# Development of a Hydrokinetic Turbine Backwater Prediction Model for Inland Flow through Validated CFD Models

^{1}

^{2}

^{3}

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Background

_{T}= 0.4 m), and observed a clear difference in the water surface once energy was extracted. Water depths increased immediately upstream of the rotor and decreased downstream for about 2 d

_{T}. Details of the water surface profiles can be seen in Figure 1. The results observed a standing wave 7–8 d

_{T}downstream (it should be noted that this was for the high-freestream-velocity case).

_{t}is the turbine diameter) may be more useful, which has also been found to govern the free-surface effects [13].

- Free-surface effects in the form of a possible standing wave formed, or decreased water surface above the turbine (due to decreasing pressure).
- Potential backwater effects caused (e.g., damming upstream).

#### 2.1. Free-Surface Effects of HK Turbines

_{D}. Additionally, the blockage ratio also affects this free-surface change, albeit not as strongly as Fr

_{D}[13]

_{.}

_{t}downstream of the turbine rotor (as shown in Figure 2). In addition, due to the wake expansion coincident with the free surface, cumulative turbine placement at intervals smaller than the recovery length may cause the flow to approach critical depth, causing severe undulations in the water surface profiles (WSPs). Turbine operation and efficiency may also vary due to decreasing fluid velocity over the blades during operation. Accurate quantification of the WSPs around an array may be a challenge due to the multiple effects of turbulence, wake mixing, and superposition of WSP effects [12].

_{h}, defined as follows:

#### 2.2. Backwater Effect

_{T}), channel flow area (A

_{o}) ($BR\left(\%\right)=\raisebox{1ex}{${A}_{T}$}\!\left/ \!\raisebox{-1ex}{${A}_{o}$}\right.$), and additional constrictions [16], as well as the theoretical to actual efficiency [17].

_{D}) of the flow can influence the backwater effect. A previous study analysing this effect drew the following conclusions [13]:

- The upstream free-surface deformation increased with Fr
_{D}. - The location of maximum damming (i.e., the highest water level) moved closer to the turbine as Fr
_{D}increased.

#### 2.3. Backwater Calculations

^{1/3}), R

_{h}is the hydraulic radius of the channel (m), and U is the velocity of the water (m/s). The change in water levels (Δz) between two sections can then be determined between two significant cross-sections (e.g., 0 and 1) and calculated as shown in Equation (7), where $\alpha $ is the Coriolis coefficient and U

_{0}and U

_{1}are the average velocities over distance $\Delta L$:

_{t}is the total power of the turbine (W). The formulation of the stress term ${\tau}_{f}$ is shown in Equation (11), where f is equal to the Darcy–Weisbach coefficient (unitless) and U is the velocity of water (m/s).

#### 2.4. Summary of Literature

## 3. Validation of CFD Models

#### 3.1. CFD Models

#### 3.2. RM1 Model Validation

_{t}. These were used previously to validate the CFD procedure for a single-phase analysis [40].

_{t}(diameters) in the streamwise direction, and 0.4 d

_{t}in the cross-stream direction. The measurement zone was −5 dt to 10 dt downstream. Elevation data were sampled at 50 Hz for 120 seconds at each location using a Massa ultrasonic range sensor, allowing for both time-averaged and fluctuating water surface elevation analysis and CFD validation.

_{t}upstream to 16 d

_{t}downstream of the axis of rotation. The specified inlet length allowed full flow development prior to reaching the turbine axis of rotation. The outlet length ensured that no effects from the downstream boundary condition affected the near-wake behaviour. Previous studies have found that around 15 d

_{t}is usually adequate for the outlet boundary length [43,44].

^{+}wall treatment on the turbine and turbine structure (y

^{+}< 1).

_{t}) proved adequate in the far-wake region.

