Uncertainty Analysis for Ship-Bank Interaction Tests in A Circulating Water Channel

This paper presents a systematic model test program to assess the uncertainty of the ship-bank interaction forces, using the planar motion mechanism (PMM) system in a circulating water channel (CWC). Therefore, the uncertainties due to ship-bank distance and water depth are considered, and they are calculated via the partial differentials of the regression formulae based on the test data. The general part of the uncertainty analysis (UA) is performed according to the ITTC recommended procedure 7.5-02-06.04, while the uncertainty of speed is identified as the bias limit due to the flow velocity maldistribution in the CWC. In each example test for the UA of ship-bank interaction forces, 12 repeated measurements were conducted. Results from the UA show that the contribution of water depth error and flow velocity maldistribution to the total uncertainty is noticeable, and the paper explains how they increase with the change of the test conditions. The present study will be useful in understanding the uncertainty regarding the ship-bank interaction force measurement in a CWC.


Introduction
For a ship that sails close to banks, the decrease of local cross-section of the flow results in the increase of flow velocity and concurrent decrease of pressure. A lateral force and a yaw moment pushing the bow away the bank act on the ship hull. The so-called ship-bank interaction forces pose great influences on ships' manoeuvrability in confined waterways and the forces depend on the water depth, the ship's distance to the bank, the forward speed and the shape of the bank. Norrbin (1974) proposed the empirical formulations of the bank induced forces with respect to water depth and ship-bank distance according to his experimental study on a tanker ship model. Later Ch'ng et al. (1993), Li et al. (2003) and Lataire and Vantorre (2008) developed different types of empirical formulations regressed from respective model tests. On the other hand, the increasing number of applications of Computational Fluid Dynamics (CFD) to the prediction of ship hydrodynamics in restricted waters, e.g. Zou and Larsson (2013), Kaidi et al. (2017) and van Hoydonck et al. (2019) motivated the requirement of suitable bench-mark test data for verification and validation. Measurements of squat and bank effects for a VLCC ship were presented by Lataire et al. (2012). Similarly, Mucha et al. (2018) conducted the experimental study on resistance, propulsion and bank effect with an inland waterway ship. The experimental studies above are vitally valuable. However, the investigation on the uncertainty of the measurements for the benchmark models is needed in order to validate the accuracy. Therefore, uncertainty analysis (UA) for the ship-bank interaction tests becomes important to both Experimental Fluid Dynamics (EFD) study and CFD study.
The standard procedures of UA were initiated in the fields of mechanical engineering (ASME, 1998). Soon UA methodology was adopted in ship hydrodynamics in the 22nd International Towing Tank Conference (ITTC) (ITTC, 1999). The application of UA methodology has expanded to cases such as wave profile and elevations (Longo and Stern, 2005), captive manoeuvring tests including PMM tests (Simonsen, 2004), circular motion tests (Ueno et al., 2009) and added resistance experiment (Park et al., 2015), and the discussion of model parameters for UAs according to test modes (Wu, 2007). The precious experience of UA for PMM tests was summarized in the ITTC Recommended Procedure 7.5- 02-06.04 in 200802-06.04 in (CuraHochbaum et al., 2008Yoon et al., 2015). Actually, this procedure shows an example of how to make UA for systematic PMM tests, which are carried out in infinite open water. Except (Park et al. 2015) which is based on ISO's methodology (1995), the methods utilized in the rest of the work are based on the ASME (1998) or AIAA (1999) procedures of UA. In the ASME or AIAA method, the uncertainty components, systematic and random uncertainties, are determined by the effect on the final uncertainty results. The essence of UA is that a simulation program can be used to determine the sensitivity of the outcome to uncertainties in the model (Tonelli and Quadvlieg, 2015). As to PMM tests, UA procedures mainly aim to determine the final outcome, the hydrodynamic forces on the model, to a variation in the inputs (e.g. model geometry, speed, PMM motion and gauge measurement). For the test in restricted water conditions, additional uncertainty sources such as ship-bank distance and water depth should be considered. The method to quantify the uncertainties from ship-bank distance and water depth is very helpful since the ITTC Recommended Procedure 7.5-02-06.04 does not consider restricted water conditions. The circulating water channel (CWC) has been used to carry out captive model tests for over 30 years (Tamashima et al., 1983). Shoji et al. (2002) validated the accuracy of the hydrodynamic coefficients via CWC tests for predicting ship turning motions. Hashizume et al. (2016) compared the PMM test results from a CWC with the PMM test results from a towing tank and found that they agree well. CWCs have the advantage of long run time which allows the change of various parameters (like propeller revolutions) to be studied with ease (Dand, 1981) and long oscillation time in the forced swaying or yawing tests, and under some of the test conditions it is more convenient to construct a situation of confined water in a CWC. Liu et al. (2017) carried out PMM tests with a KVLCC2 model in a CWC to investigate the influence of ship-bank interaction on the hydrodynamic derivatives. However, it should be noted as well that the CWC has non-negligible shortcomings, such as the surging and rolling motions which are naturally generated in free running tests cannot be addressed. The limitation of size to the choice of model length for CWC tests arouses the issue of scale effects, compared with the trend towards larger model lengths to reduce scale effects in towing tank tests. But the CWC test is capable as a method to study the ship hydrodynamic characteristics, instead of the choice as accurate as the large towing tanks for commercial test programs.
This paper presents a study on UA for the measurement of ship-bank interaction. Therefore, the uncertainties due to ship-bank distance and water depth are considered. The process of UA basically follows ITTC Recommended Procedure 7.5-02-06.04, while the contributions of water depth and ship-bank distance to the uncertainty are quantified through the partial differentials of the regressive formulae. To obtain the regressive formulae, the measurements were carried out in the CWC at Shanghai Jiao Tong University, with a series of ship-bank distances for two water depths, h=1.5T and h=1.25T. For the validation of the present experimental results, the results predicted via the regressive formulae are also compared with those of other experiments. This paper does not assess the potential uncertainty of ship manoeuvring in restricted waters, but it hopefully helps with understanding the uncertainty for ship-bank interaction tests, especially those carried out in CWCs.

