Predicting the Effect of Hull Roughness on Ship Resistance Using a Fully Turbulent Flow Channel

Abstract

The consequences of poor hull surface conditions on fuel consumption and emissions are well-known. However, their rationales are yet to be thoroughly understood. The present study investigates the hydrodynamics of fouling control coatings and mimicked biofouling. Novel experimental roughness function data were developed from the “young” fully turbulent flow channel facility of the University of Strathclyde. Different surfaces, including a novel hard foul-release coating, were tested. Finally, the performance of a benchmark full-scale containership was predicted using Granville’s similarity law scaling calculations. Interestingly, the numerical predictions showed that the novel hard foul-release coating tested had better hydrodynamic performance than the smooth case. A maximum 3.79% decrease in the effective power requirements was observed. Eventually, the results confirmed the practicality of flow channel experiments in combination with numerical-based methods to investigate hull roughness effects on ship resistance and powering. The present study can also serve as a valuable guide for future experimental campaigns using the fully turbulent flow channel facility of the University of Strathclyde.

1. Introduction

A ship’s hull surface condition is crucial to its hydrodynamic performance [1]. Hence, choosing the right fouling control coating (FCC) and drydock strategies for a vessel can offer significant economic and environmental advantages. Furthermore, improving hull performances associated with surface conditions enables the vessels to comply with IMO regulations [2], such as the operating expense index (OPEX), Energy Efficiency Existing Ship Index (EEXI) and the novel Carbon Intensity Indicator (CII). Although extensive literature was dedicated to assessing the effect of hull roughness on ship resistance, as summarised by [3], the understanding of the hydrodynamics behind the problem is still limited.

Granville method [4,5] can accurately predict the hull roughness effect on ship resistance, provided that the roughness function is known. Such method is based on the turbulent boundary layer similarity law scaling technique. The roughness function is the difference between the velocities in the boundary layer between a rough surface and a hydraulically smooth reference surface [6]. Furthermore, Granville’s method owes its merit to its robustness and practicality [7]. Additionally, it allows to predict the roughness effect on the frictional resistance for ships of arbitrary lengths and speeds [8].

The specific roughness function models can effectively represent hull surface conditions. However, no universal roughness function can represent all surfaces. Therefore, the roughness function can be seen as the hydrodynamic fingerprint of any given surface. Consequently, several theoretical and experimental methods have been developed for determining the roughness function of rough surfaces. Ref. [9] gave a comprehensive overview of these experimental methods and their advantages and disadvantages, and [10] presented a historical overview of the experimental facilities used in hull coating hydrodynamic tests. Among the literature, a recurrent successful alternative to determine the roughness functions of given surfaces is a fully turbulent flow channel (FTFC) facility. By offering rapid experimental turn-around times combined with high Reynolds numbers, FTFCs can provide reliable results combined with significant financial savings.

Therefore, several investigators have studied turbulent channel flow experimentally. For example, Dean et al. [11] provided us with a widely adopted reference equation. However, much of this research had focused on the Reynolds-number dependence of the skin friction and the mean flow, as reported in [12]. In [13], an investigation was conducted on the skin-friction behaviour in the transitionally rough regime using a turbulent channel flow facility installed in the United States Naval Academy using Granville’s indirect method for pipes [14]. Results were analogous to the Nikuradse-type roughness function [15], which were obtained when investigating the effect of wall roughness on turbulent flows by measuring the pressure drop across a pipe. Additionally, in a recent investigation on the effect of hull roughness on ship resistance using a FTFC, ref. [16] recommended a procedure to estimate the effect of roughness on ship hull resistance based on Granville’s procedure [14] by using the experimentally determined database for roughness functions of rough surfaces. Recently, ref. [17] conducted skin friction measurements with an FTFC on two different sizes of silicon carbide particles (i.e., F220 silicon carbide particles with an average grain size of 53–75 μm and F80 silicon carbide particles with an average grain size of 150–212 μm), proving that roughness amplitude parameters alone are not enough to explain the hydrodynamic performance of surfaces. Furthermore, ref. [18] developed roughness functions for different antifouling coatings by conducting flow cell experiments and predicted the frictional performance of a KVLCC2 hull (Korean Very Large Crude Carrier) model case.

Within the framework of the above literature review, it is clear that the most rational current approach to tackling the effect of ship hull roughness, including biofouling, is to combine experimental and numerical methods. This would require determining the roughness functions using experimental methods, such as cost-effective and practical FTFCs. Therefore, this study aims to obtain new roughness functions for a novel hard foul-release coating, other commonly used marine coatings, and mimicked biofouled hull conditions. Furthermore, the purpose of this paper was to demonstrate the advantages of FTFC experiments to predict the effect hull roughness on full-scale ship resistance and powering. In fact, an important objective was to use the FTFC of the UoS, which is a more practical facility than, e.g., a towing tank. To the best of author’s knowledge, only limited (and unpublished) research has been conducted using the FTFC of the UoS. Hence, the sophisticated new FTFC designed and custom-built [19] at the University of Strathclyde (UoS) was used in the present study. While the facility supports drag reduction studies, another aim of such a facility is to contribute to the international database of the roughness functions for different FCCs and biofouling, as recommended, e.g., by the 21st ITTC Surface Treatment Committee [20]. Therefore, different FCCs produced by Graphite Innovation and Technologies [21], including antifouling, foul-release and barrier resin coatings and the newly developed hard foul-release coating (FR02) were tested in the FTFC. Roughness functions were developed from FTFC tests for widely adopted sandpaper-like surfaces mimicking biofouled conditions (medium light slime and medium slime) as similarly done in towing tests [22]. Furthermore, the roughness function developed for a sandpaper-like surface (Sand 220) from the FTFC experiments was compared with previous towing tank tests. Finally, the present study also aims to confirm the robustness of Granville’s method to predict the effect of hull roughness on ship resistance and powering.

