# 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

- Niebuhr, C.M.; van Dijk, M.; Bhagwan, J.N. Development of a design and implementation process for the integration of hy-drokinetic devices into existing infrastructure in South Africa. Water SA
**2019**, 45, 434–446. [Google Scholar] [CrossRef] [Green Version] - Riglin, J.D. Design, Manufacture and Prototyping of a Hydrokinetic Turbine Unit for River Application. Master’s Thesis, Lehigh University, Bethlehem, PA, USA, 2016. [Google Scholar]
- Runge, S. Performance and Technology Readiness of a Freestream Turbine in a Canal Environment. Ph.D. Thesis, Cardiff University, Cardiff, UK, 2018. [Google Scholar]
- Kartezhnikova, M.; Ravens, T.M. Hydraulic impacts of hydrokinetic devices. Renew. Energy
**2014**, 66, 425–432. [Google Scholar] [CrossRef] [Green Version] - Gunawan, B.; Roberts, J.; Neary, V. Hydrodynamic Effects of Hydrokinetic Turbine Deployment in an Irrigation Canal. In Proceedings of the 3rd Marine Energy Technology Symposium, Washington, DC, USA, 27–29 April 2015; pp. 1–6. [Google Scholar]
- Bahaj, A.S.; Myers, L.E.; Rawlinson-Smith, R.I.; Thomson, M. The effect of boundary proximity upon the wake structure of horizontal axis marine current turbines. J. Offshore Mech. Arct. Eng.
**2011**, 134, 021104. [Google Scholar] [CrossRef] - Bachant, P.; Wosnik, M. Effects of Reynolds Number on the Energy Conversion and Near-Wake Dynamics of a High Solidity Vertical-Axis Cross-Flow Turbine. Energies
**2016**, 9, 73. [Google Scholar] [CrossRef] [Green Version] - Turnock, S.R.; Phillips, A.B.; Banks, J.; Nicholls-Lee, R. Modelling tidal current turbine wakes using a coupled RANS-BEMT approach as a tool for analysing power capture of arrays of turbines. Ocean Eng.
**2011**, 38, 1300–1307. [Google Scholar] [CrossRef] [Green Version] - Mycek, P.; Gaurier, B.; Germain, G.; Pinon, G.; Rivoalen, E. Experimental study of the turbulence intensity effects on marine current turbines behaviour. Part II: Two interacting turbines. Renew. Energy
**2014**, 68, 876–892. [Google Scholar] [CrossRef] [Green Version] - Hill, C.; Neary, V.S.; Guala, M.; Sotiropoulos, F. Performance and Wake Characterization of a Model Hydrokinetic Turbine: The Reference Model 1 (RM1) Dual Rotor Tidal Energy Converter. Energies
**2020**, 13, 5145. [Google Scholar] [CrossRef] - Lalander, E.; Leijon, M. In-stream energy converters in a river—Effects on upstream hydropower station. Renew. Energy
**2011**, 36, 399–404. [Google Scholar] [CrossRef] - Myers, L.; Bahaj, A.S. Wake studies of a 1/30th scale horizontal axis marine current turbine. Ocean Eng.
**2007**, 34, 758–762. [Google Scholar] [CrossRef] - Adamski, S.J. Numerical Modeling of the Effects of a Free Surface on the Operating Characteristics of Marine Hydrokinetic Turbines. Ph.D. Thesis, University of Washington, Washington, DC, USA, 2013. [Google Scholar]
- Henderson, F.M. Open Channel Flow; The Mcmillan Company: New York, NY, USA, 1966. [Google Scholar]
- Birjandi, A.H.; Bibeau, E.L.; Chatoorgoon, V.; Kumar, A. Power measurement of hydrokinetic turbines with free-surface and blockage effect. Ocean Eng.
**2013**, 69, 9–17. [Google Scholar] [CrossRef] - Niebuhr, C.; van Dijk, M.; Neary, V.; Bhagwan, J. A review of hydrokinetic turbines and enhancement techniques for canal installations: Technology, applicability and potential. Renew. Sustain. Energy Rev.
**2019**, 113, 109240. [Google Scholar] [CrossRef] - Whelan, J.I.; Graham, J.M.R.; Peiró, J. A free-surface and blockage correction for tidal turbines. J. Fluid Mech.
