UNCERTAINTY QUANTIFICATION OF SELF-PROPULSION ANALYSES WITH RANS-CFD AND COMPARISON WITH FULL-SCALE SHIP TRIALS

Summary RANS-CFD is a well-established tool with widespread use in maritime industry and research. Valuable information might be extracted from the results of such simulations in terms of ship resistance and flow field variables. With recent advancements in computational power, it became possible to investigate the performance of ships in self-propulsion conditions with RANS method. This paper presents the results of a study in which self-propulsion analyses of a small size product/oil tanker has been carried out at ship scale. The methodology proposed in this study makes use of open water propeller performance predictions, resistance analyses at model scale and self-propulsion computations at ship scale for a minimum of 2 different propeller loadings to obtain the self-propulsion point and respective performance parameters. In order to speed up the time-consuming self-propulsion computations, these cases have been solved with a single-phase approach. Resistance predictions have been compared with experimental findings. Uncertainty associated with prediction of resistance and thrust has been quantified. Additionally, sea trials have been conducted on the subject vessel and its two sisters and measured delivered power data have been used for evaluating the capability of the numerical method in self-propulsion predictions. Comparison of results indicate that the proposed self-propulsion computation methodology with RANS CFD at ship scale is capable of predicting delivered power with sufficient accuracy at an acceptable computational cost.


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Li, Han, Dong, and Zhao [9] have studied the light running margin (LRM) problem of the propeller. The study includes the open water performance analysis and then the trial speed and propeller revolution predictions. The results have been compared to sea trials. The theoretical results have shown about 0.03 knots lower vessel speed and 1.8 rpm higher propeller rotational speed compared to sea trials.
Hasselaar and Kaeding [10] have compared full scale RANS-BEM coupling method based CFD simulations and model tests to validate the performance energy saving devices (ESD). The full-scale CFD simulations and sea trials have been compared by means of the nondimensionalised power and speed correlation. The calculation has been performed with/without ESD by considering propeller blade roughness. The prediction of delivered power difference between sea trials and CFD simulation has been reduced to 8% by considering the propeller roughness.
Kamal, Shamsuddin and Binns [11] have compared full scale CFD simulation with sea trials and the towing tank test results by means of delivered power. In the CFD analysis of the training vessel, actuator disk modelling has been used for propeller model. Sea trials results are in good agreement with the towing tank test results, where the CFD simulation has underpredicted the delivered power compared to towing tank results.
Dogrul [12] has conducted a numerical study on the scale effects associated with the propulsion performance of Joubert BB2 submarine equipped with MARIN7371R propeller with RANS method. Body force method was used for self-propulsion computations. Scale effect on the wake fraction was found to be excessive when compared to thrust deduction.
Dai et al [13] have conducted a numerical study to obtain propulsion parameters in regular head and oblique waves with RANS method. Self-propulsion simulations in calm water and waves have been performed. The effect of waves on thrust deduction and wake fraction, open water, relative rotative, hull and propulsive efficiencies are discussed.
There are several verification studies for propulsion performance prediction of various ship types by RANS CFD in the literature. Generally, the studies are based on Richardson extrapolation [7], [8], [14], [15]. Applications utilizing the safety factor method are also present [6].
Farkas et al [16] have conducted experimental and numerical studies on a handymax bulk carrier. Numerical studies were conducted with RANS method in multi-phase configuration. Body force method has been used for including the propeller effect during computations. Verification and validation analyses were conducted by utilizing extrapolated experimental data. Effect of different turbulence modelling methods on results were also investigated. Although their results indicate superiority of Reynolds Stresses Model for turbulence modelling, additional computation time required for solving the extra equations is found as a disadvantage of the approach.
Farkas et al [17] have further investigated the capabilities of RANS based CFD in predicting the wake field of a ship. Results were validated with experimental data at model scale. Wall function approach was found to predict the ship wake satisfactorily. They have observed major scale effects for nominal wake characteristics and recommended that full scale viscous simulations are beneficial in terms of accurate prediction of stern flow characteristics.
