**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 [1–4]. 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 [3–6].

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 [11–13]. 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 [11–13]. 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].

chord Reynolds number ð Þ *Re*¼ *ρVc=μ* , where *ρ* and *μ* are the water density and viscosity, respectively. In the literature, there are several studies including experimental, numerical (computational fluid dynamics) and theoretical studies, focused on optimizing the hydrokinetic turbine geometric parameters, mainly the blade profiles, using different techniques [8, 16–23]. **Figure 1** shows the typical *power* 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 TSR values than vertical axis hydrokinetic turbines. **Figure 1** is adapted from the behavior of wind turbines; from the best of the authors' knowledge, there is no overall graph regarding the behavior of such types

*Computational Fluid Dynamic Simulation of Vertical Axis Hydrokinetic Turbines*

The objective of this chapter is to present a methodology for the efficient design and numerical simulation of a H-Darrieus vertical axis hydrokinetic turbine using CFD simulation and the commercial software ANSYS Fluent. The designed hydrokinetic turbine is expected to generate 500 W output power at 1.5 m*=*s water velocity. The hydrodynamic performance (power coefficient) was analyzed for

**2. Vertical axis hydrokinetic turbine hydrodynamic models**

[28, 30], respectively, according to the increasing order of complexity.

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 [24–32]. 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

of turbines considered as hydrokinetic turbines.

several TSR values.

**167**

**Figure 1.**

*Power coefficients for different rotor designs.*

*DOI: http://dx.doi.org/10.5772/intechopen.89184*

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 Hshaped Darrieus turbines have a lift-type rotor since lift force is the main driving force for these kinds of machines. Generally, drag-based turbines are considered to be less efficient than their lift-based counterparts. The power coefficient of Darrieus hydrokinetic turbines is comparatively higher than that of the Savonius hydrokinetic turbines. However, Darrieus hydrokinetic turbines have poor starting characteristics [15].

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 (TSR or *λ* ¼ *Rψ=V*, which is defined as the ratio between the blade tip speed and the fluid speed (*V*), being *R* the turbine radius and *ψ* the rpm); (ii) the aspect ratio ð Þ *AR* , defined as the ratio between the hydrokinetic turbine height ð Þ *H* and its diameter ð Þ *D* ; and (iii) the solidity ð Þ *σ* ¼ *Bc=*2*R* , which refers to the ratio of the product of the blade chord length ð Þ*c* and the number of blades ð Þ *B* , the blade profiles, and the

*Computational Fluid Dynamic Simulation of Vertical Axis Hydrokinetic Turbines DOI: http://dx.doi.org/10.5772/intechopen.89184*

#### **Figure 1.**

hydroelectricity, and emerging technologies such as wave and hydrokinetic systems. These technologies have unquestionably started to transform the global

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

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

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 [11–13]. 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 [11–13]. 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

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 Hshaped Darrieus turbines have a lift-type rotor since lift force is the main driving force for these kinds of machines. Generally, drag-based turbines are considered to be less efficient than their lift-based counterparts. The power coefficient of Darrieus hydrokinetic turbines is comparatively higher than that of the Savonius hydrokinetic turbines. However, Darrieus hydrokinetic turbines have poor starting

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 (TSR or *λ* ¼ *Rψ=V*, which is defined as the ratio between the blade tip speed and the fluid speed (*V*), being *R* the turbine radius and *ψ* the rpm); (ii) the aspect ratio ð Þ *AR* , defined as the ratio between the hydrokinetic turbine height ð Þ *H* and its diameter ð Þ *D* ; and (iii) the solidity ð Þ *σ* ¼ *Bc=*2*R* , which refers to the ratio of the product of the blade chord length ð Þ*c* and the number of blades ð Þ *B* , the blade profiles, and the

energy landscape in a promising way [7, 8].

*Computational Fluid Dynamics Simulations*

water flow rate is relatively limited [14].

devices [9].

communities [10].

characteristics [15].

**166**

*Power coefficients for different rotor designs.*

chord Reynolds number ð Þ *Re*¼ *ρVc=μ* , where *ρ* and *μ* are the water density and viscosity, respectively. In the literature, there are several studies including experimental, numerical (computational fluid dynamics) and theoretical studies, focused on optimizing the hydrokinetic turbine geometric parameters, mainly the blade profiles, using different techniques [8, 16–23]. **Figure 1** shows the typical *power* 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 TSR values than vertical axis hydrokinetic turbines. **Figure 1** is adapted from the behavior of wind turbines; from the best of the authors' knowledge, there is no overall graph regarding the behavior of such types of turbines considered as hydrokinetic turbines.

The objective of this chapter is to present a methodology for the efficient design and numerical simulation of a H-Darrieus vertical axis hydrokinetic turbine using CFD simulation and the commercial software ANSYS Fluent. The designed hydrokinetic turbine is expected to generate 500 W output power at 1.5 m*=*s water velocity. The hydrodynamic performance (power coefficient) was analyzed for several TSR values.

### **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 [24–32]. 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.

number flow; therefore, they are not practical for modeling the flow around the hydrokinetic turbine. An alternative for CFD simulation is the use of unsteady Reynolds-averaged Navier-Stokes (URANS) turbulence models, which are computationally economical and of great use among both industry and academia in

*Computational Fluid Dynamic Simulation of Vertical Axis Hydrokinetic Turbines*

Recently, hybrid unsteady RANS/LES method has been increasingly used for

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 *i* ¼ 1, 2, 3; *j* ¼ 1, 2, 3; *u* and *p* represent the time-averaged velocity and pressure, respectively; *μ* is the dynamic viscosity of water; *ρ* represents the density of water; *fi* expresses the body force per unit of volume, which may represent the Coriolis and centrifugal contributions; *xj* denotes the spatial coordinate component; *t* is the time; and *τij* refers to the Reynolds stress, which is required to be modeled for approximating the above equations. Nowadays, turbulent flows may be divided into four groups (**Figure 2**). The first one relies on the Boussinesq assumption according to which Reynolds stress is proportional to the mean strain rate. These models are often called eddy viscosity models. The second group includes the socalled nonlinear eddy viscosity models, which allow for the modeling of the turbulent stress as a nonlinear function of mean velocity gradients. Within this group, turbulent scales are determined by solving transport equations (usually, *k*þ one other physical quantity). The model is set to mimic response of turbulence to certain important types of strain. The third group involves modeling Reynolds stress transport equations known as a second-moment closure method. The last group concerns the DNS, LES, and the detached eddy simulation (DES). These models rely on a direct resolution of the Navier-Stokes equation for some class of the flows (DNS) or direct resolution of the Navier-Stokes equation for large scales and modeling sub-grid scale (LES, DES). DES is a hybrid model of URANS and LES

¼ 0 (1)

*<sup>τ</sup>ij* <sup>þ</sup> *<sup>ρ</sup>fi* (2)

þ *∂ ∂xj*

*∂ui ∂xj*

certain classes of simulations, including separated flows; nevertheless, the techniques that combine the near-wall RANS region with the outer, large-eddy

comparison with LES models [41, 42, 44–50].

*DOI: http://dx.doi.org/10.5772/intechopen.89184*

simulation region need further development [51].