**2009**, 624, 281–291. [Google Scholar] [CrossRef] - Polagye, B.L. Hydrodynamic Effects of Kinetic Power Extraction by In-Stream Tidal Turbines; University of Washington: Washington, DC, USA, 2009. [Google Scholar]
- Bryden, I.; Grinsted, T.; Melville, G. Assessing the potential of a simple tidal channel to deliver useful energy. Appl. Ocean Res.
**2004**, 26, 198–204. [Google Scholar] [CrossRef] - Chanson, H. Hydraulics of Open Channel Flow, 2nd ed.; Elsevier Science & Technology: Amsterdam, The Netherlands, 2004. [Google Scholar]
- Mańko, R. Ranges of Backwater Curves in Lower Odra. Civ. Environ. Eng. Rep.
**2018**, 28, 25–35. [Google Scholar] [CrossRef] [Green Version] - Garrett, C.; Cummins, P. The efficiency of a turbine in a tidal channel. J. Fluid Mech.
**2007**, 588, 243–251. [Google Scholar] [CrossRef] [Green Version] - Garrett, C.; Cummins, P. The power potential of tidal currents in channels. Proc. R. Soc. A Math. Phys. Eng. Sci.
**2005**, 461, 2563–2572. [Google Scholar] [CrossRef] - Ross, H.; Polagye, B. An experimental assessment of analytical blockage corrections for turbines. Renew. Energy
**2020**, 152, 1328–1341. [Google Scholar] [CrossRef] [Green Version] - López, Y.; Contreras, L.; Laín, S. CFD Simulation of a Horizontal Axis Hydrokinetic Turbine. Renew. Energy Power Qual. J.
**2017**, 1, 512–517. [Google Scholar] [CrossRef] - Laín, S.; Contreras, L.T.; López, O. A review on computational fluid dynamics modeling and simulation of horizontal axis hydrokinetic turbines. J. Braz. Soc. Mech. Sci. Eng.
**2019**, 41, 375. [Google Scholar] [CrossRef] - Adcock, T.A.; Draper, S.; Nishino, T. Tidal power generation—A review of hydrodynamic modelling. J. Power Energy
**2015**, 229, 755–771. [Google Scholar] [CrossRef] - Nishino, T.; Willden, R.H. Effects of 3-D channel blockage and turbulent wake mixing on the limit of power extraction by tidal turbines. Int. J. Heat Fluid Flow
**2012**, 37, 123–135. [Google Scholar] [CrossRef] - Nishino, T.; Willden, R.H.J. Two-scale dynamics of flow past a partial cross-stream array of tidal turbines. J. Fluid Mech.
**2013**, 730, 220–244. [Google Scholar] [CrossRef] [Green Version] - Gotelli, C.; Musa, M.; Guala, M.; Escauriaza, C. Experimental and Numerical Investigation of Wake Interactions of Marine Hydrokinetic Turbines. Energies
**2019**, 12, 3188. [Google Scholar] [CrossRef] [Green Version] - Sanderse, B.; van der Pijl, S.P.; Koren, B. Review of computational fluid dynamics for wind turbine wake aerodynamics. Wind Energy
**2011**, 14, 799–819. [Google Scholar] [CrossRef] [Green Version] - Whale, J.; Anderson, C.; Bareiss, R.; Wagner, S. An experimental and numerical study of the vortex structure in the wake of a wind turbine. J. Wind Eng. Ind. Aerodyn.
**2000**, 84, 1–21. [Google Scholar] [CrossRef] - Pyakurel, P.; Tian, W.; VanZwieten, J.H.; Dhanak, M. Characterization of the mean flow field in the far wake region behind ocean current turbines. J. Ocean Eng. Mar. Energy
**2017**, 3, 113–123. [Google Scholar] [CrossRef] - Masters, I.; Chapman, J.C.; Willis, M.R.; Orme, J.A.C. A robust blade element momentum theory model for tidal stream tur-bines including tip and hub loss corrections. J. Mar. Eng. Technol.