The remaining of the paper is structured as follows: Section 2 presents the methodology adopted, including the experimental setup, roughness functions development, Granville’s similarity law scaling procedure, and experimental uncertainty analysis. Section 3 of the paper discusses the current experimental and numerical investigation results. The novel roughness functions of the test surfaces are presented and used to predict the variation of resistance coefficients and effective power for the full-scale KRISO Container Ship (KCS) hull. Section 4 presents the conclusions, final remarks, and recommendations for future studies.

2. Methodology

2.1. Approach

Figure 1 shows a schematic illustration of the experimental and numerical methodology adopted to investigate the roughness effects of marine coatings and mimicked biofouling on the well-known KRISO Container Ship (KCS) [23].

Drag characterisation of arbitrary rough surfaces on flat plates can be evaluated by the indirect method for pipes [14] that uses the pressure drop $\Delta p$, which can be measured along the coatings (i.e., the pressure drop method). The FTFC was used to determine the skin friction coefficients $c_f$, by measuring the pressure drop $\Delta p$ on the test surfaces. Eventually, the roughness functions obtained for the test surfaces (i.e., roughness functions, $\Delta U^{+}$, roughness Reynolds numbers $k^{+}$, roughness length scale, $k$, etc.), were compared and validated [24], and then embedded in numerical methods to predict the effect of such surfaces on ship resistance and powering.

Consequently, the boundary layer similarity law scaling procedure of Granville [4,5] was adopted in the present study to predict the effect of the test surfaces on the full-scale KCS hull. Additionally, the resistance coefficient results of the numerical predictions were compared and validated across similar studies assessing the KCS resistance in smooth and rough conditions [25,26]. Finally, the variations in effective power, $\Delta P_E$ due to each test surface were estimated to give an immediate understanding of the effects of marine coatings and hull roughness on ship resistance and powering. Comparison and validation of the $\Delta P_E$ values obtained were conducted among similar studies [27].

2.2. Experimental Setup

2.2.1. Fully Turbulent Flow Channel

The University of Strathclyde’s Fully Turbulent Flow Channel (FTFC), as shown in Figure 2a, was designed and installed at the KHL of the NAOME Dept (UoS) to conduct a series of measurements for various types of fouling control coatings and rough surfaces in the freshly applied condition. Delivered to the UoS in 2019, the FTFC is a closed-circuit flow channel that can accommodate two opposing panels in its test section (Figure 2b) located downstream of a single centrifugal pump. The channel has a speed range of 1.5–13.5 m/s, thus able to reach high Reynolds numbers $R{e}_M\approx 3.0·{10}^5$. $R{e}_M$ is the channel height-based Reynolds number based on mean bulk velocity of the flow, $U_M$. The resulting wall shear stress over 300 Pa in the facility with smooth panels. This creates wall shear stress conditions that are similar to average conditions on a smooth ship, 150 m in length, travelling at up to 17 m s⁻¹ (33 knots), as observed by [28].

The UoS FTFC ensures the development of a two-dimensional flow at its test section located at the downstream (tail end) of the upper limb, where the flow becomes fully turbulent. This is due to its features, e.g., a relatively large test section with a channel height (22.5 mm), an aspect ratio of 8:1, water speed (13.5 m/s) and a laser-based measurement access as well as a capability for circulating seawater,

Table 1. Main particulars of the FTFC upper limb.

Name	Symbol	Unit	Value
Length (Tolerance)	l	mm	$3000 \left(\pm 0.05\right)$
Height (Tolerance)	h	mm	$22.5 \left(\pm 0.05\right)$
Beam (Tolerance)	b	mm	$180 \left(\pm 0.05\right)$
Speed range	U	m/s	1.5–13.5
Flow rate	Q	l/s	10–60
Channel height-based Reynolds number	$R{e}_M$	-	$\approx 3.0·{10}^5$
Material	-	-	Stainless steel (316 L)
Centrifugal Pump power	P	kW	22

summarises the main particulars of the FTFC upper limb section. The volume of the system (main tank, auxiliary tank, etc.) is 2.58 m³. For information on the FTFC design, operation and calibration, the reader is advised to see [19].

2.2.2. Test Panels Design and Preparation

In the present experimental campaign, four different types of FCCs were tested in the FTFC, including the newly developed hard foul-release coating (FR02) manufactured by GIT [21] and marine coatings type that are commonly used in the shipping industry manufactured at Dalhousie University (DU), i.e., a self-polishing antifouling coating (AF01), a gelcoat barrier coating (BL01), and a soft foul-release coating (FR01). Furthermore, two sandpaper-like surfaces mimicking slime biofouling, i.e., Sand 220 (medium light slime) and the coarser, Sand 60-80 (medium slime) manufactured at the UoS, were tested. The coated panels (Figure 3a) were tested along with an uncoated “control surface” or the “reference” to represent hydraulically smooth surfaces, Figure 3b. Additionally, the sanded control panels (Sand 220 and Sand 60-80) and the reference panel were of acrylic (i.e., Polymethyl Methacrylate, PMMA) sheets. On the other hand, high-density polyethene was used as the material to manufacture the test panels for marine coating applications.

Table 2. Dimensions of the FTFC test panels.

Dimension	[mm]
Inner length	599
Inner breadth	218
Inner thickness	14
Outer length	662
Outer breadth	282
Outer thickness	16
Tolerance	0.1

describes the dimensions of the test panels, while a breakdown of the type of each marine coating applied and the method of application is provided in

Table 3. Overview of each test panel set. Similar test surfaces can be found in [31].