*∂ρuiuj ∂xj*

¼ � *<sup>∂</sup><sup>p</sup> ∂xi* þ *∂ ∂xi μ ∂ui ∂xj* 

*∂ρui ∂t* þ

**Figure 2.**

**169**

*Classification of turbulence 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 TSR with a fast processing time [25, 28].

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 TSR and having a high processing time [31–35].

A number of studies have been carried out to simplify numerical models [36–40]. Nevertheless, the use of numerical methods associated with different algorithms of CFD to analyze complex fluid flow applications has grown in recent years due to the rapid development of computer technology. Therefore, the analysis of vertical axis hydrokinetic turbines is also being conducted using CFD simulation [18–23].

CFD is found in some commercial software tools such as ANSYS Fluent, FLACS, Phoenix, and COMSOL, among others. The CFD tools transform the governing equations of the fundamental physical principles of the fluid flow in discretized algebraic forms, which are solved to find the flow field values in time and space. The simulation results can be very sensitive to the wide range of computational parameters that must be set by the user; for a typical simulation, the user needs to select the variables of interest, turbulence models, computational domain, computational mesh, boundary conditions, methods of discretization, and the convergence criteria, among other simulation setup parameters [20–23]. In general, the CFD tools compute the flow field values by the equations of the fundamental physical principles of a fluid flow. The physical aspects of any fluid flow are governed by three principles: mass is conserved, Newton's second law (momentum equation) is fulfilled, and energy is conserved. These principles are expressed in integral equations or partial differential equation (continuity, momentum and energy equation), being the most common form of momentum equation, the Navier-Stokes equations for viscous flows and the Euler equations for inviscid flows [20–23].

The use of CFD simulation of the turbulent flow surrounding the hydrokinetic turbine is either done by statistical modeling of turbulence based on the Reynoldsaveraged Navier-Stokes (RANS) equations, large eddy simulation (LES), or direct numerical simulation (DNS) [41]. RANS are time-averaged equations of motion for fluid flow; therefore, when the flow is not a stationary one, RANS equation is not the same, due to the simulation must be a time-dependent one. This significantly increases the computational time over conventional RANS. The use of LES and DNS methods for this case requires a careful application and significant computational resources [42, 43]. Additionally, since LES and DNS methods require very fine grids for wall-bounded flows, they are still in development stages for high Reynolds

*Computational Fluid Dynamic Simulation of Vertical Axis Hydrokinetic Turbines DOI: http://dx.doi.org/10.5772/intechopen.89184*

number flow; therefore, they are not practical for modeling the flow around the hydrokinetic turbine. An alternative for CFD simulation is the use of unsteady Reynolds-averaged Navier-Stokes (URANS) turbulence models, which are computationally economical and of great use among both industry and academia in comparison with LES models [41, 42, 44–50].

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]:

$$\frac{\partial u\_i}{\partial \mathbf{x}\_j} = \mathbf{0} \tag{1}$$

$$\frac{\partial \rho u\_i}{\partial t} + \frac{\partial \rho u\_i u\_j}{\partial \mathbf{x}\_j} = -\frac{\partial p}{\partial \mathbf{x}\_i} + \frac{\partial}{\partial \mathbf{x}\_i} \left( \mu \frac{\partial u\_i}{\partial \mathbf{x}\_j} \right) + \frac{\partial}{\partial \mathbf{x}\_j} \left( \mathbf{r}\_{ij} \right) + \rho \mathbf{f}\_i \tag{2}$$

where *i* ¼ 1, 2, 3; *j* ¼ 1, 2, 3; *u* and *p* represent the time-averaged velocity and pressure, respectively; *μ* is the dynamic viscosity of water; *ρ* represents the density of water; *fi* expresses the body force per unit of volume, which may represent the Coriolis and centrifugal contributions; *xj* denotes the spatial coordinate component; *t* is the time; and *τij* refers to the Reynolds stress, which is required to be modeled for approximating the above equations. Nowadays, turbulent flows may be divided into four groups (**Figure 2**). The first one relies on the Boussinesq assumption according to which Reynolds stress is proportional to the mean strain rate. These models are often called eddy viscosity models. The second group includes the socalled nonlinear eddy viscosity models, which allow for the modeling of the turbulent stress as a nonlinear function of mean velocity gradients. Within this group, turbulent scales are determined by solving transport equations (usually, *k*þ one other physical quantity). The model is set to mimic response of turbulence to certain important types of strain. The third group involves modeling Reynolds stress transport equations known as a second-moment closure method. The last group concerns the DNS, LES, and the detached eddy simulation (DES). These models rely on a direct resolution of the Navier-Stokes equation for some class of the flows (DNS) or direct resolution of the Navier-Stokes equation for large scales and modeling sub-grid scale (LES, DES). DES is a hybrid model of URANS and LES

**Figure 2.** *Classification of turbulence models.*

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

A variation of the single streamtube modeling is the multiple streamtube modeling, which divides the single streamtube in several parallel adjacent

A number of studies have been carried out to simplify numerical models [36–40]. Nevertheless, the use of numerical methods associated with different algorithms of CFD to analyze complex fluid flow applications has grown in recent years due to the rapid development of computer technology. Therefore, the analysis of vertical axis hydrokinetic turbines is also being conducted using CFD

Phoenix, and COMSOL, among others. The CFD tools transform the governing equations of the fundamental physical principles of the fluid flow in discretized algebraic forms, which are solved to find the flow field values in time and space. The simulation results can be very sensitive to the wide range of computational parameters that must be set by the user; for a typical simulation, the user needs to select the variables of interest, turbulence models, computational domain, computational mesh, boundary conditions, methods of discretization, and the convergence criteria, among other simulation setup parameters [20–23]. In general, the CFD tools compute the flow field values by the equations of the fundamental physical principles of a fluid flow. The physical aspects of any fluid flow are governed by three principles: mass is conserved, Newton's second law (momentum equation) is fulfilled, and energy is conserved. These principles are expressed in integral equations or partial differential equation (continuity, momentum and energy equation), being the most common form of momentum equation, the Navier-Stokes equations

for viscous flows and the Euler equations for inviscid flows [20–23].

The use of CFD simulation of the turbulent flow surrounding the hydrokinetic turbine is either done by statistical modeling of turbulence based on the Reynoldsaveraged Navier-Stokes (RANS) equations, large eddy simulation (LES), or direct numerical simulation (DNS) [41]. RANS are time-averaged equations of motion for fluid flow; therefore, when the flow is not a stationary one, RANS equation is not the same, due to the simulation must be a time-dependent one. This significantly increases the computational time over conventional RANS. The use of LES and DNS methods for this case requires a careful application and significant computational resources [42, 43]. Additionally, since LES and DNS methods require very fine grids for wall-bounded flows, they are still in development stages for high Reynolds

CFD is found in some commercial software tools such as ANSYS Fluent, FLACS,

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

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 TSR and

other models.

high TSR with a fast processing time [25, 28].

*Computational Fluid Dynamics Simulations*

having a high processing time [31–35].

simulation [18–23].

**168**

models, being the former one for modeling the wall region and the latter one for the other regions. The motivation of researchers for utilizing DES is the high cost of LES in the boundary layer region; the strategy of using DES aims at accurately modeling the large vortices in the separated shed wake while keeping the simulation of the near-wall region at a limited cost [51].