Pena and Huang [18] have assessed the suitability of different turbulence models on ship hydrodynamic analyses. Computational effort and accuracy along with applicability have been of primary concern. They concluded that RANS approach with k-ω SST turbulence modelling is capable of predicting the ship resistance in full scale. They also point out that relatively coarse meshes adopted in RANS approach tend to create inaccuracies associated with flow around propellers.
Uncertainty quantification of self-propulsion analyses Mustafa İnsel, Serhan Gökçay with RANS-CFD and comparison with full-scale ship trials Terziev et al [19] have reviewed the sources of uncertainties associated with hydrodynamic performance prediction and claimed that although reliable data may be obtained by full-scale ship hydrodynamic simulations, the progress achieved is still not satisfactory to eliminate the need of experimental testing campaigns and extrapolation of data to ship's scale. They identify the main reasons preventing further progress in the field of full-scale computational hydrodynamics as lack of open full-scale data and lack of high computational power.
Having reviewed the literature on propulsive performance prediction of ships with CFD, it can be evaluated that RANS-CFD has become an alternative tool for the purpose along with model experiments. There exist numerous cases in literature where findings from CFD studies are compared with data from model experiments and even full-scale trials. Predictions may tend to underpredict or overpredict the actual delivered power depending on the adopted methodology indicating that a generalized best practice is not existent at this stage. It should also be noted that current and future efforts aiming to reduce the computational effort for finding the self-propulsion point and corresponding performance parameters of a ship with a verified computational approach would make notable contribution to the field of computational hydrodynamics. Full-scale measured data for validating the computational results is seen as a valuable and necessary tool for further progress.
The current research campaign aims to utilize a methodology enabling the prediction of self-propulsion characteristics by a cost-efficient approach while maintaining fidelity. The reduction in computational effort is based on conducting the final self-propulsion analyses in a single-phase configuration rather than simplifying the propeller model. To adopt this costeffective approach and maintain high-fidelity in the predictions, self-propulsion analyses are conducted at ship's scale and sensitivity of the numerical approach to mesh size and time-step are investigated. Uncertainty associated with calculation of resistance, thrust and torque have been quantified. Results of the analyses have been compared with available full-scale uncertainty data and delivered power measurements conducted during the sea-trials for the subject vessel and its sisters.

General information on the pilot ship
In this section, brief information is presented about the tested ship's principal dimensions and powertrain. The tested tanker is shown in Figure 1 and technical details of the ship are presented in Table 1. Two more sister ships have also been tested and delivered power data are available for comparison. The vessel is equipped with a single 2300 kW engine coupled to a gearbox with a reduction ratio of 6.22 and a 4 bladed CPP type propeller. The 3-d hull model is shown in Figure 2 in perspective view.

Sea-trial measurements
Sea-trial measurements on board the pilot ships have been conducted for validating numerical analyses of self-propulsion and manoeuvring under a research campaign aiming to develop a decision support system. In the current context, measurement system and methodology will be presented concerning the monitoring of propulsion parameters.
Delivered shaft power on the ship is obtained by torque and rpm measurements with strain gauge telemetry and laser sensors, respectively. The strain gauge enables the quantification of torque which is then converted to delivered power.
The strain gauges, wireless transponder and batteries and the reflectors for rpm measurement are mounted on the shaft prior to measurements. In Figure 3, installed equipment on the tankers is shown. The strain gauge is calibrated on-board prior to the measurements electrically. An adjustable external resistance is added to the bridge depending on the maximum expected torque. The resistance is varied to achieve the expected range of torque values and a linear relationship between voltage and torque is achieved.