**2014**, 10, 25–35. [Google Scholar] [CrossRef] [Green Version] - Guo, Q.; Zhou, L.; Wang, Z. Comparison of BEM-CFD and full rotor geometry simulations for the performance and flow field of a marine current turbine. Renew. Energy
**2015**, 75, 640–648. [Google Scholar] [CrossRef] - Malki, R.; Masters, I.; Williams, A.J.; Croft, N. The variation in wake structure of a tidal stream turbine with flow velocity. In Proceedings of the MARINE 2011, IV International Conference on Computational Methods in Marine Engineering, Lisbon, Portugal, 28–30 September 2011. [Google Scholar] [CrossRef]
- Edmunds, M.; Williams, A.; Masters, I.; Croft, N. An enhanced disk averaged CFD model for the simulation of horizontal axis tidal turbines. Renew. Energy
**2017**, 101, 67–81. [Google Scholar] [CrossRef] [Green Version] - Masters, I.; Williams, A.; Croft, T.N.; Togneri, M.; Edmunds, M.; Zangiabadi, E.; Fairley, I.; Karunarathna, H. A Comparison of Numerical Modelling Techniques for Tidal Stream Turbine Analysis. Energies
**2015**, 8, 7833–7853. [Google Scholar] [CrossRef] [Green Version] - Masters, I.; Malki, R.; Williams, A.J.; Croft, T.N. The influence of flow acceleration on tidal stream turbine wake dynamics: A numerical study using a coupled BEM–CFD model. Appl. Math. Model.
**2013**, 37, 7905–7918. [Google Scholar] [CrossRef] - Niebuhr, C.; Schmidt, S.; van Dijk, M.; Smith, L.; Neary, V. A review of commercial numerical modelling approaches for axial hydrokinetic turbine wake analysis in channel flow. Renew. Sustain. Energy Rev.
**2022**, 158, 112151. [Google Scholar] [CrossRef] - Mycek, P.; Gaurier, B.; Germain, G.; Pinon, G.; Rivoalen, E. Experimental study of the turbulence intensity effects on marine current turbines behaviour. Part I: One single turbine. Renew. Energy
**2014**, 66, 729–746. [Google Scholar] [CrossRef] [Green Version] - Hill, C.; Neary, V.S.; Gunawan, B.; Guala, M.; Sotiropoulos, F.U.S. Department of Energy Reference Model Program RM1: Experimental Results; University of Minnesota: Minneapolis, MN, USA, 2014. [Google Scholar]
- Nasef, M.H.; El-Askary, W.A.; AbdEL-hamid, A.A.; Gad, H.E. Evaluation of Savonius rotor performance: Static and dynamic studies. J. Wind Eng. Ind. Aerodyn.
**2013**, 123, 1–11. [Google Scholar] [CrossRef] - Franke, J.; Hirsch, C.; Jensen, A.G.; Krus, H.W.; Schatzmann, P.S.; Miles, S.D.; Wisse, J.A.; Wright, N.G. Recommendations on the use of CFD in wind engineering. In Proceedings of the CWE2006 Fourth International Symposium Computational Wind Engineering, Yokohama, Japan, 16–19 July 2006. [Google Scholar]
- Malki, R.; Williams, A.; Croft, T.; Togneri, M.; Masters, I. A coupled blade element momentum—Computational fluid dynamics model for evaluating tidal stream turbine performance. Appl. Math. Model.
**2013**, 37, 3006–3020. [Google Scholar] [CrossRef] [Green Version] - Bekker, A.; Van Dijk, M.; Niebuhr, C.M. A review of low head hydropower at wastewater treatment works and development of an evaluation framework for South Africa. Renew. Sustain. Energy Rev.
**2022**, 159, 112216. [Google Scholar] [CrossRef] - Shen, W.Z.; Mikkelsen, R.; Sørensen, J.N.; Bak, C. Tip loss corrections for wind turbine computations. Wind Energy
**2005**, 8, 457–475. [Google Scholar] [CrossRef] - Speziale, C.G.; Sarkar, S.; Gatski, T.B. Modelling the pressure-strain correlation of turbulence: An invariant dynamical systems approach. J. Fluid. Mech.
**1991**, 227, 245–272. [Google Scholar] [CrossRef] - Sarkar, S.; Lakshmanan, B. Application of a Reynolds stress turbulence model to the compressible shear layer. AIAA J.
