Impact of Hard Fouling on the Ship Performance of Di ﬀ erent Ship Forms

: The successful optimization of a maintenance schedule, which represents one of the most important operational measures for the reduction of fuel consumption and greenhouse gas emission, relies on accurate prediction of the impact of cleaning on the ship performance. The impact of cleaning can be considered through the impact of biofouling on ship performance, which is deﬁned with delivered power and propeller rotation rate. In this study, the impact of hard fouling on the ship performance is investigated for three ship types, keeping in mind that ship performance can signiﬁcantly vary amongst di ﬀ erent ship types. Computational ﬂuid dynamics (CFD) simulations are carried out for several fouling conditions by employing the roughness function for hard fouling into the wall function of CFD solver. Firstly, the veriﬁcation study is performed, and the numerical uncertainty is quantiﬁed. The validation study is performed for smooth surface condition and, thereafter, the impact of hard fouling on resistance, open water and propulsion characteristics is assessed. The di ﬀ erences in the impact of biofouling on the ship performance are noticed amongst di ﬀ erent ship forms. They are mainly inﬂuenced by the portion of viscous resistance in the total resistance, relative roughness, roughness Reynolds number and advance coe ﬃ cient for the self-propulsion point. GCI 21 KCS U T


Introduction
Although recognized as an efficient mode of transport that has steadily enhanced safety, as well as environmental performance, over the past few decades, the maritime transport industry is transforming. Lately, in order to fulfil the new regulatory requirements and market needs, ship operators and ship owners have to improve capability of their ships to enable innovative, relevant and efficient services. Several technical and operational measures are adopted for increasing energy efficiency [1], however, it is crucial to accurately measure their effects. Namely, new regulations demand an increasing level of environmental performance, while ship operators and ship owners are faced with mounting pressure to keep up the competitiveness of their ships. As a result of this, ship operators and ship owners often hesitate to implement measure for increasing the energy efficiency due to the lack of reliable data on their effect [2,3]. The optimization of the maintenance schedule related to hull and propeller cleaning presents an important operational measure for increasing energy efficiency as ship operator or ship owner has large degree of control over it [4]. The successful optimization of maintenance schedule relies on accurate prediction of the impact of cleaning on the ship performance. The presence of biofouling on ship hull and propeller is causing an increase in roughness, which leads to an increase in ship resistance and if the ship speed is kept constant, an increase in the fuel consumption [5]. The biofouling occurrence is mostly prevented through the application of antifouling (AF) coatings, while hull and performed, by comparing the numerically obtained results with the extrapolated towing tank results. Finally, the detail investigation of the impact of hard fouling on the ship resistance and propulsion characteristics is performed for six different fouling conditions. The obtained results show the impact of hard fouling on the resistance and propulsion characteristics amongst different ship types, as well as on the increase in delivered power and propeller rotation rate.

Governing Equations
In this study Reynolds-averaged Navier-Stokes (RANS) and averaged continuity equations are used as governing equations, and they read: where ρ is the density, u i is the averaged velocity vector, ρu i u j is the Reynolds stress tensor, p is the mean pressure and τ ij is the mean viscous stress tensor, given as: where µ is the dynamic viscosity coefficient. In order to close Equations (1) and (2), k − ω SST turbulence model with wall functions is applied. For the discretization of governing equations, the finite volume method (FVM) is utilized, and the volume of fluid (VOF) method with high resolution interface capturing (HRIC) is utilized for tracking and locating the free surface. After the discretization, Equations (1) and (2) are solved in a segregated manner, the second order upwind convection scheme is used for the discretization of convective terms, while temporal discretization is performed using the first order scheme.
As already noted, the impact of roughness, i.e., biofouling, can be noticed as a downward shift of the mean velocity profile within the log-law region of TBL: where κ is the von Karman constant, U + is the non-dimensional mean velocity, y + is the non-dimensional normal distance from the wall and B is the smooth wall log-law intercept.
The drag characterization of a certain roughness or fouling type means finding the relation between ∆U + and k + , where k + is defined as: where k is the roughness length scale, which cannot be directly measured. Schultz has proposed following scaling for the hard fouling [12]: where R t is the height of the largest barnacles, while %SC is the percentage of the surface covered with barnacles.
Using Equation (6), Schultz has demonstrated excellent collapse for the obtained results with the Grigson roughness function, which is given with following equation: It should be noted that Schultz has proposed Equation (6) based on the assumption that the height of the larger barnacles has the dominant influence on drag and that the effect of increase in %SC is larger for lower %SC and smaller for higher %SC, and these assumptions were deduced from the obtained results, pipe flow experiments [23] and the observations from [24] for typical roughness types.
An explanation of the approach for the determination of the impact of biofilm on the ship resistance and propulsion characteristics is presented in [8,18] and is applied within this study. Firstly, an experimental study related to towing tank measurements of fouled flat plates was carried out within [12]. Based on the obtained results, Schultz has proposed Equation (6) for the determination of roughness length scale and Equation (7) as a ∆U + model for hard fouling. This ∆U + model was implemented within the wall function of CFD solver and CFD model was validated with the comparison of the numerically obtained frictional resistance coefficients for fouled flat plates [16] with the experimentally measured ones [12]. Additionally, CFD simulations for fouled full-scale plates representing two merchant ships were carried out, and the obtained results were compared with the results obtained using Granville similarity law scaling method [16]. Once the CFD model is validated, it can be utilized for the assessment of the impact of hard fouling on the resistance and propulsion characteristics. The impact of hard fouling on the ship resistance characteristics for two merchant ships is studied in [16] using CFD simulations of a towed ship. In this paper, the impact of hard fouling on the propeller performance in open water conditions is assessed through implementation of ∆U + model for hard fouling within wall function of CFD solver and by performing CFD simulations of the open water test (OWT). CFD simulations of OWT are performed using the moving reference frame (MRF) method, and CFD simulations are performed as steady simulations. More details regarding this method can be found within [25]. The impact of hard fouling on ship propulsion characteristics is assessed utilizing the proposed ∆U + model within CFD simulations of the self-propulsion test (SPT). It should be noted that CFD simulations of SPT are performed using the body force method and more details regarding this method can be found in [25]. The change in certain hydrodynamic characteristic is calculated as follows: where ϕ R represents certain hydrodynamic characteristic for fouled condition and ϕ S represents certain hydrodynamic characteristic for smooth surface condition. The impact of hard fouling on the ship performance is studied for six different fouling conditions presented in Table 1. The presented fouling conditions are investigated considering certain fouling condition present both at the hull and propeller.

Resistance, Open Water and Propulsion Characteristics
The total resistance coefficient can be decomposed as follows: where k represents the form factor, C F represents the frictional resistance coefficient and C W represents the wave resistance coefficient. It should be noted that C T is obtained by dividing total resistance (R T ) with 1 2 ρv 2 S (where v is the ship speed and S is the wetted surface) and in that way, the non-dimensional form is obtained.
Effective power (P E ) can be obtained as a product of R T and v. Most studies related to the impact of biofouling on ship performance investigate the effect of biofouling on effective power. However, the fuel consumption and greenhouse gas (GHG) emission can be related to delivered power (P D ) and propeller rotation rate (n). The quasi-propulsive efficiency coefficient defines relation between P E and P D as follows: where η H is the hull efficiency, η O is the open water efficiency and η R is the relative rotative efficiency. These efficiencies are defined as follows: where t is the thrust deduction coefficient, w is the wake fraction coefficient, J is the advance coefficient, K TO is the thrust coefficient in open water conditions, K QO is the torque coefficient in open water conditions and K Q is the torque coefficient obtained in SPT. Delivered power can be obtained as follows: where D is the propeller diameter.

