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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](https://image.slidesharecdn.com/presentationinore2008-210311161440/85/Unsteady-Potential-Flow-Calculations-on-a-Horizontal-Axis-Marine-Current-Turbine-with-a-Boundary-Element-Method-1-320.jpg)
Unsteady Potential Flow Calculations on a Horizontal Axis Marine Current Turbine with a Boundary Element Method
- 1. I NS T I T U T O S U P E R I ORT É C NI C O U n i v e r s i d a d e T é c n i c a d eL i s b o a 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