Geometric specifications of the considered turbine.

## Abstract

Hydrokinetic turbines are one of the technological alternatives to generate and supply electricity for rural communities isolated from the national electrical grid with almost zero emission. These technologies may appear suitable to convert kinetic energy of canal, river, tidal, or ocean water currents into electricity. Nevertheless, they are in an early stage of development; therefore, studying the hydrokinetic system is an active topic of academic research. In order to improve their efficiencies and understand their performance, several works focusing on both experimental and numerical studies have been reported. For the particular case of flow behavior simulation of hydrokinetic turbines with complex geometries, the use of computational fluids dynamics (CFD) nowadays is still suffering from a high computational cost and time; thus, in the first instance, the analysis of the problem is required for defining the computational domain, the mesh characteristics, and the model of turbulence to be used. In this chapter, CFD analysis of a H-Darrieus vertical axis hydrokinetic turbines is carried out for a rated power output of 0.5 kW at a designed water speed of 1.5 m/s, a tip speed ratio of 1.75, a chord length of 0.33 m, a swept area of 0.636 m2, 3 blades, and NACA 0025 hydrofoil profile.

### Keywords

- renewable energy
- hydrokinetic turbine
- numerical simulation
- power coefficient
- tip-speed ratio

## 1. Introduction

Nowadays, both the developed and developing countries have been actively promoting renewable energy for their environmental and economic benefits such as the reduction of greenhouse gas emissions, decarbonization, and the diversification of the energy supply in order to reduce the dependence on fossil fuels, especially for the production of electricity, creating economic development and jobs in manufacturing and installation of the systems [1, 2, 3, 4]. Several countries have developed policies and programs to promote the use of renewable energy sources. In general, the policies and programs of developing countries are limited in scope, focusing mainly on regulatory measures, with little attention to other relevant aspects such as research and development, market liberalization, information campaigns, and training [3, 4, 5, 6].

The most common renewable power technologies include solar (photovoltaic and solar thermal systems), wind, biogas, geothermal, biomass, low-impact hydroelectricity, and emerging technologies such as wave and hydrokinetic systems. These technologies have unquestionably started to transform the global energy landscape in a promising way [7, 8].

The hydrokinetic systems convert kinetic energy of canal, river, and tidal or ocean water currents into mechanical energy and, consequently, in electrical energy without the use of large civil structures, being the energy provided by the hydrokinetic system more valuable and predictable than that supplied by wind and solar devices [9].

Even though the hydrokinetic systems are still in early stages of development, the referred technology is attractive as renewable energy sources, and it could be a response to the decentralization of energy markets due to the low operating costs, good predict ability, and minimal environmental impact associated [10]. Although the hydrokinetic turbines have relatively small-scale power production, they can be installed as multiunit arrays like wind farms to increase energy extraction. Therefore, these turbines can be used for rural electrification of isolated and river-side communities [10].

The two most common types of hydrokinetic turbines extract energy utilizing horizontal or vertical axis rotor with blade moving through the water [11]. The horizontal and vertical axis turbines have axes parallel and perpendicular to the fluid flow, respectively. Horizontal axis turbines are widely used in tidal energy converters [11, 12]. In contrast, the hydrokinetic turbines with vertical axis are generally used for small-scale power generation because they are less expensive and require lower maintenance than horizontal axis turbines [12, 13]. On the other hand, the rotor of vertical axis hydrokinetic turbines can rotate regardless of the flow direction, which constitutes an advantage. In turn, horizontal axis hydrokinetic turbines typically reach higher tip speeds, making them more prone to cavitation, which reduce the efficiency and create surface damage [11, 12, 13]. In spite of vertical axis turbine being not as efficient as the horizontal axis turbines because they exhibit a very low starting torque as well as dynamic stability problems, the interest in implementing vertical axis turbines is increased which drives further research into the development of improved turbine designs [11, 12, 13]. Gney and Kaygusu reported a detailed comparison of various types of hydrokinetic turbines and concluded that vertical axis turbines are more suitable for the cases where water flow rate is relatively limited [14].

