CFD ANALYSIS OF THE EFFECT OF HETEROGENEOUS HULL ROUGHNESS ON SHIP RESISTANCE

Hull roughness increases ship resistance, power, and fuel consumption significantly. Recent studies have demonstrated that Computational Fluid Dynamics (CFD) can accurately predict the effect of roughness on ship resistance by using a modified wall-function approach. Although hull roughness is often spatially heterogeneous, little research has been carried out on the heterogeneous roughness effect on ship resistance. Therefore, this study aims to investigate the heterogeneous roughness effect on ship resistance using CFD. A series of CFD simulations were conducted on a scaled model of the KRISO Container Ship (KSC) hull. Various surface coverage conditions were considered, including homogeneous (i.e., smooth, and full-rough conditions) and heterogeneous conditions (i.e., different smooth/rough wetted surface ratios). The present findings showed that increased roughness on the fore hull regions has a greater impact on ship resistance than the rough aft-hull regions. The introduction of a so-called roughness impact factor correlated the added resistance of the heterogeneous roughness scenarios to the corresponding rough wetted surface area. Accordingly, the rough fore-hull scenarios presented higher roughness impact factors than the rough aft-hull cases.


Introduction
It is well established that hull roughness is foremost responsible for the decay of ship performance over time.Biofouling accumulation, coatings failure and corrosion are the leading causes of the increase of roughness on the hull surfaces [1], [2].Naval architects, shipowners, and operators are fully aware of the economic and environmental penalties linked to a poorly maintained hull.Nevertheless, our understanding of the hull roughness effect on ship resistance is still limited.Dry-dock and fouling control coating (FCC) strategies to mitigate the roughness effect on ship resistance are arguably incomplete.The approach of ignoring the vessel's underwater hull conditions for long dry-dock intervals leads to considerable losses on a fleet's economy.
Theoretical and numerical methods based on the turbulent boundary layer theory can accurately predict the roughness effect on ship resistance, provided that the roughness function of the surface is known [3].Researchers, e.g., [4]- [10], have widely adopted Granville's similarity law scaling procedure, [11], [12], due to its robustness and cost-effectiveness [13].However, Granville's theoretical method is limited by a number of simplifications.In fact, the 2D flat plate assumption neglects the 3D effect [14], and the assumption of a constant roughness function along the flat plate may lead to scaling problems and inaccurate added resistance predictions [15].
An alternative to Granville's method is Computational Fluid Dynamics (CFD), which is cost-efficient compared to experimental approaches and can overcome related shortcomings [14].The CFD approach avoids the nonlinear problems of theoretical studies and dynamically computes the roughness function for each discretised cell [16]- [18].Furthermore, the ship resistance predictions are more accurate in CFD since the 3D effect of the hull are considered, and the ship can be modelled in full-scale.Accordingly, several studies recently adopted the CFD approach for investigating the roughness effect on ship resistance, [19]- [21].Nonetheless, despite the hull roughness distribution of in-service ships is inherently heterogeneous, most published studies have treated the hull roughness as homogeneous.