The voltage signal generated by the strain gauge is converted to a telemetry signal and transferred to a receiving unit and is then transferred to a laptop via A/D converter. The shaft torque value is acquired with a sampling frequency of 400 Hz and torque values averaged within 1 Hz intervals are logged separately. The complete schematic of the measurement system is given in Figure 4. On the bridge, an IMU containing dGPS, galvanometer and accelerometers are utilized to acquire the spatial orientation and dynamic attitude of the ship; rudder angle, pitch, wind speed and direction and depth have all been acquired from ship's equipment. Photographs of each have been saved at 1Hz intervals. An in-house OCR software has been used to digitize the data as shown in Figure 5. Sea trial for the pilot ship has been conducted on 05/12/2018. The hull of the ship was clean as the final hull cleaning prior to the measurements was reported on 29/09/2018. The ship was loaded to the EEDI draft. True wind speed was in average between 6-8 knots and sea state was reported as 1-2.

Numerical methodology
The numerical study on the pilot ship for evaluating the propulsion performance characteristics have been split into three phases. Initially, the resistance of the ship has been predicted in model scale followed by an extrapolation to full-scale. The choice of utilizing the model scale for resistance analyses is associated with the availability of model test results which are used for validation purposes. Separately, open water performance characteristics of the propeller have been investigated. Finally, self-propulsion analyses have been conducted to obtain the self-propulsion point and delivered power. Open water performance predictions and self-propulsion computations are conducted at ship's scale as the results will be compared to shaft power measurements from sea-trials of the ship and its two sisters.
Analyses have been conducted with RANS technique utilizing ANSYS CFX v 19.2 commercial CFD software. The software uses Finite Volume Method to solve the discretized continuity and momentum equations with a vertex centered approach. Advection scheme and turbulence numerics are "high resolution" which forces the solver to adjust the order of the related scheme to the highest possible. The blend factor values are varied across the solution domain depending on the local flow characteristics ensuring boundedness [20]. Throughout all the analyses, turbulence modelling has been achieved by SST model. Previous experience gained by the utilization of this method suggests that the viscous flow around the ship is well represented with this type of turbulence modelling even in the existence of flow separation [21]. Resistance and open water analyses have been conducted with steady state modelling whereas self-propulsion analyses were modelled as unsteady. Initially, resistance analyses have been conducted for a shorter variant of the current pilot tanker (100 mt in length, 6000 DWT) for which model experiment results were available. The results of these analyses have been compared to experimental data for validation of the numerical model. Following these computations, resistance analyses for the naked hull were conducted for the actual pilot ship with a length of 107.7 mt (6600 DWT). As the process is identical, modelling phases of the actual pilot ship is presented for simplification purposes.
Tanker resistance analyses have been conducted at 1/20 scale. The multi-phase problem has been solved with Volume of Fluid method. The vessel is left free to sink during the computations. Analyses have been conducted at a draft corresponding to 6.5 m at ship's scale for two speeds. This draft is identical to the draft of the model experiments.
The 3-d model of the hull is pre-processed in Rhinoceros and exported to CFD software for solution domain creation. The mesh structure around the bow region of the hull and expected free surface region is shown in Figure 6. The boundary layer modelling along the ship hull is achieved by 12 cells with a growth rate of 1.2 A total of 1,809,429 cells have been used.   diameters. The inlet is a velocity inlet with fixed inflow velocity and outlet is modelled as an opening. The top boundary is set as a free-slip wall. The boundaries with the wavy arrows in Figure 9 indicate the rotationally periodic regions. The propeller blade is rotating around the x-axis as shown in the Figure 9. The rotation is modelled via the RFR (rotating frame of reference model) by which problems involving rotation around an axis at a specified angular velocity may be solved. When this model is activated, the solver computes the additional Coriolis and centrifugal components in the momentum equations [20].

Self-propulsion analyses
The final phase of the numerical analyses is comprised of self-propulsion analyses. For this purpose, a solution domain including the hull and the propellers need to be prepared. These models are prepared in Rhinoceros and exported to CFD software for further processing. The self-propulsion problem is handled as a single-phase problem within this research campaign and hence only the underwater part has been modelled. The mesh details are given in Figure  10. Boundary layer modelling of the hull, propeller and rudder is comprised of 12 cells with a growth rate of 1.2 A total of 15,503,157 cells have been used.