There are two different types of vertical axis turbines: (i) those ones based on the drag force, which are included in the former group and (ii) those ones mainly based on the lift force, corresponding to the second group. Savonius turbines have a drag-type rotor and helical turbine (Gorlov turbine). In turn, Darrieus and

Due to the high cost of the harvesting energy from water current associated with the use of these turbines, choosing a turbine with an optimum performance at the selected site is utmost importance. The hydrodynamics and performance of these turbines are governed by several parameters such as (i) the tip-speed ratio (* V*), being

*curves for the most common types of turbine. In Figure 1, it can be observed that horizontal axis hydrokinetic turbines are typically more hydrodynamically efficient and operate at much higher*power

The objective of this chapter is to present a methodology for the efficient design and numerical simulation of a

## 2. Vertical axis hydrokinetic turbine hydrodynamic models

For the design of vertical axis hydrokinetic turbine, several numerical models have been used, each of them with their own strengths and weaknesses, to accurately predict the performance of a hydrokinetic turbine depending on their configuration. The most common models are (i) blade element momentum (BEM) models, (ii) vortex models, and (iii) computational fluid dynamics models [24, 25, 26, 27, 28, 29, 30, 31, 32]. BEM are analytical models that combine the blade element and the momentum theory in order to study the behavior of the water flow on the blades and related forces. It is a technique that was pioneered by Glauert [24], Strickland [25], and Templin [26]. BEM models can be classified into three submodels: (a) single stream tube model, (b) multiple stream tube model, and (c) double multiple stream tube (DMS) model [28, 30], respectively, according to the increasing order of complexity.

The hydrokinetic turbine is placed inside a single streamtube, and the blade revolution is translated in an actuator disk when the single stream tube model is used. The water speed in the upstream and downstream sides of the turbine is considered constant. Additionally, the effects outside the streamtube are assumed negligible. It is highlighted that this model does not deliver good accuracy due to the assumptions required to be made. The obtained results in a number of cases provide higher estimate values. In contrast, it has fast processing time compared to other models.

A variation of the single streamtube modeling is the multiple streamtube modeling, which divides the single streamtube in several parallel adjacent streamtubes that are independent from each other and have their own undisrupted, wake, and induced velocities. Although this model is not accurate, the predicted performance is close to field test values, tending to give values a little higher for high

On the other hand, a variation of the multiple streamtube modeling is the double multiple streamtube modeling. This model divides the actuator disk into two half cycles in tandem, representing the upstream and downstream sides of the rotor, respectively. This model has suffered several improvements throughout time and offers a good performance prediction and, however, presents convergence problems for high solidity turbines, giving high power prediction for high

A number of studies have been carried out to simplify numerical models [36, 37, 38, 39, 40]. Nevertheless, the use of numerical methods associated with different algorithms of

The use of

Recently, hybrid unsteady RANS/LES method has been increasingly used for certain classes of simulations, including separated flows; nevertheless, the techniques that combine the near-wall RANS region with the outer, large-eddy simulation region need further development [51].

The specific case of the URANS equations that govern incompressible and isothermal turbulent flow around the turbine blade is given by Eqs. (1) and (2) [52]:

where

Eddy viscosity models use the Boussinesq approximation, which is a constitutive relation to compute the turbulence stress tensor,

where

In Eq. (5),

Each turbulence model calculates

The turbulence models most usually utilized for the

In general, BEM and vortex models previously described must adopt several assumptions and corrections to account for the full three-dimensional, turbulent flow dynamics around the turbine blades. The use of

## 3. Geometrical modeling of a Darrieus hydrokinetic turbine

In the vertical axis category,

where

The power output of the hydrokinetic turbine given by Eq. (9) is also limited by mechanical losses in transmissions and electrical losses [72, 73]. Because of these losses and inefficiencies, one more variable is added to Eq. (9). The variable is

Adding

Eq. (10) is considered the power equation for the hydrokinetic turbine.

The swept area is the section of water that encloses the turbine in its movement. Its shape depends on the rotor configuration; thus, the swept area of horizontal axis hydrokinetic turbine is a circular shaped, while for a straight-bladed vertical axis hydrokinetic turbine, the swept area has a rectangular shape. Therefore, for a

Each rotor design has an optimal

In turn,

The factor

The optimum range for the rotor

In relation to the blade profile, a great number of hydrofoil families and thicknesses have been informed to be suitable for Darrieus turbines. Since it is impossible to analyze all of them, the choices must be narrowed in some way. Symmetrical hydrofoils have been traditionally selected because the energy capturing is approximately symmetrical throughout the turbine axis.

