An iteratively coupled solution method for the calculation of unsteady sheet cavitation on marine current turbines with a potential-based Boundary Element Method is investigated. The solution of the linear system of equations is obtained with an iterative technique which avoids a new matrix inversion at each iteration step in the prediction of the cavity planform. The solution method explores the fact that only the source strengths on the panels beneath the cavity change due to the presence of the cavity. The advantage is that the complete system matrix is identical to the matrix of the wetted flow problem and needs only to be inverted once at each time step. The numerical studies are carried out for a marine current turbine, where a significant reduction in the computational time is obtained with the iteratively coupled technique in comparison with the classical approach to the cavitating problem.

1. Prediction of Unsteady Sheet Cavitation
on Marine Current Turbines
With a Boundary Element Method
J. Baltazar∗, J.A.C. Falc˜ao de Campos
MARETEC, Department of Mechanical Engineering
Instituto Superior T´ecnico, Universidade de Lisboa, Portugal
M2D’2015 Ponta Delgada, Azores, Portugal 26-30 July 1

2. Motivations
Considerable interest in the use of horizontal axis marine current
turbines for hydro-kinetic energy extraction.
M2D’2015 Ponta Delgada, Azores, Portugal 26-30 July 2

3. Motivations
Considerable interest in the use of horizontal axis marine current
turbines for hydro-kinetic energy extraction.
The ability to predict the hydrodynamic performance is essential
for the design and analysis of such systems.
M2D’2015 Ponta Delgada, Azores, Portugal 26-30 July 2

4. Motivations
Considerable interest in the use of horizontal axis marine current
turbines for hydro-kinetic energy extraction.
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, forces and cavitation performance.
M2D’2015 Ponta Delgada, Azores, Portugal 26-30 July 2

5. 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 unsteady
sheet cavitation on the turbine blades.
M2D’2015 Ponta Delgada, Azores, Portugal 26-30 July 3

6. Mathematical Formulation
Potential Flow Problem
Undisturbed onset velocity:
U∞ = U − Ω × x
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 = pvapour on SC
V + · n = V − · n, p+ = p− on SW
φ → 0 if |r| → ∞
Kutta condition: | φ| < ∞
Ue(x0,r0,θ0)
x0≡x
y
z
r0≡r
θ
Ω
y0
z0
θ0
SC
s1
s2
s3
q
h cavity thickness
M2D’2015 Ponta Delgada, Azores, Portugal 26-30 July 4

7. Mathematical Formulation
Potential Flow Problem
Fredholm integral equation for Morino formulation:
2πφ (p, t) =
SB∪SC
G ∂φ
∂nq
− φ (q, t) ∂G
∂nq
dS −
SW
∆φ (q, t) ∂G
∂nq
dS
Green’s function: G(p, q) = −1/R(p, q)
M2D’2015 Ponta Delgada, Azores, Portugal 26-30 July 5

8. 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
M2D’2015 Ponta Delgada, Azores, Portugal 26-30 July 6

9. Mathematical Formulation
Cavitation Model
Kinematic Boundary Condition:
∂η
∂s1
[Vs1 − Vs2 cos θ] + ∂η
∂s2
[Vs2 − Vs1 cos θ] = Vs3 − ∂η
∂t
sin2
θ
Dynamic Boundary Condition:
φ = φ (s0) +
s1
s0
−U∞ · t1 ds1 +
s1
s0
(nD)2
σn + |U∞|2 − 2gy0 − 2∂φ
∂t
− V 2
u2
ds1
where σn = p∞−pvapour
1
2
ρn2D2
.
M2D’2015 Ponta Delgada, Azores, Portugal 26-30 July 7

10. Numerical Method
Discretisation
Discretisation with hyperboloidal quadrilateral panels.
Constant strength of the dipole and source distributions
on each panel.
Numerical Kutta condition: rigid wake model with
iterative pressure Kutta condition.
Integral equation solved by the collocation method:



D11 · · · D1N
...
...
...
DN1 · · · DNN






φ1
...
φN



=



S11 · · · S1N
...
...
...
SN1 · · · SNN






σ1
...
σN



,
Dij and Sij , dipole and source influence coefficients, respectively.
M2D’2015 Ponta Delgada, Azores, Portugal 26-30 July 8

11. Numerical Method
Conventional Procedure as implemented by Fine (1992) and Vaz (2005)









D11 · · · D1N
...
... Sij
...
...
DN1 · · · DNN












φ
(wet)
1
...
σ
(cav)
k
...
φ
(wet)
N



=









S11 · · · S1N
...
... Dij
...
...
SN1 · · · SNN












σ
(wet)
1
...
φ
(cav)
k
...
σ
(wet)
N



Known: φ(cav)
from DBC and σ(wet)
from KBC.
Unknowns: φ(wet)
and σ(cav)
.
M2D’2015 Ponta Delgada, Azores, Portugal 26-30 July 9

12. 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.
M2D’2015 Ponta Delgada, Azores, Portugal 26-30 July 10

13. 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 for 5◦
set
angle: TSR = ΩR
U
= 6;
Turbine tested by the
University of Southampton.
M2D’2015 Ponta Delgada, Azores, Portugal 26-30 July 11

