In this study, a Reynolds averaged Navier-Stokes solver is used for prediction of the propeller performance in open-water conditions at different Reynolds numbers ranging from 10^4 to 10^7. The k-\omega SST turbulence model and the \gamma-Re_\theta transition model are utilised and results compared for a conventional marine propeller. First, the selection of the turbulence inlet quantities for different flow regimes is discussed. Then, an analysis of the iterative and discretisation errors is made. This work is followed by an investigation of the predicted propeller flow and wake field at variable Reynolds numbers. Finally, the propeller scale-effects and the influence of the turbulence and transition models on the performance prediction are discussed. The variation of the flow regime showed an increase in thrust and decrease in torque for increasing Reynolds number. From the comparison between the turbulence model and the transition model, different flow solutions are obtained for the Reynolds numbers between 10^5 and 10^6.

1. Prediction of the Propeller Performance at
Different Reynolds Number Regimes with RANS
J. Baltazar1
, D. Rijpkema2
, J.A.C. Falcão de Campos1
1Instituto Superior Técnico, Universidade de Lisboa, Portugal
2Maritime Research Institute Netherlands, Wageningen, the Netherlands
smp’19 Rome, Italy May 26-30 1

2. Introduction
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3. Introduction
Full-scale prediction propellers mostly based on simple
extrapolation methods from model-scale experiments
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4. Introduction
Full-scale prediction propellers mostly based on simple
extrapolation methods from model-scale experiments
RANS solvers may be used at both model and full scale and
offer an alternative scaling method
smp’19 Rome, Italy May 26-30 2

5. Introduction
Full-scale prediction propellers mostly based on simple
extrapolation methods from model-scale experiments
RANS solvers may be used at both model and full scale and
offer an alternative scaling method
Requires accurate prediction at both Reynolds numbers
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6. Introduction
Turbulence models (k − ω, SST, k −
√
kL, etc.) are known to
provide a good prediction for fully developed turbulent flows
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7. Introduction
Turbulence models (k − ω, SST, k −
√
kL, etc.) are known to
provide a good prediction for fully developed turbulent flows
However, these models predict transition at lower Reynolds
number than seen in experiments
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8. Introduction
Turbulence models (k − ω, SST, k −
√
kL, etc.) are known to
provide a good prediction for fully developed turbulent flows
However, these models predict transition at lower Reynolds
number than seen in experiments
Model-scale experiments in critical Reynolds number regime
smp’19 Rome, Italy May 26-30 3

9. Introduction
Turbulence models (k − ω, SST, k −
√
kL, etc.) are known to
provide a good prediction for fully developed turbulent flows
However, these models predict transition at lower Reynolds
number than seen in experiments
Model-scale experiments in critical Reynolds number regime
Propeller performance prediction at different Reynolds number
regimes using the γ − R̃eθ transition model and compare with
the k − ω SST turbulence model
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10. Propeller Characteristics: S6368
D [m] 0.2714
c0.7R [m] 0.0694
Z 4
P/D0.7R 0.757
AE /A0 0.464
smp’19 Rome, Italy May 26-30 4

11. Propeller Performance Prediction
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12. Propeller Performance Prediction
RANSE solver ReFRESCO
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13. Propeller Performance Prediction
RANSE solver ReFRESCO
Finite-volume discretisation
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14. Propeller Performance Prediction
RANSE solver ReFRESCO
Finite-volume discretisation
Flow variables defined in cell-centres
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15. Propeller Performance Prediction
RANSE solver ReFRESCO
Finite-volume discretisation
Flow variables defined in cell-centres
Turbulence model:
- k − ω SST (Menter et al., 2003)
- Not developed for transition
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16. Propeller Performance Prediction
RANSE solver ReFRESCO
Finite-volume discretisation
Flow variables defined in cell-centres
Turbulence model:
- k − ω SST (Menter et al., 2003)
- Not developed for transition
Transition model:
- γ − R̃eθt (Langtry and Menter, 2009)
- Strong dependency to turbulence intensity Tu
and eddy-viscosity ratio µt/µ (Baltazar et al., 2017)
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17. Numerical Set-Up
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18. Numerical Set-Up
Cylindrical domain (5D)
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19. Numerical Set-Up
Cylindrical domain (5D)
Multi-block structured grids (GridPro)
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20. Numerical Set-Up
Cylindrical domain (5D)
Multi-block structured grids (GridPro)
No wall functions are used (y+
∼ 1)
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21. Numerical Set-Up
Cylindrical domain (5D)
Multi-block structured grids (GridPro)
No wall functions are used (y+
∼ 1)
Uniform inflow (open-water)
smp’19 Rome, Italy May 26-30 6