Case Study
Within this paper, the impact of hard fouling on the ship performance is presented on the example of three commercial ships: containership, oil tanker and bulk carrier. The portion of CO 2 emission from containerships, bulk carriers and tankers in total CO 2 emission from international shipping is significantly higher than for other ship types and accounts for almost 62% of CO 2 emission from international shipping [26]. The Kriso Container Ship (KCS) was designed with an aim to represent a modern panamax container ship with a bulbous bow [27]. The Korea Research Institute for Ships and Ocean Engineering (KRISO) carried out an extensive towing tank experiments, in order to determine resistance, mean flow data and free surface waves [27]. Self-propulsion tests were performed at the Ship Research Institute (now the National Maritime Research Institute, NMRI) in Tokyo, and the obtained results were reported in the Proceedings of the CFD Workshop Tokyo in 2005 [28]. Kriso Very Large Crude-oil Carrier 2 (KVLCC2) was designed with the aim to represent a large oil tanker that can transport 300,000 t of crude oil, and it represents the second variant of KRISO tanker with more U-shaped stern frame lines in comparison with KVLCC. KRISO carried out resistance and self-propulsion tests, as well as towing tank measurements for the determination of mean flow data and wave profile elevations [27]. Bulk Carrier (BC) represents a typical handymax bulk carrier. Extensive towing tank experiments, including resistance tests, self-propulsion tests, as well as nominal wake measurements were performed in Brodarski institute [29]. It should be noted that KCS, KVLCC2 and BC were only designed as models, i.e., full-scale ships have never been built. The geometry of the investigated ships is presented in Figure 1. 2005 [28]. Kriso Very Large Crude-oil Carrier 2 (KVLCC2) was designed with the aim to represent a large oil tanker that can transport 300,000 t of crude oil, and it represents the second variant of KRISO tanker with more U-shaped stern frame lines in comparison with KVLCC. KRISO carried out resistance and self-propulsion tests, as well as towing tank measurements for the determination of mean flow data and wave profile elevations [27]. Bulk Carrier (BC) represents a typical handymax bulk carrier. Extensive towing tank experiments, including resistance tests, self-propulsion tests, as well as nominal wake measurements were performed in Brodarski institute [29]. It should be noted that KCS, KVLCC2 and BC were only designed as models, i.e., full-scale ships have never been built. The geometry of the investigated ships is presented in Figure 1. From Figure 1. it is evident that all three ships have bulbous bow and transom stern. KCS has more slender form than BC and KVLCC2. The main particulars of the investigated ships are presented in Table 2.  From Figure 1. it is evident that all three ships have bulbous bow and transom stern. KCS has more slender form than BC and KVLCC2. The main particulars of the investigated ships are presented in Table 2. SPT were performed using the KP505 for KCS, the KP458 for KVLCC2 and one stock propeller from the Wageningen series (WB) for BC, and their geometry is shown in Figure 2. The main particulars of the investigated propellers are given in Table 3. Towing tank tests for all three investigated propellers are performed at Reynolds numbers (Rn) higher than Rn = 2 · 10 5 as prescribed by ITTC [30], and the obtained results are given in [29,31,32].
SPT were performed using the KP505 for KCS, the KP458 for KVLCC2 and one stock propeller from the Wageningen series (WB) for BC, and their geometry is shown in Figure 2. The main particulars of the investigated propellers are given in Table 3. Towing tank tests for all three investigated propellers are performed at Reynolds numbers ( ) Rn higher than 5 2 10 R n = ⋅ as prescribed by ITTC [30], and the obtained results are given in [29,31,32].

Computational Domain and Boundary Conditions
In this study, the impact of hard fouling on resistance, open water and propulsion characteristics is investigated using CFD simulations of resistance, open water and self-propulsion tests. It should be noted that the impact of hard fouling on resistance characteristics of KCS and KVLCC2 is already investigated in [16]. Therefore, within this paper, the impact of hard fouling on ship resistance characteristics is only briefly presented as it is important for further discussion.  [16]. It should be noted that CFD simulations of resistance tests for BC are performed using the same computational domain and boundary conditions as in [16]. CFD simulations of OWT are performed using the cylindrical computational domain. The domain boundaries are placed sufficiently far from the investigated propeller and appropriate boundary conditions are applied in order to prevent their impact on the obtained solution, Figure 3. The computational domain for CFD simulations of SPT is the same as for CFD simulations of resistance test, however within CFD simulations of SPT symmetry condition is not applied, i.e., the whole computational domain is generated ( Figure 4). In Figure 4, the applied boundary conditions are presented as well. It should be noted that the same boundary conditions are

Computational Domain and Boundary Conditions
In this study, the impact of hard fouling on resistance, open water and propulsion characteristics is investigated using CFD simulations of resistance, open water and self-propulsion tests. It should be noted that the impact of hard fouling on resistance characteristics of KCS and KVLCC2 is already investigated in [16]. Therefore, within this paper, the impact of hard fouling on ship resistance characteristics is only briefly presented as it is important for further discussion. R T of a ship is determined using CFD simulations which include free surface effects, i.e., free surface simulations (FSS). Viscous resistance (R V ) is obtained using double body simulations (DBS), which do not take free surface effects into account. In DBS, the flow around deeply immersed double body ship is simulated and thus the obtained R T is equal to R V . The frictional resistance (R F ) is obtained by integrating the tangential stresses over the wetted surface, while viscous pressure resistance (R VP ) is obtained by integrating the pressure over the wetted surface in DBS. Once R V and R F are determined, 1 + k is determined as a ratio between R V and R F . Wave resistance (R W ) is obtained as difference between R T obtained in FSS and R V obtained in DBS. For more details regarding the performed CFD simulations of resistance tests, reference may be given to [16]. It should be noted that CFD simulations of resistance tests for BC are performed using the same computational domain and boundary conditions as in [16]. CFD simulations of OWT are performed using the cylindrical computational domain. The domain boundaries are placed sufficiently far from the investigated propeller and appropriate boundary conditions are applied in order to prevent their impact on the obtained solution, Figure 3. The computational domain for CFD simulations of SPT is the same as for CFD simulations of resistance test, however within CFD simulations of SPT symmetry condition is not applied, i.e., the whole computational domain is generated ( Figure 4). In Figure 4, the applied boundary conditions are presented as well. It should be noted that the same boundary conditions are applied in CFD simulations of the resistance test, except for the symmetry boundary condition, which is applied at the symmetry plane within CFD simulations of resistance test. Possible occurrence of wave reflection is prevented by applying VOF wave damping at the inlet, outlet and side boundaries. More details regarding the applied damping function can be found in [33], and the VOF wave damping length is set to L pp . applied in CFD simulations of the resistance test, except for the symmetry boundary condition, which is applied at the symmetry plane within CFD simulations of resistance test. Possible occurrence of wave reflection is prevented by applying VOF wave damping at the inlet, outlet and side boundaries. More details regarding the applied damping function can be found in [33], and the VOF wave damping length is set to pp L .

Discretization of Computational Domain and Computational Setup
Cut-cell grids with prism layer mesh on the walls were made utilizing the surface remesher, prism layer mesher and trimmer mesher within STAR-CCM+. The unstructured hexahedral mesh is refined locally in the critical regions. Thus, within DBS and FSS of resistance test, as well as in CFD simulations of SPT, mesh is refined near the hull surface, near the bow and stern and hull surface is applied in CFD simulations of the resistance test, except for the symmetry boundary condition, which is applied at the symmetry plane within CFD simulations of resistance test. Possible occurrence of wave reflection is prevented by applying VOF wave damping at the inlet, outlet and side boundaries. More details regarding the applied damping function can be found in [33], and the VOF wave damping length is set to pp L .

