For marine current turbines under certain operation conditions cavitation on the blades may occur. Therefore, it is important from the design stage of such systems to be able to predict the presence and extent of cavitation on the blades. In this paper a boundary element method for the prediction of sheet cavitation of a horizontal axis marine current turbine is presented. The boundary element method is based on a low-order potential formulation. Dipoles and sources are placed on the rigid body surfaces either on the wetted part and beneath the cavities. Kinematic boundary conditions are applied on the wetted surfaces and kinematic and dynamic boundary conditions are applied on the surface of the cavities. The blade wakes are modeled with an empirical formulation. The method is applied to analyze a marine current turbine in steady flow conditions and results are compared with the cavitation observations available in the literature.

1. Prediction of Sheet Cavitation
on Marine Current Turbines
with a Boundary Element 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)
Technical University of Lisbon, Portugal
MARETEC
31st OMAE Rio de Janeiro, Brazil 1-6 July 2012 1 / 19

2. Motivations
Considerable interest in the use of horizontal axis marine current
turbines for 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.
31st OMAE Rio de Janeiro, Brazil 1-6 July 2012 2 / 19

3. Objectives
Application of BEM code PROPAN (Baltazar & Falc˜ao de
Campos, 2011) to horizontal axis marine current turbines.
Extension of the method to include prediction of sheet cavitation
on both suction and pressure sides of the turbine blades.
Comparison of the numerical results with experimental
observations (Bahaj et al, 2007).
31st OMAE Rio de Janeiro, Brazil 1-6 July 2012 3 / 19

4. Mathematical Formulation
Potential Flow Problem
Undisturbed onset velocity:
U∞ = Uex + Ωreθ
Velocity field: V = U∞ + φ
Laplace equation: 2
φ = 0
Boundary conditions:
∂φ
∂n = −n · U∞ on SB
D
Dt [s3 − η (s1, s2)] = 0,
p = pvapor on SC
V + · n = V − · n, p+ = p− on SW
φ → 0 if |r| → ∞
Kutta condition: | φ| < ∞
U
x
y
z
r
θ
Ω
SC
s1
s2
s3
q
h cavity thickness
31st OMAE Rio de Janeiro, Brazil 1-6 July 2012 4 / 19

5. Mathematical Formulation
Potential Flow Problem
Fredholm integral equation for Morino formulation:
2πφ (p) =
SB∪SC
G ∂φ
∂nq
− φ (q) ∂G
∂nq
dS −
SW
∆φ (q) ∂G
∂nq
dS
Green’s function: G(p, q) = −1/R(p, q)
31st OMAE Rio de Janeiro, Brazil 1-6 July 2012 5 / 19

6. Mathematical Formulation
Cavitation Model
KBC and DBC applied on the blade surface SC and wake surface
SW beneath the cavity.
SW
SC
V
S'W
S'C
n
n
η
ηs 3
SB
31st OMAE Rio de Janeiro, Brazil 1-6 July 2012 6 / 19

7. Mathematical Formulation
Cavitation Model
Kinematic Boundary Condition:
∂η
∂s1
[Vs1 − Vs2 cos θ] + ∂η
∂s2
[Vs2 − Vs1 cos θ] = Vs3 sin2
θ
Dynamic Boundary Condition:
φ =
φ (s0) +
s1
s0
(nD)2
σn + |U∞|2 − V 2
u2
ds1 +
s1
s0
−U∞ · t1 ds1
where σn = p∞−pvapor
1
2
ρn2D2
.
31st OMAE Rio de Janeiro, Brazil 1-6 July 2012 7 / 19

8. Numerical Method
Surface Discretisation
Structured surface grid.
Hyperboloidal quadrilateral panels.
Turbine blade surface: cosine spacing in the radial
and chordwise directions.
Blade wake surface: half-cosine spacing along the
streamwise direction.
Hub surface: elliptical grid generator (E¸ca, 1994).
31st OMAE Rio de Janeiro, Brazil 1-6 July 2012 8 / 19

9. Numerical Method
Panel Method
Integral equation solved by the collocation method.
Constant dipole and source distributions on each panel.
Influence coefficients from Morino & Kuo (1974).
Iterative pressure Kutta condition.
Rigid wake model with prescribed wake geometry:
Constant vortex pitch distribution obtained from
lifting line theory with optimum circulation
distribution (Falc˜ao de Campos, 2007).
Expansion neglected.
31st OMAE Rio de Janeiro, Brazil 1-6 July 2012 9 / 19