In this study, British method for obtaining the self-propulsion point has been utilized. With this method, the loading of the propeller is varied at the specified ship speed. At least two runs need to be conducted, one where the propeller is under-loaded and one over-loaded [22]. The thrust and variation of the hull resistance hence thrust deduction is monitored and the loading (rpm) at which total resistance and delivered thrust balance is sought. The boundary conditions and the solution domain are shown in Figure 11. The inlet is positioned 1 ship length away from the hull whereas the outlet is 3 ship lengths away. Sides and bottom boundaries are placed 0.75 ship length away from the hull. Sides and sea bottom are modelled as free-slip walls whereas the top boundary is modelled as a symmetry plane. Outlet of the domain is modelled as an opening. Inlet is modelled as a velocity inlet with fixed inflow speed.
Interface modelling between the rotating and stationary domains are shown in Figure 12. Self-propulsion analyses of the tanker have been conducted at ship's scale for 12.5 knots. As the analyses are conducted with unsteady (transient) modelling, the meshes in the rotating regions are sliding with the specified angular velocity and the boundaries shown with arrows in Figure 12 serve as interfaces between the rotating and stationary models. Transient rotorstator interface modelling has been used enabling proper interaction between stationary and rotating domains which is vital for self-propulsion analyses as the interaction between the hull and the propellers need to be captured with high accuracy. By using this type of interface between rotating and stationary domains, fluid particles can proceed to the rotating domain via the interface at their "corresponding position", ensuring continuous particle tracks. This is achieved by accounting for the comparative orientation of the sliding and stationary domains at the specific time step [20].

Results and discussions
In this section, the results obtained from the numerical analyses for prediction of the resistance, open water propeller performance and self-propulsion performance of the tanker will be presented alongside sensitivity analyses results. Comparisons are also going to be made with sea trial results of the pilot ship and two sister ships.

Resistance analyses
As previously stated, a shorter variant of the pilot tanker has been tested in a model basin and naked hull resistance computations for the mentioned variant have been conducted accordingly. The experiments have been conducted with a scale of 1/24.5 on a towing tank with dimensions of 160m, 6m, 4.5m.
The resistance analyses for the shorter tanker variant have been conducted for speeds corresponding to 11.5 and 12.5 knots at ship's scale. The resistance data obtained from the analyses of the shorter variant of the pilot tanker at 1/20 scale have been extrapolated to ship's scale and effective power predictions have been obtained as given in Figure 13. ITTC-1978 method has been used for extrapolation of resistance data [23]. In this methodology, the total resistance coefficient of a ship is expressed as: where CFS is the frictional resistance coefficient of the ship, k is the form factor, CW is the wave resistance coefficient, CA is the correlation allowance and CAAS is the air resistance coefficient respectively.
In Figure 13, the predicted effective power values are compared with experimental results, and it is seen that a good agreement exists between the experimental and predicted values. The overprediction by CFD is 1.07% and 2.91% for 11.5 and 12.5 knots respectively. As the agreement between the data is suitable for proceeding to pilot tanker analyses, the resistance analyses of the actual pilot tanker have been conducted with analogous approach for speeds corresponding to 11.5 and 12.5 knots at ship's scale.

Open water analyses
Open-water performance predictions have been conducted for advance coefficients ranging from 0.4 to 0.8. Non-dimensional thrust and torque coefficients as well as open water efficiency have been obtained for further use in self propulsion analyses.