Discretization of Computational Domain and Computational Setup
Cut-cell grids with prism layer mesh on the walls were made utilizing the surface remesher, prism layer mesher and trimmer mesher within STAR-CCM+. The unstructured hexahedral mesh is refined locally in the critical regions. Thus, within DBS and FSS of resistance test, as well as in CFD simulations of SPT, mesh is refined near the hull surface, near the bow and stern and hull surface is

Discretization of Computational Domain and Computational Setup
Cut-cell grids with prism layer mesh on the walls were made utilizing the surface remesher, prism layer mesher and trimmer mesher within STAR-CCM+. The unstructured hexahedral mesh is refined locally in the critical regions. Thus, within DBS and FSS of resistance test, as well as in CFD simulations of SPT, mesh is refined near the hull surface, near the bow and stern and hull surface is discretized very fine, i.e., the cell size at the hull surface is set to 1/1000 L pp . Within CFD simulations including free surface effects, mesh is refined in the region where free surface is expected, as well as in order to capture Kelvin wake around free surface. Additionally, mesh for CFD simulations of SPT is refined in the region where virtual disk is located. It should be noted that refinements are made in the same way within [8,25,34]. The mesh for CFD simulations of OWT is refined in the region around the propeller.
Additionally, mesh is particularly refined along the leading and trailing edges of propeller in order to allow proper demarcation between the suction and pressure sides. The thickness of the first cell on the wall surfaces within all CFD simulations is chosen in a way that y + values are higher than 30 and k + values, as recommended by [15]. As a result of this, near wall mesh for smooth and fouled surfaces is not the same since investigated surface conditions represent very severe fouling conditions with high k values. The obtained mesh for CFD simulations of OWT is presented in Figure 5, while the obtained mesh for CFD simulations of SPT is shown in Figure 6. Within these two figures, the above mentioned refinements can be seen. discretized very fine, i.e., the cell size at the hull surface is set to 1/1000 pp L . Within CFD simulations including free surface effects, mesh is refined in the region where free surface is expected, as well as in order to capture Kelvin wake around free surface. Additionally, mesh for CFD simulations of SPT is refined in the region where virtual disk is located. It should be noted that refinements are made in the same way within [8,25,34]. The mesh for CFD simulations of OWT is refined in the region around the propeller. Additionally, mesh is particularly refined along the leading and trailing edges of propeller in order to allow proper demarcation between the suction and pressure sides. The thickness of the first cell on the wall surfaces within all CFD simulations is chosen in a way that y + values are higher than 30 and k + values, as recommended by [15]. As a result of this, near wall mesh for smooth and fouled surfaces is not the same since investigated surface conditions represent very severe fouling conditions with high k values. The obtained mesh for CFD simulations of OWT is presented in Figure 5, while the obtained mesh for CFD simulations of SPT is shown in Figure 6. Within these two figures, the above mentioned refinements can be seen.   discretized very fine, i.e., the cell size at the hull surface is set to 1/1000 pp L . Within CFD simulations including free surface effects, mesh is refined in the region where free surface is expected, as well as in order to capture Kelvin wake around free surface. Additionally, mesh for CFD simulations of SPT is refined in the region where virtual disk is located. It should be noted that refinements are made in the same way within [8,25,34]. The mesh for CFD simulations of OWT is refined in the region around the propeller. Additionally, mesh is particularly refined along the leading and trailing edges of propeller in order to allow proper demarcation between the suction and pressure sides. The thickness of the first cell on the wall surfaces within all CFD simulations is chosen in a way that y + values are higher than 30 and k + values, as recommended by [15]. As a result of this, near wall mesh for smooth and fouled surfaces is not the same since investigated surface conditions represent very severe fouling conditions with high k values. The obtained mesh for CFD simulations of OWT is presented in Figure 5, while the obtained mesh for CFD simulations of SPT is shown in Figure 6. Within these two figures, the above mentioned refinements can be seen.   CFD simulations of OWT are performed for full-scale KP505, KP458 and WB in a way that n = 1.5 rps is kept constant and advance velocity varies with J. CFD simulations for KP505 are performed for range of J from 0.1 to 0.8, with a step equal to 0.1, for KP458 for range of J from 0.1 to 0.7 with step equal to 0.1 and for WB for range of J from 0.08 to 0.88 with step equal to 0.08. CFD simulations of SPT are performed without discretization of propeller geometry, as the body force method is applied. Therefore, a virtual disk model is placed at the propeller location with the inner radius of the virtual disk set to the propeller hub radius and the outer radius set to the propeller radius (R). Thickness of virtual disk model is set as propeller thickness, the inflow plane radius is set as 1.1R and the inflow plane offset is set as 2.2R towards the bow from the half of virtual disk thickness.
CFD simulations without free surface effects, i.e., DBS of resistance test and CFD simulations of OWT, are performed as steady simulations. The remaining CFD simulations include free surface effect, and they are performed with time step equal to T/200, where T is the ratio between L pp and ship speed (v). FSS of resistance test and CFD simulations of SPT are stopped once R T and thrust (T) force became steady, i.e., once they oscillate around averaged value with oscillation amplitude lower than 0.5% of R T or T value.