10. Numerical Method
Detachment and Reattachment Conditions
Initial detachment and reattachment conditions are obtained
based on the fully wetted pressures.
Determination of the detachment point
(Mueller & Kinnas, 1999):
η < 0 ⇒ detachment point moves downstream.
pupstream < pvapor ⇒ detachment point moves upstream.
Determination of the reattachment point:
η < 0 ⇒ reattachment point moves upstream.
η > 0 ⇒ reattachment point moves downstream.
31st OMAE Rio de Janeiro, Brazil 1-6 July 2012 10 / 19

11. Results
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.
In the present work, 10◦
pitch setting angle is considered.
31st OMAE Rio de Janeiro, Brazil 1-6 July 2012 11 / 19

12. Results
Panel Arrangement
X
Y
Z
X
Y
Z
Blade 60×31, Wake 180×30, Hub 80×36.
31st OMAE Rio de Janeiro, Brazil 1-6 July 2012 12 / 19

13. Results
TSR=8.1 and σn = 4.6 - Cavity Extent and Cavity Thickness
X
Z
Y
0.0027
0.0024
0.0021
0.0018
0.0015
0.0012
0.0009
0.0006
0.0003
0.0000
η/RPressure
Side
X
Y
Z
0.0009
0.0008
0.0007
0.0006
0.0005
0.0004
0.0003
0.0002
0.0001
0.0000
η/R
Suction
Side
31st OMAE Rio de Janeiro, Brazil 1-6 July 2012 13 / 19

14. Results
TSR=8.1 and σn = 4.6 - Convergence and Pressure Distribution
Iteration
0 5 10 15 20 25 30 35 40 45 50
0.0
0.1
0.2
0.3
0.4
0.5
lc max /R
Ac
/R2
η
s/c
-Cpn
0.0 0.2 0.4 0.6 0.8 1.0
-12.0
-8.0
-4.0
0.0
4.0
8.0
r/R=0.950
Detachment on the Suction Side
Detachment on the Pressure Side
31st OMAE Rio de Janeiro, Brazil 1-6 July 2012 14 / 19

15. Results
TSR=8.1 and σn = 4.6 - Comparison with Experiments (Bahaj et al, 2007)
0.0009
0.0008
0.0007
0.0006
0.0005
0.0004
0.0003
0.0002
0.0001
0.0000
η/R
31st OMAE Rio de Janeiro, Brazil 1-6 July 2012 15 / 19

16. Results
TSR=7.2 and σn = 3.7 - Comparison with Experiments (Bahaj et al, 2007)
0.0045
0.0040
0.0035
0.0030
0.0025
0.0020
0.0015
0.0010
0.0005
0.0000
η/R
31st OMAE Rio de Janeiro, Brazil 1-6 July 2012 16 / 19

17. Results
TSR=7.2 - Influence of the Cavitation Number
Pressure Distribution:
s/c
-Cpn
0.0 0.2 0.4 0.6 0.8 1.0
-12.0
-8.0
-4.0
0.0
4.0
8.0
σn
=6.0
σn
=5.0
σn
=4.5
σn=3.7
r/R=0.95
Inviscid Forces:
σn CT CP
Wetted 0.530 0.405
6.0 0.510 0.366
5.0 0.503 0.347
4.5 0.499 0.327
3.7 0.388 0.019
31st OMAE Rio de Janeiro, Brazil 1-6 July 2012 17 / 19

18. Results
Axial Force and Power Predictions
TSR
2 4 6 8 10
0.0
0.2
0.4
0.6
Experiments (Bahaj et al, 2007)
Present Method: Wetted
Present Method: σn
=3.7
Present Method: σn
=3.9
Present Method: σn
=4.6
Young et al (2010): σn=3.9
CT
CP
31st OMAE Rio de Janeiro, Brazil 1-6 July 2012 18 / 19

19. Concluding Remarks
A low-order potential-based BEM for modelling of sheet
cavitation on marine current turbines is presented.
Calculations for two operation conditions are made in
steady flow conditions and results are compared with
cavitation observations from cavitation tunnel tests.
Results show mid-chord cavitation, leading-edge partial
cavitation and super-cavitation.
A considerable effect on the axial force and power predictions
due to the presence of cavitation is found.
Further work:
Effect of the wake model on the cavitation predictions.
Further validation studies.
31st OMAE Rio de Janeiro, Brazil 1-6 July 2012 19 / 19