This study addresses some aspects of the numerical solution with the Boundary Element Method of the three-dimensional potential flow on marine propellers with partial cavitation. An alternative iteratively coupled procedure is proposed to solve the linear system of equations resulting from the formulation of the cavitating flow problem. The alternative procedure is aimed to avoid solving anew the large system of equations at each iteration step for the determination of the cavity extension. The numerical studies are carried out for the MARIN S-Propeller. A large reduction in computational time is achieved with the alternative procedure for the cavitating potential flow solution.
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A Numerical Study on the Iterative Techniques to Solve Partial Cavitation on Marine Propellers Using BEM
1. A Numerical Study on the Iterative
Techniques to Solve Partial Cavitation 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)
Lisbon, Portugal
MARETEC
MARINE 2009 Trondheim, Norway 15-17 June 1 / 17
2. Introduction
Motivations
BEM potential flow calculations have been used in the analysis
of marine propellers with sheet cavitation.
Calculations may be time consuming due to solution of new
system of equations during cavity extent iterative determination.
Objectives
Investigate an alternative iterative technique to solve the linear
system of equations for the prediction of the cavity planform.
MARINE 2009 Trondheim, Norway 15-17 June 2 / 17
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,
−Cp = σ 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
MARINE 2009 Trondheim, Norway 15-17 June 3 / 17
4. Mathematical Formulation
Integral Equation
Fredholm integral equation for Morino formulation:
2πφ (p) =
SB∪SH∪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)
MARINE 2009 Trondheim, Norway 15-17 June 4 / 17
5. Numerical Method
Cavitation Model
KBC and DBC applied on the blade surface SB beneath the
cavity.
KBC: ∂η
∂s1
[Vs1 − Vs2 cos θ] + ∂η
∂s2
[Vs2 − Vs1 cos θ] = Vs3 sin2
θ.
DBC: φ = φ0+
s2
s1
V 2
ref σ + V∞
2
− 2gy − V 2
u2
ds1 +
s2
s1
−V∞ · t1 ds1.
Pressure recovery model: smooth transition from vapour pressure
to the pressure on the wet part immediately downstream.
MARINE 2009 Trondheim, Norway 15-17 June 5 / 17
6. Numerical Method
Surface discretisation
Hyperboloidal quadrilateral panels.
Propeller blade surface: cosine spacing in the radial and
chordwise directions.
Hub surface: elliptical grid generator (E¸ca, 1994).
Blade wake surface: half-cosine spacing along the streamwise
direction.
MARINE 2009 Trondheim, Norway 15-17 June 6 / 17
7. Numerical Method
Complete System of Equations
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.
MARINE 2009 Trondheim, Norway 15-17 June 7 / 17
9. Solution Method
Reduced System of Equations
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
MARINE 2009 Trondheim, Norway 15-17 June 9 / 17
10. Solution Method
Iteratively Coupled Procedure (ICP)
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 all domain):
[D] {φ} = [S] {σ}.
6 Estimation of cavity thickness and length from KBC.
MARINE 2009 Trondheim, Norway 15-17 June 10 / 17
11. Test Case
S-Propeller (Kuiper, 1981)
X
Y
Z
X
Y
Z
Discretisation: 100×20 Blade, 150×20 Wake, 100×36 Hub.
MARINE 2009 Trondheim, Norway 15-17 June 11 / 17
17. Conclusions
The new iteratively coupled procedure converged for all
cases to the solution of the conventional coupled system.
Small differences are seen near the re-attachment region.
A large reduction in computational time is achieved with
the iteratively coupled procedure.
Similar cavity extents and thicknesses are seen between
the present method and the results of Vaz (2005).
Some differences are found near the blade tip.
Comparison with experiments:
Cavity inception is under-predicted for J = 0, 6;
Reasonable to good agreement of the cavity extent
for J = 0, 4.
MARINE 2009 Trondheim, Norway 15-17 June 17 / 17