Eddy viscosity models use the Boussinesq approximation, which is a constitutive relation to compute the turbulence stress tensor, *τij*, defined in Eqs. (3) and (4):

$$\overline{\rho}\pi\_{\vec{\eta}} = 2\mu\_t \left( \tilde{\mathbf{S}}\_{\vec{\eta}} - \frac{1}{3} \mathbf{S}\_{kk}^{\tilde{}} \delta\_{\vec{\eta}} \right) - \frac{2}{3} \rho k \delta\_{\vec{\eta}} \tag{3}$$

$$
\pi\_{ij} \stackrel{def}{=} \overline{-u\_i' u\_j'} \tag{4}
$$

For this reason, an SST turbulence model is adopted in the design of this vertical axis hydrokinetic turbine proposed here in the same way used by other authors

*Computational Fluid Dynamic Simulation of Vertical Axis Hydrokinetic Turbines*

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 CFD simulation eliminates the need for many, but not all, of these assumptions. It also has the advantage of resolving the full time-dependent flow field allowing for a better understanding of the flow at the blade wall as well as in the wake of the turbine. Although a much greater computation cost is associated with the use of CFD simulation, it has demonstrated an ability to generate results that can be favorably compared with

In the vertical axis category, H-Darrieus turbines are the most common options because their installation is simple, they are less expensive, and they require less maintenance than other types of vertical axis hydrokinetic turbines [17, 20]. During the design of Darrieus turbines, a high rotor efficiency is desirable for increased water energy extraction and should be maximized within the limits of affordable production. H-Darrieus turbine consists of two or more hydrofoil blades around a shaft perpendicular to the direction of the water flow. The available hydrokinetic power ð Þ *P* of this turbine can be computed from the water flow and the turbine

**3. Geometrical modeling of a Darrieus hydrokinetic turbine**

*<sup>P</sup>* <sup>¼</sup> <sup>1</sup> 2

where *A* is the swept area and *Cp* refers to the power coefficient of the turbine. It is widely known that a physical limit exists to quantify the amount of energy that can be extracted, which is independent on the turbine design. The energy extraction is maintained in the flow process through the reduction of kinetic energy and subsequent velocity of the water. The magnitude of energy harnessed is a function of the reduction water speed over the turbine. Total extraction, i.e., 100% extraction, would imply zero final velocity and, therefore, zero flow. The zero-flow scenario cannot be achieved; hence all the water kinetic energy may not be utilized. This principle is widely accepted and indicates that hydrokinetic turbine efficiency cannot exceed 59.3%. This parameter is commonly known as *Cp*, being the maximum *Cp* equal to 0.593 is referred to as the Betz limit. It is important to note that the Betz theory assumes constant linear velocity, inviscid, and without swirl flow. Therefore, any rotational forces such as wake rotation and turbulence caused by drag or vortex shedding (tip losses) will further reduce the maximum efficiency. Efficiency losses are generally reduced by avoiding low tip-speed ratios which increase wake rotation, selecting hydrofoils which have a high lift-to-drag ratio and specialized tip geometries. Making an analysis of data from literature, the value of maximum *Cp* has been found to be usually ranging between 0.15 and 0.586 [22]. This coefficient only considers the mechanical energy converted directly from water energy, i.e., it does not consider the mechanical conversion into electrical

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 *η*,

*ρCpAV*<sup>3</sup> (9)

[22, 61–65].

energy [70, 71].

**171**

experimental data [19, 58, 66–69].

*DOI: http://dx.doi.org/10.5772/intechopen.89184*

dimension, as represented by Eq. (9) [11, 12]:

where ~*Sij* is given by Eq. (5)

$$\tilde{S}\_i \dot{j} = \frac{1}{2} \left( \frac{\partial \tilde{u}\_i}{\partial \mathbf{x}\_j} + \frac{\partial \tilde{u}\_j}{\partial \mathbf{x}\_i} \right) \tag{5}$$

In Eq. (5), *k* is the turbulent kinetic energy, *μ<sup>t</sup>* refers to the dynamic eddy viscosity, *ρ* is the water density, *τij* denotes the shear stress, *δij* is the Kronecker delta, and *u* expresses the velocity. Based on dimensional analysis, *μ<sup>t</sup>* can be determined from a turbulence time scale (or velocity scale) and a length scale: ið Þ turbulent kinetic energy ð Þ*k* , represented by Eq. (6); (ii) turbulence dissipation rate ð Þϵ , described by Eq. (7); and (iii) specific dissipation rate ð Þ *ω* , expressed in Eq. (8):

$$k = \overline{-u\_i'u\_i'}/2\tag{6}$$

$$\epsilon = \nu \partial u\_i' / \partial \mathfrak{x}\_j \left( \partial u\_i' / \partial \mathfrak{x}\_j + \partial u\_j' / \partial \mathfrak{x}\_j \right) \tag{7}$$

$$
\rho\_- = \epsilon/k
\tag{8}
$$

Each turbulence model calculates *μ<sup>t</sup>* differently. Within the group of eddy viscosity models, it is possible to identify the following one- and two-equation turbulence models: Spalart-Allmaras, standard *k* � ϵ model, renormalization group (RNG) *k* � ϵ model, realizable *k* � ϵ model, standard *k* � *ω* model, and shear stress transport (SST) *k* � *ω* model. The choice of turbulence model will depend on several considerations such as the physics encompassed in the flow, the established practice for a specific class of problem, the level of accuracy required, the available computational resources, and the amount of time available for the simulation. To select the most appropriate model for each application, the capabilities and limitations of such as options are needed to be understood [20, 43, 52–60].

The turbulence models most usually utilized for the CFD analysis of a vertical axis hydrokinetic turbine are RANS approach because LES models require a highly refined mesh and are, therefore, extremely expensive from a computational point of view [53]. For vertical axis hydrokinetic design, the most commonly used turbulence models include the referred models mentioned previously, e.g., *k* � ϵ model, realizable *k* � ϵ model, standard *k* � *ω* model, and shear stress transport (SST) *k* � *ω* model. The SST model was developed to provide an answer to the necessity of models which can handle aeronautical flows with strong adverse pressure gradients and separation of the boundary layers. For this purpose, SST model combines the *k* � *ω* model in regions near to solid walls, with *k* � ϵ model for the free stream flow out of the boundary layers. SST model has been found to give very good results for CFD analysis of horizontal and vertical axis hydrokinetic turbines [20, 54–60].

*Computational Fluid Dynamic Simulation of Vertical Axis Hydrokinetic Turbines DOI: http://dx.doi.org/10.5772/intechopen.89184*

For this reason, an SST turbulence model is adopted in the design of this vertical axis hydrokinetic turbine proposed here in the same way used by other authors [22, 61–65].

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 CFD simulation eliminates the need for many, but not all, of these assumptions. It also has the advantage of resolving the full time-dependent flow field allowing for a better understanding of the flow at the blade wall as well as in the wake of the turbine. Although a much greater computation cost is associated with the use of CFD simulation, it has demonstrated an ability to generate results that can be favorably compared with experimental data [19, 58, 66–69].