Recently, Song et al., [22], [23], investigated the heterogeneous hull roughness effect on ship hydrodynamics by means of experimental fluid dynamics (EFD) and CFD.In our previous study, [22], the Wigley hull was modelled with different hull roughness conditions by applying sand-grit on the hull surface in different configurations (i.e., smooth and full-rough, ¼-bow-rough, ¼-aft-rough, ½-bow-rough and ½-aft-rough).In [23], Song et al. performed CFD simulations using the modified wall-function approach and the roughness function model of [18].In [18] the CFD model was validated to predict the effect of roughness on ship resistance for 3D hulls.The findings showed good agreement between the CFD modified wall-function approach, Granville's similarity law and measurements from towing tests of a ship model test with a rough surface [24].These studies showed that the increased hull roughness in the forward wetted area of the hull causes further added resistance than the hull roughness in other parts.The findings suggest the possibility of prioritising a partial hull cleaning depending on the impact of the roughness in different hull regions.
Vargas et al. [25] conducted CFD simulations on a surface combatant exposed to different roughness scenarios.The effects of heterogeneous distribution of roughness and the benefits of partial hull cleaning were investigated dividing the hull into sections.The findings indicated that the increase in the skin friction resulting from localised roughness is highest at the bow, followed by sides, flat bottom, stern, and transom.Östman et al. [26] carried out a CFD analysis investigating the potential of a selective application of different quality coatings on a full-scale tanker.Only the regions where high skin friction is concentrated were modelled with high-quality coating (low roughness), while the rest of the hull was modelled with a low-quality coating.The results confirmed the expectations: this selective approach can reduce the ship resistance compared to when the inferior coating is applied on the entire hull.
The promising findings of the few studies addressing the effect of heterogeneous distribution of hull roughness are limited to the ship types considered (the Wigley hull, a tanker, and a surface combatant).Despite these recent studies, our understanding on the effect of heterogeneous hull roughness is still limited.The different impacts of hull roughness according to the different hull regions need to be investigated for better comprehension.The present study aims to fill this research gap by investigating the effect of heterogeneous roughness on the hydrodynamic resistance of the well-known Kriso Container Ship (KCS) hull using CFD, and supporting a targeted strategy for hull maintenance.Furthermore, the CFD simulations provide complete and accurate data to assess how the heterogeneous hull roughness affects the flow regime around the hull.
The present study is a CFD investigation conducted on the benchmark KCS hull in heterogeneous coverage conditions, namely: Bulbous Bow, Fore Hull, Midship, Flat Bottom, Aft Hull, and Stern.The paper is structured as follows: the methodology is explained in Section 2, including the approach and numerical modelling.Details of the modified wall-function approach, mathematical formulations, geometry and boundary conditions, and mesh generations are given in this section.In Section 3 are presented the results of the current CFD investigation.The effects of the heterogeneous roughness on the hydrodynamics of the ship were assessed.The various hull roughness conditions were correlated with the predicted resistance coefficients, roughness Reynolds number values, and boundary layer distributions and the findings compared with the homogeneous full rough and full smooth cases.The conclusions in Section 4 summarise and discuss the findings and suggest recommendations for future studies.

