# Sensitivity Analysis of Adjustable River Surf Waves in the Absence of Channel Drop

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## Abstract

**:**

## 1. Introduction

#### Necessity of the Research and Novelty

## 2. Methodology

#### 2.1. Mathematical Model and Numerical Solver

#### 2.2. Calibration

#### 2.3. Parametric Analysis

**ψ**). Wave height (${W}_{h}$), wave slope ($S$) and depth averaged velocity at the kicker tip (${V}_{kt}$) are characteristics of the surf wave. Flowrate per unit width ($q$) and tailwater depth (${T}_{w}$) are flow features (hydraulic boundary conditions). Gravitational acceleration ($g$), dynamic viscosity of the fluid ($\mu $), and fluid density ($\rho $) are ambient and fluid properties. Equation (6) shows all the influential variables in this study in functional form:

^{3}/s/m [36] which has been covered in the experiments of this research (${q}_{max}=4{\mathrm{m}}^{3}/\mathrm{s}/\mathrm{m}$) and could be achieved in practice by constricting the river width. The tailwater depth can be changed from the initial value, normal depth of the river, to a maximum value in which the surf wave breaks into a conventional hydraulic jump. In this research, tailwater depth ranged from the normal depth of the Kananaskis River and iteratively increased to meet the maximum value. Tailwater depth adjustment takes place by stop-logs or coffer dams in practice which is not shown in Figure 1. While the modelling has been parameterized using this case study, the dimensionless results are generalizable to other locations.

## 3. Results

#### 3.1. Ramp

^{0.5}/s) when ramp slope is between $R/H$ = 6.23 (ϕ = 9°) and 3.30 (ϕ = 18°) (Figure 5). The ratio of water depth on weir crest to critical depth decreased from 1.12 to 1.02 as $R/H$ decreased from 6.23 (ϕ = 9°) to 3.30 (ϕ = 18°) which implies that the ramp crest cannot be considered as a control section.

#### 3.2. Transition

#### 3.3. Tailwater Depth

^{3}/s (${Q}_{max}$) to 16 m

^{3}/s ($0.5{Q}_{max}$). The Froude number calculated using these normal depths varies from 0.374 to 0.356, respectively, which will roughly be referred to as 0.37 hereafter due to its slight variation. Figure 8 shows how increasing tailwater depth from normal to maximum value affects the wave. Note that Figure 8a shows only the surf wave while omitting the downstream water surface fluctuations, but tailwater elevation (downstream boundary) can be observed in Figure 8b. For the range of flowrate from $0.5{Q}_{max}$ to ${Q}_{max}$ the ratio of maximum tailwater depth to normal tailwater depth is approximately 1.7. At maximum tailwater depth, where the wave is at the threshold of breaking, downstream Froude number varies from 0.161 to 0.148 for flowrate changing from ${Q}_{max}$ to $0.5{Q}_{max}$. Due to slight variation of Froude number at maximum tailwater depth, it will roughly be referred to as 0.16 hereafter regardless of the flowrate. At ${Q}_{max}$ and maximum tailwater condition, tailwater depth is 60% larger than ramp height, yet the wave is not broken. Wave height as the most important parameter in designing a surf wave showed no significant change by flowrate when tailwater depth is normal. However, in maximum tailwater condition, increase in flowrate considerably increases the wave height. In other words, increasing the tailwater depth not only increases the wave height, but also makes the wave height sensitive to flowrate. The tailwater depth as the major parameter affecting the wave height could increase the wave height by up to 2.2 times the initial value. This increase in wave height is also associated with increase in wave steepness.

#### 3.4. Kicker Geometry

**ψ**) constant and increasing the length ($K$) or keeping the length constant and increasing the angle. Figure 9 shows that increasing kicker length increases both wave height and steepness. At the constant angle

**ψ**= 25°, increasing $K/H$ from 0.75 to 1.75 increased the wave height by 30% and steepness by 80%. The wave breaks if $K/H$ exceeds 1.75. Kicker geometry also significantly affects the second wave. In certain range of geometric and hydraulic boundary conditions the first wave (over the kicker) is not a hydraulic jump; rather, the flow leaving the kicker section remains supercritical leading to formation of a wave train in the downstream pool. Figure 10a shows an example wave train, naturally existing in Ottawa River, with a surfer on the second wave. Figure 9 shows how elongation of the kicker augments the second wave and brings it to break. Figure 10b shows 3-D velocity fields, cut along the longitudinal axis, for shorter and longer kickers. A triangular pattern around the broken apex of the second wave, resembling the Ottawa River wave, can be seen in the flow field with the larger kicker, vividly demonstrating the effect of kicker enlargement on increasing height and steepness of the first and second wave.

**ψ**). A set of experiments at $K/H=0.75$ with 5-degree increments of kicker angle was conducted to find the maximum kicker angle without wave breaking. Figure 11a compares waves generated by the kicker of the maximum length versus the kicker of the maximum angle. The “first wave” generated by maximum kicker length ($K/H=1.75$,

**ψ**= 25°) has almost the same height and steepness as the first wave generated by maximum kicker angle ($K/H=0.75$,

**ψ**= 35°). However, the longer kicker has a higher vertical projected height than the steepest kicker and the second wave is significantly higher and steeper in case of using the longer kicker.

**ψ**= 25°) reduction in the tailwater from maximum to normal reduced the wave height more than 60% (Figure 8), but with the extreme kicker the corresponding value is about 30% (Figure 11b).

**ψ**= 10° (extremely long) and extremely steep: $K/H=0.75$,

**ψ**= 25°. These two extreme kickers have almost similar vertical projected height, but the resultant wave of extremely long kicker is higher and less steep than that of extremely steep kicker (Figure 11c).