## **3. Geometrical modeling of a Darrieus hydrokinetic turbine**

In the vertical axis category, H-Darrieus turbines are the most common options because their installation is simple, they are less expensive, and they require less maintenance than other types of vertical axis hydrokinetic turbines [17, 20]. During the design of Darrieus turbines, a high rotor efficiency is desirable for increased water energy extraction and should be maximized within the limits of affordable production. H-Darrieus turbine consists of two or more hydrofoil blades around a shaft perpendicular to the direction of the water flow. The available hydrokinetic power ð Þ *P* of this turbine can be computed from the water flow and the turbine dimension, as represented by Eq. (9) [11, 12]:

$$P\_{\perp} = \frac{1}{2} \rho C\_p A V^3 \tag{9}$$

where *A* is the swept area and *Cp* refers to the power coefficient of the turbine. It is widely known that a physical limit exists to quantify the amount of energy that can be extracted, which is independent on the turbine design. The energy extraction is maintained in the flow process through the reduction of kinetic energy and subsequent velocity of the water. The magnitude of energy harnessed is a function of the reduction water speed over the turbine. Total extraction, i.e., 100% extraction, would imply zero final velocity and, therefore, zero flow. The zero-flow scenario cannot be achieved; hence all the water kinetic energy may not be utilized. This principle is widely accepted and indicates that hydrokinetic turbine efficiency cannot exceed 59.3%. This parameter is commonly known as *Cp*, being the maximum *Cp* equal to 0.593 is referred to as the Betz limit. It is important to note that the Betz theory assumes constant linear velocity, inviscid, and without swirl flow. Therefore, any rotational forces such as wake rotation and turbulence caused by drag or vortex shedding (tip losses) will further reduce the maximum efficiency. Efficiency losses are generally reduced by avoiding low tip-speed ratios which increase wake rotation, selecting hydrofoils which have a high lift-to-drag ratio and specialized tip geometries. Making an analysis of data from literature, the value of maximum *Cp* has been found to be usually ranging between 0.15 and 0.586 [22]. This coefficient only considers the mechanical energy converted directly from water energy, i.e., it does not consider the mechanical conversion into electrical energy [70, 71].

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 *η*,

models, being the former one for modeling the wall region and the latter one for the other regions. The motivation of researchers for utilizing DES is the high cost of LES in the boundary layer region; the strategy of using DES aims at accurately modeling the large vortices in the separated shed wake while keeping the simulation

Eddy viscosity models use the Boussinesq approximation, which is a constitutive relation to compute the turbulence stress tensor, *τij*, defined in Eqs. (3) and (4):

> <sup>3</sup> <sup>~</sup>*Skkδij*

> > *∂u*~*i ∂xj* þ *∂u*~*<sup>j</sup> ∂xi*

In Eq. (5), *k* is the turbulent kinetic energy, *μ<sup>t</sup>* refers to the dynamic eddy viscosity, *ρ* is the water density, *τij* denotes the shear stress, *δij* is the Kronecker delta, and *u* expresses the velocity. Based on dimensional analysis, *μ<sup>t</sup>* can be determined from a turbulence time scale (or velocity scale) and a length scale: ið Þ turbulent kinetic energy ð Þ*k* , represented by Eq. (6); (ii) turbulence dissipation rate ð Þϵ , described by Eq. (7); and (iii) specific dissipation rate ð Þ *ω* , expressed in Eq. (8):

> *k* ¼ �*ui* 0 *ui* 0

Each turbulence model calculates *μ<sup>t</sup>* differently. Within the group of eddy viscosity models, it is possible to identify the following one- and two-equation turbulence models: Spalart-Allmaras, standard *k* � ϵ model, renormalization group (RNG) *k* � ϵ model, realizable *k* � ϵ model, standard *k* � *ω* model, and shear stress transport (SST) *k* � *ω* model. The choice of turbulence model will depend on several considerations such as the physics encompassed in the flow, the established practice for a specific class of problem, the level of accuracy required, the available computational resources, and the amount of time available for the simulation. To select the most appropriate model for each application, the capabilities and limita-

The turbulence models most usually utilized for the CFD analysis of a vertical axis hydrokinetic turbine are RANS approach because LES models require a highly refined mesh and are, therefore, extremely expensive from a computational point of view [53]. For vertical axis hydrokinetic design, the most commonly used turbulence models include the referred models mentioned previously, e.g., *k* � ϵ model, realizable *k* � ϵ model, standard *k* � *ω* model, and shear stress transport (SST) *k* � *ω* model. The SST model was developed to provide an answer to the necessity of models which can handle aeronautical flows with strong adverse pressure gradients and separation of the boundary layers. For this purpose, SST model combines the *k* � *ω* model in regions near to solid walls, with *k* � ϵ model for the free stream flow out of the boundary layers. SST model has been found to give very good results for CFD analysis of horizontal and vertical axis hydrokinetic turbines [20, 54–60].

tions of such as options are needed to be understood [20, 43, 52–60].

*<sup>=</sup>∂xj* <sup>þ</sup> *<sup>∂</sup>uj*

0 *=∂xj* (7)

<sup>ϵ</sup> <sup>¼</sup> *<sup>v</sup>∂ui* 0 *=∂xj ∂ui* 0 � 2 3

*ρkδij* (3)

(5)

*<sup>j</sup>* (4)

*=*2 (6)

*ω* ¼ ϵ*=k* (8)

*ρτij* <sup>¼</sup> <sup>2</sup>*μ<sup>t</sup>* <sup>~</sup>*Sij* � <sup>1</sup>

<sup>~</sup>*Si <sup>j</sup>* <sup>¼</sup> <sup>1</sup> 2

*τij* ¼ *def* �*u*<sup>0</sup> *i u*0

of the near-wall region at a limited cost [51].

*Computational Fluid Dynamics Simulations*

where ~*Sij* is given by Eq. (5)

**170**

which is a measure of the efficiency of the gearbox, the electrical inverter, and the generator. It takes into account all the friction, slippage, and heat losses associated with the interior mechanical and electrical components. Values for *η* can greatly differ among the turbine models. Some smaller turbines do not have transmission; therefore, the electric generator is directly moved by the turbine shaft, while larger turbines have transmission [74]. The range of values of *η* is presented in the literature, highlighting the study conducted by Hagerman et al., which states a range of efficiencies between 95 and 98%. However, for the design of the blade, a reasonable and conservative value of *η* around 70% was used in this work [75].

