# Simulation of Horizontal-Axis Tidal Turbine Wakes Using a Coupled Approach With Rankine-Froude Actuator Disk Model and a Weakly-Compressible Finite Volume Solver

Baptiste Elie, Guillaume Oger, Pierre Emmanuel Guillerm, Bertrand Alessandrini

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Baptiste Elie, Guillaume Oger, Pierre Emmanuel Guillerm, Bertrand Alessandrini. Simulation of Horizontal-Axis Tidal Turbine Wakes Using a Coupled Approach With Rankine-Froude Actuator Disk Model and a Weakly-Compressible Finite Volume Solver. 32nd International Conference on Ocean, Offshore and Arctic Engineering, 2013, Nantes, France. 10.1115/OMAE2013-11576. [hal-01158863]

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# SIMULATION OF HORIZONTAL-AXIS TIDAL TURBINE WAKES USING A COUPLED APPROACH WITH RANKINE-FROUDE ACTUATOR DISK MODEL AND A WEAKLY- COMPRESSIBLE FINITE VOLUME SOLVER

B. Elie

UNAM, Ecole Centrale Nantes
LHEEA lab, UMR6598

France

G. Oger

UNAM, Ecole Centrale Nantes
LHEEA lab, UMR6598

France

P.-E. Guillerm

UNAM, Ecole Centrale Nantes
LHEEA lab, UMR6598

France

B. Alessandrini

UNAM, Ecole Centrale Nantes
LHEEA lab, UMR6598

France

## ABSTRACT

This paper presents the early work that has been achieved in the perspective of developing a tidal farm simulation tool. First a horizontal-axis tidal turbine (HATT) simulation model coupled with a Finite Volume solver is introduced. This model is based on the classical actuator disk theory (Rankine and Froude, 19th century). Then the fluid model adopted in our approach is presented and discussed. This model relies on a Finite Volume solver applied to a fully-cartesian grid, and is based on weakly-compressible explicit Navier-Stokes equations. The first results obtained on the wake are presented and discussed in order to quantify the accuracy and the validity of this approach.

## INTRODUCTION

In the context of renewable marine energies, tidal turbine technology is one of the solutions that are currently under development in an industrial context. The purpose of such energy extraction devices is to be settled in farms so the global energy extracted is high enough to be noticeable and the cost of electrical transport and conversion components decreases. However, wake flow interactions between devices occur, and their behavior and final outputs depend on the spatial pattern within the farm. As observed in the past for wind turbine researches, dedicated simulation tools will be needed sooner or later so the engineers can optimize the overall outputs and study the environmental impacts of such farms.

Generally, fluid solvers are diffusive and simulating wake interactions remains a difficult task. This fact motivates the choice of a Finite Volume formulation applied on a Cartesian Grid, for which the diffusion is sufficiently low. Furthermore, the simulations will possibly involve large tidal turbine farms, requiring several hundred million cells on massively parallel calculations. The choice of an explicit formalism is then judicious for such a framework.

Several approaches are possible for modeling tidal turbines, from the simplest approach using analytical methods (Lissaman [1]) to fully-meshed methods (direct modeling). Various other methods can be found in the literature, such as the vortex wake method (Maganga et al. [2]). Another possibility consists in using actuator models (Sanders [3], Ivanell [4]). This technique consists in representing the rotor by a simple geometry, calculating the force which would be actually applied by the real turbine on its surrounding environment and finally apply this force on the fluid flow as a momentum sink. Different variants are available, such as actuator disk, actuator line or actuator surface. In each model, a distribution of source terms is made on surfaces representing the blades. These methods offer a good trade-off between accuracy and performance.

In this paper, first the tidal turbine model is introduced. The fluid solver adopted in our approach is then presented. Finally, preliminary results obtained with our model are presented and discussed.
## TIDAL TURBINE MODEL

The HATT is modeled using the actuator disk theory. In the current application, the actuator disk models a tidal turbine and thus slows down the flow. The model has been developed in 2-D and 3-D and can also be used in multi-model simulations.