Panel Set Name	Description (Prepared/Manufactured by)	Panel Material	Coating Type (Topcoat, Underlayers)	Method of Application (Topcoat, Underlayers)	Colour	Arithmetic Mean Roughness (Ra) [µm]
Reference	Smooth reference panel (UoS)	Acrylic	N/A	N/A	Transparent	0.04
AF01	Self-Polishing antifouling coating (DU)	High Density Polyethylene	Self-polishing antifouling, anticorrosive primer	Airless spray, Airless spray	Red matt	0.96
BL01	Gelcoat barrier coating (DU)	High Density Polyethylene	Vinyl ester resin barrier	Airless spray	Green matt	1.44
FR01	Soft foul-release coating (DU)	High Density Polyethylene	Fluoropolymer/silicone foul-release, elastomeric tie coat, anticorrosive primer	Roller, Roller, Airless spray	Blue lucid	0.10
FR02	Hard foul-release coating (DU/GIT)	High Density Polyethylene	Hard foul-release, anticorrosive primer	Airless spray, Airless spray	White lucid	0.22
Sand 220	Medium light slime surface (UoS)	Acrylic	Sanded rough, Aluminium oxide sand grit 220	Scattering, Roller resin	Gray matt	294
Sand 60-80	Medium slime surface (UoS)	Acrylic	Sanded rough, Aluminium oxide sand grit 60-80	Scattering, Roller resin	Brown matt	509

. It is of note that in Table 3, the arithmetic mean roughness, $R_a$, for the FCCs was measured using the Surtronic 25 gauge by Taylor & Hobson over a cut-off length of 0.8 mm, while the $R_a$ for the sanded rough surfaces was measured using the TQC roughness measurement gauge. The filtering is often carried out because the long wavelength component of the roughness is not expected to contribute significantly to the frictional drag. However, it is worth noting that the selection of the optimum cut-off length in order to characterise a surface in a hydrodynamic sense is still unresolved [28]. A review article by [29] discusses this issue in the context of ship hull paints. E.g., the Gaussian filter with a 2.5 mm cut-off length is used by [30] whilst [28] use a 5 mm cut-off length for similar hull paints. Further surface statistics studies could be carried in future studies. Different cut-off lengths could be used to filter the measured data. The surfaces were profiled with an OSP100 optical profilometer (Uniscan Instruments Ltd., Buxton, UK) utilising laser interferometry. The optical laser sensor was adjusted on the two-axis traverse with positioning range of 90 mm × 40 mm. Eighty linear profiles were measured at a scanning speed of 15 mm/s giving a total of 3600 points on the x axis. Representative images of the surface topography for the test surfaces are presented in Appendix A.

2.2.3. Pressure Drop Measurements

The UoS’ FTFC facility is fitted with six pressure taps on the side opposite the laser window to measure the pressure drop (Figure 4).

Pressure taps 2–5 were chosen for pressure drop measurements to avoid pressure waves and noise disturbances at the ends of the measuring section. In fact, taps 2–5 also provided the lowest uncertainty in pressure drop values at the mid-range pump frequency (16 Hz) [19]. Each pair of taps can be connected to a differential pressure transducer with a range of 0–400 mbar via plastic hoses. It is of note that the pressure taps are 120 mm apart from each other, and the pressure drop $\Delta p$ is used in relation to the streamwise linear distance $\Delta x$ to assess the skin friction of the surfaces, according to the following formulae from Equations (1)–(4) [11]:

$$\mathrm{Skin} \mathrm{friction} \mathrm{coefficient}: c_f=\frac{{\tau }_w}{\frac{1}{2}\rho U_M^2}$$

(1)

$$\mathrm{Wall} \mathrm{shear} \mathrm{stress}: {\tau }_w=-\frac{D_h}{4}\frac{\Delta p}{\Delta x}$$

(2)

$$\mathrm{Hydraulic} \mathrm{diameter}: D_h=\frac{2 hb}{h+b}$$

(3)

$$\mathrm{Reynolds} \mathrm{number}: R{e}_M=\frac{U_M h}{\nu }$$

(4)

where $h$ and $b$ are, respectively, the channel height $\left(h=22.5 \mathrm{mm}\right)$ and channel width $\left(b=180 \mathrm{mm}\right)$, $\rho$, the water density, and $U_M$, the mean bulk velocity of the flow in the test section. The water density, $\rho$, is specified based on the formulae provided by [31], including the correction for the temperature of the channel flow, which is continuously recorded by the channel sensor. Furthermore, the pressure drop measurements were conducted for a range of mean bulk velocities, $U_M$, calculated by using the data obtained from the magnetic flowmeter. For each set of test panels, the full range of pump frequencies was assessed (5–40 Hz) to give 36 different mean bulk velocity values (approx. 1.5–13.5 m/s). The variation of the mean bulk velocity at the test section of the FTFC with the smooth reference (uncoated) panel against the pump frequency (rotation per second) is shown in Figure 5. Future studies could consider the kinetic Reynolds number (i.e., Valensi number) to correlate the oscillating frequency with the flow velocity.

2.3. Roughness Functions Determination

Different surfaces are characterised by different roughness functions to be modelled experimentally [5]. For Roughness Function (or velocity loss function), $\Delta U^{+}$, is intended further retardation of flow in the boundary layer over a rough surface due to the physical roughness of that surface, which manifests itself as additional drag relative to a smooth surface. Therefore, 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 (i.e., variance in the local velocities). The non-dimensional velocity profile ($U^{+}$) in the log-law region for a rough surface can be written by (5) [32]:

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

(5)

where, $\kappa$ is the von-Karman constant, $y^{+}$ is the non-dimensional normal distance from the boundary ($y^{+}=y{U}_{\tau }/\nu )$, $B$ is the smooth wall log-law intercept.

