**1. 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 propeller cleaning are usually performed in drydock. It should be noted that both of those measures are

costly [6]. Consequently, an accurate assessment of the impact of biofouling on the ship performance is required for the proper selection of AF coatings and scheduling of hull cleaning [7].

There are different approaches for the assessment of this impact which can be classified into statistical studies, performance monitoring and approaches, based on the wall similarity hypothesis [8]. Approach based on the wall similarity hypothesis allows estimation of the fouling effect if the drag characterization of certain fouling type is performed. Drag characterization of a rough surface implies assessing the velocity decrement caused by the frictional drag of the surface as a function of the roughness Reynolds number (*k*+). This velocity decrement, i.e., downward shift of the mean velocity in the log-law region of turbulent boundary layer (TBL) is called the roughness function (Δ*U*+). There is no universal roughness function, however, once Δ*U*<sup>+</sup> for a certain fouling type is assessed, it can be used for the determination of frictional drag of any arbitrary body covered with that fouling type [9]. Over the last few decades, Granville similarity law scaling method has been imposed for the assessment of the impact of biofouling on the ship resistance with Δ*U*<sup>+</sup> = *f*(*k*+) known and it has been widely used in the literature [10–14]. Nevertheless, this method has several important drawbacks, as claimed by [15]. Namely, this method can be used for the prediction of the frictional resistance coefficient of the fouled flat plate having the same length as an investigated ship, and other resistance components of fouled ship are considered to be the same as for smooth ship. What is more, this method assumes only one *k*<sup>+</sup> value and thus one Δ*U*<sup>+</sup> value over the entire flat plate. Since the *k*<sup>+</sup> value depends on friction velocity (*u*τ), this assumption may lead to certain errors, as, even on a flat plate *u*τ, it is not constant over the entire plate. Lastly, using Granville similarity law scaling method only increase in effective power can be estimated. As shown in [16], due to the presence of biofilm the increase in the delivered power is significantly higher than the increase in effective power.

Recently, there have been an increasing number of studies using a computational fluid dynamics (CFD) approach based on the implementation of certain Δ*U*<sup>+</sup> model within the wall function [17–20]. This approach can calculate *u*τ for each discretized cell and, in that way, can obtain the distribution of *u*<sup>τ</sup> values along the investigated surface. Consequently, *k*<sup>+</sup> distribution along the investigated surface will be obtained, and various Δ*U*<sup>+</sup> values will be used along the surface. Furthermore, the fouling effects on the other resistance components can be investigated, as well as the impact of biofouling on the open water and propulsion characteristics. This approach for the assessment of the impact of hull roughness on the ship's total resistance has been recently validated within [21]. Namely, within [21], it was demonstrated that CFD wall function approach can precisely determine not only the impact of roughness on the skin friction, but on the total resistance of 3D hull as well. The investigations related to the impact of barnacle and biofilm fouling on the ship propulsion performance have been presented in [8,22]. These studies demonstrated the impact of biofouling on the propulsion characteristics using CFD approach. However, both studies were performed on the example of Kriso Container Ship (KCS). Since ship resistance and propulsion characteristics can significantly vary amongst different ship forms, it would be beneficial to investigate the fouling effect on the ship performance of different ship forms.

In this study, the impact of biofouling on the ship performance of three merchant ships is analyzed. As already noted, the obtained increases due to the presence of biofouling in effective and delivered power are not equal. Therefore, it is more accurate to study the impact of biofouling on the ship performance through the analysis of the increase in delivered power and propeller rotation rate, than through analysis of the increase in effective power solely. To the best of the authors' knowledge, the impact of biofouling on the ship performance of different hull forms is investigated in this paper for the first time. This investigation is performed utilizing the CFD simulations and a Colebrook-type Δ*U*<sup>+</sup> of Grigson which is implemented within the wall function of CFD solver. Drag characterization study of hard fouling was performed by Schultz [12]. CFD model for the assessment of the impact of hard fouling on the ship resistance has been proposed in [16], where the CFD model is validated. This study can be considered as a continuation of study [16]. A verification study is carried out in order to assess grid and temporal uncertainty. A validation study for smooth surface conditions is 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.

#### **2. Materials and Methods**

#### *2.1. Governing Equations*

In this study Reynolds-averaged Navier–Stokes (RANS) and averaged continuity equations are used as governing equations, and they read:

$$\frac{\partial(\rho \overline{u}\_i)}{\partial t} + \frac{\partial}{\partial \mathbf{x}\_j} (\rho \overline{u}\_i \overline{u}\_j + \rho \overline{u'}\_i \overline{u'}\_j) = -\frac{\partial}{\partial \mathbf{x}\_i} + \frac{\partial \overline{\tau}\_{ij}}{\partial \mathbf{x}\_j} \tag{1}$$

$$\frac{\partial(\rho \overline{u}\_i)}{\partial \mathbf{x}\_i} = 0 \tag{2}$$

where ρ is the density, *ui* is the averaged velocity vector, ρ*u iu <sup>j</sup>* is the Reynolds stress tensor, *p* is the mean pressure and τ*ij* is the mean viscous stress tensor, given as:

$$
\overline{\pi}\_{ij} = \mu \left( \frac{\partial \overline{u}\_i}{\partial \mathbf{x}\_j} + \frac{\partial \overline{u}\_j}{\partial \mathbf{x}\_i} \right) \tag{3}
$$

