This document summarizes a study that used an unsteady panel method to model the hydrodynamic performance of a horizontal axis marine current turbine in straight and yawed inflow conditions. The panel method code PROPAN was applied and results were compared to experimental data. Including viscous corrections in the model improved agreement with experimental performance coefficients for the turbine, capturing the effects of yaw angle near the turbine's design operating point. Future work could involve predicting the geometry of the turbine's vortex wake.

1. Unsteady Analysis of a
Horizontal Axis Marine Current Turbine
in Yawed Inflow Conditions With a Panel Method
J. Baltazar and J.A.C. Falc˜ao de Campos
Marine Environment and Technology Center (MARETEC)
Department of Mechanical Engineering
Instituto Superior T´ecnico (IST)
Lisbon, Portugal
MARETEC
SMP09 Trondheim, Norway 22-24 June 1 / 15

2. Motivations
There has been a growing interest in the utilisation of horizontal
axis marine current turbines for electrical power production from
renewable tidal energy.
The ability to predict the hydrodynamic performance is essential
for the design and analysis of such systems.
BEM potential flow models may be used to predict pressure
distributions, integrated forces and cavitation performance.
SMP09 Trondheim, Norway 22-24 June 2 / 15

3. Objectives
Application of BEM code PROPAN (Falc˜ao de Campos, 2000)
for marine current turbines in straight and yawed inflow
conditions.
Comparison of the numerical results with experimental data
(Bahaj et al., 2007).
SMP09 Trondheim, Norway 22-24 June 3 / 15

4. 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| → 0 if |r| → ∞
Ue(x0,r0,θ0)
x0≡x
y
z
r0≡r
θ
Ω
y0
z0
θ0
SMP09 Trondheim, Norway 22-24 June 4 / 15

5. Mathematical Formulation
Wake model:
∂(∆φ)
∂t
+ Ω∂(∆φ)
∂θ
= 0 ⇒ ∆φ (r, θ, t) = ∆φ r, t − θ−θTE
Ω
Kutta condition: | φ| < ∞
Fredholm integral equation for Morino formulation:
2πφ (p, t) =
SB ∪SH
G (p, q) ∂φ
∂nq
− φ (q, t) ∂G
∂nq
dS −
SW
∆φ (q, t) ∂G
∂nq
dS
where G (p, q) = −1/R (p, q)
SMP09 Trondheim, Norway 22-24 June 5 / 15

6. Numerical Method
Time Discretisation:
Time step n = t/∆t, where ∆t = 2π/(ΩNθ).
Surface Discretisation:
Structured grid with quadrilateral hyperboloidal elements.
Panel Method:
Integral equation solved in space by the collocation method
for the key blade.
Constant source and dipole distributions. Linear dipole
distribution at the first radial wake strip.
Influence coefficients calculated using the formulations of
Morino and Kuo (1974).
Rigid wake model with iterative pressure Kutta condition.
SMP09 Trondheim, Norway 22-24 June 6 / 15

7. Vortex Wake Model
Rigid Wake Model:
Helicoidal vortex wake.
Pitch distribution determined from the lifting line theory with
optimum circulation distribution (Falc˜ao de Campos, 2007).
Expansion of vortex lines is neglected.
SMP09 Trondheim, Norway 22-24 June 7 / 15

8. Viscous Effects on Blade Forces
Hydrodynamic pitch angle:
tan βi = dQi
rdTi
Angle of attack: α = βi − ψ
Viscous lift force: dL = dLi
CLv
CLi
Drag force: dD = CD
1
2
ρU2
e cdr
Axial force and torque with viscous
effects:
dT = dL cos βi + dD sin βi
dQ/r = dL sin βi − dD cos βi
rΩ
U
Ue
βα
ψ
iv
dL
dD
dT
dQ
r
iβ
iβ
CLi
(α) from 2D BEM computations (Vaz, 2005)
CLv (α),CD(α) from XFoil computations (Bahaj et al., 2007)
SMP09 Trondheim, Norway 22-24 June 8 / 15

9. Turbine Performance Data
Tip-speed-ratio: TSR = ΩR
U
Axial force coefficient: CT = T
1/2ρU2πR2
Power coefficient: CP = ΩQ
1/2ρU3πR2
SMP09 Trondheim, Norway 22-24 June 9 / 15

10. Test Case
Turbine rotor: Bahaj et al. (2007)
Three-bladed turbine with NACA 63-8XX sections.
Standard geometry has a pitch angle at blade root equal
to 15◦
, corresponding to 0◦
pitch setting.
Design condition: 5◦
pitch setting at TSR = 6.
In the present work, 5◦
and 10◦
pitch setting angles were
considered.
Yaw angles: 0◦
, 15◦
and 30◦
.
SMP09 Trondheim, Norway 22-24 June 10 / 15