22. Numerical Set-Up
Cylindrical domain (5D)
Multi-block structured grids (GridPro)
No wall functions are used (y+
∼ 1)
Uniform inflow (open-water)
Discretisation of convective flux:
- Momentum: QUICK
- Turbulence/Transition: Upwind
smp’19 Rome, Italy May 26-30 6

23. Grid Generation
Volume 1.0M 2.2M 4.3M 8.0M 17.8M 34.8M
Blade 4k 6k 10k 15k 25k 39k
Re = 1×104
max y+
0.04 0.04 0.03 0.03 0.02 0.02
mean y+
0.01 0.01 0.01 0.00 0.00 0.00
Re = 5×105
max y+
0.74 0.67 0.54 0.45 0.36 0.31
mean y+
0.23 0.18 0.13 0.11 0.08 0.06
Volume 1.4M 3.2M 6.1M 11.4M 25.0M 39.0M
Blade 4k 6k 10k 15k 25k 39k
Re = 1×107
max y+
0.34 0.26 0.20 0.16 0.12 0.10
mean y+
0.10 0.08 0.06 0.05 0.04 0.03
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24. Results Summary
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25. Results Summary
RANS simulations at Re = 104
to 107
(J = 0.568)
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26. Results Summary
RANS simulations at Re = 104
to 107
(J = 0.568)
Selection of turbulence inlet quantities
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27. Results Summary
RANS simulations at Re = 104
to 107
(J = 0.568)
Selection of turbulence inlet quantities
Estimation of the numerical errors: round-off error (negligible),
iterative error and discretisation error
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28. Results Summary
RANS simulations at Re = 104
to 107
(J = 0.568)
Selection of turbulence inlet quantities
Estimation of the numerical errors: round-off error (negligible),
iterative error and discretisation error
Blade flow analysis
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29. Results Summary
RANS simulations at Re = 104
to 107
(J = 0.568)
Selection of turbulence inlet quantities
Estimation of the numerical errors: round-off error (negligible),
iterative error and discretisation error
Blade flow analysis
Prediction of scale-effects
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30. Selection of Turbulence Inlet Quantities
Re = 5 × 105
, suction side (Baltazar et al., 2017)
γ − R̃eθt Paint
Tu=2.5% Tests
µt/µ = 500
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31. Selection of Turbulence Inlet Quantities
Re = 5 × 105
, pressure side (Baltazar et al., 2017)
γ − R̃eθt Paint
Tu=2.5% Tests
µt/µ = 500
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32. Selection of Turbulence Inlet Quantities
Re = 5 × 105
, suction side (Baltazar et al., 2017)
k − ω SST Paint
Tu=1.0% Tests
µt/µ = 1 (LER)
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33. Selection of Turbulence Inlet Quantities
Re = 5 × 105
, pressure side (Baltazar et al., 2017)
k − ω SST Paint
Tu=1.0% Tests
µt/µ = 1 (LER)
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34. Comparison with Experiments (Boorsma, 2000)
J = 0.568, Re = 5 × 105
KT 10KQ
k − ω SST 0.1119 0.1653
γ − R̃eθt 0.1166 0.1631
Experimental (LER) 0.118 0.176
Experimental 0.129 0.174
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35. Selection of Turbulence Inlet Quantities
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36. Selection of Turbulence Inlet Quantities
Turbulence models predict strong decay of turbulence quantities
from the inlet along the streamwise direction.
smp’19 Rome, Italy May 26-30 14

37. Selection of Turbulence Inlet Quantities
Turbulence models predict strong decay of turbulence quantities
from the inlet along the streamwise direction.
Analytical solution for uniform axial flow:
- k∗ = k∗
inlet
1 + β(x∗ − x∗
inlet)
k∗
inlet
(µtinlet/µ) Re
−β∗/β
- ω∗ = ω∗
inlet
1 + β(x∗ − x∗
inlet)
k∗
inlet
(µtinlet/µ) Re
−1
smp’19 Rome, Italy May 26-30 14

38. Selection of Turbulence Inlet Quantities
Turbulence models predict strong decay of turbulence quantities
from the inlet along the streamwise direction.
Analytical solution for uniform axial flow:
- k∗ = k∗
inlet
1 + β(x∗ − x∗
inlet)
k∗
inlet
(µtinlet/µ) Re
−β∗/β
- ω∗ = ω∗
inlet
1 + β(x∗ − x∗
inlet)
k∗
inlet
(µtinlet/µ) Re
−1
In this study (based on Re = 5 × 105
):
smp’19 Rome, Italy May 26-30 14

