Recent Developments in Computational Methods for the Analysis of Ducted Propellers in Open Water

Recent Developments in Computational Methods for
the Analysis of Ducted Propellers in Open Water
J. Baltazar1
, J.A.C. Falcão de Campos1
, J. Bosschers2
, D. Rijpkema2
1Instituto Superior Técnico, Universidade de Lisboa, Portugal
2Maritime Research Institute Netherlands, Wageningen, the Netherlands
32nd SNH Hamburg, Germany August 5-10, 2018 1

Outline
Motivation
Computational Methods
Estimation of Numerical Errors
Ducted Propeller Flow Modelling with BEM
Comparison Between BEM and RANS
Open-Water Prediction
Conclusions

Motivation
Ducted propellers have been widely used for marine applications
(tugs, trawlers, etc.)

Motivation
The complex interaction which occurs between propeller and
duct makes the hydrodynamic design a complicated task!

Motivation
Selection of the computational simulation tool:

Motivation
Simplified tool that predicts the main features of the flow field.

Motivation
Simplified tool that predicts the main features of the flow field.
Complex tool that provides detailed information in viscous
dominated areas (gap region, tip vortex).

Motivation
Presently, there is interest in the development of an accurate
and cost-effective numerical tool for design studies:

Motivation
BEM (Boundary Element Method)
Simplicity
Computational efficiency
Direct relation to simpler design tools
(lifting line and lifting surface)

Motivation
Simplicity
RANS
Increase of computational resources
Provide more detailed information of the flow field

Motivation
Simplicity
RANS
Increase of computational resources
Provide more detailed information of the flow field
RANS-BEM coupling

BEM Codes PROPAN (IST) and PROCAL (MARIN/CRS)
Low-order potential-based panel method
Fredholm integral equation solved by the collocation method
Structured surface grids

BEM Codes PROPAN (IST) and PROCAL (MARIN/CRS)
Low-order potential-based panel method
Fredholm integral equation solved by the collocation method
Structured surface grids
RANS Code ReFRESCO
Viscous flow CFD code (development led by MARIN)
Solves the incompressible RANS equations, complemented
with turbulence models
Finite volume discretisation

Test Case
Propeller Ka4-70 with P/D = 1 inside duct 19A
Duct 19A
Straight Part
Cylindrical

Grid Sizes
BEM RANS
Blade Duct Hub Volume Blade Duct
G1 80×41 280×280 110×128 61.4M 59.0K 127.9K
G2 70×36 250×240 98×112 26.8M 34.0K 71.9K
G3 60×31 220×200 87×96 12.9M 20.7K 43.5K
G4 50×26 190×160 75×80 7.7M 14.8K 32.0K
G5 40×21 160×120 63×64 2.6M 7.1K 14.2K
G6 30×16 130×80 51×48 1.1M 4.1K 8.0K
G7 20×11 100×40 39×32 – – –

Grids
BEM (left) and RANS (right)
Blade Panels: 50×26 Volume Cells: 13M
Duct Panels: 190×160 Blade Faces: 21K

Error Estimation
Numerical errors involved on a numerical flow method:

Error Estimation
Round-off error: due to the finite precision of computers.
Considered to be low.

Error Estimation
Iterative error: related to the non-linearity of the transport
equations, integral equation, etc. Monitored from the residuals.

Error Estimation
Iterative error: related to the non-linearity of the transport
equations, integral equation, etc. Monitored from the residuals.
Discretisation error: discrete representation (space and time)
of a (partial) differential equation, integral equation, etc.
Monitored from grid refinement studies.

Numerical Errors Involved in BEM Calculations
Contributions to the iterative error:

Kutta Condition: equal pressure at the trailing edge solved
iteratively by the Newton-Raphson method.

Gap Flow Model: a closed (sealed) gap is assumed.

Gap Flow Model: a closed (sealed) gap is assumed.
Wake Alignment Scheme: iterative alignment of the blade
wake pitch with local flow.