14. Experimental Results (Bahaj et al, 2007)
TSR = 7.2 and σn = 3.7
M2D’2015 Ponta Delgada, Azores, Portugal 26-30 July 12

15. Numerical Method
Solution Method
Decomposition: {φ} = {φw } + {φc}
{σ} = {σw } + {σc}
with: w wetted solution
c cavity perturbation to the wetted solution
From [D] {φ} = [S] {σ},
we write [D] ¨¨¨
{φw } + {φc} = [S] ¨¨¨{σw } + {σc} .
Note that {σc} = 0 on the wetted part.
We introduce [S] {σc} = [D] {φc}
Nc ×Nc Nc ×1 Nc ×N N×1
where: Nc is the number of cavitating panels
N total number of panels
M2D’2015 Ponta Delgada, Azores, Portugal 26-30 July 13

16. Numerical Method
Solution Method
1 {φc} known from DBC.
2 Estimation of {σc} (on the cavity):
[S] {σc} = [D] {φc}.
3 {σw } known from KBC.
4 Source distribution: {σ} = {σw } + {σc}.
5 Calculation of {φ} (in the whole domain):
[D] {φ} = [S] {σ}.
6 Estimation of cavity thickness and length from KBC.
Advantage: In step 5 [D] is not changed and
needs only to be inverted once!
M2D’2015 Ponta Delgada, Azores, Portugal 26-30 July 14

17. Panel Arrangement
Discretisation: Blade 80×31, Wake 630×30, Hub 90×16
X
Y
Z
X
Y
Z
LE
TE
M2D’2015 Ponta Delgada, Azores, Portugal 26-30 July 15

18. Steady Cavitating Flow
TSR = 7.0 and σn = 6.5
X
Y
Z
0.0013
0.0012
0.0010
0.0009
0.0007
0.0006
0.0004
0.0003
0.0001
0.0000
η/R
leading
edgetrailing
edge
Iteration
0 2 4 6 8 10 12 14 16 18
0.0
0.2
0.4
0.6
0.8
1.0
0.000
0.002
0.004
0.006
0.008
0.010
lc max
/R
Ac
/A0
η*
η*lc max
/R, AC
/A0
Relative differences are of 0.2% for the perturbation potential.
Relative computational time of 19% per iteration.
M2D’2015 Ponta Delgada, Azores, Portugal 26-30 July 16

19. Unsteady Cavitating Flow
Velocity profile across the
marine current turbine based
on a 10 m radius turbine in a
d0 = 30 m deep sea and a
tidal velocity of u0 = 3.5 m/s;
Calculations at TSR = 4.28
and assuming the vapour
pressure to be 1230 Pa.
u/u0
d/d0
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(d)=u0(d/d0)
1/7
M2D’2015 Ponta Delgada, Azores, Portugal 26-30 July 17

20. Unsteady Cavitating Flow
TSR = 4.28 and σn = 21.47 - Cavity extent and thickness
0.0004
0.0003
0.0002
0.0001
0.0000
0
=-32º
/R
0.0004
0.0003
0.0002
0.0001
0.0000
0
=-20º
/R
0.0004
0.0003
0.0002
0.0001
0.0000
0
=-8º
/R
0.0004
0.0003
0.0002
0.0001
0.0000
0
=0º
/R
0.0004
0.0003
0.0002
0.0001
0.0000
0
=8º
/R
0.0004
0.0003
0.0002
0.0001
0.0000
0
=20º
/R
0.0004
0.0003
0.0002
0.0001
0.0000
0
=32º
/R
M2D’2015 Ponta Delgada, Azores, Portugal 26-30 July 18

21. Concluding Remarks
An iteratively coupled solution method implemented in a
low-order BEM is presented for unsteady potential flow
calculations on marine current turbines with sheet cavitation.
M2D’2015 Ponta Delgada, Azores, Portugal 26-30 July 19

22. Concluding Remarks
An iteratively coupled solution method implemented in a
low-order BEM is presented for unsteady potential flow
calculations on marine current turbines with sheet cavitation.
The iteratively coupled solution method converged to the
solution of the usual complete coupled procedure.
M2D’2015 Ponta Delgada, Azores, Portugal 26-30 July 19

23. Concluding Remarks
An iteratively coupled solution method implemented in a
low-order BEM is presented for unsteady potential flow
calculations on marine current turbines with sheet cavitation.
The iteratively coupled solution method converged to the
solution of the usual complete coupled procedure.
The maximum differences are in the cavity closure region,
which are related to the uncertainty in the cavity extents
due to panel discretisation error.
M2D’2015 Ponta Delgada, Azores, Portugal 26-30 July 19

24. Concluding Remarks
An iteratively coupled solution method implemented in a
low-order BEM is presented for unsteady potential flow
calculations on marine current turbines with sheet cavitation.
The iteratively coupled solution method converged to the
solution of the usual complete coupled procedure.
The maximum differences are in the cavity closure region,
which are related to the uncertainty in the cavity extents
due to panel discretisation error.
A significant reduction in computational time is achieved
with the iterative coupled procedure, which makes the
method especially attractive to unsteady computations,
where the cavity has to be iterated for each time step.
M2D’2015 Ponta Delgada, Azores, Portugal 26-30 July 19