### PRINCIPLE

Starting from the simplistic concept of mass flow rate conservation along a stream-tube (Figure 1) (Burton [5], Espana [6], Crespo [7]) $ρA_∞U_∞ = ρA_dU_d = ρA_ωU_ω$ (1), around a tidal turbine which induces a velocity variation $U_d = U_∞(1 - a)$ (2). Using the momentum theory which describes the rate of change of momentum while the fluid passes through the disk, and then using Bernoulli's equation on each side of the actuator disk, we end up with $U_ω = (1 - 2a)U_∞$ (3), where a is the axial flow induction factor, $U_ω$ is the speed downward the tidal turbine and $U_∞$ is the speed of the flow away from the machine. As we also need to consider a rotating wake, we add up the angular momentum theory which introduces a tangential flow induction factor $a'$, we obtain the tangential velocity downward the disk which is $2Ωra'$.

<figure><img src="image_2.png" /><figcaption>Figure 1: Stream-tube around a tidal turbine</figcaption></figure>

From this, one can easily get the thrust and torque coefficients depending on the upstream velocity:

$$C_T = \frac{F}{\frac{1}{2}\rho U_\infty^2 \frac{\pi D^2}{4}} \quad (4)$$

$$C_Q = \frac{Q}{\frac{1}{2}\rho U_\infty^2 \frac{\pi D^3}{4}} \quad (5)$$

where D is the diameter of the actuator disk, F the force of the actuator disk on the fluid and Q its torque.

Then, we introduce the advance ratio parameter:

$$J_0 = \frac{U_\infty}{n \cdot D} \quad (6)$$

where n is the rotational speed of the blades.

Hence $J_0$ is used to define $K_t$ and $K_q$, which are the thrust and torque coefficients depending on the rotational speed of the blades as a function of their upstream velocity form:

$$K_T = \frac{\pi C_T J_0^2}{8} \quad (7)$$

$$K_Q = \frac{\pi C_Q J_0^2}{8} \quad (8)$$

As we know $K_T$ and $K_Q$ evolution when we change the rotational speed for given tidal turbine parameters, we calculate $T$, and $Q$, then $C_T$ and $C_Q$ and finally $a$ and $a'$, leading to the source terms applied into the fluid.

In order to regulate the turbines, for instance in the case of wakes interactions, the advance ratio term is taken into account by using the actual incident fluid velocity entering the turbine.

This way, playing on their rotational speed, the tidal turbines can regulate each other.

## FLUID SOLVER

### GENERAL VIEW

The fluid solver used here is a cell-centered finite volume solver based on the compressible Navier-Stokes equations in their Euler form, within an explicit formalism (Leroy [8]).

The idea is to use compressible Navier-Stokes equations in a weakly-compressible approach, so that a fluid that is usually considered as incompressible is simulated with a low Mach number ($Ma \approx 0.1$). The equations are resolved on a Cartesian grid and are generalized 2-D/3-D for our kind of applications.

### GENERAL EQUATIONS

Here are Navier-Stokes equations for compressible flows on Newtonian fluids in a weakly-compressible approach.

**Continuity equation**

$$\frac{\partial \rho}{\partial t} + \nabla \cdot (\rho u) = 0 \quad (9)$$

**Momentum equation**

$$\rho \left( \frac{\partial v}{\partial t} + v \cdot \nabla v \right) = - \nabla p + \mu \nabla^2 v + f \quad (10)$$

An additional equation is required to close the system. We introduce the Tait equation of state, which describes the fluid pressure as a function of density as follows

$$P - P_0 = \frac{\rho_0 c_0^2}{\gamma} \left[ \left( \frac{\rho}{\rho_0} \right)^\gamma - 1 \right] \quad (11)$$

where $ρ$ is the density, $γ$ the polytrophic constant usually set as 7 for water applications, $P_0$ a reference pressure, $ρ_0$ the nominal density and $C_0$ the nominal speed of sound.
## Characteristic flux schemes

