A fully three-dimensional potential flow Boundary Element Method for the analysis of the unsteady flow around marine current turbines is presented. An empirical vortex model independent of the induced velocities is assumed for the turbine wake. The application of the method to the analysis of a controllable pitch horizontal axis marine current turbine is illustrated for straight and yawed inflow conditions at two different pitch settings in a wide range of tip-speed-ratios. Comparison of the numerical calculations with experimental data available in the literature is presented. The effect of a tidal velocity profile on the unsteady turbine blade is illustrated.

1. A BOUNDARY ELEMENT METHOD FOR THE
UNSTEADY HYDRODYNAMIC ANALYSIS
OF MARINE CURRENT TURBINES
J. Baltazar and J.A.C. Falc˜ao de Campos
baltazar@marine.ist.utl.pt, fcampos@hidro1.ist.utl.pt
IST/MARETEC
Department of Mechanical Engineering
Instituto Superior T´ecnico, Lisbon, Portugal
INSTITUTO SUPERIORTÉCNICO
UniversidadeTécnicadeLisboa
Introduction
• There has been a growing interest in the utilisation of
horizontal axis marine current turbines for electrical power
generation.
• The ability to predict the hydrodynamic performance is
fundamental for the design and analysis of such systems.
• Marine current turbines are subject to a non-uniform inflow
due to variations on the tidal direction and velocity profile.
Mathematical Formulation
• Undisturbed inflow velocity field:
V∞ (x, r, θ, t) = Ue (x, r, θ − Ωt) − Ω × x
• Velocity field: V = V∞ + φ
• Laplace equation: 2
φ = 0
• Boundary conditions:
∂φ
∂n = −V∞ · n on SB ∪ SH
V +
· n = V −
· n, p+ = p− on SW ⇒ ∂(∆φ)
∂t + Ω∂(∆φ)
∂θ = 0
φ → 0, if |r| → ∞
• Kutta condition: φ < ∞ ⇒ ∆φ = φ+
− φ−
or ∆pte = 0
• Fredholm integral equation for Morino formulation:
2πφ (p, t) =
SB∪SH
G ∂φ
∂nq
− φ (q, t) ∂G
∂nq
dS −
SW
∆φ (q, t) ∂G
∂nq
dS
Numerical Method
• Time discretisation: ∆t = 2π/ΩNt.
• Surface discretisation:
Turbine blade: cosine spacing in the radial and chordwise
directions.
Hub surface: elliptical grid generator.
Blade wake surface: half-cosine spacing along the
streamwise direction.
• Solution of the integral equation: integral equation solved in
space by the collocation method.
• Numerical Kutta condition: rigid wake model with optimum
pitch distribution [1] and iterative pressure Kutta condition.
Effect of Tidal Velocity Profile
U/U
y0/d
0.0 0.2 0.4 0.6 0.8 1.0
0.0
0.2
0.4
0.6
0.8
1.0
Tidal Velocity Profile
U(y0
)=U(y0
/d)1/7
--
-θ0 [º]
CT
0 120 240 360
0.20
0.22
0.24
0.26
0.28
0.30
Blade A
Blade B
Blade C
Axial Force Coefficient
• Velocity profile across
the marine current
turbine, based on a
10 m radius turbine in
a 30 m deep sea and a
tidal speed of 2 m/s.
• 5 degrees set angle at
design condition
TSR = 6.
-θ0 [º]
CP
0 120 240 360
0.10
0.12
0.14
0.16
0.18
0.20
Blade A
Blade B
Blade C
Power Coefficient
References
1. J.A.C. Falc˜ao de Campos, 2007. “Hydrodynamic Power
Optimization of a Horizontal Axis Marine Current Turbine
With Lifting Line Theory”. In Proceedings of the 17th
International Offshore and Polar Engineering Conference, 1,
pp. 307–313.
2. A.S. Bahaj, A.F. Molland, J.R. Chaplin, W.M.J. Batten,
2007. “Power and Thrust Measurements of Marine Current
Turbines Under Various Hydrodynamic Flow Conditions in a
Cavitation Tunnel and a Towing Tank”. Renewable Energy,
32(3), pp. 407–426.
Comparison With Experimental Data
TSR
2 4 6 8 10
0.0
0.2
0.4
0.6
0.8
1.0
1.2
0.0º Yaw - Experiments [2]
15.0º Yaw
22.5º Yaw
30.0º Yaw
0.0º Yaw - Present Method
15.0º Yaw
22.5º Yaw
30.0º Yaw
Axial Force Coefficient
CT
TSR
2 4 6 8 10
0.0
0.2
0.4
0.6
0.8
1.0
0.0º Yaw - Experiments [2]
15.0º Yaw
22.5º Yaw
30.0º Yaw
0.0º Yaw - Present Method
15.0º Yaw
22.5º Yaw
30.0º Yaw
Power Coefficient
CP
TSR
2 4 6 8 10
0.0
0.2
0.4
0.6
0.8
1.0
0.0º Yaw - Experiments [2]
15.0º Yaw
30.0º Yaw
0.0º Yaw - Present Method
15.0º Yaw
30.0º Yaw
Axial Force Coefficient
CT
TSR
2 4 6 8 10
0.0
0.2
0.4
0.6
0.8
1.0
0.0º Yaw - Experiments [2]
15.0º Yaw
30.0º Yaw
0.0º Yaw - Present Method
15.0º Yaw
30.0º Yaw
Power Coefficient
CP
Figure: Comparison with experimental data [2] for 5o
(top) and 10o
(bottom) set angles.
Conclusions
• A reasonable to good agreement with the experimental data is achieved near the design condition
where viscous effects are small.
• Viscous effects need to be taken into account for predictions at off-design conditions.
• The effect of yaw on the decrease of axial force and power coefficients seems to be reasonably
captured near design condition.
• A considerable time-dependent effect on the turbine blade loadings due to tidal velocity profile has
been found.
Surface Grids
LE
TE
ICOE 2008 Brest, France October 15-17