Advance coefficient is defined as the ratio of advance velocity over diameter and rps of the propeller such as nD where VA is the speed of advance, n is the revolution rate of the propeller and D is the diameter of the propeller. The thrust (T) and torque (Q) produced by the propeller is expressed with non-dimensional form as thrust coefficient and torque coefficient. The open-water performance of the propeller is assessed by using thrust coefficient, torque coefficient and the open water efficiency which are denoted as KT, KQ, and η0, respectively: The predicted open water performance data for the propeller of the pilot tanker have been plotted in Figure 14. subject to propeller action have been evaluated. The uncertainty analysis has been based on the procedure recommended by ITTC [24]. The details of the procedures have been provided in the study of uncertainty quantification by Ayyıldız and others [1]. Grid sensitivity analyses have been performed on three different grid sizes for a time step of 0.01 s. The grid size has been changed by the refinement ratio ri which has been determined as √2. The time sensitivity has also been studied with three different time steps for the medium grid size. The refinement ratio for time sensitivity has been taken as 0.5. The total number of elements for the grid sensitivity and the time steps for the time sensitivity analyses have been provided in Table 2. Both time and grid sensitivity analyses have been performed on the hull with rudder and propeller. The vessel speed has been set to 12.5 knots and the propeller rpm has been set at 118. The resistance of the hull, rudder, and shaft, the thrust and the torque of propeller have been monitored separately. The numerical simulation results for the grid sensitivity (SG1, SG2, and SG3) and time sensitivity (ST1, ST2, and ST3) have been provided in Table 3. The convergence ratios for this study has been evaluated based on the numerical simulations results by using recommended formula as below [24]: 32 , 21 , The convergence ratio provides the convergence type according to following conditions: (i) Monotonic convergence for 0 < Ri < 1, (ii) Oscillatory convergence for -1 < Ri < 0, (iii) Monotonic divergence for 1 < Ri, (iv) Oscillatory divergence for Ri < -1.
For cases with monotonic convergence, the procedure also recommends evaluating the error (δ * i,1), the order of accuracy (pi), the correction factor (Ci) by using the generalized Richardson Extrapolation (RE) as follows: Where Pi,est is the estimated order of accuracy which is 2 in this study. The uncertainty (Ui) and the corrected uncertainty (Uci) have been calculated as: The verification uncertainty for simulations (USN) has been evaluated as: where UG and UT are the grid and time uncertainty, respectively. The uncertainty quantification parameters as Ri, δ * i,1, δ * REi,1 , pi, Ci, Uci have been provided in Tables 4, 5 and 6 for resistance and thrust, and torque respectively.  Monotonic convergence has been achieved for force parameters (resistance and thrust) with respect to grid and time as obtained Ri values are between 0 and 1. Results have been compared with findings of Degiuli et al [25]. They have assessed the uncertainty of resistance for grid and time step for 3 different motor yacht hull models. The results of both studies indicate that numerical uncertainties are lower than 1.1 %. However, torque convergence with respect to grid was observed to be oscillatory. Therefore, the uncertainty has been based on oscillation maximum (SU) and minimum (SL) as shown below: The corrected simulation results (Sc=Si1-δ * i,1) and the uncertainties have been given in the Table 7. For grid sensitivity on torque, corrected results are not provided, and uncertainty has been given as percentage of simulation result.

Self-propulsion analyses
Prior to the analyses required for obtaining the self-propulsion point, an initial case needs to be run without any propeller rotation. While this serves as a better initial condition to actual analyses as the boundary layer develops along the hull, it enables the quantification of the force component that needs to be accounted for obtaining the self-propulsion point. As the analyses have been conducted with single-phase modelling, the resistance increments due to wave action and wind has been added manually.