Starting from the Navier-Stokes equations for inviscid flows:

$$ \overline{W}_t + \overline{\Psi}(W)_{x,y,z} = \vec{0} \quad (12) $$

$$ \text{with } \overline{W}_t = \begin{pmatrix} \rho \\ \rho u \\ \rho v \\ \rho w \end{pmatrix}_t $$

$$ \text{and } \overline{\Psi(W)}_{x,y,z} = \begin{pmatrix} \rho u \\ \rho u^2 + P \\ \rho uv \\ \rho uw \end{pmatrix}_x + \begin{pmatrix} \rho v \\ \rho vu \\ \rho v^2 + P \\ \rho vw \end{pmatrix}_y + \begin{pmatrix} \rho w \\ \rho wu \\ \rho ww \\ \rho w^2 + P \end{pmatrix}_z $$

which can be rewritten in integral form as:

$$ \iiint_{V(t)} \frac{\partial \overline{W}}{\partial t} dV + \iiint_{V(t)} \vec{\nabla} \cdot \Psi(W) dV = 0 \quad (13) $$

Then using the Green-Ostrogradsky theorem and the Leibniz integral rule, we end up with:

$$ \frac{d}{dt} \iiint_{V(t)} \overline{W} dV - \iint_{S(t)} W U_{int} \vec{n} dS + \iint_{S(t)} \Psi(W, \vec{n}) dS = 0 \quad (14) $$

Applying the characteristic flux formalism introduced by Ghidaglia et al. [9], fluxes at cell interfaces are calculated as follows:

$$ \Psi(\overline{W}, \vec{n}) = W(\vec{u}, \vec{n}) + P(0, \vec{n}) \quad (15) $$

where $\vec{u}$, $\vec{n}$ and $\overline{W}$ corresponds respectively to the velocity vector, the normal vector to the interface and the vector of conservative variables.

Decomposing the surface terms into parts applied on fixed edges $\Gamma$ and moving edges $\Sigma(t)$ ($S(t) = \Gamma + \Sigma(t)$), the integral form of equation 15 is obtained:

$$ \iint_{\Sigma} \Psi(W, \vec{n}) dS = \iint_{\Sigma} W U_{int} \vec{n} dS + \iint_{\Sigma} P \begin{pmatrix} 0 \\ \vec{n} \end{pmatrix} dS \quad (16) $$

Then, using this last equation into equation 14 leads to:

$$ \frac{d}{dt} \iiint_{V(t)} \overline{W} dV + \iint_{\Sigma} P \begin{pmatrix} 0 \\ \vec{n} \end{pmatrix} dS + \iint_{\Gamma} \Psi(W, \vec{n}) dS = 0 \quad (17) $$

Hence, considering the hyperbolic part of the Navier-Stokes equations, system (12) can be rewritten as follows:

$$ \frac{\partial W}{\partial t} + \sum_{i=1}^{nb \text{ of dim } \Gamma_{\text{int}}} \bar{J}(W, \vec{e}_i) \frac{\partial W}{\partial x_i} = 0 \quad (18) $$

where $\vec{e}_i$ is a vector of the Cartesian base and $\bar{J}(W, \vec{e}_i)$ is the Jacobian Matrix of our system expressed in the Cartesian base.

Expressing $W$ at the mid interface through a simple linear interpolation between the left and right states leads to $W_{int}$. Then, using the Jacobian for barotropic fluids (19), we finally compute the eigensystem (20) giving eigenvalues $\lambda_i$ (with $i = [1, ..., nb \text{ of dim}$] and left and right eigenvectors $L$ and $R$ of the Jacobian matrix, leading to the upwind flux at the interface:

$$ \bar{J} = \frac{\partial \Psi(W, \vec{n})}{\partial W} = \begin{pmatrix} 0 & \vec{n} \\ c^2 \vec{n} - (\vec{u} \cdot \vec{n}) \vec{u} & \vec{u} \otimes \vec{n} + \vec{u} \cdot \vec{n} \mathrm{Id}_{\mathrm{dim}+2} \end{pmatrix} \quad (19) $$