Verification Study
A verification study is carried out in order to estimate sufficient grid spacings and adequate time steps. This study is carried out using three different meshes and three different time steps. Verification study for grid size is made with fine time step and verification study for time step is made with fine mesh. Thereafter, numerical uncertainty, which is consisted of both spatial and temporal uncertainties, is calculated using the grid convergence index (GCI) method. This method is recommended by the American Society of Mechanical Engineers, as well as by the American Institute of Aeronautics and Astronautics for the assessment of grid uncertainty (U G ) [35], but can be used for the assessment of temporal uncertainty (U T ) as well [35][36][37]. More details regarding the GCI method and numerical uncertainty can be found in [18].
For the purposes of verification study three meshes are generated for smooth surface condition and fouling condition R1. Since all mesh parameters, except prism layer mesh, are set to be relative to cell base size, mesh is refined by changing cell base size. It should be noted that all remaining CFD simulations, i.e., for the fouling conditions R2, R3, R4, R5 and R6 are performed using fine mesh. In Table 4, the number of cells used in the verification study is shown. Three different time steps, i.e., T/50, T/100 and T/200 are used in the verification study for time step. It should be noted that the verification study for CFD simulations of resistance tests of KCS and KVLCC2 is carried out in [16]. Numerical uncertainties in the prediction of R F and R V consisted of grid uncertainties solely, and R T consisted of grid and temporal uncertainties, which are calculated using the GCI method. The obtained numerical uncertainties in the prediction of R F were below 1.3% for both ships and for all analyzed fouling conditions (Table 1). Numerical uncertainties in the prediction of R V were slightly higher, however, the highest obtained numerical uncertainty was equal to 2.86%. Finally, the highest numerical uncertainties are obtained for the prediction of R T . Nevertheless, these grid and time step uncertainties were relatively low, i.e., the highest obtained grid uncertainty in the prediction of R T was equal to 2.99%, while the highest time step uncertainty in the prediction of R T was equal to 0.1%. Within this paper, the numerical uncertainty in the prediction of K TO and 10K QO from CFD simulations of OWT are calculated for one J value and the obtained results are presented in Tables 5 and 6. Additionally, numerical uncertainty in the prediction of P D , n, T and J from CFD simulations of SPT are calculated. As can be seen from Tables 5 and 6, relatively low numerical uncertainties are obtained, and are in line with numerical uncertainties of other CFD studies regarding open water tests [38,39]. Thus, the highest U G in the prediction of K TO and 10K QO is obtained for the WB propeller with smooth surface condition, and it is equal to 3.565% and 2.815%, respectively. It should be noted that numerical uncertainties obtained for smooth and fouled propellers are relatively close, i.e., numerical uncertainty has not raised due to the roughness effects.
From the results of verification study of SPT, Tables 7-9, it can be concluded that U T are lower than U G . Generally, the obtained U G related to the prediction of P D for smooth and fouled ships are slightly higher than for the other investigated key variables and the highest GCI 21 f ine for KCS is equal to 3.123%, for KVLCC2 is equal to 1.174% and for BC is equal to 7.318%. The obtained U T related to the prediction of P D for smooth and fouled ships are lower and the highest GCI 21 f ine for KCS is equal to 1.366%, for KVLCC2 is equal to 1.502% and for BC is equal to 3.390%. The obtained U G related to the prediction of n for smooth and fouled ships are the lowest amongst investigated key variables and the highest GCI 21 f ine for KCS is equal to 0.255%, for KVLCC2 is equal to 0.164% and for BC is equal to 1.661%. Interestingly, the obtained U T values related to the prediction of n for smooth and fouled ships are higher than U G values and the highest U T for KCS is equal to 0.401%, for KVLCC2 is equal to 0.701% and for BC is equal to 2.909%. The obtained U G values related to the prediction of T for smooth and fouled ships are low and the highest GCI 21 f ine for KCS is equal to 3.273%, for KVLCC2 is equal to 1.478% and for BC is equal to 4.717%. The obtained U T values related to the prediction of T for smooth and fouled ships are lower or similar to U G and the highest GCI 21 f ine for KCS is equal to 0.807%, for KVLCC2 is equal to 1.529% and for BC is equal to 3.499%. Finally, the obtained U G values related to the prediction of J for smooth and fouled ships are low and the highest GCI 21 f ine for KCS is equal to 0.452%, for KVLCC2 is equal to 1.257% and for BC is equal to 2.041%. The obtained U T values related to the prediction of J for smooth and fouled ships are low as well, and the highest GCI 21 f ine for KCS is equal to 0.451%, for KVLCC2 is equal to 0.703% and for BC is equal to 2.719%. The obtained U T , U Q , U n , U P D and U J , which consist of both U G and U T , are shown in Table 9. As can be seen from Table 9, the lowest U SN values for smooth and fouled ships are obtained for KCS, which was expected, since U G values are higher than U T values and the mesh for KCS had more cells than for KVLCC2 and BC. The highest U SN is obtained for the prediction of U P D for BC fouled with R1 and it is equal to 7.421% and other obtained U SN values are lower than 5.5%. Higher U P D were expected, since, for the prediction of P D , both n and the propeller torque should be determined. It should be noted that the obtained U P D are in line with the previously published studies [8,25]. From Table 9, it can be seen that higher numerical uncertainties are obtained for the prediction of P D and T, than for n and J, which was also obtained in [8]. Additionally, it can be seen that U SN in the prediction of key variables for R1 are mostly below U SN for smooth surface condition. Higher U SN obtained for R1 than for smooth surface condition can be ascribed to the lower cell number used in CFD simulations of SPT for rough surface condition (Table 4). Therefore, it can be concluded that the implementation of ∆U + within the wall function did not cause higher uncertainties in the prediction of the key variables.  Table 9. The obtained simulation uncertainties (U SN ) in the prediction of P D (U P D ), n (U n ), T (U T ) and J U J .

Validation Study
Relative deviations between numerically obtained and extrapolated results are calculated using the following equation: where φ CFD is the certain hydrodynamic characteristic obtained using CFD and φ EX is the certain hydrodynamic characteristic obtained using the ITTC 1978 Performance Prediction Method (PPM) and experimental results [30]. The obtained C T for full-scale KCS and KVLCC2 is validated within [16] through comparison of the obtained numerical results with extrapolated values using original ITTC 1978 PPM, based on Equation (9). Within ITTC 1978 PPM, C F is determined using the ITTC 1957 model-ship correlation line. In Table 10, the validation of the numerically obtained C T for the smooth surface condition is presented. As can be seen from Table 10, the obtained results are in satisfactory agreement with the extrapolated results, i.e., the highest RD is obtained for BC and it is equal to −4.338%. The numerically obtained open water characteristics for all three propellers have been validated, with the towing tank results published in the literature [29, 31,32]. It should be noted that CFD simulations of OWT are performed in full-scale, while experimental OWT are performed in model scale. Towing tank tests for all three investigated propellers are performed at Rn above Rn = 2 · 10 5 , as prescribed by ITTC [30]. In Figure 7, the comparison between the numerically and experimentally obtained open water characteristics is presented. From this figure, it can be seen that numerically obtained K TO , 10K QO and η O are in satisfactory agreement with the experimentally obtained ones. Slightly higher RD between numerically and experimentally obtained K TO and especially 10K QO is obtained at lower J values, however, at higher J values, these RD are significantly lower.
The obtained results of the validation study for P D and n are presented in Table 11, from which it can be concluded that satisfactory agreement is obtained. The highest obtained RD between numerical and extrapolated P D is obtained for KVLCC2 and it is equal to −5.701%, while the highest obtained RD for n is obtained for BC and it is equal to −1.786%. The validation study for ship propulsion characteristics is presented in Table 12. From Table 12, it can be seen that the obtained RD for 1 − t are lower than 3.7%, for 1 − w are lower than 7.4% and for η H are lower than 5.6% for all analyzed ships. It should be noted that slightly higher RD for 1 − w is obtained only for BC, and this can be attributed to the application of body force method. However, this RD is in line with previously published studies dealing with CFD simulations of SPT where the virtual disk model is applied [40,41]. The obtained RD for η O are lower than 3.1%, for propeller efficiency behind ship (η B ) are lower than 3.8%, for η R are lower than 2.9% and for η D is lower than 6.2%. It should be noted that slightly higher RD for η D is obtained only for KCS. However, in [42] where the authors carried out full-scale SPT for KCS using discretized propeller, η D was equal to 0.766, which is also lower than the extrapolated result. From this result, the obtained η D in this paper has RD equal to −3.394%. In Table 12, the validation for the obtained J, K T and K Q for self-propulsion point is shown as well. It can be seen that the obtained RD for J are lower than 5.7%, for K T are lower than 4.1% and for K Q are lower than 3.4% for all analyzed ships. Generally, the obtained RD presented in Tables 11 and 12 can be ascribed to different reasons. For example, insufficiently precise assessment of the nominal wake, as well as the propeller performance in OWT can be related to the inaccurate assessment of J for self-propulsion point, which then leads to inaccurate assessment of other propulsion characteristics. In addition to this, the modelling error should also be taken into account, as, in the body force method, the effect of propeller is modelled, rather than propeller itself. Furthermore, there is a numerical error as well, which is related to the applied mesh and time step. Lastly, there are also aspects regarding the applied PPM for the extrapolation of towing tank results. Namely, in [25] four different PPM are compared, and it was shown that extrapolated values can significantly vary with respect to the applied PPM. Thus, it was shown that for BC, extrapolated value of P D can vary up to 1.5%, for n up to 0.4%, for 1 − t up to 0.5%, for 1 − w up to 6.3%, for η R up to 1.1% and for η B up to 2.6%. In addition to these variations, experimental uncertainty should also be considered. Considering all above mentioned aspects, it can be concluded that satisfactory agreement is achieved for P D , n and all propulsion characteristics. The obtained results of the validation study for D P and n are presented in Table 11, from which it can be concluded that satisfactory agreement is obtained. The highest obtained RD between numerical and extrapolated D P is obtained for KVLCC2 and it is equal to −5.701%, while the highest obtained RD for n is obtained for BC and it is equal to −1.786%. The validation study for ship propulsion characteristics is presented in Table 12. From Table 12, it can be seen that the obtained RD for 1 t − are lower than 3.7%, for 1 w − are lower than 7.4% and for H η are lower than 5.6% for all analyzed ships. It should be noted that slightly higher RD for 1 w − is obtained only for BC, and this can be attributed to the application of body force method. However, this RD is in line with previously published studies dealing with CFD simulations of SPT where the virtual disk model is applied [40,41]. The obtained RD for O η are lower than 3.1%, for propeller efficiency behind ship ( ) B η are lower than 3.8%, for R η are lower than 2.9% and for D η is lower than 6.2%.