Contributions to the Iterative Error
Kutta Condition:
J = 0.2 J = 0.5
Tol. KTP
KTD
10KQ KTP
KTD
10KQ
10−1 0.3826 0.1601 0.5538 0.2655 0.0537 0.4041
10−2 0.3826 0.1601 0.5537 0.2655 0.0537 0.4041
10−3 0.3827 0.1601 0.5539 0.2656 0.0537 0.4043
10−4 0.3827 0.1601 0.5539 0.2656 0.0537 0.4043
10−5 0.3827 0.1601 0.5539 0.2656 0.0537 0.4043
Wake Alignment Scheme:
J = 0.2 J = 0.5
Iter. KTP
KTD
10KQ KTP
KTD
10KQ
0 0.3827 0.1601 0.5539 0.2656 0.0537 0.4043
1 0.2854 0.1782 0.4307 0.2045 0.0562 0.3196
2 0.2965 0.1761 0.4455 0.2091 0.0560 0.3261
3 0.2903 0.1778 0.4372 0.2085 0.0562 0.3253
4 0.2935 0.1765 0.4415 0.2086 0.0562 0.3255
5 0.2908 0.1778 0.4379 0.2086 0.0562 0.3254

Discretisation Error
Grid refinement study for BEM calculations
J = 0.2 J = 0.5
Grid ∆KTP
∆KTD
∆KQ ∆KTP
∆KTD
∆KQ
20×11 -3.50% -0.62% -3.30% -4.76% -7.84% -6.06%
30×16 0.39% -1.06% 0.36% -0.60% -3.86% -1.52%
40×21 -0.26% -0.25% -0.22% -0.79% -1.75% -1.33%
50×26 -0.05% -0.25% -0.07% -0.45% -0.94% -0.79%
60×31 0.05% -0.19% 0.02% -0.15% -0.68% -0.34%
70×36 0.16% -0.19% 0.11% 0.00% -0.55% -0.10%
80×41 – – – – – –

Grid refinement study for BEM calculations
J = 0.2 J = 0.5
Grid ∆KTP
∆KTD
∆KQ ∆KTP
∆KTD
∆KQ
20×11 -3.50% -0.62% -3.30% -4.76% -7.84% -6.06%
30×16 0.39% -1.06% 0.36% -0.60% -3.86% -1.52%
40×21 -0.26% -0.25% -0.22% -0.79% -1.75% -1.33%
50×26 -0.05% -0.25% -0.07% -0.45% -0.94% -0.79%
60×31 0.05% -0.19% 0.02% -0.15% -0.68% -0.34%
70×36 0.16% -0.19% 0.11% 0.00% -0.55% -0.10%
80×41 – – – – – –

Numerical Errors Involved in RANS Calculations
Related to the non-linearity of the transport equations
(RANS + Turbulence Model).
Monitored from the residuals:
L∞(φ) = max |res(φi )|,
L2(φ) =
s
Ncells
P
i=1
res2(φi )

Ncells
where φ represents a local flow quantity.

Iterative Error
RANS calculations at J = 0.2
Iteration
0 15000 30000 45000 60000 75000
10
-7
10-6
10
-5
10
-4
10-3
10-2
10
-1
100
VX
VY
VZ
p
k
ω
L
∞
Iteration
0 15000 30000 45000 60000 75000
10
-8
10-7
10
-6
10
-5
10-4
10-3
10
-2
10-1
VX
VY
VZ
p
k
ω
L
2
Iteration
0 15000 30000 45000 60000 75000
0.20
0.25
0.30
0.35
0.40
0.45
KT
10KQ

Grid refinement study for RANS calculations
J = 0.2 J = 0.5
Grid ∆KTP
∆KTD
∆KQ ∆KTP
∆KTD
∆KQ
1.1M 5.45% -0.92% 5.35% 3.40% 2.63% 3.68%
2.6M 3.44% -0.18% 3.31% 2.93% 3.16% 2.88%
7.7M 1.52% 0.24% 1.40% 0.82% 2.33% 0.80%
12.9M 0.90% 0.12% 0.88% 0.36% 1.29% 0.42%
26.8M 0.33% 0.12% 0.33% 0.10% 0.34% 0.14%
61.4M – – – – – –

Verification Procedure (Eça and Hoekstra, 2014)
The goal of verification is to estimate the uncertainty of a given
numerical prediction Unum:
φi − Unum ≤ φexact ≤ φi + Unum

Numerical Uncertainty:
Unum = Fs||

Numerical Uncertainty:
Unum = Fs||
Estimation of the discretisation error :
= φi − φ0 = αhp
i
Unknowns: φ0, α and p.