## 4. Methods

#### 4.1. Assumptions and Exclusions

- Subcritical flow regime (Fr < 1);
- 5000 < Re < 1,500,000;
- Typical operational velocities of channels (0.8–2.8 m/s);
- Manning n-value around 0.016–0.023 s/m
^{1/3}(lined channel).

#### 4.2. Mathematical Formulation

_{e}), included as an energy loss:

_{L}is the loss coefficient predefined for typical losses in a channel. The drop in water level due to a particular loss can be quantified/included by applying either the momentum or energy equation over a channel section, and the upstream and downstream sections (in which the energy loss exists). Additionally, an empirical approach may be used, where experimental results are used to determine an empirical relationship, such as that done by Yarnell in 1934 for bridge piers [54].

#### 4.2.1. Approach 1: Momentum Approach

_{1}) and downstream (F

_{2}) of the device (water level change), as well as the friction from the channel bed and walls (F

_{f}) and the force due to the turbine (F

_{D}). This can then be rewritten to Equation (14), in terms of the drag force (F

_{D}) due to the presence of the turbine.

#### 4.2.2. Approach 2: Energy Approach

_{f}) or local losses (h

_{l}).

_{t}) can be written as a function of a loss coefficient ($\alpha $), the freestream velocity (U), and the blockage ratio of the turbine, as shown in Equation (16).

_{t}) may also be quantified as a pressure drop, which is then directly converted to an energy loss as follows:

#### 4.2.3. Validation of Pressure Drop Measurement in CFD Results

_{t}shown in Equation (17) holds true, the $\Delta {P}_{t}$ was measured in the CFD model for the RM1 validation case. The subsequently calculated loss (h

_{t}) was then compared to the measured backwater effects in the laboratory tests (as well as multiphase CFD analysis). Inclusion of the support structure blockage was incorporated using the Yarnell approximation. The Yarnell approximation for a single circular bridge pier (similar to the support stanchion) was implemented:

_{r}is the downstream Froude number, and α is the ratio of the flow area obstructed by the pier to the total flow area downstream of the pier (also referred to as the blockage ratio). K is used as a coefficient reflecting the pier’s shape. To ensure that the Yarnell approximation and pressure loss ($\Delta {P}_{t})$calculation work independently, the RM1 model free-surface deformation was measured with and without the stanchion structure (Figure 11), and the results were compared to the backwater calculation using only the pressure drop, as well as including the stanchion through the Yarnell approximation.

_{t}, as would be the result of the RM1 device in channel flow.

#### 4.2.4. Lambda Approximation

_{T}was selected as the energy loss coefficient used in the energy equation:

_{t}is included as a loss in the energy equation (Equation (15)), and λ

_{T}is calculated as a function of the thrust coefficient (C

_{t}):

_{t}is a value that can be obtained from the manufacturer, calculated, or assumed in the pre-feasibility stage. For HAHTs, these thrust coefficients (C

_{t}) usually range from 0.52 to 0.89 [9,62,63,64]. According to the actuator disk theory, C

_{t}may be written in terms of the induction factor a [65]. It is also known that ideally, according to the Betz limit, a = $\frac{1}{3}$; therefore, the ideal and highest attainable C

_{t}would be 0.88. Theoretically, according to the BEM theory, this should result in the highest velocity deficit in the near wake and, therefore, the “worst case” scenario for the operational conditions. Realistically, the values lie at an upper limit of C

_{t}= 0.8. The thrust coefficient can be calculated directly if the thrust force (T), inlet velocity (U), and swept area (A) are known:

_{T}approximation, the validated CFD models were analysed, the pressure drop/total thrust was measured, and the subsequent backwater effect was determined. The calculated h

_{t}(through Equation (19)) was then compared to the h

_{t}determined through the $\Delta {P}_{t}$ (Equation (17)) results, as validated in Section 4.2.3.

_{T}were included (calculated and assumed C

_{t}). The model should be usable with only basic knowledge of the turbine installation and operating parameters; therefore, simple available metrics could be used to obtain a conservative result. Acceptable correlation between the experimental and calculated values created confidence to proceed with the model and build a larger dataset to analyse the model’s accuracy at a larger operational variance from optimal conditions.