Overview of UA methodology
The uncertainty components, systematic and random uncertainties, are categorized with bias limits (B R ) and precision limits (P R ), respectively. As indicated in Longo and Stern (2005), the bias limits are estimated with consideration of elemental error sources for individual variables, whereas the precision limits are usually estimated end to end for experimental results. The uncertainty following the ITTC Procedures 7.5-02-06-04 is indicated by the 95% confidence uncertainty U RSS , which is the root sum square (RSS) as follows: In the ship-bank interaction test, R refers to the non-dimensional lateral force and yaw moment, Y′ or N′. Y′ and N′ are determined after the measurement of forces and several variables, including ship length L PP , draft T, ship speed U and water density , with the data reduction equations R=R(X 1 , X 2 , ···, X J ). Specifically, the DREs for Y′ and N′ are (2) The bias limit, B R , is determined with an error propagation equation where is the bias of the elemental variables identified from data reduction Eqs. (2) and (3), and refers to ( ), where F is either Y or N. is defined as . Fig. 1 shows the propagation of errors in the ship-bank interaction test. Herein, is from model fabrication. B T is from the ship ballasting error with respect to the draft markers.
is from the error of measured temperature for density. B U is from the water flow velocity maldistribution in the measuring section of the CWC, which is different from the bias caused by the fluctuating carriage speed during runs for towing tank tests. According to ITTC Procedures 7.5-02-06-04, the error sources of B F in steady tests include the gauge calibration weights errors (from which the bias is denoted as B F,calib ) and the gauge volt-to-force conversion (from which the bias is denoted as B F,acquis ). Other error sources are regarding the variables that the measured hydrodynamic forces directly depend on. In general, the relationships between the force (F) and the variable (x) can be defined as dF/dx. For example, in oblique towing tests, the bias for the measured force due to the deviation from the designated drift angle is expressed as: As the ship-bank interaction forces depend on the water depth, h, and the ship-bank distance, y bank , the bias of water depth, B F,h , and the bias of lateral ship-bank distance, are introduced to B F . The former is linked to the roughness of the false bottom that is used for the tests, while the latter is mainly from the model misalignment with respect to the centreline of the CWC. The bias limit for measured forces with the propagation of the four uncorrelated errors is expressed as: The precision limits, P R , are determined from repeated measurements of the forces (Y and N) and the estimation is given by In the present study, M=12 for each test condition that is selected to conduct UA.

Estimation of bias limits in CWC
The procedures to estimate , , B T , B F,calib and B F,acquis as well as the expression for each in the CWC are identical to the ones currently applied to towing tank tests. Readers are suggested to refer to Simonsen (2004) for the full procedures. Sections 3.1−3.3 present the estimation procedures of B U , B F,h and because of the estimation methods for the biases are not considered or different in the method used in towing tanks.