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:

$$
\delta \mathcal{U}^+ = \frac{1}{\kappa} \ln \mathcal{y}^+ + B - \Delta \mathcal{U}^+ \tag{4}
$$

where κ is the von Karman constant, *U*<sup>+</sup> is the non-dimensional mean velocity, *y*<sup>+</sup> 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 Δ*U*<sup>+</sup> and *k*+, where *k*<sup>+</sup> is defined as:

$$k^{+} = \frac{ku\_{\text{tr}}\rho}{\mu} \tag{5}$$

where *k* is the roughness length scale, which cannot be directly measured.

Schultz has proposed following scaling for the hard fouling [12]:

$$k = 0.059 R\_t \sqrt[n]{\% \text{SC}},\tag{6}$$

where *Rt* 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:

$$
\Delta l I^{+} = \frac{1}{\kappa} \ln \left( 1 + k^{+} \right) \tag{7}
$$

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 Δ*U*<sup>+</sup> model for hard fouling. This Δ*U*<sup>+</sup> 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 Δ*U*<sup>+</sup> 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 Δ*U*<sup>+</sup> 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:

$$
\Delta \varphi = \frac{\varphi\_R - \varphi\_S}{\varphi\_S} \cdot 100\% \tag{8}
$$

where ϕ*<sup>R</sup>* represents certain hydrodynamic characteristic for fouled condition and ϕ*<sup>S</sup>* 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.


**Table 1.** Studied fouling conditions.

#### *2.2. Resistance, Open Water and Propulsion Characteristics*

The total resistance coefficient can be decomposed as follows:

$$\mathbf{C}\_{T} = (1+k)\mathbf{C}\_{F} + \mathbf{C}\_{W} \tag{9}$$

where *k* represents the form factor, *CF* represents the frictional resistance coefficient and *CW* represents the wave resistance coefficient. It should be noted that *CT* is obtained by dividing total resistance (*RT*) with <sup>1</sup> <sup>2</sup>ρ*v*2*<sup>S</sup>* (where *<sup>v</sup>* is the ship speed and *<sup>S</sup>* is the wetted surface) and in that way, the non-dimensional form is obtained.

Effective power (*PE*) can be obtained as a product of *RT* 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 (*PD*) and propeller rotation rate (*n*). The quasi-propulsive efficiency coefficient defines relation between *PE* and *PD* as follows:

$$
\eta\_D = \frac{P\_E}{P\_D} = \eta\_H \eta\_O \eta\_R \tag{10}
$$

where η*<sup>H</sup>* is the hull efficiency, η*<sup>O</sup>* is the open water efficiency and η*<sup>R</sup>* is the relative rotative efficiency. These efficiencies are defined as follows:

$$
\eta\_H = \frac{1-t}{1-w} \tag{11}
$$

$$
\eta\_{\text{O}} = \frac{I}{2\pi} \frac{K\_{\text{TO}}}{K\_{\text{QO}}} \tag{12}
$$

$$
\eta\_R = \frac{K\_{\text{QO}}}{K\_{\text{Q}}} \tag{13}
$$

where *t* is the thrust deduction coefficient, *w* is the wake fraction coefficient, *J* is the advance coefficient, *KTO* is the thrust coefficient in open water conditions, *KQO* is the torque coefficient in open water conditions and *KQ* is the torque coefficient obtained in SPT.

Delivered power can be obtained as follows:

$$P\_D = 2\pi \,\rho K\_Q n^3 D^5 \tag{14}$$

where *D* is the propeller diameter.

#### **3. Computational Model**

#### *3.1. 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 CO2 emission from containerships, bulk carriers and tankers in total CO2 emission from international shipping is significantly higher than for other ship types and accounts for almost 62% of CO2 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.

**Figure 1.** Geometry of the Kriso Container Ship (KCS) (**upper**), Kriso Very Large Crude-oil Carrier 2 (KVLCC2) (**middle**) and Bulk Carrier (BC) (**lower**).

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.


**Table 2.** The main particulars of KCS, KVLCC2 and BC.

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* = <sup>2</sup> · <sup>10</sup><sup>5</sup> as prescribed by ITTC [30], and the obtained results are given in [29,31,32].

**Figure 2.** KP505 (**left**), KP458 (**middle**) and Wageningen series (WB) (**right**) propeller.


**Table 3.** The main particulars of KP505, KP458 and WB.