The roughness function, $\Delta U^{+}$ is a function of the roughness Reynolds number, $k^{+}$, which is defined by Equation (6) [33]:

$$k^{+}=\frac{k{U}_{\tau }}{\nu }$$

(6)

where, $k$ is the roughness lengths scale of the surface, and $U_{\tau }$ is the friction velocity based on wall shear stress defined by Equation (7) [34]:

$$U_{\tau }=\sqrt{{\tau }_w/\rho }$$

(7)

where, ${\tau }_w$ is the wall shear stress.

For this study, the indirect method for characterizing the drag of an arbitrarily rough surface on a fully developed pipe flow proposed by [14] is used to calculate the roughness function $\Delta U^{+}$ and roughness Reynolds number $k^{+}$ for each coating as follows:

$$\mathrm{Roughness} \mathrm{function}: \Delta U^{+}=\sqrt{\frac{2}{c_{f,s}}}-\sqrt{\frac{2}{c_{f,r}}}$$

(8)

$$\mathrm{Roughness} \mathrm{Reynolds} \mathrm{number}: k^{+}=\frac{1}{\sqrt{2}}R{e}_{M,r}\sqrt{c_{f,r}}\frac{k}{D_h}$$

(9)

where, $c_{f,s}$ and $c_{f,r}$ are the skin friction factors measured in the smooth and rough pipes, respectively, at the same value of $R{e}_M\sqrt{c_f}$, and $k$ is the roughness length scale. It is of note that the roughness length scale values, $k$, were selected so that the roughness function models obtained were in agreement with the Nikuradse [15,33] or Colebrook type [35]. Furthermore, the hydraulic diameter, $D_h$ of the channel was calculated by Equation (3) as in [14]. Finally, Granville’s method was adopted to predict the effect of the test surfaces on the KCS full-scale hull. Notably, the process for Granville’s scale-up method [4,5] is explained in detail in [6]. An in-house code was developed to conduct Granville’s scaling procedure based on the roughness function data obtained in this study.

2.4. Experimental Uncertainty Analysis

The uncertainties of the measurements in the FTFC tests,

Table 4. Uncertainty in the $c_f$ with 95% confidence level at the highest speed.

		${\mathit{P}}_{\mathit{A}}$	${\mathit{B}}_{\mathit{A}}$	${\mathit{U}}_{\mathit{A}}$
$c_f$	Absolute	$1.23·{10}^{-5}$	$1.44 ·{10}^{-5}$	$1.89 ·{10}^{-5}$
	Relative	±0.34%	±0.4%	±0.52%

, were assessed following the ITTC-recommended procedures [36]. The standard errors for the coefficient of friction were calculated based on four to six replicate runs of the FR01 panel at the minimum and maximum flow velocities, respectively. The precision uncertainty in the skin friction coefficient was calculated at a 95% confidence interval by multiplying the standard error by the two-tailed t values (t = 3.182, 2.571) for three to five degrees of freedom, according to [37]. The accuracy of the differential pressure sensor is ±0.075% and the accuracy of the magnetic flow meter was ±0.2% according to the manufacturer’s specifications,

Table 5. Manufacturer specification of all measuring instruments.

Measuring and Control Instrument	Accuracy
Magnetic flow meter	±0.2%
Differential pressure sensor	±0.075%
Pressure sensor	±0.5%
Temperature transmitter with resistance thermometer	±0.1%

. The total uncertainty was calculated using Equation (10) and typical error propagation techniques [38]:

$${(U_A)}^2={\left(B_A\right)}^2+{\left(P_A\right)}^2$$

(10)

where $B_A$ is the bias uncertainty limit, $P_A$ is the precision uncertainty limit and $U_A$ is the total uncertainty. The overall uncertainty in the roughness function, $\Delta U^{+}$ is ±14.4% or 0.04 (whichever is larger) at the lowest $R{e}_M$, ±6.5% or 0.04 (whichever is larger) at the highest $R{e}_M$. The total bias limit and precision limit for the skin friction coefficients $c_f$ were combined using Equation (10) to give a total uncertainty of ±0.74% at the lowest $R{e}_M$ and ±0.52% at the highest $R{e}_M$, Figure 6. For comparison, the high Reynolds number turbulent flow facility at the US Naval Academy achieved a relatively similar level of uncertainty, with their skin friction data being ±1.2% at $R{e}_M$ between 4.0·10⁴–3.0·10⁵ [28].

3. Results and Discussion

3.1. Fully Turbulent Flow Channel Experiments

3.1.1. Wall Shear Stress, ${\tau }_w$

The wall shear stress of each panel set was calculated by Equation (2) based on the hydraulic diameter of the channel defined in Equation (3) and the resulting longitudinal pressure drop ($\Delta p/\Delta x$). In Figure 7 a plot of ${\tau }_w$ vs. flow speed for each panel set is shown, including the 1957 ITTC skin friction formulation for a 232.5 m long flat plate. As shown, the wall shear stress of a 232.5 m long flat plate representing the full-scale KCS at ship speed of 24 knots (12.35 m/s) can be achieved in the FTFC at a considerably low flow speed (4.8–7.2 m/s). In fact, the horizontal dashed red line represents the constant ${\tau }_w$ achieved by KCS at 24 knots. This line crosses the ${\tau }_w$ curves of the FTFC tested panels at considerably lower velocities than the ITTC curve. In other words, at a constant speed of 24 knots (indicated by the vertical dashed red line at 12.35 m/s), the ${\tau }_w$ values of the tested panels in the FTFC are much higher than the value from the ITTC formulation. Therefore, the results from the FTFC can be accurately analogised to the turbulent boundary layer formed on a ship’s hull at cruising speed. In fact, the FTFC enables the measurement of much higher flow speeds and ${\tau }_w$ values that would not be otherwise achievable in a typical towing tank with flat friction test plates.