Adding *η* into Eq. (9), Eq. (10) is obtained:

$$P\_- = \frac{1}{2} \rho C\_p A V^3 \eta \tag{10}$$

*AR* <sup>¼</sup> *<sup>H</sup>*

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

The optimum range for the rotor *AR* for a vertical axis wind turbine was obtained using the double multiple streamtube analytical approach by Ahmadi-Baloutaki et al. [77], and it was found to be 1< *H=R*<4. The hydrokinetic turbine of

*Computational Fluid Dynamic Simulation of Vertical Axis Hydrokinetic Turbines*

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 fourdigit series of 0012, 0015, 0018, and 0025). In the current work, NACA 0025 hydrofoil profile was selected [66, 76, 79]. The maximum *Cp* of a turbine with NACA 0025 hydrofoil profile was numerical and experimentally analyzed by Dai and Lam [101]. The results showed that a maximum average *Cp* equal to 25.1% was obtained for a TSR of 1745. Consequently, this *Cp* value was used for the design

Nevertheless, before using Eq. (10), *V* must be determined or assumed. In this research, a *V* value of 1.5 m*=*s was assumed since it constitutes an average speed of the great rivers in Colombia. In turn, *P*, *η*, and *ρ* equal to 500 W, 70%, and 997 kg*=*m3ð Þ at 25°C were calculated. *<sup>R</sup>* and *<sup>H</sup>* were equal to 1.13 m and 0.75 m, respectively. It is important to note that long blades can cause many natural frequency of

Subsequently, given the rotor design parameters (e.g., *R*, *H*, hydrofoil profile and water current velocity, among others), the main task during the blade design procedure is to determine *c*, which can be obtained by equaling the thrust on the rotor, determined from the actuator disk theory, with the thrust obtained from the BEM theory, considering the drag different from zero [72, 81]. From the actuator disk theory, the axial thrust (*I*) on the disk can be written as represented by Eq. (15):

where *a* is the axial induction factor, which is defined as the fractional water velocity decrease between the free stream and the rotor plane. On the other hand, because the Darrieus turbine is primarily a lift force drive turbine, the resultant tangential components of the lift and the drag forces on the hydrofoil provide the driving force to the turbine. The value of *a* can be determined from Eq. (16) that

*Cp* <sup>¼</sup> <sup>4</sup>*a*ð Þ <sup>1</sup> � *<sup>a</sup>*

be obtained (1.1916, 0.76, 0.0483). The thrust coefficient for an ideal turbine is equal to zero when *a* is equal to 1. On the other hand, the momentum theory is no longer valid for axial induction factors greater than 0.5 because V in the far wake would be negative. In practice, as the axial induction factor increases above 0.5, the flow patterns through the hydrokinetic turbine become much more complex than those predicted by the momentum theory. However, when axial induction is equal

*η*

Therefore, for *Cp* equal to 0.25 and *η* of 70%, three values for axial induction can

2

*<sup>I</sup>* <sup>¼</sup> <sup>2</sup>*ρAa*ð Þ <sup>1</sup> � *<sup>a</sup> <sup>V</sup>*<sup>2</sup> (15)

(16)

vibration, which must be avoided during the operation [78, 80, 81].

the mentioned work was designed for a rotor with an *AR* of 1.5.

approximately symmetrical throughout the turbine axis.

*DOI: http://dx.doi.org/10.5772/intechopen.89184*

of the blade.

relates *Cp* with *a* [82]:

**173**

*<sup>R</sup>* (14)

Eq. (10) is considered the power equation for the hydrokinetic turbine. P is the net power derived from the water after accounting for losses and inefficiencies. The hydrokinetic turbine parameters considered in the design process mentioned previously are (i) ð Þ *A* , (ii) ð Þ *P* and *Cp*, (iii) TSR or *λ*, (iv) blade profile and *c*, (v) *B*, and (vi) *σ* and *AR*.

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 H-Darrieus vertical axis hydrokinetic turbine, the swept area depends on both ð Þ *D* and ð Þ *H* [76, 77]. Therefore, A is given by Eq. (11):

$$A = DH = \mathcal{Z}RH\tag{11}$$

*A* limits the volume of water passing throughout the turbine. The rotor converts the energy contained in the water in a rotational movement, so as the area is larger, the power output under the same water operating conditions is also larger. Furthermore, *Cp* is strongly dependent on *λ*, defined in Eq. (12). This parameter is the ratio between the tangential speed at the blade tip and the water speed:

$$
\lambda = \frac{R\mu}{V} \tag{12}
$$

Each rotor design has an optimal TSR at which the maximum power extraction is achieved. On the other hand, the number of blades ð Þ *B* has a direct effect in the smoothness of the rotor operation as they can compensate cycled hydrodynamic loads [76–78]. For easiness of the turbine manufacturing, three blades have been considered in the rotor design. In general, three blades are used for the turbine system to keep the dynamic balance and minimize the fatigue effect [75–77].

In turn, *σ* is defined as the ratio between the total blade area and the projected turbine area [76, 77]. It is a crucial non-dimensional parameter affecting selfstarting capabilities. For a straight-bladed hydrokinetic turbine, *σ* is calculated from Eq. (13):

$$
\sigma = \frac{Bc}{2R} \tag{13}
$$

The factor *AR* is defined as the ratio between *H* and *R* [76, 77]. It can be defined as represented by Eq. (14):

*Computational Fluid Dynamic Simulation of Vertical Axis Hydrokinetic Turbines DOI: http://dx.doi.org/10.5772/intechopen.89184*

$$AR = \frac{H}{R} \tag{14}$$

The optimum range for the rotor *AR* for a vertical axis wind turbine was obtained using the double multiple streamtube analytical approach by Ahmadi-Baloutaki et al. [77], and it was found to be 1< *H=R*<4. The hydrokinetic turbine of the mentioned work was designed for a rotor with an *AR* of 1.5.

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 fourdigit series of 0012, 0015, 0018, and 0025). In the current work, NACA 0025 hydrofoil profile was selected [66, 76, 79]. The maximum *Cp* of a turbine with NACA 0025 hydrofoil profile was numerical and experimentally analyzed by Dai and Lam [101]. The results showed that a maximum average *Cp* equal to 25.1% was obtained for a TSR of 1745. Consequently, this *Cp* value was used for the design of the blade.

Nevertheless, before using Eq. (10), *V* must be determined or assumed. In this research, a *V* value of 1.5 m*=*s was assumed since it constitutes an average speed of the great rivers in Colombia. In turn, *P*, *η*, and *ρ* equal to 500 W, 70%, and 997 kg*=*m3ð Þ at 25°C were calculated. *<sup>R</sup>* and *<sup>H</sup>* were equal to 1.13 m and 0.75 m, respectively. It is important to note that long blades can cause many natural frequency of vibration, which must be avoided during the operation [78, 80, 81].

Subsequently, given the rotor design parameters (e.g., *R*, *H*, hydrofoil profile and water current velocity, among others), the main task during the blade design procedure is to determine *c*, which can be obtained by equaling the thrust on the rotor, determined from the actuator disk theory, with the thrust obtained from the BEM theory, considering the drag different from zero [72, 81]. From the actuator disk theory, the axial thrust (*I*) on the disk can be written as represented by Eq. (15):

$$I = 2\rho A a (1 - a) V^2 \tag{15}$$

where *a* is the axial induction factor, which is defined as the fractional water velocity decrease between the free stream and the rotor plane. On the other hand, because the Darrieus turbine is primarily a lift force drive turbine, the resultant tangential components of the lift and the drag forces on the hydrofoil provide the driving force to the turbine. The value of *a* can be determined from Eq. (16) that relates *Cp* with *a* [82]:

$$C\_p = \frac{4a(1-a)^2}{\eta} \tag{16}$$

Therefore, for *Cp* equal to 0.25 and *η* of 70%, three values for axial induction can be obtained (1.1916, 0.76, 0.0483). The thrust coefficient for an ideal turbine is equal to zero when *a* is equal to 1. On the other hand, the momentum theory is no longer valid for axial induction factors greater than 0.5 because V in the far wake would be negative. In practice, as the axial induction factor increases above 0.5, the flow patterns through the hydrokinetic turbine become much more complex than those predicted by the momentum theory. However, when axial induction is equal

which is a measure of the efficiency of the gearbox, the electrical inverter, and the generator. It takes into account all the friction, slippage, and heat losses associated with the interior mechanical and electrical components. Values for *η* can greatly differ among the turbine models. Some smaller turbines do not have transmission; therefore, the electric generator is directly moved by the turbine shaft, while larger turbines have transmission [74]. The range of values of *η* is presented in the literature, highlighting the study conducted by Hagerman et al., which states a range of efficiencies between 95 and 98%. However, for the design of the blade, a reasonable

and conservative value of *η* around 70% was used in this work [75].

