectively.

## 5. Conclusions

## Acknowledgments

## Author Contributions

## Conflicts of Interest

## Abbreviations

A | jet area on the impact surface; |

B_{g} | thickness of the jet due to gravity effect; |

B_{i} | thickness of the jet in initial condition; |

B_{j} | thickness of the jet in the impingement conditions; |

C_{D,i} | drag coefficient for sediment species i; |

C_{p} | mean dynamic pressure coefficient; |

C_{p}′ | fluctuating dynamic pressure coefficient; |

C_{r} | coefficient of relative density; |

c_{rough} | proportional constant of local mean grain diameter in packed sediment (default value 1.0); |

c_{s,i} | concentration of the suspended sediment, |

D_{s} | scour depth below tailwater level; |

d | characteristic particle diameter; |

d_{i} | characteristic size of bed material in which i % is smaller in weight; |

d_{50},_{packed} | local mean grain diameter in packed sediment: |

d_{s,i} | diameter of sediment species i; |

d_{m} | average particle size of the bed material; |

d_{∗} | dimensionless particle diameter; |

f | residual friction angle of the granular earth material; |

f_{b,i} | volume fraction of sediment i in the bed-load layer; |

f_{s} | total volume fraction of sediment; |

f_{s,i} | volume fraction of sediment species i; |

F | reduction factor of the fluctuating dynamic pressure coefficient; |

F | body and viscous forces; |

g | gravitational acceleration; |

$\Vert g\Vert $ | magnitude of the gravitational vector; |

H | fall height; |

H_{n} | net energy head; |

h | energy head at the crest weir; |

h_{0} | vertical distance between the outlet exit and the tailwater level; |

J_{a} | join wall alteration number; |

J_{n} | join set number; |

J_{r} | joint wall roughness number; |

J_{s} | number of structure relative of the grain; |

J_{x}, J_{y}, J_{z} | discontinuity spacing; |

K | erodibility index; |

K_{b} | number of the block size; |

K_{d} | number of resistance to shear strength on the discontinuity contour; |

K_{i} | coefficient of quadratic drag for species i; |

K_{φ} | experimental parameter; |

k_{s} | Nikuradse roughness of the bed surface; |

L_{b} | disintegration height; |

M_{s} | number of resistance of the mass; |

n_{s} | outward pointing normal to the packed bed interface; |

P | relative capacity of the material to resisting erosion; |

P_{jet} | stream power per unit of area; |

P_{r} | pressure; |

Q | flow; |

Q_{i} | flow with return period i; |

q | specific flow; |

q_{bi} | volumetric bed-load transport rate per unit width; |

R_{e} | Reynolds number; |

${R}_{i}^{*}$ | dimensionless parameter to computing the critical Shields number; |

RQD | rock quality designation; |

SD | Standard deviation; |

SPT | standard penetration test; |

TR | return period; |

t_{0} | energy loss in the duct; |

U_{i} | velocity of the jet in the initial condition; |

U_{j} | velocity of the jet in the impingement conditions; |

$\overline{\mathbf{u}}$ | mean velocity of the fluid-sediment mixture; |

$\overline{\mathbf{u}}/\Vert \underset{\_}{\overline{\mathbf{u}}}\Vert $ | direction of the fluid-sediment mixture adjacent to the packed interface; |

u_{bedload,i} | velocity magnitude of bed-load; |

u_{bedload,i} | vector velocity of bed-load; |

u_{drift,i} | velocity of the sediment due to drift; |

u_{f} | fluid velocity; |

u_{lift,i} | entrainment lift velocity of sediment; |

u_{r,i} | relative velocity between the velocity of sediment species i and the fluid velocity; |

u_{settling,i} | velocity magnitude of settling; |

u_{s,i} | velocity of sediment species i; |

${\mathbf{u}}_{r,i}^{eff}$ | drift velocity to account for particle/particle interactions; |