Overall, the wall shear stress trend of all the FCCs surfaces tested is quite similar. The very subtle differences are related to minor differences in surface roughness that likely arise from application rather than being inherent in any differences in the coatings themselves. Similar observations were made in [28].

3.1.2. Skin Friction Coefficients, $c_f$

Figure 8 shows the skin friction coefficient, $c_f$ of each test surface plotted against the Reynolds number, $R{e}_M$ compared to the hydraulically smooth acrylic panel and reference data taken from [39]. It is of note that the almost unitary $R^2$ value in Figure 8 refers to the polynomial trendline fitted for the experimental reference data. Interestingly, all the test surfaces had skin friction coefficient values beneath the smooth friction line at low Reynolds numbers except for the sanded surfaces. It can be noted that the AF01 displayed unique frictional coefficients behaviour below values of $R{e}_M<{10}^5$, which its surface condition appearance could not explain. The FR01 and BL01 coatings had skin friction curves that followed the behaviour of the smooth acrylic reference panel up until $R{e}_M=2·{10}^5$ where the surfaces showed an increase in skin friction compared to the reference surface. Furthermore, all the surfaces had skin friction coefficient values above the smooth friction line at higher Re values ($R{e}_M>2·{10}^5$) except for the AF01 and FR02 surfaces. In fact, AF01 and FR02 were the only surfaces to maintain a lower skin friction coefficient than the smooth reference surface over the entire Reynolds number range $\left(3·{10}^4<R{e}_M<3·{10}^5\right)$. These coatings (AF01 and FR02) probably have lower friction than the smooth reference because their surface is amphiphilic and hydrophobic, while the reference is neutral. It is of note that each panel set separated from the hydraulically smooth condition at slightly different values of Reynolds numbers. On the other hand, Sand 220 and Sand 60-80 had a neat increase in skin friction from the smooth reference panel. As expected, the increase in skin friction is greater for the coarsest of the two surfaces, Sand 60-80, than the smoothest, Sand 220. It is of note that each panel set separated from the hydraulically smooth condition at slightly different values of Reynolds numbers. On the other hand, Sand 220 and Sand 60-80 had a neat increase in skin friction from the smooth reference panel. As expected, the increase in skin friction is greater for the coarsest of the two surfaces, Sand 60-80, than the smoothest, Sand 220.

As discussed, the AF01 and FR02 coatings had an interesting decrease in skin friction from the smooth reference panel. The authors believe that (regardless of experimental errors) this virtuous behaviour may illustrate the impact of the application method as well as each coating’s ability to behave like a smoother surface under high flow conditions than the smooth reference surface. In fact, the chemistry of the surface and the application method affect the surface roughness and hence the frictional resistance. In other words, as discussed in the methodology section, it is also important to note that these surfaces were applied in a largely isolated (i.e., laboratory) environment which is not representative of the conditions of a real-world coating application in a dockyard. In fact, a dockyard environment can be subject to a variety of external factors, including high winds, temperature, and pre-existing hull roughness (macro roughness). Therefore, the coating surfaces presented in this study and those compared in other studies, especially the coatings that were airless sprayed (FR02 & AF01) should therefore be taken as a better finish than one that would be achieved on the surface of a ship in drydock [28,40].

It is worth noting that all the FCCs tested exhibited decreasing $c_f$ with increasing Reynolds number until $R{e}_M<{10}^4$. This is indicative of surfaces that have not yet reached the fully rough flow regime. For $R{e}_M>{10}^4$ the $c_{f}s$ of the FCCs become independent of Reynold number. This could be a proof of the Reynolds number of these tests being high enough to achieve fully rough behaviour. On the other hand, for the Sand 220 and Sand 60-80 that were tested at the same Reynolds numbers, this was not the case. In fact, it could be indicative of a fundamental change in flow regimes. Sand 220 and Sand 60-80 do not display typical fully rough behaviour at least over the range of Reynolds number assessed here. Instead, $c_f$ continues to increase with Reynolds number over the entire Reynolds number range.

3.1.3. Roughness Function Models

As discussed in the methodology section (Section 2), provided that the roughness functions of the test surfaces are known, Granville’s similarity law can be used to predict the effect of hull roughness on ship resistance. Once the roughness functions have been calculated, they were directly compared with both Colebrook-type [41] and Nikuradse-type [15] roughness functions. Furthermore, the roughness functions of the sandpaper-like surfaces were compared for validation purposes with results obtained from other studies [24]. In fact, previous flat plate towing tank experiments conducted for the same surface roughness (Sand 220) were used for comparison to the present results, Figure 9. These towing tank experiments, Figure 10, were conducted on a stainless steel flat plate (dimensions 1.495 × 0.588 m) at a range of speed (1.5–4.5 m/s) as used in [24,42]. The towing tank facilities of Kelvin Hydrodynamics Laboratory (KHL) of the University of Strathclyde were used. In that occasion, the same $k=1.532·{10}^{-4} \mathrm{m}$ was used to collapse the roughness function on the Nikuradse reference.

Similar to Figure 9, Figure 11 shows the experimental roughness functions, $\Delta U^{+}$, vs. roughness Reynolds numbers, $k^{+}$ obtained from the FTFC pressure drop measurements following Granville’s approach [14]. Notably, the experimental roughness functions of the FCCs tested (Figure 11b) show a deviation between the experimental results with Nikuradse in the vicinity of 0.4 to 5 $\left(k^{+}\right)$. This is probably due to the special amphiphilic and hydrophobic characteristics of the coatings, which exhibited lower friction than the smooth reference. To overcome the deviation from Nikuradse’s model, the roughness functions were modelled by adapting the roughness function model of [43] with the curve fitting coefficients in

Table 6. Curve fitting coefficients of the roughness functions for the test surfaces.