$$ l_i(\overline{W_{\text{int}}, \vec{n}}).\emptyset_S = \begin{cases} l_i(\overline{W_{\text{int}}, \vec{n}}).\Psi(W_r, \vec{n}) & \text{if } \lambda_{\text{ik}}(\overline{W_{\text{int}}, \vec{n}}) > 0 \\ l_i(\overline{W_{\text{int}}, \vec{n}}).\frac{\Psi(W_r, \vec{n}) + \Psi(W_l, \vec{n})}{2} & \text{if } \lambda_{\text{ik}}(\overline{W_{\text{int}}, \vec{n}}) = 0 \\ l_i(\overline{W_{\text{int}}, \vec{n}}).\Psi(W_l, \vec{n}) & \text{if } \lambda_{\text{ik}}(\overline{W_{\text{int}}, \vec{n}}) < 0 \end{cases} \quad (20) $$

$$ \psi_S = \frac{\psi(W_L, \vec{n}) + \psi(W_R, \vec{n})}{2} - \operatorname{sign}(\bar{J}(W_{\text{int}}, \vec{n})) \frac{\psi(W_R, \vec{n}) + \psi(W_L, \vec{n})}{2} \quad (21) $$

$$ \text{with } \operatorname{sign}(\bar{J}(W_{\text{int}}, \vec{n})) = R(W_{\text{int}}, \vec{n}) \cdot \operatorname{diag}(\operatorname{sign}(\lambda(W_{\text{int}}, \vec{n}))) \cdot L(W_{\text{int}}, \vec{n}). $$

The Finite Volume formalism is finally obtained as follows:

$$ \frac{d(WV)}{dt} + \sum_{k=1}^{nb \text{ of interfaces}} \psi_S \cdot \vec{n}_k \cdot \overline{dS} = 0 \quad (22) $$

then advanced in time through an explicit time integration scheme such as Runge-Kutta of second or any higher order.

## CFL (Courant-Friedrichs-Lewy) condition

In order to ensure the stability of our explicit system, timesteps are defined by the following CFL condition:

$$ dt \le \min_j \left( \frac{\mathrm{Vol}_j}{\max(\Gamma_{\text{int}}) \cdot \max_i |\lambda_j)_i} \right) \quad (23) $$

where $\mathrm{Vol}_j$ and $\Gamma_{\text{int}}$ are respectively the volume of the considered cell, and the area of interface between two adjacent cells.

## Boundary conditions

The fluid domain used in the simulations presented in this paper is illustrated in Figure 2. In a first approach, free-slip boundary conditions are applied at the top and at the bottom of the tank. Inlet and outlet conditions are applied on the left and right tank boundaries respectively. An inflow velocity is imposed at the inlet boundary, while a hydrostatic pressure field is imposed as outflow pressure.
<figure><img src="image_3.png" /><figcaption>Figure 2: Domain geometry and boundary conditions</figcaption></figure>

<figure><img src="image_4.png" /><figcaption>Figure 3: Axial-velocity field and stream-lines around a single Tidal Turbine</figcaption></figure>

## FIRST RESULTS

We present here the first results obtained with this implementation.

The calculations are performed in 2-D, so that the tangential terms normally induced by the actuator disk are neglected here. These features are exclusively developed in the 3-D version.

The results presented here involve a rough hub model, as displayed in figures 3 and 4. Actually, for now we simply assume the presence of source terms modeling an actuator disk settled around a rough structure that represents a hub.

## TIDAL TURBINE ALONE

The two figures in this part display the fluid flow behavior around the tidal turbine.

The figure 3 shows the axial component of the velocity field obtained. As expected, the average velocity downstream the machine is decreased. Upstream the actuator disk, the presence of the turbine causes the approaching water velocity to slow down gradually so the stream tube starts to expand.

Figure 4 shows that the pressure rises upstream the actuator disk in order to absorb the decrease in kinetic energy. As expected, a drop of pressure appears as the water passes through the actuator disk. Furthermore, low pressures are observed behind the hub. Far downstream, the pressure return to the nominal level. The rise of static pressure decreases the kinetic energy, slowing down the water. This is a classical result of the simplified theory of flows around wind/tidal turbines [2].