Approach
Figure 1 shows a schematic illustration of the CFD methodology adopted to investigate the effect of heterogeneous distributions of hull roughness on the well-known KRISO Container Ship (KCS), [27].The model-scale numerical towing tests investigated the effect of various heterogeneous hull roughness conditions (different smooth/rough wetted surface coverage ratios) and effects on ship resistance.These roughness scenarios were designed with the aim of investigating the potential of low-cost targeted hull maintenance.
The simulations were developed in the StarCCM+ software package (Version 15.06.007-R8), adopting the Unsteady Reynolds Averaged Navier-Stokes (URANS)-based CFD with the modified wall-function model recently validated by Song et al. [18].
Finally, the skin friction coefficients, !!, the roughness Reynolds number, " " , and the boundary layer on the hull surfaces were examined and correlated with the findings.The report figures for these parameters will support a better comprehension of the impact of the increased roughness of different hull conditions on ship resistance.

Mathematical Formulations
The governing equations of this hydrodynamics study are given in tensor notation and Cartesian coordinates by equation ( 1) and ( 2), [28]: where, % is the density, & ' # is the averaged velocity vector, '''''' is the Reynolds stress, 1̅ is the averaged pressure, 3̅ #$ is the mean viscous stress tensor components.Newtonian fluid's viscous stress can be expressed as in equation ( 3): where, 4 is the dynamic viscosity.Using the Boussinesq hypothesis, the Reynolds stress can be written as in equation ( 4): where, 4 ( is the turbulent eddy viscosity, " is turbulent kinetic energy, and ; #$ is the Kronecker delta. The above averaged continuity and momentum equations for incompressible flows were solved by the commercial CFD software STAR-CCM+ using URANS and finite volume methods.The CFD solver used a second order upwind convection scheme and a first order temporal discretisation for the momentum equations.On the other hand, the continuity equations were solved in a segregated manner and linked to the momentum equations with a predictorcorrector algorithm.The "-< SST (Shear Stress Transport) turbulence model [29] was used with a second-order convection scheme.This turbulent model combines "-< and "-ε formulations for an accurate near wall treatment of the effects of turbulence, and an overall enhanced prediction in adverse pressure gradients and separating flow.The free surface effects were modelled with the Volume of Fluid (VOF) using High Resolution Interface Capturing (HRIC).

Modified Wall-Function Approach
An increase in surface roughness reduces the boundary layer thickness and causes an increase in flow turbulence.Hence, the turbulent stress, wall shear stress and ultimately the skin friction increase.The roughness effect can also be seen as a downward shift of the non-dimensional velocity profile in the turbulent boundary layer log-law region.
The non-dimensional velocity profile (> " ) in the log-law region for a rough surface can be written as equation ( 5): where @ is the von-Karman constant, C " is the non-dimensional normal distance from the boundary, (C " = C> * /G), D is the smooth wall log-law intercept.The downward shift of the non-dimensional velocity profile (E> " ), also known as "roughness function", is a unique characteristic of a rough surface.In other words, different rough surfaces are characterised by different roughness functions to be modelled experimentally [12].The roughness function model of the surface can then be implemented in the wall-function of the CFD model.The roughness function, E> " is a function of the roughness Reynolds number, " " , which is defined as (6): in which, " is the roughness height of the surface, and > * is the velocity based on wall shear stress defined as (7): where 3 + is the wall shear stress and % is the water density.

Roughness function model
Figure 2. Experimental roughness function of Song et al. [18] and the roughness function model adopted in this study [24].Roughness Function Model, [26] Proceedings Figure 2 and equation (8) show the roughness function model employed in this study.Song et al. [18] developed this model from the towing tests of a flat plate covered with sand (aluminium oxide, 60/80 grit) [24].One may notice that this is a very rough case (roughness height over a 50 mm interval was " = J, ,-= 353 µm).

Geometry and Boundary Conditions
The CFD simulations were carried out on the well-known container ship KCS modelled on the scale factor of 75, as used for [18].Table 1 presents the particulars of the full-scale and model KCS adapted from Kim et al. [30] and [31].3 depict the characteristics of the different hull roughness scenarios of the CFD simulations.It is of note that the total wetted surface area without rudder, b, of the KCS model used by Song et al. [18], and the actual one used in the present simulations (b FG(HI = 1.7415 ] 3 ) differs by (3.8%).In other words, the actual b FG(HI was used instead of the reference value, b, for consistency with the exact rough wetted surface areas of the heterogeneous scenarios read from the simulations.The effect of the difference between actual and reference values on the results has been considered negligible.Accordingly, the modified-wall function approach previously demonstrated by Song et al. [18] was assumed valid for this study.Figure 4 shows the computational domain, a towing tank with size chosen in accordance with ITTC recommendations [32] and similar studies [18], [22], [23].A pressure outlet was chosen for the outlet boundary condition while a velocity inlet was applied for all the other surfaces of the domain (inlet, side walls, bottom and top).These boundary conditions simulated the deep water and infinite air conditions.Bottom, top and side walls of the tank were selected as slip-walls whilst for free-surface modelling, the no-slip wall type boundary condition was used on the hull surfaces.A symmetry boundary condition was applied on the vertical centre plane of the domain to shorten the computational time.No constraints were given to the model ship which was free to sink and trim.