Model geometry and test setup
The specific model is the KVLCC2 Moeri tanker at a scale of 1:128.77, which is often used as a benchmark vessel in manoeuvring-related studies. The principal dimensions of full scale and model scale hull are listed in Table 1.
The tests were conducted with the bare hull.
The Shanghai Jiao Tong University Circulating Water Channel is a vertical water channel with the transversal arrangement of flow-uniformizing deflectors, as shown in Fig. 2. The main dimensions of the measuring section are 8.0 m×3.0 m×1.6 m and the velocity arrangement is 0.1− 3.0 m/s. In this experimental study the water velocity was set at U =0.351 m/s, corresponding to a full-scale speed of 7.8 knots (Fr= 0.071).
The CWC is equipped with a planar motion carriage. The model's motion in the horizontal plane is determined by  the carriage, and the model is free in heave and pitch motions. The ship's towing speed in the towing tank test is now substituted by the water flow velocity in the CWC. Fig. 3 shows the ship's transverse position during the ship-bank interaction test. The model hull is positioned parallel to the bank with different . y bank is defined as: where W=3 m.
To simulate the shallow water conditions, a false bottom that is 6 m long and 2.5 m wide is assembled in the measured section. The bottom legs are length-adjustable to make the water depth to draft ratio, h/T, change from 1.1 to 3.5 for the KVLCC2 model ship. The scene of the ship-bank interaction test with the false bottom is presented in Fig. 4.
The measurement system for the test consists of a 3component strain-gauge type 20-kg load cell (Nissho Electric Works Co., Ltd., 2009), a DPM-911B amplifier with 10-Hz low-pass filter (Kyowa Electronic Instruments Co., Ltd., 2017), and 16-bit AD conversion card with 16 channels embedded into PC (Interface Corporation, 2002). Data acquisition for the measurement was done through a collection of over 30 s at 50 Hz. The tolerance of the weights used for gauge calibration is verified as ±5.0 mg/100 g by the Shanghai Institute of Measurement and Testing Technology.

Bias of flow velocity
The flow velocity distribution across the cross section of the flow field is measured by a pitometre that can move under the control of the program-encoded motion mechanism. The measurement was conducted on the cross section where the midship of the model ship would be positioned. As shown in Fig. 5, a set of measuring points that can cover the range of the model's travel was selected. The model always moved on the right side of the center line during the test. The reference point for the velocity comparison was set at (y=0, z=80 mm). The target speed is 0.351 m/s, which was achieved after a trial-and-error process of adjusting the turbines' rotation rate. The deviation is the difference between the velocity at each measuring point and the velocity at (y=0, z=80 mm), where the measured velocity matches 0.351 m/s. Figs. 6 and 7 show the velocity deviation for h=1.5T and h=1.25T, respectively. As shown in Figs. 6 and 7, the velocity deviations are within 2%. The distribution of the velocity for h=1.5T features better uniformity along the transverse axis than that for h=1.25T. B U is quantified with the uncertainty factor u95 value, which is according to Tonelli and Quadvlieg (2015) and cal- where y B is defined as (see Fig. 3): .
The regression coefficients in Eqs. (9) and (10) are lin-early dependent on the parameter C B B/T. As to the KVL-CC2 model used for this study, the coefficients should be obtained by regressing the present test results rather than the linear functions in terms of the three models in Ch'ng et al. (1993). Moreover, since only one speed is involved in this experiment, Froude number is not considered as an independent variable in the regression. Finally, the modified non-dimensional formulae for the test results are written as: (12) The regression coefficients ( and , i=1, 2, 3) are determined from the test results at h=1.5T and 1.25T, corresponding to T/(h−T) =2 and 4, respectively. For each h, η varies from 0.0 m to 0.9 m with an interval of 0.1 m, correspondingly y B ranging from 0.28 to 0.47. The non-dimensional forces Y′ and N′, y B and T/(h−T) are input to the Curve Fitting Tools from MATLAB, in which we use the Custom Equation option to specify the function to be fitted as the form of Eqs. (12) and (13). The nonlinear optimization algorithm for curve fitting is based on the two-dimensional subspace restricted trust-region method (Byrd et al., 1988). The function is a two-dimensional plane dependent from y B and T/(h−T) as illustrated in Fig. 8. The positive sign of Y′ means a force attracting the ship towards the wall and the negative sign of N′ means a moment turning the bow away from the wall. Beyond the toolbox provides the fitting goodness statistics like the coefficient of determination (R-square) or the root mean square error (RMSE). The resultant regression coefficients and the fitting goodness statistics are shown in Fig. 8 In Eqs. (14) and (15), is estimated according to the uncertainty factor u95 value of R fb , which is used to reflect the roughness of the false bottom. R fb is defined as the difference between the depth of the measured point and the pre-set depth, here h=1.5, T=242.3 mm. The roughness measurement covered the bottom area where the model travelled through during the test, from 2 m to 4.5 m with an interval of 0.5 m along the length of the false bottom and from = −0.2 m to 1.0 m in the transverse direction. Table 2 lists the mean values, the standard deviations and the u95 values that are calculated as two times the standard deviations. The values are calculated with respect to different transverse positions. By averaging the u95 values from the measurements, can be assumed to be about 0.6 mm. In Eqs. (16) and (17), is the combination of the model's installation error with respect to the center line and the lead error of the ball screw that is used in the PMM carriage. The installation error with respect to the center line in the transverse direction is estimated to be 1 mm. The lead accuracy for the current ball screw is ±0.023 mm/300 mm and therefore the highest lead error in the present test cases is about 0.07 mm.