*<sup>P</sup>* <sup>¼</sup> <sup>1</sup> 2

*ρCpAV*<sup>3</sup>

Eq. (10) is considered the power equation for the hydrokinetic turbine. P is the net power derived from the water after accounting for losses and inefficiencies. The hydrokinetic turbine parameters considered in the design process mentioned previously are (i) ð Þ *A* , (ii) ð Þ *P* and *Cp*, (iii) TSR or *λ*, (iv) blade profile and *c*, (v) *B*, and

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 H-Darrieus vertical axis hydrokinetic turbine, the swept area depends on both ð Þ *D*

*A* limits the volume of water passing throughout the turbine. The rotor converts the energy contained in the water in a rotational movement, so as the area is larger, the power output under the same water operating conditions is also larger. Furthermore, *Cp* is strongly dependent on *λ*, defined in Eq. (12). This parameter is the

*<sup>λ</sup>* <sup>¼</sup> *<sup>R</sup><sup>ψ</sup>*

In turn, *σ* is defined as the ratio between the total blade area and the projected

*<sup>σ</sup>* <sup>¼</sup> *Bc*

The factor *AR* is defined as the ratio between *H* and *R* [76, 77]. It can be defined

turbine area [76, 77]. It is a crucial non-dimensional parameter affecting selfstarting capabilities. For a straight-bladed hydrokinetic turbine, *σ* is calculated from

ratio between the tangential speed at the blade tip and the water speed:

Each rotor design has an optimal TSR at which the maximum power extraction is achieved. On the other hand, the number of blades ð Þ *B* has a direct effect in the smoothness of the rotor operation as they can compensate cycled hydrodynamic loads [76–78]. For easiness of the turbine manufacturing, three blades have been considered in the rotor design. In general, three blades are used for the turbine system to keep the dynamic balance and minimize the fatigue effect

*η* (10)

*A* ¼ *DH* ¼ 2*RH* (11)

*<sup>V</sup>* (12)

<sup>2</sup>*<sup>R</sup>* (13)

Adding *η* into Eq. (9), Eq. (10) is obtained:

*Computational Fluid Dynamics Simulations*

and ð Þ *H* [76, 77]. Therefore, A is given by Eq. (11):

(vi) *σ* and *AR*.

[75–77].

Eq. (13):

**172**

as represented by Eq. (14):

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

*DOI: http://dx.doi.org/10.5772/intechopen.89184*

*Computational Fluid Dynamic Simulation of Vertical Axis Hydrokinetic Turbines*

velocity vector is, the larger the tip speed ratio is.

*FN* <sup>¼</sup> <sup>1</sup> 2 *ρw*<sup>2</sup>

*FT* <sup>¼</sup> <sup>1</sup> 2 *ρw*<sup>2</sup>

*Angle of attack variation in a blade revolution for different tip-speed ratios.*

location can be expressed as Eqs. (25) and (26), respectively:

Eq. (23):

is obtained:

**Figure 4.**

**175**

*<sup>w</sup>* <sup>¼</sup> *<sup>V</sup>*ð Þ <sup>1</sup> � *<sup>a</sup> Sin*ð Þ*<sup>θ</sup>*

From the geometry of **Figure 3**, the local angle of attack can be expressed as

*tan* ð Þ¼ *<sup>α</sup> viSin*ð Þ*<sup>θ</sup>*

*<sup>α</sup>* <sup>¼</sup> *tan* �<sup>1</sup> ð Þ <sup>1</sup> � *<sup>a</sup> Sin*ð Þ*<sup>θ</sup>*

As shown in Eq. (24), *α*, is affected by *λ*. In **Figure 4**, the variation of *α* as the blade rotates according to *λ* different values is illustrated. It is noteworthy that *α* is the angle between the vector of the relative velocity and the direction of *c*. As the vector of the relative velocity changes, a positive and a negative values in the upstream and downstream region of the rotor, respectively, are observed. The larger *λ*, the smaller *α* throughout the blade rotation. The larger the blade tip

The normal ð Þ *FN* and tangential ð Þ *FT* force for a single blade at a single azimuthal

where *CL* is the lift coefficient and *C*<sup>D</sup> is the drag coefficient for *α*. The instantaneous thrust force, which is the force of the water on the turbine

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

ð Þ 1 � *a Cos*ð Þþ *θ λ*

*Sin*ð Þ *<sup>α</sup>* (22)

*viCos*ð Þþ *<sup>θ</sup> <sup>ψ</sup><sup>R</sup>* (23)

(24)

*Hc C*ð Þ *LCos*ð Þþ *α C*D*Sin*ð Þ *α* (25)

*Hc C*ð Þ *LSin*ð Þ� *α C*D*Cos*ð Þ *α* (26)

**Figure 3.** *Vertical axis hydrokinetic turbine flow velocities and blades.*

to a small value, the chord length values can be also smaller, generating a decrease in the surface where the water acts and subsequently complicating the structural integrity of the blade; therefore, a value of axial induction equal to 0.76 was used for the design of the vertical axis hydrokinetic turbine proposed here.

**Figure 3** shows a basic schematic representation of the hydrodynamic design of a H-Darrieus turbine. A feature of H-Darrieus turbine is that the angle of attack (*α*) on a blade is a function of the blade azimuth angle (*θ*). 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):

$$\omega = \sqrt{\left(\upsilon\_i \text{Sin}(\theta)\right)^2 + \left(\upsilon\_i \text{Cos}(\theta) + \psi R\right)^2} \tag{17}$$

where *vi* is the induced velocity through the rotor. *w* can be written in a nondimensional form using free stream velocity. Therefore, Eq. (17) can be expressed as Eq. (18):

$$\frac{w}{V} = \sqrt{\left(\frac{v\_i}{V}\text{Sin}(\theta)^2\right) + \left(\frac{v\_i}{V}\text{Cost}(\theta) + \frac{\wp R}{V}\right)^2} \tag{18}$$

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

$$v\_i = V(\mathbf{1} - \mathbf{a}) \tag{19}$$

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

$$\frac{w}{V} = \sqrt{\left(\left(1 - a\right) \text{Sin}(\theta)^2\right) + \left(\left(1 - a\right) \text{Cos}(\theta) + \lambda\right)^2} \tag{20}$$

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

$$w = \frac{v\_i \text{Sin}(\theta)}{\text{Sin}(a)}\tag{21}$$

*Computational Fluid Dynamic Simulation of Vertical Axis Hydrokinetic Turbines DOI: http://dx.doi.org/10.5772/intechopen.89184*

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

$$w = \frac{V(1-a)\text{Sin}(\theta)}{\text{Sin}(a)}\tag{22}$$

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

$$\tan\left(a\right) = \frac{v\_i \text{Sin}(\theta)}{v\_i \text{Cost}(\theta) + \text{y}R} \tag{23}$$

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

$$a = \tan^{-1} \left( \frac{(\mathbf{1} - a) \text{Sim}(\theta)}{(\mathbf{1} - a) \text{Cov}(\theta) + \lambda} \right) \tag{24}$$

As shown in Eq. (24), *α*, is affected by *λ*. In **Figure 4**, the variation of *α* as the blade rotates according to *λ* different values is illustrated. It is noteworthy that *α* is the angle between the vector of the relative velocity and the direction of *c*. As the vector of the relative velocity changes, a positive and a negative values in the upstream and downstream region of the rotor, respectively, are observed. The larger *λ*, the smaller *α* throughout the blade rotation. The larger the blade tip velocity vector is, the larger the tip speed ratio is.