Mesh Generation
Figure 5 shows the volume mesh of this CFD analysis.The built-in automated mesher of Star-CCM+ software was used to generate the trimmed hexahedral-dominant finite element mesh.Further near-wall mesh refinements were made applying mesh refinement on the critical regions such as the free surface, the bulbous bow, the stern, and the rough boundaries regions.For these simulations, the wall C " values were kept between 30 and 300 and higher than " " values, as recommended by Siemens [33].All the simulations used the same mesh regardless of the hull roughness scenarios.

Effect of Heterogeneous Roughness on Ship Resistance
Heterogeneous hull roughness affects the ship performances in a complicated, non-proportional way.Therefore, in this section the results of the towing tests conducted in CFD on the KCS model in heterogeneous hull roughness conditions are discussed and compared with the homogeneous fully smooth and fully rough cases.As reported above in Table 1, the simulations were carried out at a towing speed of 1.425 ]/g, Froude number he = 0.26, and Reynolds number Jn A = 3.74 × 10 E , which correspond to the full-scale design speed of KCS (24 knots) and Jn A = 2.43 × 10 D .According to Froude's method, the total ship resistance coefficient, !F , can be decomposed into the frictional resistance coefficient, !!, and the residuary resistance coefficient, !J , given by equation ( 9): where the resistance coefficients !, are a function of the corresponding drag, J, the dynamic pressure, 1/2 %d 3 , and the total wetted surface area of the ship hull, b.The total resistance coefficient, !F , is defined as in equation (10): in which, % is the water density and d is the towing speed (i.e., the inlet velocity).
In Figure 6 and Table 3 are compared the resistance coefficients, total (! F ), frictional (! ! ), residuary (! J ), and added (∆! ! ), of the KCS hull in heterogeneous roughness configurations.The added resistance coefficient, (or roughness allowance, ∆! ! ), is the variation in the total resistance coefficient between the rough and smooth conditions.Different ∆! !values were found across the fore-rough conditions (Bulbous Bow, Fore Hull), the midship-rough conditions (Midship, Flat Bottom), and the aft-rough conditions (Aft Hull, Stern) due to the different local increased hull roughness and hence locally increased skin friction of the hull.It would be expected that larger rough/smooth wetted surface area ratios would correspond to more significant resistance coefficients.
On the other hand, the present CFD simulations discredited this proportional assumption.For example, the Bulbous Bow case accounts for a smaller rough/smooth wetted surface area ratio than the Stern configuration, but its corresponding added resistance is greater.In fact, the Bulbous Bow rough/smooth wetted surface area ratio is 2.57% and the added resistance coefficient is 9.90 • 10 N, while for the Stern case these parameters are 8.14% and 7.83 • 10 N, , respectively.In other words, the results of this study showed that the position of the selected area of the hull with increased surface roughness affects its impact on the ship hydrodynamics.    4 highlight this non-proportional impact of increased roughness of the heterogeneous configurations.For the purpose of weighting the effect of increased heterogeneous hull roughness on added resistance a so-called "roughness impact factor", h JO , has been defined as in equation ( 11): (11) In which (∆! ! ) # and (b JGKLM ) # are, respectively, the added resistance and the rough wetted surface area of the i-th heterogeneous hull roughness case in exam.(∆! ! ) !KII JGKLM is the added resistance of the full rough hull surface condition, and b FG(HI is the total wetted surface area of the hull.Evidently, h JO is unitary for the Full Rough scenario.For each case, the roughness impact factor, h JO , correlates the added resistance coefficient to the rough wetted surface area. It can be noted that while being the smallest area with increased hull roughness, the Bulbous Bow scenario has the most significant impact on ship resistance (impact factor of 2.49) of all the scenarios.The Fore Hull case has the second-largest impact (impact factor of 1.36).On the other hand, despite being the largest rough wetted surface area tested, the Midship configuration only gives an impact factor of 1.21.Similarly, the Aft Hull, Stern and Flat Bottom configurations showed the lowest impact factors of the heterogeneous scenarios tested.For these configurations, h JO is lower than unity (0.97, 0.62, 0.89, respectively).