**1991**, 29, 743–749. [Google Scholar] [CrossRef] [Green Version] - Roache, P.J. Perspectvie: A method for Uniform Reporting of Grid Refinement Studies. J. Fluids Eng. Trans. ASME
**1994**, 116, 405–413. [Google Scholar] [CrossRef] - Silva, P.A.S.F.; De Oliveira, T.F.; Brasil Junior, A.C.P.; Vaz, J.R.P.P.; Oliveira, T.F.D.E.; Junior, A.C.P.B.; Vaz, J.R.P.P. Numerical Study of Wake Characteristics in a Horizontal-Axis Hydrokinetic Turbine. Ann. Braziian Acad. Sci.
**2016**, 88, 2441–2456. [Google Scholar] [CrossRef] [Green Version] - Gibson, M.M.; Launder, B.E. Ground effects on pressure fluctuations in the atmospheric boundary layer. J. Fluid Mech.
**1978**, 86, 491–511. [Google Scholar] [CrossRef] - Neary, V.S.; Gunawan, B.; Hill, C.; Chamorro, L.P. Near and far field flow disturbances induced by model hydrokinetic tur-bine: ADV and ADP comparison. Renew. Energy
**2013**, 60, 1–6. [Google Scholar] [CrossRef] - Yarnell, D. Bridge Piers as Channel Obstructions; United States Department of Agriculture: Washington, DC, USA, 1934. [Google Scholar]
- Azinfar, H.; Kells, J.A. Backwater Prediction due to the Blockage Caused by a Single, Submerged Spur Dike in an Open Channel. J. Hydraul. Eng.
**2008**, 134, 1153–1157. [Google Scholar] [CrossRef] - Martin-Vide, J.; Prio, J. Backwater of arch bridges under free and submerged conditions. J. Hydraul. Res.
**2005**, 43, 515–521. [Google Scholar] [CrossRef] - Follett, E.; Schalko, I.; Nepf, H. Momentum and Energy Predict the Backwater Rise Generated by a Large Wood Jam. Geophys. Res. Lett.
**2020**, 47, e2020GL089346. [Google Scholar] [CrossRef] - Kocaman, S. Prediction of Backwater Profiles due to Bridges in a Compound Channel Using CFD. Adv. Mech. Eng.
**2014**, 6, 905217. [Google Scholar] [CrossRef] - Azinfar, H.; Kells, J.A. Drag force and associated backwater effect due to an open channel spur dike field. J. Hydraul. Res.
**2011**, 49, 248–256. [Google Scholar] [CrossRef] - Raju, K.R.; Rana, O.; Asawa, G.; Pillai, A. Rational assessment of blockage effect in channel flow past smooth circular cylinders. J. Hydraul. Res.
**1983**, 21, 289–302. [Google Scholar] [CrossRef] - Morandi, B.; Di Felice, F.; Costanzo, M.; Romano, G.; Dhomé, D.; Allo, J. Experimental investigation of the near wake of a horizontal axis tidal current turbine. Int. J. Mar. Energy
**2016**, 14, 229–247. [Google Scholar] [CrossRef] - Jeffcoate, P.; Whittaker, T.; Boake, C.; Elsaesser, B. Field tests of multiple 1/10 scale tidal turbines in steady flows. Renew. Energy
**2016**, 87, 240–252. [Google Scholar] [CrossRef] - Stallard, T.; Collings, R.; Feng, T.; Whelan, J. Interactions between tidal turbine wakes: Experimental study of a group of three-bladed rotors. Philos. Trans. R. Soc. A Math. Phys. Eng. Sci.
**2013**, 371, 20120159. [Google Scholar] [CrossRef] [PubMed] - Lam, W.-H.; Chen, L. Equations used to predict the velocity distribution within a wake from a horizontal-axis tidal-current turbine. Ocean Eng.
**2014**, 79, 35–42. [Google Scholar] [CrossRef] - Sandia National Laboritories: Refernce Model Porject (RMP). Available online: https://energy.sandia.gov/programs/renewable-energy/water-power/projects/reference-model-project-rmp/ (accessed on 21 January 2022).

**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 |

Publisher’s Note: MDPI stays neutral with regard to jurisdictional claims in published maps and institutional affiliations. |

© 2022 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (https://creativecommons.org/licenses/by/4.0/).

## 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