UCS | unconfined compressive strength; |

x, y, z, v, w | empirical exponents defined by regression or optimization; |

Y | water cushion depth; |

Y_{0} | tailwater depth; |

Y_{s} | scour depth below the original bed; |

α_{i} | entrainment parameter (recommended value 0.018); |

β | air-water relationship; |

β_{i} | proportionality constant of Meyer-Peter and Müller equation; |

Γ | experimental coefficient; |

γ | specific weight of water; |

γ_{r} | reference unit weight of rock (27·10 ^{3} N/m^{3}); |

δ_{i} | bed-load thickness; |

ζ_{0} | Richardson-Zaki coefficient; |

ζ_{user} | coefficient of Richardson-Zaki coefficient (default value 1.0); |

ζ | exponent of Richardson-Zaki relation; |

θ_{i} | local Shields number based on the local shear stress, angle of the jet in the initial conditions; |

θ_{j} | angle of the jet in the impingement conditions; |

${\theta}_{cr,i}$ | dimensionless critical Shields parameter; |

${\theta}_{cr,i}^{\prime}$ | dimensionless critical Shields parameter for sloping surfaces to include the angle of repose; |

μ_{f} | dynamic viscosity of fluid; |

ξ | jet lateral spread distance due to the turbulence effect; |

ρ | water density; |

$\overline{\rho}$ | density of fluid-sediment mixture; |

ρ_{r} | mass density of the rock; |

ρ_{s} | density of sediment; |

ρ_{s,i} | density of the sediment species i; |

τ | shear stress; |

Φ_{i} | dimensionless bed-load transport rate; |

φ | parameter (φ = K _{φ}T_{u}); |

φ | residual friction angle of the granular earth material. |

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**Figure 4.**Mean dynamic pressure coefficient (C

_{p}) for the nappe flow case in function of ratios Y/B

_{j}and H/L

_{b}(Castillo et al. [27]).

**Figure 5.**Fluctuating dynamic pressure coefficient (C

_{p}') for the nappe flow case in function of ratios Y/B

_{j}and H/L

_{b}(Castillo et al. [27]).

**Figure 6.**Reduction factor F of fluctuating dynamic pressure coefficient in function of ratio H/L

_{b}and the jet type (Castillo et al. [27]).

**Figure 7.**Stream power of the jet for different flows as a function of the erodibility. Alluvial, weathered rock and intact rock indexes (Y

_{s}= 18 m, Y

_{0}= 6 m) for the free surface weir.

**Figure 8.**Stream power of the jet for different flows as a function of the erodibility. Alluvial, weathered rock and intact rock indexes (Y

_{s}= 18 m, Y

_{0}= 6 m) for the half-height outlet.

**Figure 9.**Pressure signal on the stagnation point for the free surface weir calculated with FLOW-3D, considering a water cushion Y

_{0}= 2 m.

**Figure 11.**Scour bowl due to the free surface weir jet. Bilayer simulation: alluvial 24 m and weathered rock 10 m (Units in m/s. Froude scale 1:50. Prototype impingement velocity = 6.1 × 50

^{1/2}= 43.13 m/s).

**Figure 12.**Scour bowl due to the half-height outlet. Bilayer simulation: alluvial 24 m and weathered rock 10 m. (Units in m/s. Froude scale 1:50. Prototype impingement velocity = 5.4 × 50

^{1/2}= 38.18 m/s).

**Figure 14.**Lateral and spatial views of the free surface weir jets in the air and in the pre-excavated stilling basin (Prototype scale. Units in m/s and in Pascal): (

**a**) Velocities; (

**b**) Pressures (Q = 700 m

^{3}/s).

**Figure 15.**Flow velocity in the air and in the pre-excavated stilling basin (Prototype scale. Units in m/s): (

**a**) Spatial view of the half-height outlet; (

**b**) Lateral view of the half-height outlet; (

**c**) Top view of the velocity vectors with the half-height outlet; (

**d**) Lateral view of the velocity vectors with the half-height outlet.

**Figure 16.**Spatial views of pressures in the pre-excavated stilling basin (Prototype scale. Units in Pascal): (

**a**) Working two ducts of half-height outlet (Q = 1760 m

^{3}/s); (

**b**) Working one duct of half-height outlet (Q = 880 m

^{3}/s).