The normal ð Þ *FN* and tangential ð Þ *FT* force for a single blade at a single azimuthal location can be expressed as Eqs. (25) and (26), respectively:

$$F\_N = \frac{1}{2}\rho w^2 Hc(\mathbf{C}\_L \text{Cost}(a) + \mathbf{C}\_D \text{Sim}(a))\tag{25}$$

$$F\_T = \frac{1}{2}\rho w^2 Hc(\mathbf{C}\_L \text{Sim}(a) - \mathbf{C}\_\mathcal{D} \text{Cos}(a))\tag{26}$$

where *CL* is the lift coefficient and *C*<sup>D</sup> is the drag coefficient for *α*. The instantaneous thrust force, which is the force of the water on the turbine

**Figure 4.** *Angle of attack variation in a blade revolution for different tip-speed ratios.*

to a small value, the chord length values can be also smaller, generating a decrease in the surface where the water acts and subsequently complicating the structural integrity of the blade; therefore, a value of axial induction equal to 0.76 was used for

**Figure 3** shows a basic schematic representation of the hydrodynamic design of a H-Darrieus turbine. A feature of H-Darrieus turbine is that the angle of attack (*α*) on a blade is a function of the blade azimuth angle (*θ*). 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):

> ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ð Þ *viSin*ð Þ*<sup>θ</sup>* <sup>2</sup> <sup>þ</sup> ð Þ *viCos*ð Þþ *<sup>θ</sup> <sup>ψ</sup><sup>R</sup>* <sup>2</sup>

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

*<sup>V</sup> Cos*ð Þþ *<sup>θ</sup> <sup>ψ</sup><sup>R</sup>*

<sup>þ</sup> ð Þ ð Þ <sup>1</sup> � *<sup>a</sup> Cos*ð Þþ *<sup>θ</sup> <sup>λ</sup>* <sup>2</sup>

� �<sup>2</sup>

*V*

*vi* ¼ *V*ð Þ 1 � *a* (19)

*Sin*ð Þ *<sup>α</sup>* (21)

where *vi* is the induced velocity through the rotor. *w* can be written in a nondimensional form using free stream velocity. Therefore, Eq. (17) can be expressed

> þ *vi*

(17)

(18)

(20)

the design of the vertical axis hydrokinetic turbine proposed here.

*w* ¼

*Vertical axis hydrokinetic turbine flow velocities and blades.*

*Computational Fluid Dynamics Simulations*

*w V* ¼

*w V* ¼ r

as Eq. (18):

**174**

**Figure 3.**

q

*vi <sup>V</sup> Sin*ð Þ*<sup>θ</sup>* <sup>2</sup> � �

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

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

ð Þ <sup>1</sup> � *<sup>a</sup> Sin*ð Þ*<sup>θ</sup>* <sup>2</sup> � �

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

*<sup>w</sup>* <sup>¼</sup> *viSin*ð Þ*<sup>θ</sup>*

s

experienced by one-blade element in the direction of the water flow can be calculate using Eq. (27):

$$I\_i = F\_N \text{Sim}(\theta) - F\_T \text{Co}(\theta) \tag{27}$$

On the other hand, the blade aspect ratio (*AB*) is defined as the ratio between

*Computational Fluid Dynamic Simulation of Vertical Axis Hydrokinetic Turbines*

*AB* <sup>¼</sup> *<sup>H</sup> c*

For blades with smaller *AB*, the lift-to-drag ratio can be reduced. Any decrease in the lift-to-drag ratio will reduce tangential forces. This would decrease the turbine overall torque and, consequently, the turbine output power. Some researchers have demonstrated an increase in the maximum *Cp* with the blade *AB* increasing a value of about 15. In spite of this fact, utilizing blades with relatively high *AB* has some drawbacks. Relatively long blades add a great amount of weight to the turbine, which increases the manufacturing and maintenance costs and creates a need for a more complex bearing. Moreover, such blades are exposed to larger bending

After finding out the value of *c* for every section, the next step was to multiply this value by the non-dimensional coordinates of NACA 0025 hydrofoil profile. The values of x and y profile coordinates for each section were exported to parametric 3D design software. From the cross sections of the blade and using the Loft command, a 3D model of the whole blade was produced. The resulting image is shown in **Figure 6**. In general, the geometric specifications of the turbine are given in

**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 (*σ*) 0.66 Turbine height (H) 1.13 m Turbine radius (R) 0.75 m

(30)

the blade length and *c* [76], as expressed by Eq. (30):

*DOI: http://dx.doi.org/10.5772/intechopen.89184*

moments. In this work, the blade *AB* parameter was fixed at 3.42.

**Figure 6.**

**Table 1.**

**177**

H*-Darrieus vertical axis hydrokinetic rotor with three blades.*

*Geometric specifications of the considered turbine.*

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

$$I\_i = \frac{1}{2}\rho\nu^2 \text{Hc}[(\text{C}\_L\text{C}\text{os}(a) + \text{C}\_D\text{Sim}(a))\text{Sim}(\theta) - (\text{C}\_L\text{Sim}(a) - \text{C}\_D\text{C}\text{os}(a))\text{Cos}(\theta)] \tag{28}$$

By equating the thrust value from Eq. (15) and Eq. (28) and substituting Eq. (22), *c* can be found using Eq. (29):

$$\mathcal{L} = \frac{8aR\text{Sin}^2\alpha}{(1-a)\text{Sin}^2\theta(\mathcal{C}\_D\text{Cos}(a-\theta)-\mathcal{C}\_L\text{Sin}(a-\theta))}\tag{29}$$

*CL* and *C*<sup>D</sup> depend on the shape of the blade, *α*, and Reynolds number under a given operating condition. *CL* and *C*<sup>D</sup> values of 0.5097 and 0.0092 were used. These values were found in the literature for the hydrofoil NACA 0025 for *α* equal to 5° [79].