- Inlet velocity changes (0.4 < U < 2.8);
- Blockage ratio changes (Swept area to flow area) (4% < BR < 23%);
- Tip speed ratio changes (lower or higher load applied) (3 < TSR < 6);
- Froude number (0.18 < Fr < 0.34) (within the subcritical flow regime);
- Froude number based on turbine diameter (0.15 < Fr
_{D}< 0.9).

_{t}). Additionally, as the sample size changed in the analysis, the strength of the sample size effect was minimized when comparing MAE. The variance was also included to give an indication of the test conditions with greater variability, and under which test conditions the model (and assumptions) performed best.

_{t}for variations in blockage ratio (BR) velocity (U) and Fr

_{D}are shown in Figure 14.

- At turbine optimal operational points, a maximum deviation of 13% from the predicted backwater was obtained when using the correct C
_{t}value. This deviation increased to 19% for the C_{t}= 0.8 approximation. - When utilizing the C
_{t}assumption of 0.8, a conservative result was obtained, with the backwater estimation generally overestimating the measured blockage. - Calculating C
_{t}based on the turbine thrust (measured thrust) lowered the h_{t}approximation. However, for test cases operating close to the optimal performance and highest C_{t}value, the backwater was underestimated by up to 20%. - Test cases at low operational velocities (low Froude numbers) resulted in larger errors in approximating h
_{t}; however, it is important to note that these are unfavourable installation conditions and far from typical installations. The turbines may have low performance at these low operational velocities and, therefore, pose an unrealistic scenario. Here, the C_{t}calculation resulted in a more realistic value, due to the reduced performance. - The C
_{t}approximation resulted in large overestimations of the h_{t}at lower TSRs. However, the C_{t}equation (Equation (22)) performed well in these scenarios, as the turbine thrust was significantly lower, and the C_{t}assumption did not hold. - The C
_{t}approximation gave significantly better results for the three-bladed turbines. The two-bladed (T1) case predicted better results with the C_{t}calculation, which was also higher than the 0.8 approximation, indicating that the turbine operates closer to the Betz limit and ideal induction factor (a), which could be further tested and calibrated. The C_{t}calculation performed better in this case, predicting C_{t}= 0.89. Therefore, utilizing this assumption may be favourable for avoiding errors—especially when turbines with higher operational tip speed ratios are used.

_{t}value for each turbine can be determined empirically, which could be improved with a larger dataset.

## 5. Conclusions

## Author Contributions

## Funding

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

## References

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**Figure 1.**Water surface profile through a scaled turbine operating at 2 different velocities, compared to the no-energy-extraction stage [12].

**Figure 2.**Wake expansion effect with free surface [12].

**Figure 3.**Backwater effect due to turbine blockages [1].

**Figure 4.**Influence of artificial energy extraction on speed and depth of flow [19].

**Figure 6.**Computational domain with grid refinements: (

**A**) near wake, (

**B**) blades, and (

**C**) free surface.

**Figure 7.**Comparison of experimental and computational water surface profiles for the RM1 tests: (

**a**) experiment and CFD water surface graphics; (

**b**) lateral WSE comparison; (

**c**) longitudinal centreline WSE comparison.

**Figure 9.**Momentum approach schematic (adapted form [55]).

**Figure 11.**(

**a**,

**b**) Velocity and (

**c**,

**d**) surface water measurements graphics for the RM1 full model vs. the RM1 rotor and nacelle only.

**Figure 12.**Pressure measurements over the horizontal and vertical planes (at the turbine hub height centerline).

**Figure 13.**Pressure measurements over the disk and planes upstream and downstream of the RM1 turbine and retaining structure.