11. Panel Arrangement
X
Y
Z
X
Z
Y
Discretisation: 60×31 Blade, 180×30 Wake, 80×36 Hub.
SMP09 Trondheim, Norway 22-24 June 11 / 15

12. Results
Straight Inflow Condition (Yaw Angle: 0◦
)
TSR
2 4 6 8 10
0.0
0.2
0.4
0.6
0.8
1.0
1.2
0º Yaw - Experiments (Bahaj et al., 2007)
Lifting Line Inviscid
Lifting Line Viscous
Panel Method
Panel Method with Drag Corrections
Panel Method with Viscous Corrections
CT
Set Angle 5º
TSR
CT
2 4 6 8 10
0.0
0.2
0.4
0.6
0.8
1.0
0º Yaw - Experiments (Bahaj et al., 2007)
Lifting Line Inviscid
Lifting Line Viscous
Panel Method
Panel Method with Drag Corrections
Panel Method with Viscous Corrections
Set Angle 10º
TSR
2 4 6 8 10
0.0
0.2
0.4
0.6
0.8
1.0
0º Yaw - Experiments (Bahaj et al., 2007)
Lifting Line Inviscid
Lifting Line Viscous
Panel Method
Panel Method with Drag Corrections
Panel Method with Viscous Corrections
CP
Set Angle 5º
TSR
2 4 6 8 10
0.0
0.2
0.4
0.6
0.8
0º Yaw - Experiments (Bahaj et al., 2007)
Lifting Line Inviscid
Lifting Line Viscous
Panel Method
Panel Method with Drag Corrections
Panel Method with Viscous CorrectionsCP
Set Angle 10º
SMP09 Trondheim, Norway 22-24 June 12 / 15

13. Results
Set Angle 5◦
in Yawed Inflow Conditions
TSR
2 4 6 8 10
0.0
0.2
0.4
0.6
0.8
1.0
1.2
Experiments (Bahaj et al., 2007)
Panel Method
Panel Method with Drag Corrections
Panel Method with Viscous Corrections
CT
15º Yaw Angle
TSR
2 4 6 8 10
0.0
0.2
0.4
0.6
0.8
1.0
1.2
Experiments (Bahaj et al., 2007)
Panel Method
Panel Method with Drag Corrections
Panel Method with Viscous Corrections
CT
30º Yaw Angle
TSR
2 4 6 8 10
0.0
0.2
0.4
0.6
0.8
Experiments (Bahaj et al., 2007)
Panel Method
Panel Method with Drag Corrections
Panel Method with Viscous Corrections
CP
15º Yaw Angle
TSR
2 4 6 8 10
0.0
0.2
0.4
0.6
0.8
Experiments (Bahaj et al., 2007)
Panel Method
Panel Method with Drag Corrections
Panel Method with Viscous Corrections
CP
30º Yaw Angle
SMP09 Trondheim, Norway 22-24 June 13 / 15

14. Results
Set Angle 10◦
in Yawed Inflow Conditions
TSR
CT
2 4 6 8 10
0.0
0.2
0.4
0.6
0.8
1.0
Experiments (Bahaj et al., 2007)
Panel Method
Panel Method with Drag Corrections
Panel Method with Viscous Corrections
15º Yaw Angle
TSR
CT
2 4 6 8 10
0.0
0.2
0.4
0.6
0.8
1.0
Experiments (Bahaj et al., 2007)
Panel Method
Panel Method with Drag Corrections
Panel Method with Viscous Corrections
30º Yaw Angle
TSR
2 4 6 8 10
0.0
0.2
0.4
0.6
0.8
Experiments (Bahaj et al., 2007)
Panel Method
Panel Method with Drag Corrections
Panel Method with Viscous Corrections
CP
15º Yaw Angle
TSR
2 4 6 8 10
0.0
0.2
0.4
0.6
0.8
Experiments (Bahaj et al., 2007)
Panel Method
Panel Method with Drag Corrections
Panel Method with Viscous Corrections
CP
30º Yaw Angle
SMP09 Trondheim, Norway 22-24 June 14 / 15

15. Conclusions
A fair to good agreement with the experimental performance
data is found near the design condition.
The effect of yaw in the performance coefficients seems to be
reasonably captured near design condition.
The inclusion of the viscous corrections is seen to significantly
decrease the performance coefficients, bringing the calculations
to a better agreement with the experimental data.
Future Developments:
Prediction of the vortex wake geometry.
SMP09 Trondheim, Norway 22-24 June 15 / 15