Bed Material | D_{16} (m) | D_{50} (m) | D_{84} (m) | D_{90} (m) | D_{m} (m) |
---|---|---|---|---|---|

Alluvial (820 MASL to 796 MASL) | 0.006 | 0.150 | 0.225 | 0.240 | 0.124 |

Weathered rock (796 MASL to 786 MASL) | 0.045 | 0.160 | 0.500 | 0.550 | 0.235 |

Author | Γ | v | w | x | y | z | d |
---|---|---|---|---|---|---|---|

Hartung [12] | 1.400 | 0 | 0 | 0.64 | 0.360 | 0.32 | d_{85} |

Chee and Padiyar [13] | 2.126 | 0 | 0 | 0.67 | 0.180 | 0.063 | d_{m} |

Bisaz and and Tschopp [14] | 2.760 | 0 | 0 | 0.50 | 0.250 | 1.00 | d_{90} |

Martins-A [15] | 1.500 | 0 | 0 | 0.60 | 0.100 | 0.00 | - |

Machado [16] | 1.350 | 0 | 0 | 0.50 | 0.3145 | 0.0645 | d_{90} |

Author (Year) | Formulae | Parameters |
---|---|---|

Jaeger [17] | ${D}_{s}=0.6{q}^{1/2}{H}_{n}^{1/4}{\left(h/{d}_{m}\right)}^{1/3}$ | d_{m} = average particle size of the bed material d _{90} = bed material size, 90% is smaller in weight θ _{T} = impingement jet angle g = gravitational acceleration (9.81 m/s ^{2}) β = air-water relationship ρ = water density ρ _{s} = density of sediment Γ = experimental coefficient |

Rubinstein [18] | ${D}_{s}=h+0.19{(\frac{{H}_{n}+h}{{d}_{90}})}^{3/4}\left(\frac{{q}^{6/5}}{{H}_{n}^{23/49}{h}^{1/3}}\right)$ | |

Mirskhulava [19] | ${D}_{s}=\left(\frac{0.97}{{d}_{90}^{1/2}}-\frac{1.35}{{H}_{n}^{1/2}}\right)\frac{q\xb7\mathrm{sin}{\theta}_{T}}{1-0.175\xb7\mathrm{cot}{\theta}_{T}}+0.25h$ | |

Mason [7] | ${D}_{s}=3.39\frac{{q}^{3/5}{(1+\beta )}^{3/10}{h}^{4/25}}{{g}^{1/3}{d}^{3/50}}$ | |

Bombardelli and Gioia [9] | Axisymmetric jet: ${D}_{s}=\Gamma \frac{{q}^{2/5}{H}_{n}^{2/5}}{{g}^{1/5}{d}^{2/5}}{\left[\frac{\rho}{{\rho}_{s}-\rho}\right]}^{-3/5}$ |

**Table 4.**Erodibility indexes parameters (adapted from Annandale [31]).

Material | Formulae | Parameters |
---|---|---|

Rock | ${M}_{s}=0.78{C}_{r}UC{S}^{1.05}whenUCS\le 10MPa$ ${M}_{s}={C}_{r}UCSwhenUCS10MPa$ ${C}_{r}=g{\rho}_{r}/{\gamma}_{r}$ | UCS = unconfined compressive strength C _{r} = coefficient of relative densityρ _{r} = mass density of the rock g = gravitational acceleration γ _{r} = reference unit weight of rock (27·10^{3} N/m^{3}) |

Non-cohesive granular soil | The relative magnitude is obtained by means of the standard penetration test (SPT). When the SPT value exceeds 80, the non-cohesive granular material is taken as rock. | |

Rock | ${K}_{b}=RQD/{J}_{n}$ | RQD = rock quality designation RQD = values range between 5 and 100 J _{n} = values range between 1 and 5 K _{b} = values range between 1 and 100 J _{n} = join set number |

Non-cohesive granular soil | ${K}_{b}=1000{d}^{3}$ | d = characteristic particle diameter (m) |

Rock | ${K}_{d}={J}_{r}/{J}_{a}$ | J_{r} = joint wall roughness number J _{a} = join wall alteration number |