Rationale behind the Effect of Heterogeneous Roughness
The effect of hull roughness on ship resistance is closely heterogeneous of the increased increased roughness affects the local skin friction coefficients, the Roughness Reynolds number values (" " ) and the boundary layer characteristics.A discussion and comparison of the local skin friction, the roughness Reynolds number, and the boundary layer of the KCS hull with heterogeneous hull roughness scenarios is presented.
Figure 8 compares the local skin friction coefficients, !Q , on the KCS hull in heterogeneous hull roughness conditions with the homogeneous full smooth and full rough cases (scalar field distribution on the hull limited to !Q = 0.01).The local skin friction coefficients, !Q , were obtained as in equation ( 12): Where 3 + is the wall shear stress, % is the water density and d is the towing speed (i.e. the inlet velocity).The Bulbous Bow and Fore Hull roughness conditions in Figure 8-1 and Figure 8-2 show similar !Q distributions as that of the Full Rough condition.The most significant increases in the local !Q values were observed for the upstream regions of the Midship configuration.In this case, the effect of the heterogeneous increase of hull roughness is dramatic, as shown in Figure 8  Figure 9 shows the distributions of the roughness Reynolds number, " " , on the KCS hull in heterogeneous hull roughness conditions (scalar field distribution on the hull surfaces limited to " " = 45.0).The roughness Reynolds numbers were as in equation (13): Where " is the roughness height, 3 + is the wall shear stress, and G is the kinematic viscosity.As shown in equation ( 8), the fully rough regime is reached when " " value is higher than 25.The distributions of " " on the heterogeneous rough surfaces is similar to the bow regions of the homogeneous Full Rough case.Accordingly, the configurations 1, 2 and 3 show larger " " values than the scenarios 4, 5 and 6 due to the observed roughness effect.
The observation in Figure 8 and Figure 9 on the local skin friction coefficients, !Q , and the roughness Reynolds numbers, " " , are strictly related to the previous findings shown in Table 3 and Table 4.It is well-known that the wall shear stress, 3 + , is more significant in the bow region of ship hulls due to the active transition behaviours, and it decreases as the flow develops along the hull.As the wall shear stress, 3 + , increases, it results in larger local skin friction coefficients and roughness Reynolds numbers in the bow regions.Accordingly, the roughness effect in the bow regions becomes more critical than in the stern regions.
Figure 10 shows the boundary layer on the KCS hull in heterogeneous hull roughness conditions.The boundary layer is represented by portions of transversal planes limited to the axial velocity, d R /d SM#T = 0.9.The roughness increase affects the boundary layer thickness around the hull in different ways, and the present results are in agreement with previous studies [6], [15], [20], [24], [34].For the homogeneous cases, the Full Rough scenario shows a thicker boundary layer than the Full Smooth condition.shows the boundary layer on the KCS hull in heterogeneous hull roughness conditions.The boundary layer is represented by portions of transversal planes limited to the axial velocity, d R /d SM#T = 0.9.The roughness increase affects the boundary layer thickness around the hull in different ways, and the present results are in agreement with previous studies [6], [15], [20], [24], [34].For the homogeneous cases, the Full Rough scenario shows a thicker boundary layer than the Full Smooth condition.Furthermore, the differences between Full Rough and Full Smooth configurations become apparent in the bow regions where the increased roughness causes a thicker boundary layer, peak-shaped on the symmetry plane.Interestingly, the boundary layer of the Bulbous Bow case (Figure 10-1) is similar to that of the Full Rough condition.While, despite a similar thickness of the boundary layer, the Fore Hull case shows a much pointy shape (Figure 10-2).On the other hand, as shown in Figure 10-3,4,5,6, the boundary layer thickness around the hull showed almost no differences compared to that of the Full Smooth case.
It could be assumed that a rough bulbous bow region is responsible for the sharp characteristic of the boundary layer, culminating on the symmetry plane.Perhaps the rationale behind the significant roughness impact factor of the bulbous bow region lies in its extensive influence on the boundary layer characteristics.