Self-propulsion analyses for the pilot tanker have been conducted at 12.5 knots for 3 different propeller loadings. The convergence history of thrust from the analysis with coarse grid size and time-step of 0.01s at 118 rpm has been shown in the Figure 15. All analyses have been performed for a simulation time of 24 seconds to achieve the required convergence in monitored parameters. Forces in the x-direction on the hull, shaft, rudder, and propeller were observed during the simulations. Each force component has different fluctuation tendencies. Additionally, the mentioned force components behave differently in terms of the required time for achieving force convergence. Whilst the forces on the rudder and shaft converge easily around 15 seconds, the forces on the hull start to converge at after 20 seconds as more time is required for the boundary layer to develop and the interaction between the hull and propeller to converge. As the numerical model is sensitive to the rpm of the propeller, any variation introduced during the computation would results in additional computational time as the propeller-hull interaction is resolved slowly. In the current study, this is eliminated by conducting two runs in under-loaded and over-loaded propeller conditions without introducing rpm variation during the computation. The Y-plus contours along the hull are depicted in Figure 16. The first cells along the majority of the hull have Y-plus values between 300 and 600. Higher values -approximately up to 700-are seen in the bow region where the boundary layer is developing. Fig. 15 Convergence history of thrust of the propeller in the analysis of medium grid size and time-step 0.01 for 118 rpm As previously mentioned, the medium grid has been used with a time-step of 0.01s for the self-propulsion computations. For all runs, the total force in the x-direction has been monitored. By adding the roughness allowance, wave and wind resistance components to the monitored total force, net force in the x-direction is obtained. As the hull has recently been cleaned prior to launching, a new ship hull with a roughness value of 120 microns has been assumed. As depicted in Figure 17, the self-propulsion point is obtained at 118.3 rpm. The utilized approach incorporates certain engineering assumptions and simplifications as the actual operation on board the ship is different to what has been applied in the numerical approach. Normally, the rpm would be kept constant, and pitch would be varied on board a ship with a CPP type propeller for varying the load on the propeller. In the current numerical approach, a fixed pitch propeller has been utilized and rpm has been varied to change the loading on the propeller. It is assumed that any change in the calculated thrust deduction, wake fraction and relative rotative efficiency values would be negligible depending on the method of varying the loading of the propeller, i.e., changing the pitch or rpm. The change in the calculated performance parameters such as t,w, eta-r is also assumed to be negligible for computation with the numerical and actual propeller as long as the geometric characteristics of the propellers are comparable. This enables the calculations to be conducted with a numerical propeller with similar geometric characteristics when compared to the actual propeller. During post processing, the advance coefficient and open water efficiency of the actual propeller may be calculated for the required thrust and hence thrust coefficient, and power predictions may be obtained for the case with the actual propeller. It is also assumed that the resistance due to wind and wave action may be added manually while obtaining the self-propulsion point with a singlephase simulation, which bears the assumption that the propeller-free surface interaction is negligible.
The increment in the resistance is used to obtain required thrust and actual obtained thrust is equal to the required thrust at the rpm at which the net force in the x-direction is zero. The resistance, thrust and propeller torque for the final self-propulsion point is obtained by interpolation between the results of the closest two runs. As thrust and rpm is known, KT is calculated. Once KT is known, J, Va, w, eta-h and eta-0 may easily be obtained. The ratio of the open water torque to propeller torque at self-propulsion point at the same advance coefficient yields eta-r and hence delivered power Pd may be obtained.
The wake profile has also been investigated alongside power prediction. In Figure 18, nominal and effective wake contours are depicted for 12.5 knots of ship speed. With the aid of the post-processing software, the average velocity on the propeller plane at self-propulsion The good agreement shows that the data obtained from the proposed calculation method is very close to the actual operating conditions of the propeller predicted by the proposed method. The results obtained from self-propulsion analyses have been used to predict KT, J, KQ, eta-0, eta-r, w, t, eta-h and Pd. The delivered power with the stock propeller is predicted as 1459 kW. Replacing the stock propeller with the actual CPP propeller, the predicted delivered power becomes 1595 kW. Also considering the transmission losses, the brake power required to propel the ship at 12.5 knots is predicted as 1512 kW with the stock propeller. When the stock propeller is replaced with the actual propeller, the required brake power becomes 1653 kW. Results are given in Table 8. The delivered power requirement predicted from self-propulsion computations have been compared to sea-trial data obtained for the pilot ship and two more sisters ships, as given in Figure 19. It is seen that the delivered power values of two sisters are very close to each other across the speed range. As the sea trials of the third ship have been conducted in relatively adverse weather conditions, even the results that have been corrected for adverse weather condition effects are not in agreement with the results of the other two sister ships. The difference between the delivered power values is at the order of 5%. To properly evaluate the differences in these results, uncertainties associated with measurements in actual sea conditions need to be taken into account. Zaho & Wang and Zhou [26] have made towing tests of a large model in real sea and obtained an uncertainty of 4% for the total resistance. Jang et al [27] have assessed the performance of the propeller in pitch motion with CFD and found that the thrust and torque coefficients tend to fluctuate substantially after 10 degrees of tilt angle. Additionally, slight variations were observed as the immersed depth of the propeller reduces below half of the propeller diameter. Insel [28] states that uncertainties associated with Beaufort scale is the most significant contributor to the overall uncertainty associated with shaft power in sea trials. He has further quantified that the total uncertainty may reach up to 7-8% at the design speed.The differences observed in the sea trial results of the current campaign are in agreement with Insel's findings obtained from sea trials of a set of 12 sister ships.