Test Surface	$\mathbf{Roughness} \mathbf{Length} \mathbf{Scale}, \mathit{k} \left[\mathit{m}\right]$
AF01	$9.598 ·{10}^{-6}$
BL01	$1.822 ·{10}^{-5}$
FR01	$1.544·{10}^{-5}$
FR02	$5.840·{10}^{-6}$
Sand 220	$1.532·{10}^{-4}$
Sand 60-80	$3.530·{10}^{-4}$

. Finally, as previously mentioned, the roughness length scale values, $k$, were selected so that the roughness function models obtained were in agreement with the Nikuradse [15,33] or Colebrook type [35], Table 6. An exaggerated difference between the rough-ness functions of the smoothest and roughest coated panels is given in Figure 12.

3.2. Numerical Prediction on Full-Scale KCS Hull

3.2.1. Ship Resistance Coefficients

Numerical predictions were conducted on the benchmark KRISO containership hull at a towing speed of 24 knots $\left(12.35 \mathrm{m}/\mathrm{s}\right)$, Froude number $Fn=0.26$. The Reynolds number based on the ship speed and length was in the range of $R{e}_L=2.72\times {10}^9$, which corresponds to the design speed of the full-scale KCS hull. Table 7 presents the particulars of the full-scale and model KCS adapted from Kim et al. [44].

The variance of resistance and powering requirements due to different test surfaces were calculated using Granville’s similarity law. It is of note that he newly developed roughness functions were incorporated into the procedure. Finally, the following Equations (11)–(18) were used to csalculate the ship resistance coefficients similar to [45]. The total ship resistance coefficient, $C_T$, is defined in Equation (11) as a function of the total drag, $R_T$, the dynamic pressure, $1/2 \rho V^2$, and the hull wetted surface area, $S$:

$$C_T=\frac{R_T}{1/2 \rho S V^2 }$$

(11)

where, $V$ is the towing speed (i.e., the inlet velocity). It is well-known that the total ship resistance coefficient, $C_T$, can be decomposed into the frictional, $C_F$, and the residuary, $C_R$ resistance coefficients, as given by Equation (12):

$$C_T=C_F+C_R$$

(12)

In the present study, the $C_F$ values in Granville’s approach are calculated iteratively solving the Schoenherr smooth friction line, Equation (13):

$$0.242 C_F=Log\left(Re·C_F\right)$$

(13)

where, $Re$ is the Reynolds number based on ship speed and ship length at the waterline.

Consequently, the residuary resistance coefficients, $C_R$ are calculated as in Equation (14):

$$C_R=C_T-C_F$$

(14)

The variation of the frictional resistance coefficient $\Delta C_F$ is the difference between the rough, $C_{F_{rough}}$, and smooth, $C_{F_{smooth}}$, conditions at the same Froude number can be given by Equation (15):

$$\Delta C_F=C_{F_{rough}}-C_{F_{smooth}}$$

(15)

Hence, the variation of the frictional resistance due to the presence of roughness can also be expressed in percentage, as in Equation (16)

$$\%\Delta C_F=\frac{C_{F_{rough}}-C_{F_{smooth}}}{C_{F_{smooth}}} ·100$$

(16)

The total resistance for the rough ship, $C_{T_{rough}}$, is determined by:

$$C_{T_{rough}}=C_{T_{smooth}}+\Delta C_T$$

(17)

where the total roughness allowance, $\Delta C_T$ is the variation in the total resistance coefficient between the rough, $C_{T_{rough}}$, and smooth, $C_{T_{smooth}}$, conditions, and can be given by Equation (18):

$$\Delta C_T=C_{T_{rough}}-C_{T_{smooth}}$$

(18)

Table 7. KRISO Container Ship (KCS) full-scale principal characteristics.

Parameters
Scale factor	$\lambda$	1
Length between the perpendiculars	$L_{PP} \left(\mathrm{m}\right)$	230
Length of waterline	$L_{WL} \left(\mathrm{m}\right)$	232.5
Beam at waterline	$B_{WL} \left(\mathrm{m}\right)$	32.2
Depth	$D \left(\mathrm{m}\right)$	19.0
Design draft	$T \left(\mathrm{m}\right)$	10.8
Wetted surface area w/o rudder	$WS{A}_{Total} ({\mathrm{m}}^2)$	9424
Displacement	$\nabla \left({\mathrm{m}}^3\right)$	52,030
Block coefficient	$C_B$	0.6505
Design speed	$V \left(\mathrm{kn}; \mathrm{m}/\mathrm{s}\right)$	24; 12.35
Froude number	$F_n$	0.26
Reynolds number	$Re$	$2.72·{10}^9$
Centre of gravity	$KG \left(\mathrm{m}\right)$	7.28
Metacentric height	$GM \left(\mathrm{m}\right)$	0.6

Table 7. KRISO Container Ship (KCS) full-scale principal characteristics.

Parameters
Scale factor	$\lambda$	1
Length between the perpendiculars	$L_{PP} \left(\mathrm{m}\right)$	230
Length of waterline	$L_{WL} \left(\mathrm{m}\right)$	232.5
Beam at waterline	$B_{WL} \left(\mathrm{m}\right)$	32.2
Depth	$D \left(\mathrm{m}\right)$	19.0
Design draft	$T \left(\mathrm{m}\right)$	10.8
Wetted surface area w/o rudder	$WS{A}_{Total} ({\mathrm{m}}^2)$	9424
Displacement	$\nabla \left({\mathrm{m}}^3\right)$	52,030
Block coefficient	$C_B$	0.6505
Design speed	$V \left(\mathrm{kn}; \mathrm{m}/\mathrm{s}\right)$	24; 12.35
Froude number	$F_n$	0.26
Reynolds number	$Re$	$2.72·{10}^9$
Centre of gravity	$KG \left(\mathrm{m}\right)$	7.28
Metacentric height	$GM \left(\mathrm{m}\right)$	0.6