Numerical Uncertainty Estimation
BEM calculations
hi
/h1
0 1 2 3 4 5
0.24
0.30
0.36
0.42
J=0.2 : α1
h+α2
h
2
, Unum
= 4.63 %
J=0.5 : αh2
, Unum
= 1.26 %
K
T
P
Propeller Thrust
hi
/h1
0 1 2 3 4 5
0.03
0.06
0.09
0.12
0.15
0.18
J=0.2 : α1
h+α2
h
2
, Unum
= 3.20 %
J=0.5 : αh
2
, Unum
= 1.53 %
K
T
D
Duct Thrust
hi
/h1
0 1 2 3 4 5
0.35
0.40
0.45
0.50
0.55
0.60
J=0.2 : α1
h+α2
h2
, Unum
= 4.46 %
J=0.5 : αh2
, Unum
= 1.39 %
10K
Q
Propeller Torque

Numerical Uncertainty Estimation
RANS calculations
hi
/h1
0 1 2 3 4
0.18
0.20
0.22
0.24
0.26
J=0.2 : αh
1.54
, Unum
= 1.18 %
J=0.5 : αh1.45
, Unum
= 1.36 %
K
T
P
Propeller Thrust
hi
/h1
0 1 2 3 4
0.03
0.06
0.09
0.12
0.15
0.18
J=0.2 : αh2
, Unum
= 0.55 %
J=0.5 : α1
h+α2
h2
, Unum
=12.42 %
K
T
D
Duct Thrust
hi
/h1
0 1 2 3 4
0.34
0.36
0.38
0.40
0.42
0.44
0.46
J=0.2 : αh1.62
, Unum
= 1.00 %
J=0.5 : αh
1.60
, Unum
= 1.13 %
10K
Q
Propeller Torque

Ducted Propeller
Gap flow

Ducted Propeller
Flow around duct trailing edge (Bosschers et al., 2015)

Modelling of Ducted Propeller Flow with BEM
“If we place the vorticity at the correct location, we will be
able to reproduce the real (viscous) flow within the limitations
of the potential flow theory”

BEM:

BEM:
Closed gap model (sealed)

BEM:
Wake alignment model

BEM:
Correction due to duct boundary layer

BEM:
Correction due to duct boundary layer
New Kutta condition for thick duct t.e.

Flow around duct trailing edge

Duct Kutta condition. J = 0.2
r/R
∆φ/(ΩR
2
)
0.2 0.4 0.6 0.8 1.0
0.00
0.02
0.04
0.06
0.08
0.10
100.0%
Blade Circulation
KTP=0.1911
Position between blade wakes [deg]
0.0 30.0 60.0 90.0
0.15
0.20
0.25
0.30
0.35
0.40
100%
Duct Circulation
∆φ/(ΩR
2
)
KTD=0.1374

r/R
∆φ/(ΩR
2
)
0.2 0.4 0.6 0.8 1.0
0.00
0.02
0.04
0.06
0.08
0.10
100.0%
99.9%
Blade Circulation
KTP=0.1911
0.0 30.0 60.0 90.0
0.15
0.20
0.25
0.30
0.35
0.40
100%
99.9%
Duct Circulation
∆φ/(ΩR
2
)
KTD=0.1374

r/R
∆φ/(ΩR
2
)
0.2 0.4 0.6 0.8 1.0
0.00
0.02
0.04
0.06
0.08
0.10
100.0%
99.9%
99.8%
Blade Circulation
KTP=0.1911
0.0 30.0 60.0 90.0
0.15
0.20
0.25
0.30
0.35
0.40
100%
99.9%
99.8%
Duct Circulation
∆φ/(ΩR
2
)
KTD=0.1374