The Impact of Hard Fouling on the Ship Performance
Within this section, the impact of hard fouling on the resistance, open water and propulsion characteristics is presented for three investigated ships. While detail investigation of the impact of hard fouling on resistance characteristics for KCS and KVLCC2 is presented in [16], within this study this impact is only briefly mentioned as emphasis is given to the impact of hard fouling on the ship performance, which is defined by propeller operating point.

The Impact of Hard Fouling on Resistance Characteristics
As demonstrated within [16,18] the impact of biofouling on each resistance component is different. Thus, the presence of biofouling causes the increase in C F , decrease in C W , while the impact of biofouling on 1 + k value is almost negligible. Consequently, it is valuable to study the increase in R T , due to the presence of hard fouling through analysis of decomposed R T and the portion of each resistance component in R T for certain fouling condition. In Figure 8, decomposition of R T for three investigated ships and fouling conditions is presented. Additionally, within Figure 8 the portions of R F , R VP and R W in R T are given. From Figure 8, it is clear that, for all analyzed ships, the portion of R F in R T increases, due to the presence of hard fouling, and this increase is the highest for KCS, which can be attributed to the ship speed. Namely, KCS is investigated at the highest speed and therefore u τ values along the KCS hull are higher than u τ values along the KVLCC2 and BC hulls. Since k + values and consequently ∆U + values for given fouling condition and fluid properties depend only on u τ values, those values are higher for KCS than for KVLCC2 and BC resulting in higher increases in C F [16]. Additionally, C F for rough surface condition at high Rn value depends solely on k/L value, i.e., relative roughness [16]. The portion of R VP in R T due to the presence of hard fouling has increased for KCS and BC, while for KVLCC2 this portion has decreased. Regardless of this, from Figure 8, it is clear that the absolute value of R VP , due to the presence of hard fouling, has increased, which is expected, since the impact of biofouling on 1 + k value is minimal [16]. Finally, the portion of R W in R T due to the presence of hard fouling decreases for all analyzed ships and this decrease is the highest for KCS, which can be also attributed to ship speed. What is more, from Figure 8 it is clear that absolute values of R W due to the presence of hard fouling have decreased for all analyzed ships [16]. Generally, KVLCC2 is the most affected, due to the presence of hard fouling in terms of the increase in R T , which can be seen from Figure 9. Thus, the increase in R T due to the presence of hard fouling for KVLCC2 ranges from 63.8% (R6) to 120.9% (R1), for BC ranges from 59.5% (R6) to 114.6% (R1) and for KCS ranges from 49.9% (R6) to 95.8% (R1). This can be mostly attributed to the portion of R V in R T , since, due to the presence of biofouling R V , significantly increases. The portion of R V in R T is the highest for KVLCC2 and for smooth surface condition this portion is equal to 99.46%, as R W of KVLCC2 is negligible [28]. However, beside the portion of R V in R T , the ship speed also affects the increase in R T , as already explained. Thus, the increase in R T due to the presence of hard fouling is only slightly lower for BC than for KVLCC2 and the portion of R V in R T for smooth surface condition is equal to 83.6%. It should be noted that the significantly lower increase in R T is obtained for KCS, as KCS has relatively large portion of R W in R T (for smooth surface condition this portion is equal to 24.7%). Due to the presence of hard fouling, R W decreases, and, therefore, the increase in R T for KCS is lower.

The Impact of Hard Fouling on Open Water Characteristics
The impact of hard fouling (R1) on the propeller performance in open water conditions is presented in Figure 10. The obtained changes in TO K , QO K and O η , due to the presence of hard fouling, are presented in Table 13. As can be seen from Figure 10 and range from 3.77% (R6) to 11.19% (R1). Fouling penalties on the propeller performance in open water conditions can be ascribed to fouling impact on the skin friction and the pressure field. Thus, due to the presence of hard fouling on propeller surfaces wall shear stress ( ) w τ increases, while the pressure difference between pressure and suction sides of propeller is reduced, which can be seen from Figures 11 and 12. In Figure 11

The Impact of Hard Fouling on Open Water Characteristics
The impact of hard fouling (R1) on the propeller performance in open water conditions is presented in Figure 10. The obtained changes in K TO , K QO and η O , due to the presence of hard fouling, are presented in Table 13. As can be seen from Figure 10 and Table 13, due to the presence of hard fouling K TO has decreased and K QO has increased resulting in significant reduction in η O . As fouling severity increases (i.e., from R6 to R1), fouling penalties related to decrease in K TO and increase in K QO increase as well. Additionally, at higher J the fouling penalty related to decrease in η O is higher. Therefore, it can be concluded that the ships operating at higher J values will experience a greater reduction in η O , i.e., propeller fouling penalty on the ship performance will be greater. Thus, due to the presence of hard fouling ∆K TO values for KP505 at J = 0.6 range from −6.22% (R6) to −12.05% (R1), for KP458 at J = 0.4 range from −7.44% (R6) to −14.45% (R1) and for WB at J = 0.48 range from −7.86% (R6) to −12.09% (R1). An increase in ∆K QO values for KP505 at J = 0.6 range from 4.66% (R6) to 11.37% (R1), for KP458 at J = 0.4 range from 2.59% (R6) to 7.46% (R1) and for WB at J = 0.48 range from 3.77% (R6) to 11.19% (R1). Fouling penalties on the propeller performance in open water conditions can be ascribed to fouling impact on the skin friction and the pressure field. Thus, due to the presence of hard fouling on propeller surfaces wall shear stress (τ w ) increases, while the pressure difference between pressure and suction sides of propeller is reduced, which can be seen from Figures 11 and 12. In Figure 11, the obtained τ w distributions at KP505 surface at J = 0.7 for both smooth and R1 surface condition are shown. It is clear that due to the presence of hard fouling τ w values at KP505 surface are significantly increased resulting in increase in drag coefficient of the blade section and consequently in K QO . In Figure 12 the obtained pressure distribution shown as distribution of pressure coefficient (C P ), which is defined as a ratio between pressure and 1 2 ρv 2 A , at KP505 surface is presented. Since the magnitudes of C P at both pressure and suction sides of fouled KP505 are significantly reduced, the pressure difference between pressure and suction sides is reduced as well, resulting in a decrease in the lift coefficient of the blade section and, consequently, in K TO .

The Impact of Hard Fouling on Propulsion Characteristics
After CFD simulations of resistance and open water tests are carried out, CFD simulations of SPT for smooth and fouled ships are performed. As said before, the fouling penalty on the ship performance should be considered through the change in D P and n . The obtained increases in D P and n due to the presence of hard fouling are presented in Figure 13. η . This highlights the importance of the assessment of the impact of biofouling on D P rather than on E P . The increase in D P due to the presence of biofouling is dependent on many parameters. Thus, besides the portion of V R in T R , / k L and ship speed, which are important for the increase in E P , it is also important at which J propeller operates and the way the propeller loading defined with 2 / T K J is affected due to the presence of hard fouling. Namely, due to change in propeller loading, J value at which propeller operates changes as well. Thus, the change in J at which propeller operates as well as the absolute value of J is important, as, for ships which operate at higher J values, the fouling penalty on the propeller performance is higher. Figure 12. The obtained C P distribution on KP505 surface for suction (upper) and pressure (lower) side of propeller.