Figure 13 and Figure 14, shows the resistance coefficients of the test cases obtained from the Granville’s similarity law compared to a hydrodynamically smooth ship hull. Interestingly, the test cases AF01 and FR02 show a negative $\Delta C_T$ of 2.3% and 3.8%, respectively. As expected, the phenomena of reduced $\Delta C_T$ values are due to the negative roughness functions, $\Delta U^{+}$ observed from the experimental measurements. On the other hand, the BL01 and FR01 cases lead to light $\Delta C_T$ increases (0.7% for BL01 and 0.2% for FR01) compared to the total added resistance due to mimicked slime (27.0% for Sand 220 and 38.9% Sand 60-80 cases). It can be noted that the FR02 is the best performing FCCs tested while the sanded surface, Sand 60-80, leads to a higher increase in the total resistance coefficients.

It is notable that the total resistance coefficient results are in good agreement and show similar trends to the frictional resistance coefficients. Table 8 and Table 9 present the frictional and total resistance coefficients obtained for the test surfaces. Furthermore, the results are reasonably in agreement with other studies found in the literature such as [16,22] found that a foul-release coating as applied measured an added frictional resistance $\%\Delta C_F$ equal to 2.6%, and for a 150 m flat plate at 12 knots coated with Sand 60-80 calculated $\%\Delta C_F=59\%$.

Table 8. Frictional resistance results $\left(C_F\right)$ results on the full-scale KCS hull at 24 knots ($Fn=0.26$ ).

Test Surface	Granville Similarity Law
	${\mathit{C}}_{\mathit{F}}$	$\mathbf{\Delta }{\mathit{C}}_{\mathit{F}}$	$\mathbf{\%}\mathbf{\Delta }{\mathit{C}}_{\mathit{F}}$
Reference	$1.358·{10}^{-3}$	-	-
AF01	$1.312·{10}^{-3}$	$-4.620·{10}^{-5}$	−3.40%
BL01	$1.378·{10}^{-3}$	$1.387·{10}^{-5}$	1.02%
FR01	$1.368·{10}^{-3}$	$1.013 ·{10}^{-5}$	0.75%
FR02	$1.282·{10}^{-3}$	$-7.557·{10}^{-5}$	−5.57%
Sand 220	$1.897·{10}^{-3}$	$5.390·{10}^{-4}$	39.70%
Sand 60-80	$2.135·{10}^{-3}$	$7.765·{10}^{-4}$	57.18%

Table 8. Frictional resistance results $\left(C_F\right)$ results on the full-scale KCS hull at 24 knots ($Fn=0.26$ ).

Test Surface	Granville Similarity Law
	${\mathit{C}}_{\mathit{F}}$	$\mathbf{\Delta }{\mathit{C}}_{\mathit{F}}$	$\mathbf{\%}\mathbf{\Delta }{\mathit{C}}_{\mathit{F}}$
Reference	$1.358·{10}^{-3}$	-	-
AF01	$1.312·{10}^{-3}$	$-4.620·{10}^{-5}$	−3.40%
BL01	$1.378·{10}^{-3}$	$1.387·{10}^{-5}$	1.02%
FR01	$1.368·{10}^{-3}$	$1.013 ·{10}^{-5}$	0.75%
FR02	$1.282·{10}^{-3}$	$-7.557·{10}^{-5}$	−5.57%
Sand 220	$1.897·{10}^{-3}$	$5.390·{10}^{-4}$	39.70%
Sand 60-80	$2.135·{10}^{-3}$	$7.765·{10}^{-4}$	57.18%

Table 9. Total resistance coefficients of the full-scale KCS at 24 knots ($Fn=0.26$ ).

Test Surface	Granville Similarity Law
	${\mathit{C}}_{\mathit{T}}$	$\mathbf{\Delta }{\mathit{C}}_{\mathit{T}}$	$\mathbf{\%}\mathbf{\Delta }{\mathit{C}}_{\mathit{T}}$
Reference	$1.996·{10}^{-3}$	-	-
AF01	$1.950·{10}^{-3}$	$-4.096·{10}^{-5}$	−2.31%
BL01	$2.010·{10}^{-3}$	$1.860·{10}^{-5}$	0.69%
FR01	$2.006·{10}^{-3}$	$4.668·{10}^{-6}$	0.51%
FR02	$1.920·{10}^{-3}$	$-7.096·{10}^{-5}$	−3.79%
Sand 220	$2.535·{10}^{-3}$	$5.320 ·{10}^{-4}$	27.01%
Sand 60-80	$2.772·{10}^{-3}$	$7.210·{10}^{-4}$	38.90%

Table 9. Total resistance coefficients of the full-scale KCS at 24 knots ($Fn=0.26$ ).

Test Surface	Granville Similarity Law
	${\mathit{C}}_{\mathit{T}}$	$\mathbf{\Delta }{\mathit{C}}_{\mathit{T}}$	$\mathbf{\%}\mathbf{\Delta }{\mathit{C}}_{\mathit{T}}$
Reference	$1.996·{10}^{-3}$	-	-
AF01	$1.950·{10}^{-3}$	$-4.096·{10}^{-5}$	−2.31%
BL01	$2.010·{10}^{-3}$	$1.860·{10}^{-5}$	0.69%
FR01	$2.006·{10}^{-3}$	$4.668·{10}^{-6}$	0.51%
FR02	$1.920·{10}^{-3}$	$-7.096·{10}^{-5}$	−3.79%
Sand 220	$2.535·{10}^{-3}$	$5.320 ·{10}^{-4}$	27.01%
Sand 60-80	$2.772·{10}^{-3}$	$7.210·{10}^{-4}$	38.90%

Figure 14. Percentage bar diagram of the resistance coefficients in different hull roughness conditions.