In this sense, the majority of the previously conducted research works on vertical axis hydrokinetic turbines has been focused on straight-bladed vertical hydrokinetic turbines equipped with symmetric hydrofoils (such as NACA four-digit series of 0012, 0015, 0018, and 0025). In the current work, NACA 0025 hydrofoil profile was selected [66, 76, 79]. The maximum

Nevertheless, before using Eq. (10),

Subsequently, given the rotor design parameters (e.g., * I*) on the disk can be written as represented by Eq. (15):

where

Therefore, for

Figure 3 shows a basic schematic representation of the hydrodynamic design of a * α*) on a blade is a function of the blade azimuth angle (

*). In Figure 3, the tip velocity vector and the lift and drag vectors generated by the rotation of the turbine blade are illustrated. As it can be seen from Figure 3, the relative velocity (w) can be obtained from the tangential and normal velocity components given by Eq. (17):*θ

where * v*is the induced velocity through the rotor.

_{i}

*can be written in a non-dimensional form using free stream velocity. Therefore, Eq. (17) can be expressed as Eq. (18):*w

The * v*can be written in terms of a, as described by Eq. (19):

_{i}

Using Eqs. (12) and (19), Eq. (18) can be rewritten as Eq. (20):

From the geometry of Figure 3, * W*can also be represented by Eq. (21):

Using Eq. (19), Eq. (21) can be rewritten as Eq. (22):

From the geometry of Figure 3, the local angle of attack can be expressed as Eq. (23):

Eq. (23) can be put in a non-dimensional form using Eq. (19); therefore, Eq. (24) is obtained:

As shown in Eq. (24), * α*, is affected by

The normal

where

Substituting Eqs. (25) and (26) in Eq. (27), Eq. (28) is obtained, which represents the instantaneous thrust force:

By equating the thrust value from Eq. (15) and Eq. (28) and substituting Eq. (22),

Although

Using Eq. (13),

On the other hand, the blade aspect ratio (

For blades with smaller

After finding out the value of

Parameter | Value (units) |
---|---|

Power output (P) | 500 W |

Blade’s profile NACA | 0025 |

Chord length (c) | 0.33 m |

Number of blades (N) | 3 |

Solidity () | 0.66 |

Turbine height (H) | 1.13 m |

Turbine radius (R) | 0.75 m |

## 4. Computational fluid dynamics of a Darrieus hydrokinetic turbine

A computational domain is a region in the space in which the numerical equations of the fluid flow are solved by

The mesh can be structured or unstructured. In the first case, the mesh consists of quadrilateral cells in

The

The vertical hydrokinetic turbine studied was a three-bladed Darrieus rotor with a NACA 0025 blade profile.

Unstructured meshes were applied to both the rotor away from the near surface region and the outer grids. Finer meshes were used around the blades and regions in the wake of the blades. Particularly, regions at the leading and trailing edge and in the middle of blade were finely meshed in order to capture the flow field more accurately. The outer mesh was coarsened in regions expanding away from the rotor in order to minimize the central processing unit (CPU) time. The different mesh zones used for the present simulations are illustrated in Figure 8, while various mesh details are shown in Figure 9a and b. A number of simulations were carried out in order to determine how the mesh quality affected the

Symmetrical boundary conditions were defined at the top plane and in the bottom plane. Additionally, a uniform pressure on the outlet boundary was set, and a uniform velocity on the inlet boundary with a magnitude of 1.5

λ | 0.5 | 1.00 | 1.75 | 2.00 | 2.50 |
---|---|---|---|---|---|

0.997 | 1.994 | 3.489 | 3.987 | 4.984 |

To solve the viscous sublayer of the turbulence model used, the values of

Mesh | Nodes | ^{+} | Maximum power coefficient _{pmax} | Error (%) |
---|---|---|---|---|

Mesh 1 | 143,672 | 0.246 | 0.56 | 10.71 |

Mesh 2 | 167,896 | 0.221 | 0.64 | 14.28 |

Mesh 3 | 187,630 | 0.283 | 0.63 | 1.56 |

Mesh 4 | 298,878 | 0.261 | 0.62 | 1.58 |

The modeling of vertical axis hydrokinetic turbines can be done in steady state or transient modes depending on the objectives of the research and the available computational resources. If computational resources are scarce, relatively simple, steady flow models can be used to model the turbine blades in different azimuthal positions [96]. A more common approach is the transient modeling of the moving blades through the use of URANS approach [19, 97, 98, 99].