The Impact of Hard Fouling on Propulsion Characteristics
After CFD simulations of resistance and open water tests are carried out, CFD simulations of SPT for smooth and fouled ships are performed. As said before, the fouling penalty on the ship performance should be considered through the change in P D and n. The obtained increases in P D and n due to the presence of hard fouling are presented in Figure 13. From this figure, it is clear that for surface conditions R1, R2 and R3 KVLCC2 is most affected due to the presence of hard fouling, while for surface conditions R4, R5 and R6 the fouling penalties for KVLCC2 and BC are almost the same and higher than fouling penalties for KCS. The obtained increases in P D due to the presence of hard fouling for KVLCC2 range from 90.7% (R6) to 213.4% (R1), for BC range from 90.6% (R6) to 201.9% (R1) and for KCS range from 75.0% (R6) to 163.2% (R1), while the obtained increases in n for KVLCC2 range from 16.7% (R6) to 32.6% (R1), for BC range from 16.6% (R6) to 30.7% (R1) and for KCS range from 9.4% (R6) to 18.2% (R1). It is clear that the obtained increases in P D are significantly higher than the obtained increases in P E due to the presence of hard fouling, which can be related with the decrease in η D . This highlights the importance of the assessment of the impact of biofouling on P D rather than on P E . The increase in P D due to the presence of biofouling is dependent on many parameters. Thus, besides the portion of R V in R T , k/L and ship speed, which are important for the increase in P E , it is also important at which J propeller operates and the way the propeller loading defined with K T /J 2 is affected due to the presence of hard fouling. Namely, due to change in propeller loading, J value at which propeller operates changes as well. Thus, the change in J at which propeller operates as well as the absolute value of J is important, as, for ships which operate at higher J values, the fouling penalty on the propeller performance is higher. In order to study the differences in the obtained fouling penalties more detailly, the impact of hard fouling on propulsion characteristics should be investigated. Within Tables 14-16, the obtained impact of hard fouling on propulsion characteristics is presented. From the obtained results, it is clear that most of the propulsion characteristics are affected by the presence of hard fouling on the hull and propeller surfaces. However, from Tables 14-16, it is clear that the impact of hard fouling on R η is minimal, i.e., it is lower than 0.45% for all analyzed fouling conditions and ships. What is more, the impact of hard fouling on 1 t − is present, however, it is relatively low. Thus, due to the presence of hard fouling, the 1 t − value for KCS and KVLCC2 decreases, while for BC, it increases. It should be noted that the 1 t − value depends on many different parameters, i.e., on the fouling penalty related to increase in T R , to propeller performance, as well as hull and propeller interaction. Obviously, the assessment of the effect of biofouling on 1 t − value is very complex. It should be noted that the obtained impact of hard fouling on 1 t − is within the obtained numerical uncertainty in the assessment of T R and T . Additionally, within the assessment of 1 t − , a modelling error is present as well, and it is related to turbulence modelling, modelling of the effect of ship propeller with body force method etc. Consequently, in order to assess this impact more accurately, numerical uncertainty as well as modelling error should be reduced through the application of more dense grids and lower time steps, as well as through the discretization of the propeller itself. Thus, a more accurate prediction of the impact of biofouling on 1 t − would be assessed. Therefore, based on the obtained results, it can be concluded that the impact of hard fouling on 1 t − is present, however, it is minimal. On the other hand, the impact of hard fouling on 1 w − is significant and detrimental, since it causes a decrease in the 1 w − value. Due to the presence of hard fouling, the obtained decreases in 1 w − values range from −6.99% (R6) to −11.7% (R1) for KCS, from −6.29% (R6) to −10.1% (R1) for KVLCC2 and from −8.46% (R6) to −12.0% (R1) for BC. The decrease in 1 w − can be attributed to slower flow around the propeller location for fouled ship, due to thicker boundary layer. The decrease in 1 w − has beneficial effect on H η (Equation (11)). Thus, due to the presence of hard fouling the obtained  In order to study the differences in the obtained fouling penalties more detailly, the impact of hard fouling on propulsion characteristics should be investigated. Within Tables 14-16, the obtained impact of hard fouling on propulsion characteristics is presented. From the obtained results, it is clear that most of the propulsion characteristics are affected by the presence of hard fouling on the hull and propeller surfaces. However, from Tables 14-16, it is clear that the impact of hard fouling on η R is minimal, i.e., it is lower than 0.45% for all analyzed fouling conditions and ships. What is more, the impact of hard fouling on 1 − t is present, however, it is relatively low. Thus, due to the presence of hard fouling, the 1 − t value for KCS and KVLCC2 decreases, while for BC, it increases. It should be noted that the 1 − t value depends on many different parameters, i.e., on the fouling penalty related to increase in R T , to propeller performance, as well as hull and propeller interaction. Obviously, the assessment of the effect of biofouling on 1 − t value is very complex. It should be noted that the obtained impact of hard fouling on 1 − t is within the obtained numerical uncertainty in the assessment of R T and T. Additionally, within the assessment of 1 − t, a modelling error is present as well, and it is related to turbulence modelling, modelling of the effect of ship propeller with body force method etc. Consequently, in order to assess this impact more accurately, numerical uncertainty as well as modelling error should be reduced through the application of more dense grids and lower time steps, as well as through the discretization of the propeller itself. Thus, a more accurate prediction of the impact of biofouling on 1 − t would be assessed. Therefore, based on the obtained results, it can be concluded that the impact of hard fouling on 1 − t is present, however, it is minimal. On the other hand, the impact of hard fouling on 1 − w is significant and detrimental, since it causes a decrease in the 1 − w value. Due to the presence of hard fouling, the obtained decreases in 1 − w values range from −6.99% (R6) to −11.7% (R1) for KCS, from −6.29% (R6) to −10.1% (R1) for KVLCC2 and from −8.46% (R6) to −12.0% (R1) for BC. The decrease in 1 − w can be attributed to slower flow around the propeller location for fouled ship, due to thicker boundary layer. The decrease in 1 − w has beneficial effect on η H (Equation (11)). Thus, due to the presence of hard fouling the obtained ∆η H values range from 6.13% (R6) to 11.3% (R1) for KCS, from 6.11% (R6) to 10.2% (R1) for KVLCC2 and from −11.3% (R6) to 16.9% (R1) for BC. Regardless of the fact that the decrease in 1 − w has beneficial effect on η H , in general, the decrease in 1 − w has detrimental effect on η D and P D . Namely, the decrease in 1 − w points out that the flow around propeller is slower and consequently propeller operating point is changed when compared with the smooth hull surface. Additionally, due to the presence of hard fouling, the nominal wake field behind the fouled ship is more inhomogeneous than nominal wake field behind the smooth ship, and because of this, the operating point is changed as well. Therefore, J for self-propulsion point decreases since v A is lower. What is more, J for self-propulsion point decreases because of the increase in n as well. Due to the presence of hard fouling the obtained ∆J values for self-propulsion point range from −15.0% (R6) to −25.3% (R1) for KCS, from −19.7% (R6) to −32.2% (R1) for KVLCC2 and from −21.5% (R6) to −32.6% (R1) for BC. The decrease in the J value is unfavorable, as KP 505, KP 458 and WB operate at J lower than J, for which the η O function has a maximum value, which is common for all marine propellers. Finally, from Tables 14-16, it is clear that the presence of hard fouling on the hull and propeller surfaces causes a significant decrease in η D , since decreases in η B are higher than increases in η H . The obtained decreases in η D values due to the presence of hard fouling range from −14.4% (R6) to −25.6% (R1) for KCS, from −16.1% (R6) to −31.0% (R1) for KVLCC2 and from −16.3% (R6) to −28.9% (R1) for BC. Since the impact of biofouling on η D value is not negligible, the increases in P E and P D are not the same, and it is therefore necessary to investigate the impact of biofouling on P D rather than on P E . It should be noted that the results presented in this subsection are obtained for the presence of biofouling on both propeller and hull surfaces. For clean propeller surfaces and fouled ship hull the obtained results, i.e., trends may not be the same. Thus, Song et al. [22], have obtained slight increases in η D values due to the presence of barnacles at hull surfaces, i.e., with a clean propeller. This can be attributed to the fact that the authors have obtained higher increases in η H due to the presence of barnacles than decreases in η B due to change in operating point. As a result of all this, the analysis of the impact of biofouling on propulsion characteristics is very important, i.e., the assessment of biofouling on the resistance characteristics and P E is not sufficient. From the results presented in Tables 14-16, it can be concluded that the impact of hard fouling on the propulsion characteristics is the most pronounced for BC. Namely, the obtained changes in 1 − t, 1 − w, J, η H , η O and η B due to the presence of hard fouling are largest for BC. What is more, the obtained changes in η D due to the presence of hard fouling for fouling conditions R4, R5 and R6 are the largest for BC as well. However, for fouling conditions R1, R2 and R3 the obtained decreases in η D are larger for KVLCC2 than for BC. For these fouling conditions, larger increase in η H which is obtained for BC has surpassed the larger decrease in η B , which has also been obtained for BC and because of this the obtained decreases in η D are larger for KVLCC2. The largest changes in ∆K T and ∆K Q are obtained for KCS and this can be attributed to the fact that KCS operates at a higher J value than KVLCC2 and BC. The largest decrease in the ratio between K T and K Q has been noticed, due to the presence of hard fouling for KCS as well. Nevertheless, amongst the investigated ships, the decrease in η O is the lowest, which can be attributed through the lowest obtained decrease in J for KCS. Namely, J for self-propulsion point decreases due to the increases in n and 1 − w. As can be seen from Figure 13, the obtained increases in n due to the presence of hard fouling are significantly lower for KCS than for KVLCC2 and BC, while increases in 1 − w due to the presence of hard fouling are relatively similar for all analyzed ships, Tables 14-16.