r/R
∆φ/(ΩR
2
)
0.2 0.4 0.6 0.8 1.0
0.00
0.02
0.04
0.06
0.08
0.10
100.0%
99.9%
99.8%
99.5%
Blade Circulation
KTP=0.1911
0.0 30.0 60.0 90.0
0.15
0.20
0.25
0.30
0.35
0.40
100%
99.9%
99.8%
99.5%
Duct Circulation
∆φ/(ΩR
2
)
KTD=0.1374

r/R
∆φ/(ΩR
2
)
0.2 0.4 0.6 0.8 1.0
0.00
0.02
0.04
0.06
0.08
0.10
100.0%
99.9%
99.8%
99.5%
99.0%
Blade Circulation
KTP=0.1911
0.0 30.0 60.0 90.0
0.15
0.20
0.25
0.30
0.35
0.40
100%
99.9%
99.8%
99.5%
99.0%
Duct Circulation
∆φ/(ΩR
2
)
KTD=0.1374

r/R
∆φ/(ΩR
2
)
0.2 0.4 0.6 0.8 1.0
0.00
0.02
0.04
0.06
0.08
0.10
100.0%
99.9%
99.8%
99.5%
99.0%
98.0%
Blade Circulation
KTP=0.1911
KTP=0.3827
0.0 30.0 60.0 90.0
0.15
0.20
0.25
0.30
0.35
0.40
100%
99.9%
99.8%
99.5%
99.0%
98.0%
Duct Circulation
∆φ/(ΩR
2
)
KTD=0.1374
KTD=0.1601

r/R
∆φ/(ΩR
2
)
0.2 0.4 0.6 0.8 1.0
0.00
0.02
0.04
0.06
0.08
0.10
100.0%
99.9%
99.8%
99.5%
99.0%
98.0%
97.0%
Blade Circulation
KTP=0.1911
KTP=0.3827
0.0 30.0 60.0 90.0
0.15
0.20
0.25
0.30
0.35
0.40
100%
99.9%
99.8%
99.5%
99.0%
98.0%
97.0%
Duct Circulation
∆φ/(ΩR
2
)
KTD=0.1374
KTD=0.1601

Wake modelling. J = 0.2
X
Y

X
Y

Influence of wake model
J = 0.2
Wake ∆KTP
∆KTD
∆KQ
Rigid Wake 0.3827 0.1601 0.5539
δ/R = 0% 0.3239 0.1718 0.4809
δ/R = 1% 0.3051 0.1741 0.4566
δ/R = 2% 0.2966 0.1768 0.4454
δ/R = 3% 0.2922 0.1778 0.4397
δ/R = 4% 0.2908 0.1778 0.4379
P/D = 0.85 0.2698 0.1676 0.4021
Prediction of the ducted propeller loading is critically dependent on the
blade wake pitch, especially at the tip!

Numerical options
BEM:

Numerical options
BEM:
Duct Kutta condition: pressure-equality at 98% of duct length.

Numerical options
BEM:
Closed gap model.

Numerical options
BEM:
Closed gap model.
Wake alignment model (δ/R = 4%).

Numerical options
BEM:
Closed gap model.
Reduced gap pitch: P/D = 0.85.

Numerical options
BEM:
Closed gap model.
ReFRESCO:

Numerical options
BEM:
Closed gap model.
ReFRESCO:
Turbulence model: k − ω SST 2-equation model
by Menter et al. (2003).

Numerical options
BEM:
Closed gap model.
ReFRESCO:
Discretisation of the convective flux:
QUICK for momentum and Upwind for turbulence.

Numerical options
BEM:
Closed gap model.
ReFRESCO:
Cylindrical domain: size 5D.

Numerical options
BEM:
Closed gap model.
ReFRESCO:
Cylindrical domain: size 5D.
No wall functions are used (y+ ∼ 1).