39. Selection of Turbulence Inlet Quantities
Turbulence models predict strong decay of turbulence quantities
from the inlet along the streamwise direction.
Analytical solution for uniform axial flow:
- k∗ = k∗
inlet
1 + β(x∗ − x∗
inlet)
k∗
inlet
(µtinlet/µ) Re
−β∗/β
- ω∗ = ω∗
inlet
1 + β(x∗ − x∗
inlet)
k∗
inlet
(µtinlet/µ) Re
−1
In this study (based on Re = 5 × 105
):
- Same Tu or k
smp’19 Rome, Italy May 26-30 14

40. Selection of Turbulence Inlet Quantities
Turbulence models predict strong decay of turbulence quantities
from the inlet along the streamwise direction.
Analytical solution for uniform axial flow:
- k∗ = k∗
inlet
1 + β(x∗ − x∗
inlet)
k∗
inlet
(µtinlet/µ) Re
−β∗/β
- ω∗ = ω∗
inlet
1 + β(x∗ − x∗
inlet)
k∗
inlet
(µtinlet/µ) Re
−1
In this study (based on Re = 5 × 105
):
- Same Tu or k
- Same decay rate:
smp’19 Rome, Italy May 26-30 14

41. Selection of Turbulence Inlet Quantities
Turbulence models predict strong decay of turbulence quantities
from the inlet along the streamwise direction.
Analytical solution for uniform axial flow:
- k∗ = k∗
inlet
1 + β(x∗ − x∗
inlet)
k∗
inlet
(µtinlet/µ) Re
−β∗/β
- ω∗ = ω∗
inlet
1 + β(x∗ − x∗
inlet)
k∗
inlet
(µtinlet/µ) Re
−1
In this study (based on Re = 5 × 105
):
- Same Tu or k
- Same decay rate:
µtinlet
µ
=
Re
5 × 105
·
µtinlet
µ

45. Re=5×105
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46. Selection of Turbulence Inlet Quantities
Model k − ω SST γ − R̃eθt
Re Tu µt/µ Tu µt/µ
1×104
5×104
1×105
5×105
1.0% 1 2.5% 500
1×106
5×106
1×107
smp’19 Rome, Italy May 26-30 15

47. Selection of Turbulence Inlet Quantities
Model k − ω SST γ − R̃eθt
Re Tu µt/µ Tu µt/µ
1×104
1.0% 2.5%
5×104
1.0% 2.5%
1×105
1.0% 2.5%
5×105
1.0% 1 2.5% 500
1×106
1.0% 2.5%
5×106
1.0% 2.5%
1×107
1.0% 2.5%
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48. Selection of Turbulence Inlet Quantities
Model k − ω SST γ − R̃eθt
Re Tu µt/µ Tu µt/µ
1×104
1.0% 0.02 2.5% 10
5×104
1.0% 0.1 2.5% 50
1×105
1.0% 0.2 2.5% 100
5×105
1.0% 1 2.5% 500
1×106
1.0% 2 2.5% 1000
5×106
1.0% 10 2.5% 5000
1×107
1.0% 20 2.5% 10000
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49. Iterative Errors
Monitored from the residuals
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50. Iterative Errors
Monitored from the residuals
Turbulence model:
- residuals 10−6 for Re = 104 to 5 × 105
- residuals ∼ 10−4 to 10−6 for Re = 107
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51. Iterative Errors
Monitored from the residuals
Turbulence model:
- residuals 10−6 for Re = 104 to 5 × 105
- residuals ∼ 10−4 to 10−6 for Re = 107
Transition model:
- residuals 10−6 for Re = 104
- residuals ∼ 10−3 to 10−6, γ ∼ 10−1 for Re = 5 × 105 to 107
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52. Iterative Errors
Monitored from the residuals
Turbulence model:
- residuals 10−6 for Re = 104 to 5 × 105
- residuals ∼ 10−4 to 10−6 for Re = 107
Transition model:
- residuals 10−6 for Re = 104
- residuals ∼ 10−3 to 10−6, γ ∼ 10−1 for Re = 5 × 105 to 107
Fast iterative convergence of the propeller forces
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53. Discretisation Errors
Estimated from a numerical uncertainty analysis (Eça and Hoekstra, 2014)
hi /h1
K
T
0 1 2 3 4
0.072
0.074
0.076
0.078
0.080
0.082
0.084
0.086
0.088
k-ω SST : p=1.17, Unum
=4.24 %
γ-Reθ
: p=1.19, Unum
=4.07 %
Re =
hi /h1
10K
Q
0 1 2 3 4
0.188
0.192
0.196
0.200
0.204
0.208
0.212
0.216
k-ω SST : p=1.56, Unum
=2.06 %
γ-Reθ
: p=1.59, Unum
=1.95 %
1 × 10
4
smp’19 Rome, Italy May 26-30 19