Conclusions
An investigation on the effect of heterogeneous hull roughness on ship resistance components and characteristics of the flow around the hull was carried out.URANS-based CFD simulations were carried out on the well-known KCS hull model in different hull roughness conditions.A modified wall-function approach was adopted to implement the roughness characteristics of the surfaces in the CFD model.The different scenarios studied were intended to assess the roughness effect of different parts of the hull on the ship hydrodynamics.The observations on the effects of heterogeneous hull roughness were correlated with the rough wetted surface areas, the distributions of the local skin friction coefficients, the roughness Reynolds number values, and the boundary layer characteristics.Furthermore, comparisons with the homogeneous full rough-smooth cases were presented.
The CFD towing tests showed that the rough regions tested (namely Bulbous Bow, Fore Hull, Midship, Aft Hull, Stern and Flat Bottom) had a different impact on the ship resistance due to their position on the hull.A roughness impact factor, h JO , was defined to predict this impact.The evidence confirmed that the added resistance observed for the fore-rough regions is proportionately greater than for the aft-rough regions.In other words, the roughness conditions of the fore regions proportionally affect the ship hydrodynamics more than the aft regions.The present study supported similar observations of other researchers [6], [15], [20], [24], [34].
Interestingly, the present findings showed that the rough Bulbous Bow condition presents the greater roughness impact factor (h JO = 2.49), despite the smaller percentage of rough wetted surface area.Thus, the rough bulbous bow scenario led to the proportionately greater added resistance than other rough regions of the hull.On the other hand, the rough Stern case present the smaller roughness impact factor (h JO = 0.62), confirming that the surface conditions of the aft regions of the hull have a minor impact on ship resistance.
The numerical investigation presented in this study provides valuable results from a practical point of view.The roughness impact of different hull regions has been investigated, adopting a widely accepted and validated CFD approach.Naval architects, ship owners, and operators could benefit from this study's insight and target limited-time maintenance on the fore-hull regions affecting the ship resistance the most.When complete maintenance on the entire hull is not feasible, it could be worth cleaning the fore hull parts first.
Future studies could focus on comparing the numerical results presented in this study with measurements obtained from tow testing CFD simulations could be conducted to investigate further heterogeneous hull roughness configurations and their effect on ship resistance.

Figure 1 .
Figure 1.Schematic illustration of the methodology adopted.

Figure 3 .
Figure 3. Test scenarios of the KCS model simulations in heterogeneous hull roughness conditions.

Figure 4 .
Figure 4. Computational domain and boundary conditions of the KCS model simulations.

5 Figure 5 .
Figure 5. Volume mesh used for the KCS model simulations.

Figure 6 .
Figure 6.Resistance coefficients for the KCS model simulations in different hull roughness conditions.

Figure 7
Figure7and Table4highlight this non-proportional impact of increased roughness of the heterogeneous configurations.For the purpose of weighting the effect of increased heterogeneous hull roughness on added resistance a so-called "roughness impact factor", h JO , has been defined as in equation (11):

Figure 7 .
Figure 7. Roughness impact factors and wetted surface area ratios of the roughness scenario tested.

- 3 .
On the other hand, the Aft Hull, Stern and Flat Bottom scenarios (Figure 8-4,5,6) are less impactful on the skin friction coefficient distribution.Hence, !Q distributions are more similar to the Full Smooth homogeneous condition than to the Full Rough.

Figure 10 .
Figure 10.Boundary layer on the KCS hull in different hull roughness conditions at test speed (he = 0.26).

Table 2 and
Figure

Table 2 .
Test scenarios of the KCS model simulations in heterogeneous hull roughness conditions.

Table 3 .
Resistance coefficients for the KCS model simulations in different hull roughness conditions.

Table 4 .
Roughness impact factors and wetted surface area ratios of the roughness scenario tested.