This paper focus on the effect of the gap flow modelling in ducted propeller performance predictions with a panel method. Different models for the potential flow in the gap region are investigated: non-zero and zero gap width and a tip leakage model with a vortex sheet shed along the blade tip. A low order panel method for the calculation of steady potential flow on ducted propellers is used. The analysis is carried out for two configurations: propeller Ka4-70 inside duct 19A and propeller 4902 inside duct 37. The inviscid results are compared with experimental data. A strong influence of the gap model on the propeller and duct loads is found. The results indicate that the modelling of flow through the gap region is of the utmost importance for an accurate prediction of the forces on ducted propellers.

1. June 24 2009 SMP’09 Trondheim 1
SMP’09-2009
ON THE MODELLING OF THE FLOW IN DUCTED PROPELLERS WITH A
PANEL METHOD
João Baltazar & José Falcão de Campos
Marine Environment and Technology Center (MARETEC)
Department of Mechanical Engineering
Instituto Superior Técnico (IST)
Lisbon, Portugal

2. June 24 2009 SMP’09 Trondheim 2
Motivation
Panel Methods (or Boundary Element Method - BEM) still provide a most
useful computational tool for analysis and design of marine propulsors.
Main reasons for using Panel Methods:
Simplicity
Computational efficiency
Direct relation to simpler design tools (lifiting line and lifting surface)
Application to Ducted Propellers involves additional modelling issues:
Complex interaction of propeller blades and duct surface – Tip Leakage Flow
Thick duct trailing edges in practical applications – Kutta Condition for round
trailing edges

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Modelling Issues
Viscous effects are dominant in the tip leakage flow and the flow around
the duct trailing edge
RANS methods:
Sanchez-Caja (2000): Ka 5 series in duct 19A
Abdel-Maksoud & Heinke (2003): Ka 5 series in duct 19A
Hoekstra (2006) with actuator disk: Duct 19A and 37
Extremely valuable information on:
Flow separation from duct surface
Flow features in the tip leakage flow

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Objectives
How does a current state-of-the-art panel method perform in predicting
duct and propeller forces and duct pressure distributions?
Focus on the effect of Tip Leakage Model:
Zero gap width
Non-zero gap width
Non-zero gap width with vortex shedding: Gu (2006), Baltazar (2008)
Simple Kutta Condition applied at the duct trailing edge
Duct 19A : Effect on duct forces
Duct 37 with a round thick trailing edge: Effect on the pressure distribution

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Mathematical Formulation
Inviscid and incompressible steady
potential flow in the rotating reference
frame
Velocity Field
Perturbation Potential
Undisturbed velocity
Laplace equation
Boundary Conditions
Infinity
Boundaries
Wakes
Kutta at sharp t.e.
U
Ω x
y
z
r
θ
V U φ∞= + ∇
U Ui reθ∞ = + Ω
2
0φ∇ =
( , , )x rφ θ
0 if andx xφ∇ → → ∞ ≠ +∞
on B D Hn U S S S
n
φ
∞
∂
= − ⋅ ∪ ∪
∂
, = on Wn U p p S
n n
φ φ+ −
∞ + −
∂ ∂
= = − ⋅
∂ ∂
φ∇ < ∞

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Mathematical Formulation
Fredholm integral equation
where
2 ( )
( ) ( ) ,
B D H Wq q qS S S S
B D H
p
G G
G q dS q dS
n n n
p S S S
πφ
φ
φ φ
∪ ∪
=
⎡ ⎤∂ ∂ ∂
− − ∆⎢ ⎥
∂ ∂ ∂⎢ ⎥⎣ ⎦
∈ ∪ ∪
∫∫ ∫∫
1
( , )
G
R p q
= −

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Velocity, Pressure and Forces
Velocity by surface differentiation
Pressure from Bernoulli equation
Forces by integration of pressure distributions
Total Thrust
Thrust ratio
Ducted propeller advance ratio
2
2
1
1/ 2
p
Vp p
C
UUρ
∞
∞∞
⎛ ⎞− ⎜ ⎟≡ = −
⎜ ⎟
⎝ ⎠
2 4 2 4 2 5
, ,P D
P D
T T Q
T T Q
K K K
n D n D n Dρ ρ ρ
= = =
T P DT T TK K K= +
P
P D
T
T T
τ =
+
U
J
nD
=

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Numerical Method
(Code PROPAN)
Surface Discretization:
Structured surface grid with quadrilateral hyperboloidal elements
Panel Method
Integral equation solved by collocation method
Constant source and dipole distributions
Influence coefficients from formulation of Morino and Kuo (1974)
Iterative pressure Kutta condition (IPK)

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Vortex Wake Models
Rigid wake model
Propeller blades:
Constant radius and pitch of vortex lines
Geometrical blade pitch is used in the present study
Duct surface:
Constant radius vortex sheet
Shedding line from specified location at the duct trailing edge
Gap flow models
Non zero-gap width: Tip vortex shed from blade tip at the trailing edge
Zero gap width: Blade tip is on the duct surface (with matching grids)
Tip Leakage Model: Vortex sheet shed from the blade tip along the chord

10. June 24 2009 SMP’09 Trondheim 10
Tip Leakage Model (TLM)
Vortex sheet shed from the blade tip
along the chord :
Strength from Morino Kutta
condition: potential jump equal to
potential difference from each side
Pitch of vortex lines at the leading
edge (L.E.):
Undisturbed pitch at L.E.
Pitch of wake vortex at T.E.
Linear pitch variation along the chord
Transition wake from T.E. to
Ultimate wake with constant pitch
1
( )
2
TWP P P= + Leading edge
Trailing edge
Blade Tip
P =
TWP =
/ 1.0x R =
Vortex Sheet shed from Tip

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Results – Ka 4-70 inside Duct 19A
Panel arrangement

12. June 24 2009 SMP’09 Trondheim 12
Results – Ka 4-70 inside Duct 19A
Effect of Vortex Shedding Location on Circulation
Propeller Duct

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Results – Ka 4-70 inside Duct 19A
Effect of Vortex Shedding Location on Pressure and Forces
Location PTK DTK QK
Inner 0.3198 0.0183 0.0566
100% 0.2860 0.0287 0.0515
Outer 0.2420 0.0388 0.0447
Propeller Duct

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Results – Ka 4-70 inside Duct 19A
Influence of Gap Model on Circulation
Propeller Duct Propeller Tip

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Results – Ka 4-70 inside Duct 19A
Influence of Gap Model on Pressure and Forces
Propeller Duct
Open Water

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Results – Propeller 4092 inside Duct 37
Geometry and Panel Arrangement

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Results – Propeller 4092 inside Duct 37
Geometry and Panel Arrangement

19. June 24 2009 SMP’09 Trondheim 19

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21. June 24 2009 SMP’09 Trondheim 21
Concluding Remarks
Predictions of duct and propeller loading are critically dependent on the gap model.
The non-zero gap model and the tip leakage model unload the duct in comparison with the zero
gap model
The shedding location of the duct vortex wake has a strong influence on the propeller
and duct loading
For the propeller 4902 in the duct 37 none of the gap models was able to predict the
pressure distribution on the duct.
The non-zero gap model and the tip leakage model produce unrealistic pressure distributions
close to the propeller tip
The zero-gap model produces high loadings in the downstream part of the duct
Further work:
Investigate the influence of wake alignment in a duct with sharp trailing edge
Combine orifice model (Kerwin et al, 1987, Hughes, 1997, Moon et al, 2002) with tip leakage
model