Non-cohesive granular soil | ${K}_{d}=\mathrm{tan}\phi $ | $\phi $ = residual friction angle of the granular earth material |

Variable | Value |
---|---|

Angle of rock friction, SPT (°) | 38 |

Specific weight (kN/m^{3}) | 27.64 |

Unconfined compress. resistant, UCS (MPa) | 50 |

Relative density coefficient, C_{r} | 1.024 |

RQD (calculated) | 82.66 |

Number of join system (calculated), J_{n} | 1.83 |

Discontinuity spacing, J_{x}, J_{y}, J_{z} (m) | 0.5 |

Average block diameter (calculated), (m) | 0.5 |

Roughness degree, J_{r} | 2 |

Alteration degree, J_{a} | 1 |

**Table 6.**Values of the parameters to estimate the Erodibility Index (K) and then the Power Threshold (P

_{rock}) for bed layers (water cushion depth D

_{s}= 24 m).

Variable | Alluvial | Weathered Rock | Intact Rock | Concrete |
---|---|---|---|---|

M_{s} | 0.19 | 0.41 | 51.19 | 20.47 |

K_{b} | 11.39 | 125 | 49.18 | 49.18 |

K_{d} | 0.78 | 0.78 | 2.00 | 5.33 |

J_{s} | 1.00 | 1.00 | 0.60 | 1.00 |

Erodibility index, K | 1.69 | 40 | 3021 | 7280 |

Stream power, P_{rock} (kW/m^{2}) | 1.50 | 16 | 408 | 788 |

Variable | Parametric Methodology | FLOW-3D |
---|---|---|

Net drop height (m) | 120.00 | 120.00 |

Mean dynamic pressure (m) | 30.56 | 33.44 |

Mean dynamic pressure coefficient, C_{p} | 0.28 | 0.31 |

Relation | Formulae | Parameters |
---|---|---|

Drag function | ${K}_{i}=\frac{3}{4}\frac{{f}_{s,i}}{{d}_{s,i}}\left({\rho}_{f}{C}_{D,i}\Vert {\mathbf{u}}_{r,i}\Vert +24\frac{{\mu}_{f}}{{d}_{s,i}}\right)$ | d_{s,i} and C_{D,i} = diameter and drag coefficient for sediment species i μ _{f} = fluid dynamic viscosityu_{r,j} = drift velocityf _{s} = sediment total volume fractionζ = ζ _{user}ζ_{0}; ζ_{user} = 1${R}_{e}={\rho}_{f}{d}_{i}\Vert {\mathbf{u}}_{r,i}\Vert /{\mu}_{f}$ = Reynolds number on the particle d _{i} ρ _{f} = fluid density${R}_{i}^{*}={d}_{s,i}\frac{\sqrt{0.1({\rho}_{s,i}-{\rho}_{f}){\rho}_{f}\Vert g\Vert {d}_{s,i}}}{{\mu}_{f}}$ ρ _{s,i} = density of sediment species iβ = slope bed angle ϕ _{i} = repose angle for sediment species i (default is 32°)Ψ = angle between the flow and the upslope direction (flow directly up a slope Ψ = 0°) τ = local shear stress $\Vert g\Vert $ = gravitational vector α _{i} = entrainment parameter (~0.018)n_{s} = outward pointing normal to the packed bed interfacef _{b,i} = volume fraction of sediment i in the bed-load layerΦ _{i} = dimensionless bed-load transport (MPM) ** d _{*} = dimensionless particle diameter ${\theta}_{i}$ = local Shields number |

Drift velocity correction | ${\mathbf{u}}_{r,i}^{eff}={\mathbf{u}}_{r,i}{(1-{f}_{s})}^{\zeta}$ | |

Richardson-Zaky coefficient | ${\zeta}_{0}=4.35$ for ${R}_{e}<0.2$ ${\zeta}_{0}=4.35/{R}_{e}^{0.03}$ for $0.2<{R}_{e}<1.0$ ${\zeta}_{0}=4.45/{R}_{e}^{0.1}$ for $1.0<{R}_{e}<500$ ${\zeta}_{0}=2.39$ for ${R}_{e}>500$ | |