Considering the above, two approaches may be utilized for comparison of computational data and sea trial results. The delivered power data from all sister ships may be averaged or the data from the two ships with close results may be utilized for comparison, extracting the results of the ship that was tested in adverse weather conditions. The second option has been chosen within this study. As seen in Figure in 19, the delivered power value obtained from the computation is in very good agreement with the sea trial results.

Conclusion
Resistance, open water propeller performance and self-propulsion analyses have been conducted with RANS-CFD for a single-screw tanker. Resistance analyses have been compared with experimental data and delivered power predictions have been compared with delivered power measurements at sea-trials. For the tanker ship, data from sea-trials of 3 sister ships have been used. Uncertainty on the force components and propeller torque have been quantified.
Resistance of the naked ship hull has been predicted with model scale analyses and extrapolated to ship's scale. Comparisons of predicted effective power with experimental data indicated a good agreement with a maximum error of 4.13%. It is seen that the errors tend to increase with increasing speed. Although a direct uncertainty assessment has not been made within the scope of this study, previous attempts on quantifying uncertainty for resistance computations and experiments indicate that solutions may be validated. A direct comparison of open water performance predictions has not been made within the scope of this study as the stock propellers utilized for computations differ from the actual units. The thrust deduction effect is largely correlated with hull design and positioning of the propellers on the hull. It has been assumed that the differences of the pressure variation induced on the hull by stock and actual propellers would be negligible as long as geometric properties are close to each other. Further investigations on the validity of this assumption requires selfpropulsion calculations with actual propellers.
The power predicted for the tanker has been compared to sea-trials results of 3 sister ships. The power predicted with proposed method agrees well with the sea trial data. The good agreement between the numerical results and full-scale power measurements should be treated cautiously as both include sources of uncertainties. An error of 4% is present in resistance predictions. The numerical uncertainty on forces (resistance and thrust) is at the order of 0.5%. The major contributor to the numerical uncertainty is the propeller torque which shows an oscillatory convergence characteristic. The uncertainty associated with propeller torque is calculated as 5%. The delivered power value is directly influenced by the predicted torque as it affects the relative rotative efficiency. Therefore, it may be assumed that the calculated uncertainty is directly affecting the predicted delivered power. As previous studies suggest that uncertainties on sea trials measurements may reach 8%, it can be said that the proposed methodology is capable of predicting delivered power with acceptable accuracy.
Results obtained with the proposed methodology of conduction at least two selfpropulsion runs, with an under-loaded and over-loaded propeller at ship's scale may also be used for predicting the propulsion performance of the ship under different environmental conditions such as at high wind speeds where more thrust would be required to maintain a certain speed. This method also enables the prediction of propulsion performance of the hull and propeller at reduced loading of the propeller. Such a case is experienced when the ship is propelled with both the propeller and a wind-assist device. As the ship speed will be constant and the propeller will run at a reduced loading in this condition, the propulsion performance of the ship at numerous equilibrium conditions may be predicted easily with the proposed method by a limited number of computations.
Further evaluation of the proposed methodology may be made by scrutinizing the process in more detail. This requires the assessment of the results at each consecutive stage, i.e., evaluating the resistance, thrust and torque at ship scale. This may be realized by measurement of thrust on-board the ship.