54. Discretisation Errors
Estimated from a numerical uncertainty analysis (Eça and Hoekstra, 2014)
hi /h1
K
T
0 1 2 3 4
0.110
0.112
0.114
0.116
0.118
0.120
k-ω SST : p=2.00, Unum
=1.13 %
γ-Reθ
: p=2.00, Unum
=0.88 %
Re =
hi /h1
10K
Q
0 1 2 3 4
0.160
0.164
0.168
0.172
0.176
k-ω SST : p=2.00, Unum
=1.70 %
γ-Reθ
: p=2.00, Unum
=0.84 %
5 × 10
5
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55. Discretisation Errors
Estimated from a numerical uncertainty analysis (Eça and Hoekstra, 2014)
hi /h1
K
T
0 1 2 3 4
0.119
0.120
0.121
0.122
0.123
k-ω SST : p=1.84, Unum
=0.42 %
γ-Reθ : p=2.00, Unum=0.84 %
Re =
hi /h1
10K
Q
0 1 2 3 4
0.160
0.165
0.170
0.175
k-ω SST : p=2.00, Unum
=1.76 %
γ-Reθ
: p=2.00, Unum
=1.94 %
1 × 10
7
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56. Blade Flow
Re = 1 × 104
(suction side)
k − ω SST γ − R̃eθt
Tu=1.0% Tu=2.5%
µt/µ = 0.02 µt/µ = 10
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57. Blade Flow
Re = 1 × 105
(suction side)
k − ω SST γ − R̃eθt
Tu=1.0% Tu=2.5%
µt/µ = 0.2 µt/µ = 100
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58. Blade Flow
Re = 5 × 105
(suction side)
k − ω SST γ − R̃eθt
Tu=1.0% Tu=2.5%
µt/µ = 1 µt/µ = 500
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59. Blade Flow
Re = 1 × 106
(suction side)
k − ω SST γ − R̃eθt
Tu=1.0% Tu=2.5%
µt/µ = 2 µt/µ = 1000
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60. Blade Flow
Re = 1 × 107
(suction side)
k − ω SST γ − R̃eθt
Tu=1.0% Tu=2.5%
µt/µ = 20 µt/µ = 10000
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61. Blade Flow
Re = 1 × 104
(pressure side)
k − ω SST γ − R̃eθt
Tu=1.0% Tu=2.5%
µt/µ = 0.02 µt/µ = 10
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62. Blade Flow
Re = 1 × 105
(pressure side)
k − ω SST γ − R̃eθt
Tu=1.0% Tu=2.5%
µt/µ = 0.2 µt/µ = 100
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63. Blade Flow
Re = 5 × 105
(pressure side)
k − ω SST γ − R̃eθt
Tu=1.0% Tu=2.5%
µt/µ = 1 µt/µ = 500
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64. Blade Flow
Re = 1 × 106
(pressure side)
k − ω SST γ − R̃eθt
Tu=1.0% Tu=2.5%
µt/µ = 2 µt/µ = 1000
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65. Blade Flow
Re = 1 × 107
(pressure side)
k − ω SST γ − R̃eθt
Tu=1.0% Tu=2.5%
µt/µ = 20 µt/µ = 10000
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66. Prediction of Scale-Effects
∆KT ∆KQ
Re k − ω SST γ − R̃eθt k − ω SST γ − R̃eθt
1×104
-31.5% -34.2% 17.1% 18.7%
5×104
-14.2% -19.2% -1.3% -0.9%
1×105
-7.0% -5.7% 0.1% 2.7%
5×105
– – – –
1×106
2.7% 0.6% 0.4% 2.0%
5×106
6.5% 2.1% -0.3% 1.0%
1×107
7.7% 3.0% -0.7% 0.6%
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67. Conclusions
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68. Conclusions
Selection of turbulence inlet quantities:
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69. Conclusions
Selection of turbulence inlet quantities:
- Strong sensitivity to inlet turbulence parameters (Tu and µt/µ)
for transition model
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70. Conclusions
Selection of turbulence inlet quantities:
- Strong sensitivity to inlet turbulence parameters (Tu and µt/µ)
for transition model
- Selection at model-scale (Re=5×105) based on paint-tests
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71. Conclusions
Selection of turbulence inlet quantities:
- Strong sensitivity to inlet turbulence parameters (Tu and µt/µ)
for transition model
- Selection at model-scale (Re=5×105) based on paint-tests
- Relation for µt/µ and Re is found to maintain decay rate of
turbulence quantities for all regimes
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72. Conclusions
Selection of turbulence inlet quantities:
- Strong sensitivity to inlet turbulence parameters (Tu and µt/µ)
for transition model
- Selection at model-scale (Re=5×105) based on paint-tests
- Relation for µt/µ and Re is found to maintain decay rate of
turbulence quantities for all regimes
Reynolds number variation:
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73. Conclusions
Selection of turbulence inlet quantities:
- Strong sensitivity to inlet turbulence parameters (Tu and µt/µ)
for transition model
- Selection at model-scale (Re=5×105) based on paint-tests
- Relation for µt/µ and Re is found to maintain decay rate of
turbulence quantities for all regimes
Reynolds number variation:
- Laminar flow regime is predicted by both models for lower Re
smp’19 Rome, Italy May 26-30 33