Turbine | Clearance Coefficient | |
---|---|---|

Seaflow | 2-Bladed, 300 kW | 0.18–0.64 |

SeaGen | 2-Bladed, 1.2 MW (2× 600 kW) | 0.25–0.38 |

HS300 | 3-Bladed, 300 kW | 0.75 |

AK-1000 | 3-Bladed, 1 MW | 1.02 |

Turbine | Name | Blades | Diameter (m) | CFD Model |
---|---|---|---|---|

T1 | RM1 [10] | 2-Bladed NACA4415 | 0.5 | Multiphase RSM-BEM model |

T2 | IFREMER [9] | 3-Bladed NACA63418 | 0.7 | Single-phase RSM-BEM model |

T3 | SHP [1] | 3-Bladed custom blade | 1 | Single-phase RSM-FRG model |

**Table 3.**RM1 laboratory setup details [42].

Description | Variable |
---|---|

Rotor diameter | 0.5 m |

Blade profile | NACA 4415 |

Flow depth | 1 m |

Flow rate | 2.425 m^{3}/s |

Tip speed ratios measured | 1 to 9 |

Flow velocity (U_{hub}) | 1.05 m/s |

Turbulence intensity | 5% |

Froude number | 0.28 |

Reynolds number (chord) | ~3.0 × 10^{5} |

ΔP_{t} Disk (Pa) | ΔP_{t} Plane (Pa) | $\mathbf{Calculated}{\mathit{h}}_{\mathit{t}}\left(\mathbf{mm}\right)$ | Yarnell Approx. (mm) | $\mathbf{Measured}{\mathit{h}}_{\mathit{t}}\left(\mathbf{mm}\right)$ | $\frac{\mathit{h}\mathit{t}\mathit{m}\mathit{e}\mathit{a}\mathit{s}-\mathit{h}\mathit{t}\mathit{c}\mathit{a}\mathit{l}\mathit{c}}{\mathit{y}}(\%)$ | |
---|---|---|---|---|---|---|

RM1 (no stanchion) | 570 | 57.73 | 8.30 | - | 9.60 | 0.13% |

RM1 (with stanchion) | 530 | 74.09 | 7.72 | 4.36 | 12.00 | 0.01% |

12.66 |

Test Condition | C_{t} | N | MAE | Variance |
---|---|---|---|---|

All tests conducted | Equation (21) | 14 | 1.45 | 2.26 |

0.8 | 14 | 1.42 | 2.17 | |

0.89 | 14 | 1.99 | 4.27 | |

Optimal operational point | Equation (21) | 3 | 1.26 | 1.85 |

0.8 | 3 | 1.25 | 1.83 | |

Variation of blockage ratios (BR = 4–22%) at optimal operational point | Equation (21) | 4 | 0.35 | 0.16 |

0.8 | 4 | 0.27 | 0.09 | |

Variation of inlet velocities at optimal tip speed ratios | Equation (21) | 4 | 2.4 | 7.65 |

0.8 | 4 | 1.36 | 2.47 |

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## Share and Cite

**MDPI and ACS Style**

Niebuhr, C.M.; Hill, C.; Van Dijk, M.; Smith, L.
Development of a Hydrokinetic Turbine Backwater Prediction Model for Inland Flow through Validated CFD Models. *Processes* **2022**, *10*, 1310.
https://doi.org/10.3390/pr10071310

**AMA Style**

Niebuhr CM, Hill C, Van Dijk M, Smith L.
Development of a Hydrokinetic Turbine Backwater Prediction Model for Inland Flow through Validated CFD Models. *Processes*. 2022; 10(7):1310.
https://doi.org/10.3390/pr10071310

**Chicago/Turabian Style**

Niebuhr, Chantel Monica, Craig Hill, Marco Van Dijk, and Lelanie Smith.
2022. "Development of a Hydrokinetic Turbine Backwater Prediction Model for Inland Flow through Validated CFD Models" *Processes* 10, no. 7: 1310.
https://doi.org/10.3390/pr10071310