RANS study of scale effects on ship-bank interaction
With the concern of validating the results of CWC testing, researchers will naturally have interests in the possible comparison with benchmark data obtained from towing tanks. The only published data of the bank effects on KVL-CC2 at present, as presented in van Hoydonck et al. (2019), are the results from the towing tank of Flanders Hydraulics Research (FHR) with a model at a scale of 1:75 (L PP =4.267 m). According to the data from a questionnaire distributed among all ITTC member organizations in 2015, larger ship models are used for shallow water tests in most of the towing tanks (mean ship model length of 3.6 m), compared with the test presented in this paper (ITTC, 2017). Scale effect seems to be a key factor in the discrepancy between the data of CWC testing and the benchmark data of FHR. . As a substitute for the test study in the CWC to investigate the closer-to-bank conditions of y B ≥0.48, a CFD study on the influence of model scale on the ship-bank interaction forces was carried out through the commercial CFD software STAR-CCM+. An incompressible, Reynoldsaveraged Navier-Stokes (RANS) solver that is provided by STAR-CCM+ is used in the present work. The CFD code discretizes the flow domain with the finite volume method and closes the RANS equations with the realizable two-layer model.
In Fig. 10 the domain and boundary definitions are shown. The inlet and outlet boundaries are located at 2L PP forward and 5L PP aft of the aft perpendicular of the ship, with the condition of velocity-inlet and pressure-outlet, respectively. The value of L PP refers to 4.267 m. Free surface deformation is not considered at the low speed and the top surface is given a symmetry condition. No-slip wall conditions are used on the hull and rudder surface, and fixed slip wall conditions are applied on the vertical wall, slope and bottom surface. Since the blockage ratio in the CWC  Fig. 11 shows the non-dimensional results of CFD simulation for two model scales as well as the results of EFD in FHR (van Hoydonck et al., 2019). From the comparison between the results of CFD and those of EFD for the model at the scale of 1:75, the accuracy of the CFD simulation for ship-bank interaction is acceptable. From the comparison between the two scaled models, the smaller hull experiences larger non-dimensional lateral force and yaw moment. Scale effects are evident in the forces of ship-bank interaction computations, and the scale effects lead to larger discrepancies in the magnitude as the ship-bank distance decreases. It is also evident that the regression functions (Eqs. (12) and (13)) need to be changed for the forces on a larger model, which lead to a different predicted uncertainty as a result.

Test condition for UA η η
The proposed UA procedure is done on the test conditions of y bank =2.7B and y bank =1.5B ( =0.3 m and =0.8 m, respectively) with the water depths h=1.5T and h=1.25T. The water temperature during the test is 18.5°C.

Results of bias limits and precision limits
B ρ B L PP , and B T are not dependent on y bank and h, and the difference of B U between h=1.5T and h=1.25T is negligible, therefore, the values as well as their percentages in the variables are summarized in Table 3.
With regard to the components of B F , following the procedure in Simonsen (2004) , B F,calib are estimated as 0.0047 N for the lateral force Y and 0.0036 Nm for the yaw moment N. B F,acquis is assumed to increase linearly with the measured forces. The functions of B F,acquis versus Y and N are expressed as: In the equations above, Y and N are the averages of the 12 samples. Tables 4 and 5 show the results of the repeated tests as well as the average, the standard deviation and the precision limit resulted from the measurement samples at h=1.5T and h=1.25T, respectively. Table 6 summarizes the results for the bias limits of measured forces. The component bias limits are presented as their percentages of . The sum of and accounts for the majority of , except takes 75% of in the case of h=1.25T and y bank =1.5B. The contribution of to increases rapidly with the decrease of shipbank distance as well as water depth. It can be expected that will be the dominant component of if the model   11. Non-dimensional lateral force and yaw moment from CFD and EFD for different scaled KVLCC2 models. Fig. 9. Cross section of the tank geometry of FHR, adopted from Zou and Larsson (2013). for y bank =2.7B, the maximum value of it is smaller than 5%. This implies that the water depth is a remarkable source of uncertainty for the measured forces relative to the ship-bank distance. The estimation of bottom roughness is important to the UA of ship-bank interaction forces.