<figure><img src="image_5.png" /><figcaption>Figure 4: Pressure field and stream-lines around a single Tidal Turbine</figcaption></figure>

Both figures 3 and 4 show the stream lines around the machine. Of course, these lines are affected by the presence of the actuator disk. In the stream-tube theory, the mass of water which passes through the rotor must slow down. No water penetrates across the stream tube boundary so that the mass flow rate remains the same along the stream tube. The mass conservation imposes a stream tube expansion in order to accommodate the water velocity slow down. All these effects are captured by the above simulation, which shows some qualitatively good results. Validations based on comparisons with experiments will be performed very soon, in order to evaluate the accuracy of this method.
## TIDAL TURBINES SETTLED IN FARMS

Figure 5, 6, 7 are a demonstration of multiple tidal turbines settled in farms. The difficulty of such a simulation relies on the complexity of the wake interactions. Namely, the fluid solver should necessarily display some nice conservation properties in order to minimize the numerical damping. We can see how the first row of turbines acts on the second one. The flow is decreased while getting through the successive turbines. We can also see the global wake deviation, namely for the up and down tidal turbines due to the absence of a neighbor turbine on one of their sides. These first results with several machines are very promising. It shows the ability of the models to capture complex wake interactions, with good robustness properties.

<figure><img src="image_6.png" /><figcaption>Figure 5: Axial-velocity field of a 8 Tidal Turbine farm $t=9s$</figcaption></figure>

<figure><img src="image_7.png" /><figcaption>Figure 6: Axial-velocity field of a 8 Tidal Turbine farm $t=15s$</figcaption></figure>

<figure><img src="image_8.png" /><figcaption>Figure 7: Axial-velocity field of a 8 Tidal Turbine farm $t=20s$</figcaption></figure>

## CONCLUSIONS

The development of a 2-D/3-D simulation tool based on Finite Volumes for modeling tidal turbine farms has been started, and some preliminary results have been presented in this paper. The coupling between the Finite Volume solver and the actuator disk model based on source terms has shown promising results. As the hubs present some non-negligible effects on the fluid flow and especially on the interacting wakes of the tidal turbines, a particular attention has been paid on their inclusion in the model, through the imposition of solid boundary conditions. The results presented here are only demonstrative, but the future works will be dedicated to validating this model through some comparisons with experiments. To allow such validations, a turbulence model will be developed. This model essentially targets to the simulation of 3-D tidal farms on long physical periods and possibly with very local effects, so that a specific implementation need to be developed in order to speed up the computations. To allow this, a parallel Adaptive Mesh Refinement (AMR) algorithm is currently under development.

## ACKNOWLEDGMENTS

This work is part of a PhD in the LHEEA laboratory of Ecole Centrale de Nantes. This PhD is supported by Alstom Hydro France.

## REFERENCES

[1] P.B.S Lissaman (1979). Energy Effectiveness of Arbitrary Arrays of Wind Turbines. J. Energy.

[2] Fabrice, M., Grégory, P., Grégory, G., & Elie, R. (2008). Numerical characterisation of the wake generated by marine current turbines farm. *2nd International Conference on Ocean Energy*.

[3] Sanderse, B., Pijl, S. P. V. D., & Koren, B. (n.d.). Review of CFD for wind-turbine wake aerodynamics. *Fluid Dynamics*, 1-28.

[4] Ivanell, B. S., Mikkelsen, R., Sørensen, J. N., & Henningson, D. (2008). Validation of methods using EllipSys3D, 183-221.

[5] Burton, T., Sharpe, D., Jenkins, N., & Bossanyi, E. (2001). Aerodynamics of Horizontal-Axis Wind Turbines. Wind Energy Handbook (Vol. 2).

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[8] C. Leroy et al., Preliminary developments of a two-fluid Cartesian-Grid Finite-Volume Characteristic Flux model for marine applications, 10th WCCM.

[9] J.-M. Ghidaglia, A. Kumbaro, G. Le Coq, Une méthode volumes finis à flux caractéristiques pour la résolution numérique des systèmes hyperboliques de lois de conservation, C.R. Acad. Sc. Paris, Vol.322, p. 981-988, (1996)