74. Conclusions
Selection of turbulence inlet quantities:
- Strong sensitivity to inlet turbulence parameters (Tu and µt/µ)
for transition model
- Selection at model-scale (Re=5×105) based on paint-tests
- Relation for µt/µ and Re is found to maintain decay rate of
turbulence quantities for all regimes
Reynolds number variation:
- Laminar flow regime is predicted by both models for lower Re
- Turbulent flow regime is predicted by both models for higher Re
smp’19 Rome, Italy May 26-30 33

75. Conclusions
Selection of turbulence inlet quantities:
- Strong sensitivity to inlet turbulence parameters (Tu and µt/µ)
for transition model
- Selection at model-scale (Re=5×105) based on paint-tests
- Relation for µt/µ and Re is found to maintain decay rate of
turbulence quantities for all regimes
Reynolds number variation:
- Laminar flow regime is predicted by both models for lower Re
- Turbulent flow regime is predicted by both models for higher Re
- Laminar to turbulent flow transition important on model-scale
smp’19 Rome, Italy May 26-30 33

76. Conclusions
Selection of turbulence inlet quantities:
- Strong sensitivity to inlet turbulence parameters (Tu and µt/µ)
for transition model
- Selection at model-scale (Re=5×105) based on paint-tests
- Relation for µt/µ and Re is found to maintain decay rate of
turbulence quantities for all regimes
Reynolds number variation:
- Laminar flow regime is predicted by both models for lower Re
- Turbulent flow regime is predicted by both models for higher Re
- Laminar to turbulent flow transition important on model-scale
Predicted scale-effects:
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77. Conclusions
Selection of turbulence inlet quantities:
- Strong sensitivity to inlet turbulence parameters (Tu and µt/µ)
for transition model
- Selection at model-scale (Re=5×105) based on paint-tests
- Relation for µt/µ and Re is found to maintain decay rate of
turbulence quantities for all regimes
Reynolds number variation:
- Laminar flow regime is predicted by both models for lower Re
- Turbulent flow regime is predicted by both models for higher Re
- Laminar to turbulent flow transition important on model-scale
Predicted scale-effects:
- Increase in KT : 7.7% (k − ω SST) and 3.0% (γ − R̃eθt )
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78. Conclusions
Selection of turbulence inlet quantities:
- Strong sensitivity to inlet turbulence parameters (Tu and µt/µ)
for transition model
- Selection at model-scale (Re=5×105) based on paint-tests
- Relation for µt/µ and Re is found to maintain decay rate of
turbulence quantities for all regimes
Reynolds number variation:
- Laminar flow regime is predicted by both models for lower Re
- Turbulent flow regime is predicted by both models for higher Re
- Laminar to turbulent flow transition important on model-scale
Predicted scale-effects:
- Increase in KT : 7.7% (k − ω SST) and 3.0% (γ − R̃eθt )
- Small variations for KQ
(lower than estimated numerical uncertainty)
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79. Prediction of Scale-Effects
Thrust coefficient KT
Re KTp KTf
KT ∆KT
k − ω SST
1×104 0.0839 -0.00722 0.0767 -31.5%
5×104 0.0992 -0.00323 0.0960 -14.2%
1×105 0.1070 -0.00290 0.1041 -7.0%
5×105 0.1143 -0.00242 0.1119 –
1×106 0.1171 -0.00219 0.1149 2.7%
5×106 0.1209 -0.00172 0.1192 6.5%
1×107 0.1221 -0.00154 0.1205 7.7%
γ − R̃eθt
1×104 0.0839 -0.00722 0.0767 -34.2%
5×104 0.0974 -0.00314 0.0942 -19.2%
1×105 0.1120 -0.00215 0.1099 -5.7%
5×105 0.1183 -0.00172 0.1166 –
1×106 0.1193 -0.00196 0.1173 0.6%
5×106 0.1207 -0.00174 0.1190 2.1%
1×107 0.1217 -0.00158 0.1201 3.0%
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80. Prediction of Scale-Effects
Torque coefficient KQ
Re 10KQp 10KQf
10KQ ∆KQ
k − ω SST
1×104 0.1164 0.0772 0.1935 17.1%
5×104 0.1274 0.0357 0.1632 -1.3%
1×105 0.1330 0.0325 0.1655 0.1%
5×105 0.1394 0.0260 0.1653 –
1×106 0.1428 0.0232 0.1660 0.4%
5×106 0.1468 0.0180 0.1648 -0.3%
1×107 0.1481 0.0161 0.1642 -0.7%
γ − R̃eθt
1×104 0.1164 0.0772 0.1936 18.7%
5×104 0.1272 0.0344 0.1616 -0.9%
1×105 0.1435 0.0240 0.1675 2.7%
5×105 0.1439 0.0192 0.1631 –
1×106 0.1453 0.0211 0.1664 2.0%
5×106 0.1466 0.0181 0.1648 1.0%
1×107 0.1477 0.0164 0.1641 0.6%
smp’19 Rome, Italy May 26-30 35

