Three-dimensional unsteady potential flow calculations for a horizontal axis marine current turbine with a low order potential based panel method, originally developed for marine propellers, are presented. The analysis is carried out for straight and yawed flow conditions for a turbine with controllable pitch for two different pitch settings in a wide range of tip-speed-ratios. An empirical vortex model is assumed for the turbine wake which includes the variation of pitch of the helicoidal vortices behind the blades. Comparison of numerical calculations with experimental measurements available in the literature and effect of the tidal velocity profile on the turbine blade loadings are presented.

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UNSTEADY POTENTIAL FLOW CALCULATIONS ON A HORIZONTAL AXIS
MARINE CURRENT TURBINE WITH A BOUNDARY ELEMENT METHOD
J. Baltazar and J.A.C. Falcão de Campos · Email: baltazar@marine.ist.utl.pt, fcampos@hidro1.ist.utl.pt
MARETEC/IST, Department of Mechanical Engineering, Instituto Superior Técnico, Lisbon, Portugal
Introduction
• There has been a growing interest in the utilisation of horizontal axis
marine current turbines for electrical power production.
• 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: ∇2φ = 0
• Boundary conditions:
I
∂φ
∂n
= −~
V∞ · ~
n on SB ∪ SH
I ~
V + · ~
n = ~
V − · ~
n, p+ = p− on SW ⇒
∂(∆φ)
∂t + Ω
∂(∆φ)
∂θ = 0
I ∇φ → 0, if |~
r| → ∞
• Kutta condition: ∇φ ∞ ⇒ ∆φ = φ+ − φ− or ∆pte = 0
• Fredholm integral equation for Morino formulation:
2πφ (p, t) =
RR
SB∪SH
h
G ∂φ
∂nq
− φ (q, t) ∂G
∂nq
i
dS −
RR
SW
∆φ (q, t) ∂G
∂nq
dS
Numerical Method
• Time discretisation:
∆t = 2π/ΩNt.
• Surface discretisation:
I Turbine blade: cosine spacing in the
radial and chordwise directions.
I Hub surface: elliptical grid generator.
I 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 iterative pressure Kutta
condition.
X
Y
Z
X
Y
Z
Effect of Wake Model
• Pitch β: the vortex lines are aligned with
the time-averaged axisymmetric inflow.
• Pitch βi Optimum: Optimum pitch
distribution obtained from lifting line [1].
• Pitch Ψ-β: the pitch of the vortex lines
varies from the blade geometric
distribution at the trailing edge to the
time-averaged axisymmetric inflow pitch
distribution at the ultimate wake.
• Pitch Ψ-βi Optimum: the pitch of the
vortex lines varies from the blade
geometric distribution at the trailing
edge to the optimum pitch distribution
obtained from lifting line [1] at the
ultimate wake.
TSR
2 4 6 8 10
0.0
0.2
0.4
0.6
0.8
1.0
1.2
Pitch β
Pitch βi
Optimum
Pitch Ψ-β
Pitch Ψ-βi
Optimum
Axial Force Coefficient
Set Angle 5º - 15º Yaw:
TSR
2 4 6 8 10
0.0
0.2
0.4
0.6
0.8
Pitch β
Pitch βi Optimum
Pitch Ψ-β
Pitch Ψ-βi
Optimum
Power Coefficient
Set Angle 5º - 15º Yaw:
Comparison with Experimental Data
TSR
2 4 6 8 10
0.0
0.2
0.4
0.6
0.8
1.0
1.2
0º Yaw
15º Yaw
22.5º Yaw
30º Yaw
Pitch βi Optimum - 0º Yaw
Pitch βi
Optimum - 15º Yaw
Pitch βi
Optimum - 22.5º Yaw
Pitch βi
Optimum - 30º Yaw
Axial Force Coefficient
TSR
2 4 6 8 10
0.0
0.2
0.4
0.6
0.8
1.0
0º Yaw
15º Yaw
22.5º Yaw
30º Yaw
Pitch βi
Optimum - 0º Yaw
Pitch βi
Optimum - 15º Yaw
Pitch βi
Optimum - 22.5º Yaw
Pitch βi Optimum - 30º Yaw
Power Coefficient
TSR
2 4 6 8 10
0.0
0.2
0.4
0.6
0.8
1.0
0º Yaw
15º Yaw
30º Yaw
Pitch βi
Optimum - 0º Yaw
Pitch βi
Optimum - 15º Yaw
Pitch βi
Optimum - 30º Yaw
Axial Force Coefficient
TSR
2 4 6 8 10
0.0
0.2
0.4
0.6
0.8
0º Yaw
15º Yaw
30º Yaw
Pitch βi
Optimum - 0º Yaw
Pitch βi
Optimum - 15º Yaw
Pitch βi
Optimum - 30º Yaw
Power Coefficient
Figure: Comparison with experimental data [2] for 5 degrees (up) and
10 degrees (down) set angles.
Effect of Tidal Velocity Profile
U/U0
y/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(y)=U0
(y/d)1/7
Blade Angle [º]
0 60 120 180 240 300 360
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 10m radius
turbine in a 30m deep sea
and a tidal speed of 2m/s.
• 5 degrees set angle at
design condition
TSR = 6.
• Vortex lines with optimum
pitch distribution [1].
Blade Angle [º]
0 60 120 180 240 300 360
0.10
0.12
0.14
0.16
0.18
0.20
Blade A
Blade B
Blade C
Power Coefficient
Conclusions
• A significant influence of the vortex wake geometry in the
hydrodynamic performance predictions is seen.
• A reasonable 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.
• A considerable time-dependent effect on the turbine blade loadings
due to tidal velocity profile has been found.
References
[1 ] J.A.C. Falcão 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.
2nd INORE Symposium Comrie, Scotland May 4-8, 2008