Results of uncertaintiesR
The summary of uncertainties for the ship-bank interaction forces with KVLCC2 is presented in Table 7. R refers to the non-dimensional value of in Tables 4 and 5. The maximum uncertainties for Y′ and N′ in the case of h/T=1.5 and y bank =2.7B are 7.1% and 9.7%, respectively. As the measured forces under this test condition are 0.16 N and 0.10 Nm, the uncertainty can be regarded as a small value. When y bank decreases to 1.5B, U RSS /|R| drops to the range of 3%−4.5%. The decrease of uncertainties percentage is clearly linked to the decrease of ship-bank distance, because of which the measured forces increase significantly.

RSS
Along with the decrease of ship-bank distance is the increase of the bias limit in the share of the uncertainty. When y bank equals 2.7B, the share of the two uncertainty components satisfies and . When y bank decreases to 1.5B, and appears for most of the hydrodynamic forces, except for N' under the condition of h=1.25T. Predominance of the bias limit in the uncertainty indicates that the measured condition features a better repeatability because random errors during the measurement are maintained at a stable level.
In addition, the percentages of the combined bias limits and sensitivity coefficient in the total bias limit are given in Table 7. The uncertainties due to ship length and water density are small or even negligible. The measured forces are identified as the primary source of bias limit for y bank =2.7B. When reducing the ship-bank distance, the contributions from the biases of draft and speed to show huge growths. This is mainly because of the change of and with respect to the measured forces, as B T and B U are constant. Taking the DRE for Y′ as an example, the sensitivity coefficients for B T , B U and B F are expressed as: θ T θ U For y bank =1.5B, the lateral force increases to more than two times the force for y bank =2.7B, leading to the significant increase of and . On the other hand, θ F is constant versus the decreasing y bank and the increase of B F is relat- ively small. This implies that the measurement of flow velocity maldistribution and the estimation of the resultant bias limit are of great importance to the UA for the ship-bank interaction test carried out in the CWC.

Conclusions
This study focuses on the UA in the ship-bank interaction tests. The UA is carried out for the KVLCC2 tanker in the CWC of Shanghai Jiao Tong University, based on ITTC Recommended Procedure 7.5-02-06.04. The error sources specific to the ship-bank interaction tests, water depth and ship-bank distance, are considered and quantified, using the partial differentials of the empirical formulae (Ch'ng et al., 1993) that are modified with data regression. From this study, the following conclusions were obtained: (1) In the current range of y bank and h, the primary source of the bias limits of measured force arises from the combined B F,calib and B F,acquis . The contributions from and to B F increase rapidly with the decreasing ship-bank distance or water depth, but the amount of is much larger than that of . With a closer ship-bank separation or a shallower water depth, will be the largest contribution to .
(2) In the present test, the maximum uncertainties of 7.32% for Y′ and 9.69% for N′ are identified under the condition of h/T=1.5, y bank =2.7B. For y bank =1.5B, the uncertainty fluctuates within the range of 2.5%−4%. The uncertainties can be regarded as small values because the measured forces/moments for all the test cases are smaller than 1 N or 1 Nm.
(3) When the ship-bank distance decreases, the bias limit exceeds the precision limit in percentage of the uncertainty, indicating that the results of the variables in the DREs are highly repeatable and random error uncertainty is kept at a stable level. For y bank =2.7B, the measured forces contribute to the largest portion of the bias limit. For a smaller ship-bank distance, on the other hand, the flow velocity maldistribution becomes the predominant source of bias limit.
It should be noted that the test setup based on the CWC with a false bottom is different from the conventional confined water condition in a towing tank. The results of CFD simulation imply that scale effects, which originate from the use of a smaller hull model in the CWC, make the nondimensional value of the hydrodynamic forces and the uncertainty of the forces different from those obtained from the towing tank tests. Accordingly, the regression formulae of bank effects versus y bank and h and the resultant equations for the bias limits are specific to the model tested in the CWC. With the limitations above, the conclusions obtained through the quantitive analyses do not hold for the tests conducted in a towing tank. Nevertheless, the methodology of the procedure of UA for water depth and ship-bank distance can be useful to the UA of the confined water condition in towing tanks.