81. Wake Flow
Circumferentially averaged axial velocity Vx at x/R = 0.15 (k − ω SST)
r/R
-U
X
/U-1
0.2 0.4 0.6 0.8 1.0
-0.2
-0.1
0.0
0.1
0.2
0.3
0.4
Re=1 × 104
Re=1 × 10
5
Re=5 × 10
5
Re=1 × 10
6
Re=1 × 107
smp’19 Rome, Italy May 26-30 36

82. Wake Flow
Circumferentially averaged radial velocity Vr at x/R = 0.15 (k − ω SST)
r/R
U
R
/U
0.2 0.4 0.6 0.8 1.0
-0.15
-0.10
-0.05
0.00
0.05
Re=1 × 104
Re=1 × 10
5
Re=5 × 10
5
Re=1 × 10
6
Re=1 × 107
smp’19 Rome, Italy May 26-30 37

83. Wake Flow
Circumferentially averaged tangential velocity Vθ at x/R = 0.15 (k − ω SST)
r/R
U
θ
/U
0.2 0.4 0.6 0.8 1.0
-0.4
-0.3
-0.2
-0.1
0.0
0.1
Re=1 × 104
Re=1 × 105
Re=5 × 10
5
Re=1 × 10
6
Re=1 × 10
7
smp’19 Rome, Italy May 26-30 38

84. Wake Flow
Circumferentially averaged axial velocity Vx at x/R = 0.15 (γ − R̃eθt
)
r/R
-U
X
/U-1
0.2 0.4 0.6 0.8 1.0
-0.2
-0.1
0.0
0.1
0.2
0.3
0.4
Re=1 × 104
Re=1 × 10
5
Re=5 × 10
5
Re=1 × 10
6
Re=1 × 107
smp’19 Rome, Italy May 26-30 39

85. Wake Flow
Circumferentially averaged radial velocity Vr at x/R = 0.15 (γ − R̃eθt
)
r/R
U
R
/U
0.2 0.4 0.6 0.8 1.0
-0.15
-0.10
-0.05
0.00
0.05
Re=1 × 104
Re=1 × 10
5
Re=5 × 10
5
Re=1 × 10
6
Re=1 × 107
smp’19 Rome, Italy May 26-30 40

86. Wake Flow
Circumferentially averaged tangential velocity Vθ at x/R = 0.15 (γ − R̃eθt
)
r/R
U
θ
/U
0.2 0.4 0.6 0.8 1.0
-0.4
-0.3
-0.2
-0.1
0.0
0.1
Re=1 × 104
Re=1 × 105
Re=5 × 10
5
Re=1 × 10
6
Re=1 × 10
7
smp’19 Rome, Italy May 26-30 41