Vorticity field. J = 0.2

Comparison with aligned wake (WAM with δ/R = 0%). J = 0.2

Comparison with aligned wake (WAM with δ/R = 4%). J = 0.2

Comparison with aligned wake using δ/R = 4% and P/D = 0.85. J = 0.2

Blade pressure distribution. J = 0.2
s/c
C
p
0.0 0.2 0.4 0.6 0.8 1.0
-1.0
-0.5
0.0
0.5
ReFRESCO
r/R=0.95
s/c
0.00 0.01 0.02 0.03 0.04 0.05
-2.0
-1.5
-1.0
-0.5
0.0
0.5
1.0
C
p

s/c
C
p
0.0 0.2 0.4 0.6 0.8 1.0
-1.0
-0.5
0.0
0.5
ReFRESCO
WAM (δ/R=0%)
r/R=0.95
s/c
0.00 0.01 0.02 0.03 0.04 0.05
-2.0
-1.5
-1.0
-0.5
0.0
0.5
1.0
C
p

s/c
C
p
0.0 0.2 0.4 0.6 0.8 1.0
-1.0
-0.5
0.0
0.5
ReFRESCO
WAM (δ/R=0%)
WAM with δ/R=4%
r/R=0.95
s/c
0.00 0.01 0.02 0.03 0.04 0.05
-2.0
-1.5
-1.0
-0.5
0.0
0.5
1.0
C
p

s/c
C
p
0.0 0.2 0.4 0.6 0.8 1.0
-1.0
-0.5
0.0
0.5
ReFRESCO
WAM (δ/R=0%)
WAM with δ/R=4%
WAM with δ/R=4% and P/D=0.85
r/R=0.95
s/c
0.00 0.01 0.02 0.03 0.04 0.05
-2.0
-1.5
-1.0
-0.5
0.0
0.5
1.0
C
p

Duct pressure distribution. J = 0.2
s/c
C
p
0.0 0.2 0.4 0.6 0.8 1.0
-0.2
-0.1
0.0
0.1
ReFRESCO
θ=0 deg.

s/c
C
p
0.0 0.2 0.4 0.6 0.8 1.0
-0.2
-0.1
0.0
0.1
ReFRESCO
WAM (δ/R=0%)
θ=0 deg.

s/c
C
p
0.0 0.2 0.4 0.6 0.8 1.0
-0.2
-0.1
0.0
0.1
ReFRESCO
WAM (δ/R=0%)
WAM with δ/R=4%
θ=0 deg.

s/c
C
p
0.0 0.2 0.4 0.6 0.8 1.0
-0.2
-0.1
0.0
0.1
ReFRESCO
WAM (δ/R=0%)
WAM with δ/R=4%
WAM with δ/R=4% and P/D=0.85
θ=0 deg.

RANS-BEM Coupling
RANS
BEM
Duct
Loading
Propeller Loading
Distribution

Comparison With RANS-BEM Coupling
Blade (left) and duct (right) pressure distributions. J = 0.2
s/c
C
p
0.0 0.2 0.4 0.6 0.8 1.0
-1.0
-0.5
0.0
0.5
BEM (PROPAN)
RANS (ReFRESCO)
RANS-BEM Coupling (ReFRESCO+PROCAL)
r/R=0.70
BEM (PROPAN): Cp=-1.22
RANS-BEM (ReFRESCO+PROCAL): Cp
=-0.58
s/c
C
p
0.0 0.2 0.4 0.6 0.8 1.0
-0.30
-0.25
-0.20
-0.15
-0.10
-0.05
0.00
0.05
BEM (PROPAN)
RANS (ReFRESCO)
θ=0 deg.

Ka4-70 (P/D=1.0) inside duct 19A
J
0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8
-0.1
0.0
0.1
0.2
0.3
0.4
Experiments
KT
KT
D
P
Propeller and Duct Thrust
J
0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8
0.0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
Experiments
η0
10KQ
Propeller Torque and Efficiency

J
0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8
-0.1
0.0
0.1
0.2
0.3
0.4
Experiments
BEM (PROPAN)
KT
KT
D
P
J
0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8
0.0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
Experiments
BEM (PROPAN)
η0
10KQ

J
0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8
-0.1
0.0
0.1
0.2
0.3
0.4
Experiments
BEM (PROPAN)
RANS (ReFRESCO)
KT
KT
D
P
J
0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8
0.0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
Experiments
BEM (PROPAN)
RANS (ReFRESCO)
η0
10KQ

Conclusions
Prediction of Ducted Propeller Performance with BEM:

Conclusions
Ducted propeller loading is critically dependent on the duct
Kutta condition and blade wake pitch, especially at the tip!