The Impact of Hard Fouling on the Flow Around Fouled Ship
The impact of hard fouling on the ship performance is investigated for three ships at their design speeds presented in Table 2. This resulted in different τ w distributions for smooth surface condition, Figure 14. From this figure it is clear that the highest τ w values are obtained for KCS, followed by BC and KVLCC2, which was expected as KCS is investigated at the highest design speed. As a result of this, the highest k + values are also obtained along the KCS hull, which can be seen from Figure 15. The obtained k + distributions for R1 fouling condition along the KCS, KVLCC2 and BC hull are shown. Since the highest k + values are obtained along the KCS hull, the highest ∆U + values are present as well, which resulted in more significant increase in τ w and C F for KCS than for BC and KVLCC2. The obtained τ w distributions for R1 fouling condition along the KCS, KVLCC2 and BC hull are presented in Figure 16.          The increase in τ w along the hull causes a decrease in the velocity in the turbulent boundary layer, i.e., turbulent boundary layer thickness increases due to the presence of roughness, which can be seen from Figure 17 As the boundary layer thickness increases it is obvious that the presence of hard fouling will cause the change in the nominal wake distribution. In Figure 18, the obtained contours of 1 N w − for smooth and fouled ships (R1) in the propeller disc plane are shown. It should be noted that 1 N w − is calculated as the ratio between axial velocity and ship speed [45]. From this figure, it is clear that the presence of hard fouling causes the significant reduction of the flow in the propeller disc plane for all three investigated ships. This reduction causes the change of J for self-propulsion point and in that way, it affects propeller efficiency, as already explained. As the boundary layer thickness increases it is obvious that the presence of hard fouling will cause the change in the nominal wake distribution. In Figure 18, the obtained contours of 1 − w N for smooth and fouled ships (R1) in the propeller disc plane are shown. It should be noted that 1 − w N is calculated as the ratio between axial velocity and ship speed [45]. From this figure, it is clear that the presence of hard fouling causes the significant reduction of the flow in the propeller disc plane for all three investigated ships. This reduction causes the change of J for self-propulsion point and in that way, it affects propeller efficiency, as already explained. Figure 17. The obtained boundary layers for smooth ships (upper) and fouled ships with fouling condition R1 (lower).
As the boundary layer thickness increases it is obvious that the presence of hard fouling will cause the change in the nominal wake distribution. In Figure 18, the obtained contours of 1 N w − for smooth and fouled ships (R1) in the propeller disc plane are shown. It should be noted that 1 N w − is calculated as the ratio between axial velocity and ship speed [45]. From this figure, it is clear that the presence of hard fouling causes the significant reduction of the flow in the propeller disc plane for all three investigated ships. This reduction causes the change of J for self-propulsion point and in that way, it affects propeller efficiency, as already explained. In addition to the impact of hard fouling on w τ values, the presence of hard fouling causes the change in pressure distribution along the hull. However, this change mainly occurs in the area near the stern of fouled ship [16]. In Figure 19, the obtained P C distributions are presented for the area near the stern of investigated ships for smooth and R1 fouling conditions within CFD simulations of SPT. It should be noted that P C is obtained as a ratio between pressure and it is clear that due to the presence of hard fouling pressure recovery at the stern is reduced and because of this VP R increases. Additionally, the impact of hull and propeller fouling on P C distribution at the rudder can be noticed, i.e., P C values at the rudder surface are slightly reduced. In addition to the impact of hard fouling on τ w values, the presence of hard fouling causes the change in pressure distribution along the hull. However, this change mainly occurs in the area near the stern of fouled ship [16]. In Figure 19, the obtained C P distributions are presented for the area near the stern of investigated ships for smooth and R1 fouling conditions within CFD simulations of SPT. It should be noted that C P is obtained as a ratio between pressure and 1 2 ρv 2 . From this figure, it is clear that due to the presence of hard fouling pressure recovery at the stern is reduced and because of this R VP increases. Additionally, the impact of hull and propeller fouling on C P distribution at the rudder can be noticed, i.e., C P values at the rudder surface are slightly reduced. In Figure 20, the obtained wave patterns around the hulls of the investigated ships for smooth surface condition and R1 fouling condition from CFD simulations of resistance tests are presented. From the comparison between wave pattern for smooth KCS and BC and wave pattern for KCS and BC fouled with R1, it can be noticed that due the presence of hard fouling wave elevations are reduced. On the other hand, wave elevations for KVLCC2 are almost the same for smooth and R1 fouling condition. The similar finding is noticed within [16,20]. Reductions of wave elevations and consequently W R , due to the presence of hard fouling can be related to the increase in viscosity [15].
It can be concluded that the impact of hard fouling on the wave elevations is in agreement with the obtained decreases in W R , i.e., for KCS and BC this impact is relevant, while for KVLCC2 this impact is negligible. Figure 19. The impact of hard fouling on C P distribution for the area near the stern.
In Figure 20, the obtained wave patterns around the hulls of the investigated ships for smooth surface condition and R1 fouling condition from CFD simulations of resistance tests are presented. From the comparison between wave pattern for smooth KCS and BC and wave pattern for KCS and BC fouled with R1, it can be noticed that due the presence of hard fouling wave elevations are reduced. On the other hand, wave elevations for KVLCC2 are almost the same for smooth and R1 fouling condition. The similar finding is noticed within [16,20]. Reductions of wave elevations and consequently R W , due to the presence of hard fouling can be related to the increase in viscosity [15].
It can be concluded that the impact of hard fouling on the wave elevations is in agreement with the obtained decreases in R W , i.e., for KCS and BC this impact is relevant, while for KVLCC2 this impact is negligible.
In Figure 20, the obtained wave patterns around the hulls of the investigated ships for smooth surface condition and R1 fouling condition from CFD simulations of resistance tests are presented. From the comparison between wave pattern for smooth KCS and BC and wave pattern for KCS and BC fouled with R1, it can be noticed that due the presence of hard fouling wave elevations are reduced. On the other hand, wave elevations for KVLCC2 are almost the same for smooth and R1 fouling condition. The similar finding is noticed within [16,20]. Reductions of wave elevations and consequently W R , due to the presence of hard fouling can be related to the increase in viscosity [15].
It can be concluded that the impact of hard fouling on the wave elevations is in agreement with the obtained decreases in W R , i.e., for KCS and BC this impact is relevant, while for KVLCC2 this impact is negligible.