Although *α* is a variable in Eq. (29) and depends on *θ* and *α*, in order to calculate *c*, *α* was considered as a constant and was equaled to angle that maximizes *CL* and *CD*. In this case, *α* was equal to 5° in order to find the blade *c* for any *θ*. Therefore, with the function defined in Eq. (29) and once *CL*, *CD*, *a*, and *R* were found, the *c* response was possible to be evaluated at different *θ* values (**Figure 5**). It is observed that *c* changes with azimuthal position, being a cyclic distribution every 180° with asymptotic values at 0° and 180°. For *θ* values between 0° and 60° and between 120° and 180°, *c* is very large, even much larger than *R*. Therefore, by averaging *θ* values between 60° and 120°, an optimal *c* value equal to 0.33 m is achieved.

Using Eq. (13), *σ* can be found, obtaining a value of 0.66. A lower *σ* requires less material and subsequently lower effective manufacturing cost.

**Figure 5.** *Chord length for different azimuth angles.*

*Computational Fluid Dynamic Simulation of Vertical Axis Hydrokinetic Turbines DOI: http://dx.doi.org/10.5772/intechopen.89184*

On the other hand, the blade aspect ratio (*AB*) is defined as the ratio between the blade length and *c* [76], as expressed by Eq. (30):

$$AB = \frac{H}{c} \tag{30}$$

For blades with smaller *AB*, the lift-to-drag ratio can be reduced. Any decrease in the lift-to-drag ratio will reduce tangential forces. This would decrease the turbine overall torque and, consequently, the turbine output power. Some researchers have demonstrated an increase in the maximum *Cp* with the blade *AB* increasing a value of about 15. In spite of this fact, utilizing blades with relatively high *AB* has some drawbacks. Relatively long blades add a great amount of weight to the turbine, which increases the manufacturing and maintenance costs and creates a need for a more complex bearing. Moreover, such blades are exposed to larger bending moments. In this work, the blade *AB* parameter was fixed at 3.42.

After finding out the value of *c* for every section, the next step was to multiply this value by the non-dimensional coordinates of NACA 0025 hydrofoil profile. The values of x and y profile coordinates for each section were exported to parametric 3D design software. From the cross sections of the blade and using the Loft command, a 3D model of the whole blade was produced. The resulting image is shown in **Figure 6**. In general, the geometric specifications of the turbine are given in

#### **Figure 6.**

experienced by one-blade element in the direction of the water flow can be calculate

Substituting Eqs. (25) and (26) in Eq. (27), Eq. (28) is obtained, which repre-

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

*CL* and *C*<sup>D</sup> depend on the shape of the blade, *α*, and Reynolds number under a given operating condition. *CL* and *C*<sup>D</sup> values of 0.5097 and 0.0092 were used. These values were found in the literature for the hydrofoil NACA 0025 for *α*

Although *α* is a variable in Eq. (29) and depends on *θ* and *α*, in order to calculate *c*, *α* was considered as a constant and was equaled to angle that maximizes *CL* and *CD*. In this case, *α* was equal to 5° in order to find the blade *c* for any *θ*. Therefore, with the function defined in Eq. (29) and once *CL*, *CD*, *a*, and *R* were found, the *c* response was possible to be evaluated at different *θ* values (**Figure 5**). It is observed that *c* changes with azimuthal position, being a cyclic distribution every 180° with asymptotic values at 0° and 180°. For *θ* values between 0° and 60° and between 120° and 180°, *c* is very large, even much larger than *R*. Therefore, by averaging *θ* values between 60° and 120°, an optimal *c* value equal to 0.33 m is achieved.

Using Eq. (13), *σ* can be found, obtaining a value of 0.66. A lower *σ* requires less

*<sup>c</sup>* <sup>¼</sup> <sup>8</sup>*aRSin*<sup>2</sup>

material and subsequently lower effective manufacturing cost.

ð Þ <sup>1</sup> � *<sup>a</sup> Sin*<sup>2</sup>

*Hc C*½ � ð Þ *LCos*ð Þþ *α CDSin*ð Þ *α Sin*ð Þ� *θ* ð Þ *CLSin*ð Þ� *α CDCos*ð Þ *α Cos*ð Þ*θ* (28)

*α*

*Ii* ¼ *FNSin*ð Þ� *θ FTCos*ð Þ*θ* (27)

*<sup>θ</sup>*ð Þ *CDCos*ð Þ� *<sup>α</sup>* � *<sup>θ</sup> CLSin*ð Þ *<sup>α</sup>* � *<sup>θ</sup>* (29)

using Eq. (27):

*Ii* <sup>¼</sup> <sup>1</sup> 2 *ρw*<sup>2</sup>

equal to 5° [79].

**Figure 5.**

**176**

*Chord length for different azimuth angles.*

sents the instantaneous thrust force:

*Computational Fluid Dynamics Simulations*

Eq. (22), *c* can be found using Eq. (29):

H*-Darrieus vertical axis hydrokinetic rotor with three blades.*


#### **Table 1.**

*Geometric specifications of the considered turbine.*

method to generate a hybrid mesh (coupling structured and unstructured mesh). In some occasions, the hybrid meshes consist of quadrilateral or hexahedral elements generated in layers near the wall surfaces to capture the boundary layer and

The CFD package, ANSYS Fluent version 18.0, was used for all the simulations performed in this study. 2DCFD model with less computational cost than 3D model was utilized to represent the vertical hydrokinetic turbine and the water domain. Based on the review of relevant works [78, 87–89], the use of a 2D model has been reported to be enough for revealing the factors that influence the performance of the turbine and the majority of factors affecting the flow physics surrounding the vertical axis hydrokinetic turbine, such as the hydrofoil profile, *B*, *σ*, and *D*. In the 2D numerical study of this work, the effects from supporting arms were not taken into consideration. URANS equations were solved using the SIMPLE algorithm for

The vertical hydrokinetic turbine studied was a three-bladed Darrieus rotor with a NACA 0025 blade profile. *c* was set at 0.33 m with *R* equal to 0.75 m. And central axis with a *D* of 0.025 m was placed in the rotation axis. As previously discussed, the mesh is a critical part of a CFD simulation for engineering purposes. It has to be coarse enough so that the calculation is affordable but also fine enough so that each important physical phenomenon is captured and simulated. In this sense, the domain mesh was created around each hydrofoil, and the surrounding water channel geometry was defined based on studies of the boundary extents. There is an inner circular rotating domain connected to a stationary rectangular domain via a sliding interface boundary condition that conserves both mass and momentum. The domain extents were also selected from a series of sensitivity tests to determine the appropriate distance of the walls, inlet and outlet boundaries from the *R* rotor. The domain extends three upstream and 8*R* downstream of the center of the turbine and two 5*R* laterally to either side of the turbine. The circular rotating domain has an overall diameter of two 8*R*. Because the computational domain is 2D, the turbine

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

of triangular or tetrahedral elements that fill the remainder of the domain. Several concepts related to mesh generation are found in [86], and a detailed discussion about the influence of the mesh in CFD applications is presented by

*Computational Fluid Dynamic Simulation of Vertical Axis Hydrokinetic Turbines*

Thompson et al. [83].

**Figure 8.**

**179**

*Illustration of the* 2D *numerical domain.*

pressure velocity coupling [34, 37, 90, 91].

*DOI: http://dx.doi.org/10.5772/intechopen.89184*

blades were implicitly assumed to be infinitely long.

**Figure 7.** *Design procedure of a vertical axis hydrokinetic turbine.*

**Table 1**. In order to calculate those parameters, the flowchart shown in **Figure 7** can be followed.