Conclusions
Comparison with RANS computations:

Conclusions
An influence of the duct boundary-layer on the convection of
the blade vorticity has been identified.

Conclusions
Reasonable to good agreement between RANS and BEM for
pressure distributions and wake geometries.

Conclusions
Alternative with RANS-BEM coupling:

Conclusions
Viscous effects on duct captured by RANS, and propeller
loading predicted by BEM.

Conclusions
Viscous effects on duct captured by RANS, and propeller
loading predicted by BEM.
In general, good agreement for the pressure distribution
with full BEM and RANS.

Conclusions
Open-Water Prediction:

Conclusions
BEM:
differences around 1.5% for KTD
, 3% for KTP
and 6% for KQ.
At J = 0: over-prediction around 12% for KTP
and 7% for KQ.

Conclusions
BEM:
, 3% for KTP
and 6% for KQ.
and 7% for KQ.
RANS:
under-prediction around 2% for KTD
, 4% for KTP
and 3% for KQ.

Conclusions
BEM:
, 3% for KTP
and 6% for KQ.
and 7% for KQ.
RANS:
under-prediction around 2% for KTD
, 4% for KTP
and 3% for KQ.
RANS-BEM coupling:
over-prediction around 4% for KTD
, 2% for KTP
and 1% for KQ.

Discusser: A. Sánchez-Caja
Q1 - You mention that transpiration velocities are used on the BEM
duct surface to match the loading predicted by the RANS solver.
Could you be more specific about the way you implement such
velocities?
For the BEM calculations, the duct geometry is modified such that it has a sharp trailing-edge,
to avoid the dependence on the location of the Kutta condition. Therefore, the duct loading in
the BEM is adjusted by applying surface transpiration velocities on the duct surface to match
the loading predicted by the RANS solver. These transpiration velocities can be interpreted as a
boundary-layer displacement thickness effect, contributing to a change in the thickness and
camber distributions of the duct. To simplify the procedure, the variation of the boundary-layer
displacement thickness with distance to the leading edge is prescribed and is applied on the inner
surface of the duct only. The value at the duct trailing edge is then left as the only unknown
and it is iteratively adjusted until the prescribed mean radial loading on the duct is obtained.

Q2 - From Table 7, the location of the duct Kutta condition seems to
have a strong effect on the propeller and duct forces. You are using
Kutta points at the same percentages of the duct length on both
sides of the duct. Due to the duct loading, the location of the
pressure-equality point should be expected to be different from the
suction to the pressure side. Have you considered using different
percentages instead of transpiration velocities for accommodating the
duct thrust obtained with the BEM analysis to that of the RANS?

Q3 - Table 8: a reduction of the KT , KQ with the increase of
boundary layer thickness is apparent except for J = 0.5 and
δ/R = 4%. Is there any explanation for it? Similarly in Table 9 a
reduction of KT with reduction of P/D is observed except for
J = 0.2 and P/D = 0.85.

J = 0.2 J = 0.5
Wake ∆KTP
∆KTD
∆KQ ∆KTP
∆KTD
∆KQ
RW 0.3827 0.1601 0.5539 0.2656 0.0537 0.4043
δ/R = 0% 0.3239 0.1718 0.4809 0.2342 0.0555 0.3613
δ/R = 1% 0.3051 0.1741 0.4566 0.2145 0.0552 0.3338
δ/R = 2% 0.2966 0.1768 0.4454 0.2082 0.0584 0.3248
δ/R = 3% 0.2922 0.1778 0.4397 0.2056 0.0586 0.3212
δ/R = 4% 0.2908 0.1778 0.4379 0.2086 0.0562 0.3254
δ/R = 5% 0.2887 0.1783 0.4351 0.2031 0.0425 0.3176