Conclusions
In this paper, the impact of hard fouling on ship performance for three different ship types is investigated. This impact is investigated using the CFD simulations of resistance, open water and self-propulsion tests. The impact of hard fouling is represented through the modification of wall function, i.e., through the implementation of the Grigson U + Δ model in the wall function within CFD solver. The verification study for grid size and time step is carried out, and grid and temporal uncertainties are estimated using GCI method. The verification study is performed for several key variables, i.e., TO

Conclusions
In this paper, the impact of hard fouling on ship performance for three different ship types is investigated. This impact is investigated using the CFD simulations of resistance, open water and self-propulsion tests. The impact of hard fouling is represented through the modification of wall function, i.e., through the implementation of the Grigson ∆U + model in the wall function within CFD solver. The verification study for grid size and time step is carried out, and grid and temporal uncertainties are estimated using GCI method. The verification study is performed for several key variables, i.e., K TO and K QO for open water test and for P D , n, T and J for self-propulsion test. Relatively low simulation uncertainties are obtained for all key variables. Thereafter, the obtained results of the performed CFD simulations for smooth surface condition are validated with the extrapolated towing tank results using the ITTC 1978 Performance Prediction Method. Satisfactory agreement is achieved for all resistance, open water and propulsion characteristics. After the verification and validation study, the impact of hard fouling on the ship performance is studied in terms of the impact on resistance, open water and propulsion characteristics. The obtained results demonstrated the significant impact of hard fouling on the increase in frictional resistance and viscous resistance, as well for all three ships. It should be noted that the viscous resistance of KCS is mostly affected due to the presence of hard fouling, which is ascribed to the fact that KCS is investigated at the highest speed. As a result of this, friction velocity along the KCS hull is higher than along the KVLCC2 and BC hulls. Higher ∆U + values are obtained along the KCS hull in comparison with KVLCC2 and BC hulls, since the roughness Reynolds number and therefore ∆U + are dependent on the friction velocity. However, wave resistance has decreased for KCS and BC, due to the presence of hard fouling, while for KVLCC2, it is almost negligible, and has remained almost the same as for smooth surface condition. The impact of hard fouling on the wave resistance is in agreement with the impact of hard fouling on the wave elevations, i.e., wave elevations for KCS and BC due to the presence of hard fouling are decreased, while for KVLCC2, it remained the same as for the smooth surface condition. Therefore, the most affected ship due to the presence of hard fouling, related to the fouling penalty on the ship resistance, is KVLCC2. Obviously, beside the ship speed the portion of viscous resistance in total resistance is very important for the estimation of the fouling penalty on the ship resistance, as well as the ratio k/L. Significant detrimental effects due to the presence of hard fouling on the propeller performance in open water conditions are found. Thus, due to the presence of hard fouling on the propeller surfaces, K TO decreases and K QO increases, which results in a significant decrease in η O . Namely, due to the presence of hard fouling drag coefficient of propeller blade section increases, causing the increase in K QO , and lift coefficient decreases, causing the decrease in K TO . The impact of hard fouling on the ship performance is best reflected through the impact on the delivered power and propeller rotation rate. From the obtained results, it is clear that increases in the delivered power are significantly larger than increases in the effective power, due to the presence of hard fouling for all three investigated ships. Therefore, the impact of hard fouling on propulsion efficiency must not be neglected, especially for fouled ship and propeller. The impact of hard fouling on the delivered power and propeller rotation rate is most pronounced for KVLCC2 for fouling conditions R1, R2 and R3, while for R4, R5 and R6, the obtained changes in the ship performance due to hard fouling are similar for BC and KVLCC2. This can be attributed to different impact of hard fouling on propulsion characteristics, as the fouling penalty on effective power for R4, R5 and R6 fouling conditions is higher for KVLCC2 than for BC. Namely, the additional important parameter that affects the impact of biofouling on the ship performance is the value of advance coefficient for self-propulsion point, since it is demonstrated that ships which operate at higher values of advance coefficient will be more affected in terms of propeller performance in open water conditions than ships which operate at lower values of advance coefficient. The impact of hard fouling on propulsion characteristics is presented for all three ships. From the obtained results it can be seen that propulsion characteristics of BC are mostly affected due to the presence of hard fouling, as the obtained changes in 1 − t, 1 − w, J, η H , η O and η B due to the presence of hard fouling are largest for BC. Additionally, the obtained changes in η D due to the presence of hard fouling for fouling conditions R4, R5 and R6 are the largest for BC as well, while, for R1, R2 and R3, they are the largest for KVLCC2. The largest change in K T and K Q values due to the presence of hard fouling are obtained for KCS, which is expected as KCS operate with the highest advance coefficient. Finally, the impact of hard fouling on the flow around fouled ship is studied through the analysis of the impact on wall shear stress distribution, boundary layer thickness, nominal wake distributions, wave elevations and pressure distributions.
The paper provided several valuable insights related to the impact of hard fouling on the ship performance amongst different ship forms. Future study will be focused on investigations related to the impact of biofouling for systematic series of certain ship at different speeds, which will allow more comprehensive insight into the impact of biofouling on the ship performance will be assessed. In this paper, the investigations related to the impact of hard fouling on the ship performance are performed for the presence of hard fouling on both propeller and hull surfaces. If analyzed per unit area, the impact of propeller fouling condition on the ship performance is significantly more important than the impact of hull fouling condition. Therefore, the future studies will be also focused on the investigations related to the impact of solely propeller cleaning on the ship performance. Thus, relatively cheap and effective practice for achieving significant energy saving will be demonstrated. The optimization of maintenance schedule is an important operational measure for reducing ship emissions and the successful application of this measure relies on the accurate assessment of the impact of cleaning, i.e., the impact of biofouling on the ship performance. Currently, these predictions are carried out using performance monitoring. However, performance monitoring has several important drawbacks [8], and the approach presented in this paper presents another way for this assessment. The important benefit of the proposed approach over the performance monitoring is that fouling effects on the ship performance can be analyzed independently of all other additional resistances, which may occur during sailing. However, since drag characterization studies are performed only for limited number of fouling conditions, CFD approach based on the modified wall function approach is limited to these fouling conditions. For more comprehensive assessment there is a need for further drag characterization studies. Additionally, the investigations performed in this paper are carried out for hull surface, which is treated as a uniformly rough surface with certain roughness length scale determined using Equation (6), as done in most of the conventional CFD studies dealing with biofouling. Since the fouling pattern along the immersed surface is not uniform, future studies will be focused on the investigations of the influence of fouling settlement on the ship performance. The locations of niche areas along the hull surface will be found from the literature and in that areas, wall function model for certain fouling condition will be implemented within the wall function of CFD solver. In that way, more realistic fouling conditions will be analyzed, and the investigations regarding the partial cleaning of the ship hull will be performed as well. Based on that, the proposed method can be used for the assessment of fouling penalties on the ship performance, after the fouling condition of the hull and propeller are determined by divers in the port.   smooth wall log-law intercept (-) C B block coefficient (-) C F frictional resistance coefficient (-) C P pressure coefficient (-) C T total resistance coefficient (-) C W wave resistance coefficient (-) c chord length at radius 0.75R (m) D propeller diameter (m) d shaft diameter (m) Fn Froude number (-) J advance coefficient (-) k roughness length scale (µm) k form factor (-) k + roughness Reynolds number (-) K T thrust coefficient (-) K Q torque coefficient (-) K TO thrust coefficient in open water conditions (-)