## 4. Discussion

## 5. Conclusions

^{0.5}/s was found to be valid for calculation of water head over the ramps of different slope. Results demonstrated slight sensitivity of the wave to ramp slope and the recommended value that maximizes the wave and minimizes the cost was found to be $R/H=3.88$ (ϕ = 15°). The transition, which is the horizontal distance between ramp toe and kicker, should be long enough to facilitate full development of the trough, but no longer to inhibit development of a H3 profile. Numerical experiments of this research recommend $T/H=3.55$ for a range of $\frac{q}{\sqrt{g{H}^{3}}}$ = 0.64 to 1.25. Results showed that although the ramp slope does not have significant effect on the wave profile, tailwater depth could significantly augment the wave. As an estimate of maximum tailwater depth, which could be provided by means of regulating structures downstream, the results showed maximum tailwater depth can be 60% higher than the ramp crest. Increasing the tailwater depth from normal depth to the maximum value can double the wave height. The kicker is the most convenient design element for readjustment and results showed the wave has considerable sensitivity to this geometric parameter. The surf wave can be brought to the threshold of breaking by either elongating or steepening the kicker. In the range of numerical experiments reported herein, extreme kickers were found to be $K/H=0.75$,

**ψ**= 35° (steepest), and $K/H=1.75$, ψ = 25° (longest).

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

## Nomenclature

$C$ | weir coefficient |

$d$ | distance from the wall |

$E$ | water head over the weir; |

${e}_{ij}$ | component of rate of deformation; |

${F}_{b}$ | body forces; |

$F{r}_{downstream}$ | Froude number of the flow downstream of the wave; |

$g$ | gravitational acceleration; |

$H$ | ramp height; |

$K$ | kicker length; |

$k$ | turbulent kinetic energy; |

$L$ | weir length; |

$p$ | pressure; |

$Q$ | flowrate; |

$q$ | flowrate per unit width of the wave structure; |

$R$ | ramp length; |

$S$ | wave slope; |

$T$ | transition length; |

${T}_{w}$ | tailwater depth; |

$t$ | time; |

$V$ | velocity; |

${V}_{i}$ | velocity component in each direction; |

${V}_{kt}$ | depth averaged velocity at kicker tip; |

${V}_{\tau}$ | shear velocity; |

${W}_{h}$ | wave height; |

${y}^{+}$ | dimensionless wall coordinate; |

${Y}_{C}$ | critical depth; |

${Y}_{weir}$ | depth over the weir crest; |

$\beta $ | volume fraction (water toair ration); |

$\rho $ | density; |

$\mu $ | dynamic viscocity; |

${\mu}_{t}$ | eddy viscosity; |

$\epsilon $ | turbulent dissipation rate; |

ϕ | ramp acute angle with respect to horizon; |

ψ | kicker acute angle with respect to horizon; |

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**Figure 1.**Structure under investigation: (

**a**) perspective, (

**b**) profile, (

**c**) geometry of the numerical model (this does not show the full model extent to the upstream and downstream boundaries).

**Figure 2.**Variation of the wave height versus grid size of the computational mesh. (flow per unit width of the ramp = 3.5 m

^{2}/s, tailwater depth = 0.86 m, ramp height = 1.0 m, Kicker length = 0.75 m, kicker angle = 25°).

**Figure 3.**Transition from surf wave to broken wave (conventional hydraulic jump) by tailwater manipulation—photo courtesy of Wyl [33].

**Figure 4.**Calibration of the numerical model with laboratory data: (

**a**) physical versus numerical simulated wave (images are at the same scale—physical model photo courtesy of Koch [24]), (

**b**) comparison of water surface profiles.

**Figure 6.**Effect of the ramp profile on the surf wave: (

**a**) effect of the ramp steepness on wave profile and velocity distribution, (

**b**) sensitivity of the surf wave to ramp slope.

**Figure 7.**Effect of the transition length on the surf wave: (

**a**) required transition for complete development of the trough, (

**b**) sensitivity of the surf wave to transition length.

**Figure 8.**Sensitivity of the surf wave to tailwater depth at different flowrates: (

**a**) effect of tailwater depth on wave profile, (

**b**) tailwater depth, wave height and steepness versus flowrate.

**Figure 10.**Flow field of river surf waves: (

**a**) wave train in Ottawa river, (

**b**) effect of kicker length on velocity field—cutaway view down the channel centerline.

**Figure 11.**Wave adjustment by means of the kicker: (

**a**) extremely long versus extremely steep kicker, (

**b**) sensitivity of the extreme wave to reduction in tailwater depth, (

**c**) extreme kickers of same height—longer versus steeper.

**Figure 12.**Similarity of wave profiles for ramps of different height; (

**a**) natural dimension water surface profiles, (

**b**) scaled (dimensionless) water surface profiles.

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**MDPI and ACS Style**

Asiaban, P.; Rennie, C.D.; Egsgard, N.
Sensitivity Analysis of Adjustable River Surf Waves in the Absence of Channel Drop. *Water* **2021**, *13*, 1287.
https://doi.org/10.3390/w13091287

**AMA Style**

Asiaban P, Rennie CD, Egsgard N.
Sensitivity Analysis of Adjustable River Surf Waves in the Absence of Channel Drop. *Water*. 2021; 13(9):1287.
https://doi.org/10.3390/w13091287

**Chicago/Turabian Style**

Asiaban, Puria, Colin D. Rennie, and Neil Egsgard.
2021. "Sensitivity Analysis of Adjustable River Surf Waves in the Absence of Channel Drop" *Water* 13, no. 9: 1287.
https://doi.org/10.3390/w13091287