Influence of the gap pitch
J = 0.2 J = 0.5
P/D ∆KTP
∆KTD
∆KQ ∆KTP
∆KTD
∆KQ
1.00 0.2908 0.1778 0.4379 0.2086 0.0562 0.3254
0.90 0.2775 0.1717 0.4151 0.2010 0.0556 0.3124
0.85 0.2698 0.1676 0.4021 0.1999 0.0530 0.3097
0.80 0.2715 0.1603 0.4019 0.1953 0.0523 0.3021

Iterative Error
Kutta condition
J = 0.2 J = 0.5
Tol. KTP
KTD
10KQ KTP
KTD
10KQ
10−1 0.3826 0.1601 0.5538 0.2655 0.0537 0.4041
10−2 0.3826 0.1601 0.5537 0.2655 0.0537 0.4041
10−3 0.3827 0.1601 0.5539 0.2656 0.0537 0.4043
10−4 0.3827 0.1601 0.5539 0.2656 0.0537 0.4043
10−5 0.3827 0.1601 0.5539 0.2656 0.0537 0.4043

Iterative Error
Wake alignment scheme
J = 0.2 J = 0.5
Iter. KTP
KTD
10KQ KTP
KTD
10KQ
0 0.3827 0.1601 0.5539 0.2656 0.0537 0.4043
1 0.2854 0.1782 0.4307 0.2045 0.0562 0.3196
2 0.2965 0.1761 0.4455 0.2091 0.0560 0.3261
3 0.2903 0.1778 0.4372 0.2085 0.0562 0.3253
4 0.2935 0.1765 0.4415 0.2086 0.0562 0.3255
5 0.2908 0.1778 0.4379 0.2086 0.0562 0.3254

Iterative Error
RANS calculations at J = 0.5
Iteration
0 5000 10000 15000 20000
10
-7
10-6
10
-5
10
-4
10-3
10-2
10
-1
100
VX
VY
VZ
p
k
ω
L
∞
Iteration
0 5000 10000 15000 20000
10
-8
10-7
10
-6
10
-5
10-4
10-3
10
-2
10-1
VX
VY
VZ
p
k
ω
L
2
Iteration
0 5000 10000 15000 20000
0.15
0.20
0.25
0.30
0.35
0.40
KT
10KQ

Duct Kutta condition
J = 0.2 J = 0.5
Kutta Location ∆KTP
∆KTD
∆KQ ∆KTP
∆KTD
∆KQ
100.0% 0.1911 0.1374 0.2992 0.1379 0.0584 0.2219
99.9% 0.2817 0.1650 0.4250 0.1963 0.0672 0.3073
99.8% 0.3108 0.1678 0.4633 0.2136 0.0668 0.3321
99.5% 0.3529 0.1658 0.5172 0.2395 0.0618 0.3683
99.0% 0.3837 0.1595 0.5551 0.2604 0.0533 0.3972
98.0% 0.3827 0.1601 0.5539 0.2656 0.0537 0.4043
97.0% 0.3534 0.1680 0.5178 0.2526 0.0609 0.3865

J = 0.2 J = 0.5
Wake ∆KTP
∆KTD
∆KQ ∆KTP
∆KTD
∆KQ
Rigid Wake 0.3827 0.1601 0.5539 0.2656 0.0537 0.4043
δ/R = 0% 0.3239 0.1718 0.4809 0.2342 0.0555 0.3613
δ/R = 1% 0.3051 0.1741 0.4566 0.2145 0.0552 0.3338
δ/R = 2% 0.2966 0.1768 0.4454 0.2082 0.0584 0.3248
δ/R = 3% 0.2922 0.1778 0.4397 0.2056 0.0586 0.3212
δ/R = 4% 0.2908 0.1778 0.4379 0.2086 0.0562 0.3254
P/D = 0.85 0.2698 0.1676 0.4021 0.1999 0.0530 0.3097
Prediction of the ducted propeller loading is critically dependent on the
blade wake pitch, especially at the tip!

Most commonly used ducts:

Recent Developments in Computational Methods for the Analysis of Ducted Propellers in Open Water

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