An iteratively coupled solution method for the calculation of partial and super-cavitating flow on propellers with a potential based Boundary Element Method is presented. The solution method explores the fact that only the source strength beneath the cavity changes due to the presence of the cavity. The knowledge of the source strength change is sufficient to solve the original Neumann problem. The system is iteratively coupled to the complete cavitating system in the cavity planform iteration. The advantage is that the complete system matrix for the Neumann problem is identical to the matrix of the wetted flow problem and needs only to be inverted once. The numerical studies are carried out for the INSEAN E779A propeller with predicted partial and super-cavitation.

1. An Iteratively Coupled Solution Method for
Partial and Super-Cavitation Prediction
on Marine Propellers Using BEM
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
ICHD 2012 St. Petersburg, Russia 1-4 October 1 / 18

2. Introduction
Motivations
BEM potential flow calculations have been used in the analysis
of marine propellers with sheet cavitation and induced pressure
fluctuations on the hull.
Calculations may be time consuming due to solution of new
system of equations during cavity extent iterative determination.
Objectives
Implementation of an alternative technique (Baltazar and Falc˜ao
de Campos, 2010) to solve the linear system of equations in the
presence of super-cavitation.
ICHD 2012 St. Petersburg, Russia 1-4 October 2 / 18

3. 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 ∪ SH
D
Dt [s3 − η (s1, s2)] = 0,
−Cpn = σn on SC
V + · n = V − · n, p+ = p− on SW
φ → 0 if |r| → ∞
Kutta condition: | φ| < ∞
U
y
θ
Ω
x
r
z
SB
SC
SH
SC
s1
s2
s3
q
h cavity thickness
ICHD 2012 St. Petersburg, Russia 1-4 October 3 / 18

4. 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
ICHD 2012 St. Petersburg, Russia 1-4 October 4 / 18

5. Mathematical Formulation
Cavitation Model
KBC and DBC applied on the blade surface SC beneath the
cavity.
KBC: ∂η
∂s1
[Vs1 − Vs2 cos θ] + ∂η
∂s2
[Vs2 − Vs1 cos θ] = Vs3 sin2
θ.
DBC: φ = φ0+
s2
s1
(nD)2
σn + U∞
2
− V 2
u2
ds1 +
s2
s1
−U∞ · t1 ds1.
ICHD 2012 St. Petersburg, Russia 1-4 October 5 / 18

6. Mathematical Formulation
Integral Equation
Fredholm integral equation for Morino formulation:
2πφ (p) =
SB∪SC
G ∂φ
∂nq
− φ (q) ∂G
∂nq
dS +
SW
∆ ∂φ
∂nq
GdS
−
SW
∆φ (q) ∂G
∂nq
dS, p ∈ SB ∪ SC
Green’s function: G(p, q) = −1/R(p, q)
ICHD 2012 St. Petersburg, Russia 1-4 October 6 / 18

7. Mathematical Formulation
Integral Equation
Fredholm integral equation for Morino formulation:
4πφ±
(p) =
SB∪SC
G ∂φ
∂nq
− φ (q) ∂G
∂nq
dS +
SW
∆ ∂φ
∂nq
GdS
±2π∆φ (p) −
SW
∆φ (q) ∂G
∂nq
dS, p ∈ SW
from Fine (1992).
ICHD 2012 St. Petersburg, Russia 1-4 October 7 / 18

8. Mathematical Formulation
Dipole and Source Strengths Decomposition
The potential and source distributions can be decomposed in the
wetted flow solution and in the cavity perturbation to the wetted
flow solution:
φ = φw + φc
σ = σw + σc = ∂φ
∂n w
+ ∂φ
∂n c
on SB ∪ SC
∆φ = ∆φw + ∆φc
∆σ = ∆σw + ∆σc = ∆ ∂φ
∂n w
+ ∆ ∂φ
∂n c
on SW
ICHD 2012 St. Petersburg, Russia 1-4 October 8 / 18

9. Mathematical Formulation
Dipole and Source Strengths Decomposition
Since the wetted solutions cancel and σc = 0 in the wetted part SB,
a new integral equation is obtained:
2πφc (p) +
SB
φc (q) ∂G
∂nq
dS =
SC
Gσc (q) − φc (q) ∂G
∂nq
dS+
SW
∆σc (q) GdS −
SW
∆φc (q) ∂G
∂nq
dS, p ∈ SB ∪ SC
ICHD 2012 St. Petersburg, Russia 1-4 October 9 / 18

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.
ICHD 2012 St. Petersburg, Russia 1-4 October 10 / 18

11. Solution Method
Conventional Procedure as implemented by Fine (1992) and Vaz (2006)









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)
.
ICHD 2012 St. Petersburg, Russia 1-4 October 11 / 18

12. 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
ICHD 2012 St. Petersburg, Russia 1-4 October 12 / 18

13. 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!
ICHD 2012 St. Petersburg, Russia 1-4 October 13 / 18

14. Results
INSEAN E779A Propeller Panel Arrangement
X
Y
Z
Discretisation: 200×20 Blade, 150×20 Wake, 128×12 Hub.
ICHD 2012 St. Petersburg, Russia 1-4 October 14 / 18

15. Results
INSEAN E779A Propeller
X
Y
Z
0.019
0.011
0.007
0.004
0.003
0.002
0.001
0.001
0.000
0.000
η/R
J=0.77, σn=2.38
Iteration
0 10 20 30 40 50
0.0
0.2
0.4
0.6
0.8
lc max
Ac/R2
η
J=0.77, σn=2.38
Relative differences are of 1% for the perturbation potential.
Relative computational time of 2.4%.
ICHD 2012 St. Petersburg, Russia 1-4 October 15 / 18

16. INSEAN E779A Propeller
Comparison with Experiments (Pereira et al., 2004)
X
Y
Z
0.019
0.011
0.007
0.004
0.003
0.002
0.001
0.001
0.000
0.000
η/R
J=0.77, σn=2.38
X
Y
Z
0.059
0.036
0.022
0.013
0.008
0.005
0.003
0.002
0.001
0.001
η/R
J=0.77, σn=1.78
ICHD 2012 St. Petersburg, Russia 1-4 October 16 / 18

17. INSEAN E779A Propeller
Comparison with Experiments (Pereira et al., 2004)
X
Y
Z
0.147
0.089
0.054
0.032
0.020
0.012
0.007
0.004
0.003
0.002
η/R
J=0.71, σn=2.02
s/c
-Cpn
0.0 0.2 0.4 0.6 0.8 1.0
-3.0
-2.0
-1.0
0.0
1.0
2.0
J=0.77, σn
=2.38
J=0.77, σn=1.78
J=0.71, σn
=2.02
r/R=0.975
ICHD 2012 St. Petersburg, Russia 1-4 October 17 / 18

18. Concluding Remarks
The new iteratively coupled procedure converged for all
cases to the solution of the conventional coupled system.
No additional convergence problems were met when there
is super-cavitation.
The maximum differences are in the cavity closure region,
which are related to the uncertainty in the cavity extents
due to the panel discretisation error.
A large reduction in computational time is achieved with
the iteratively coupled procedure.
The method tends to over-predict the cavity extents in
comparison with experimental observations.
ICHD 2012 St. Petersburg, Russia 1-4 October 18 / 18