Critical Shields parameter (S-W) * | ${\theta}_{cr,i}=\frac{0.3}{1+1.2{R}_{i}^{*}}+0.055\left[1-\mathrm{exp}(-0.02{R}_{i}^{*})\right]$ | |

Critical Shields parameter modified for sloping surface | $\begin{array}{l}{{\theta}^{\prime}}_{cr,i}=\\ {\theta}_{cr,i}\frac{\mathrm{cos}\Psi \mathrm{sin}\beta +\sqrt{{\mathrm{cos}}^{2}\beta {\mathrm{tan}}^{2}{\varphi}_{i}-se{n}^{2}\Psi se{n}^{2}\beta}}{\mathrm{tan}{\varphi}_{i}}\end{array}$ | |

Local Shields number | ${\theta}_{i}=\frac{\tau}{\Vert g\Vert {d}_{s,i}({\rho}_{s,i}-{\rho}_{f})}$ | |

Sediment entrainment lift velocity | ${\mathbf{u}}_{lift,i}={\alpha}_{i}{\mathbf{n}}_{s}{d}_{*}^{0.3}{\left({\theta}_{i}-{{\theta}^{\prime}}_{cr,i}\right)}^{1.5}\sqrt{\frac{\Vert g\Vert {d}_{s,i}({\rho}_{s,i}-{\rho}_{f})}{\Vert g\Vert d{\rho}_{f}}}$ | |

Dimensionless particle diameter | ${d}_{*}={d}_{s,i}{\left[\frac{{\rho}_{f}({\rho}_{s,i}-{\rho}_{f})\Vert g\Vert}{{\mu}_{f}^{2}}\right]}^{\frac{1}{3}}$ | |

Volumetric bed-load transport rate per unit width | ${q}_{b,i}={f}_{b,i}{\Phi}_{i}{\left[\Vert g\Vert \left(\frac{({\rho}_{s,i}-{\rho}_{f})}{{\rho}_{f}}\right){d}_{s,i}^{3}\right]}^{\frac{1}{2}}$ | |

Bed-load thickness | $\frac{{\delta}_{i}}{{d}_{s,i}}=0.3{d}_{*}^{0.7}{\left(\frac{{\theta}_{i}}{{{\theta}^{\prime}}_{cr,i}}-1\right)}^{0.5}$ |

**Table 9.**Comparison of scour obtained by different methods for free surface weir and half-height outlet.

Method | Free Surface Weir Q_{4} = 700 m^{3}/s | Half-Height Outlet Q_{40} = 1760 m^{3}/s | ||||
---|---|---|---|---|---|---|

Y_{s} (m) | Y_{s} + 0.50SD (m) | Y_{s} + SD (m) | Y_{s} (m) | Y_{s} + 0.50SD (m) | Y_{s} + SD (m) | |

Empirical formulations | 17 | 24 | 34 | 32 | >34 | >34 |

Erodibility Index Pressure fluctuations | 20 | - | - | >34 | - | - |

FLOW-3D v11 | 21 | - | - | >34 | - | - |

© 2016 by the authors; licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons by Attribution (CC-BY) license (http://creativecommons.org/licenses/by/4.0/).

## Share and Cite

**MDPI and ACS Style**

Castillo, L.G.; Carrillo, J.M.
Scour, Velocities and Pressures Evaluations Produced by Spillway and Outlets of Dam. *Water* **2016**, *8*, 68.
https://doi.org/10.3390/w8030068

**AMA Style**

Castillo LG, Carrillo JM.
Scour, Velocities and Pressures Evaluations Produced by Spillway and Outlets of Dam. *Water*. 2016; 8(3):68.
https://doi.org/10.3390/w8030068

**Chicago/Turabian Style**

Castillo, Luis G., and José M. Carrillo.
2016. "Scour, Velocities and Pressures Evaluations Produced by Spillway and Outlets of Dam" *Water* 8, no. 3: 68.
https://doi.org/10.3390/w8030068