In this work, transient analyses were carried out to characterize the performance of the investigated NACA 0025 hydrofoil profile. Performances are described in terms of

In the literature, optimum values of

The same tendencies of

Figure 11 shows pressure contours at

In order to perceive the effect of TSR on the turbine performance, the average * C*achieved for different TSRs can be found.

_{p}

*is not constant because the torque and velocity are not constant in a Darrieus turbine. Hence, the average*C

_{p}

*per cycle is calculated as the product of the average values of these terms per cycle. TSR and*C

_{p}

*are in a direct relationship when the TSR is between 0.5 and 1.75. However, TSR and*C

_{p}

*have an opposite tendency for TSR greater than 1.75. The maximum value of*C

_{p}

*was achieved when TSR was 1.75; the value of the average*C

_{p}

*was 44.33% per cycle. Similar results were obtained by Dai et al. [22] for NACA 0025 blade profile. Additionally, Lain and Osorio [102] used experimental data of Dai and Lams work [22] and developed numerical models. They performed analysis on CFX solver, DMS model, and Fluent solver and achieved efficiencies of 58.6, 46.3, and 52.8%, respectively, when TSR was 1.745. Results observed by using Fluent solver were quite accurate.*C

_{p}

## 5. Conclusions

The design and simulation of a vertical axis hydrokinetic turbine were presented in this work. The most common modeling method for vertical axis hydrokinetic turbine is CFD. The numerical simulations allow analyzing many different turbine design parameters, providing an optimal configuration for a given set of design parameters. Before the simulations can be performed, the determination of the optimal grid and time step sizes required must be conducted. Finer grids and smaller time steps might give a more accurate solution, but they increase the computational cost. Thus, finding optimal values is also required. For this purpose, the turbulence model generally used is * k-ω*shear stress transport (SST) turbulence model.

The TSR is a significant parameter that affects the performance of hydrokinetic turbines. Consequently, the performance of the turbine was investigated with a simplified 2D numerical model. From the 2D model, * C*was computed for various TSRs. During a turbine revolution, the blade of the turbine may experience large, as well as rapid variation, in

_{p}

*. A*C

_{p}

*maximum of 62% was achieved when TSR was equal to 1.75. The value of the average*C

_{p}

*was 44.33% per cycle. Though Darrieus turbine is very simple to be constructed, it has some disadvantages when compared to axial turbines since they exhibit a lower power coefficient and a variation in the torque produced within the cycle, leading to periodic loading on the components of the turbine.*C

_{p}

## Acknowledgments

The authors gratefully acknowledge the financial support provided by the Colombia Scientific Program within the framework of the call Ecosistema Cientifico (Contract No. FP44842-218-2018).

## Abbreviations

CFD | computational fluid dynamics |

TSR | tip-speed ratio |

BEM | blade element momentum theory |

DMS | double multiple streamtube |

RANS | Reynolds-averaged Navier-Stokes |

LES | large eddy simulation |

DNS | direct numerical simulation |

URANS | unsteady Reynolds-averaged Navier-Stokes |

DES | detached eddy simulation |

## Nomenclature

P | power required |

ρ | water density |

V | water speed |

θ | Azimuth angle |

ψ | turbine rotational speed |

λ | blade tip-speed ratio (TSR) |

α | blade angle of attack |

AR | blade aspect ratio |

H | turbine height |

σ | solidity |

R | turbine radius |

D | turbine diameter |

B | number of blades |

c | blade chord length |

a | axial induction factor |

A | swept area of the turbine |

CD | drag coefficient |

CL | lift coefficient |

CP | power coefficient |

T | turbine torque |

I | AXIAL thrust |