Figure 14. Percentage bar diagram of the resistance coefficients in different hull roughness conditions.

3.2.2. Ship Effective Power, $\mathsf{\Delta }{P}_{\mathrm{E}}$

The change in effective power, $\%\Delta P_E$ due to the different surfaces tested can be expressed by:

$$\%\Delta P_E=\frac{C_{T_{rough}}-C_{T_{smooth}}}{C_{T_{smooth}}} ·100=\frac{\Delta C_T}{C_{T_{smooth}}} ·100$$

(19)

similar to that used by [46], where $C_{T_{smooth}}$ is the total resistance coefficient of the hull in smooth conditions obtained from CFD simulations. It is of note that $\%\Delta P_E$ is equal to $\%\Delta C_T$.

Table 10. Effective power variation $\left(\%\Delta P_E\right)$ of the full-scale KCS at 24 knots ($Fn=0.26$ ).

Test Surfaces	$\mathbf{\%}\mathbf{\Delta }{\mathit{P}}_{{\mathit{E}}_{\mathit{G}\mathit{r}\mathit{a}\mathit{n}\mathit{v}\mathit{i}\mathit{l}\mathit{l}\mathit{e}}}$
AF01	−2.31%
BL01	0.69%
FR01	0.51%
FR02	−3.79%
Sand 220	27.01%
Sand 60-80	38.90%

show the change in effective power, $\%\Delta P_E$ due to the different test cases obtained from Granville’s approach. It is of note that the largest difference between coating types for powering requirements is an average of 4.48%, between FR02 and BL01. If the coatings AF01 and FR02 were applied on the ship hull, they would lead to a reduction in effective power requirements. In fact, AF01 guarantees a maximum decrease of power requirements of 2.31%, while FR02 of 3.79%.

As expected, the phenomena are due to the negative roughness functions, $\Delta U^{+}$ observed from the experimental measurements to which correspond negative $\Delta C_T$ values. On the other hand, the BL01 and FR01 cases lead to positive $\%\Delta P_E$, which translates into increases in effective power requirements of 0.69% and 0.51%, respectively. Furthermore, the total added effective power due to mimicked slime is 27.01% for Sand 220 and 38.90% Sand 60-80 cases. The FR02 is the best performing FCCs assessed ($\%\Delta P_E=-3.79\%)$, while the sanded surface, Sand 60-80, would lead to the highest increase in the effective power ($\%\Delta P_E=38.90\%)$. Finally, it can also be noted that the rate $\%\Delta P_E/\%\Delta C_F$ is in the range of $65\%÷70\%$, as would be expected [27].

4. Conclusions

An experimental and numerical study was conducted to investigate the full ship hydrodynamic performance of different fouling control coatings and mimicked biofouling. The investigation presented had five key tasks: physically conducting the pressure drop measurements experiment, calculating the skin friction coefficient, calculating the roughness functions, and implementing numerical methods.

The present study confirmed that the most rational approach to tackling the effect of ship hull roughness, including biofouling, is to combine experimental and numerical methods. The practical and sophisticated FTFC facility recently installed at the UoS was adopted for this scope. In fact, novel roughness functions for a novel hard foul-release coating, other commonly used marine coatings, and mimicked biofouled hull conditions were developed. Furthermore, this paper exploited the advantages of FTFC experiments to predict the effect hull roughness on full-scale ship resistance and powering. To the best of author’s knowledge, only limited (and unpublished) research were conducted using the FTFC of the UoS before the present study. Hence, the urgency of using the FTFC designed and custom-built [19] at the University of Strathclyde (UoS). Furthermore, the experimental data produced supports drag reduction studies, and contributes to the international database of the roughness functions for different FCCs and biofouling. Producing experimental data was indeed recommended, e.g., by the 21st ITTC Surface Treatment Committee [20]. Finally, the goal of developing transferrable expertise with the FTFC of the UoS was met.

Hence, the experimental part of the study led to the introduction of novel experimental roughness functions for the FCCs tested including GIT’s (FR02 novel hard foul-release coating). Each surface’s wall shear stress values and specific friction coefficients relative to the smooth uncoated reference surface were presented for completeness. Furthermore, the roughness function developed for a sandpaper-like surface (Sand 220) from the FTFC experiments was compared with previous towing tank tests. It is of note that this was the first time the same surface was tested in two different facilities of KHL. Therefore, the present study also confirmed the robustness of the FTFC (instead of a towing tank) combined with Granville’s method to predict the effect of hull roughness on ship resistance and powering.

On the other hand, the numerical part of the study scaled up the laboratory results to the size of a full ship length. It is of note that the numerical predictions were conducted adopting Granville’s similarity law scaling procedure based on the experimental results. In fact, the frictional resistance results for the FTFC experiments were used to determine the novel roughness functions for each test surface. The benchmark KRISO containership (KCS) hull in full-scale was chosen to calculate the variance of resistance and powering requirements due to different test surfaces at the design speed of 24 knots ($Fn=0.26$).

Among the four fouling control coatings (FCCs) that were tested in the FTFC, the FR02 coating (hard foul-release) displayed the best hydrodynamic performance across the entire Reynolds number range. In fact, the FR02 coating displayed lower frictional resistance coefficients than if the ship was considered as smooth as the acrylic reference panel (5.57% decrease). Furthermore, the FR02 surface led to a maximum decrease in effective power requirements of 3.79%. The results of the numerical prediction also show that the AF01 (self-polishing antifouling coating) have better hydrodynamic performance than the smooth reference case (maximum decrease in effective power requirements of 2.31%). In contrast, Sand 220 (medium light slime) and Sand 60-80 (medium slime) have, as expected, the highest resistance due to their rougher characteristics. In fact, a ship hull with medium light slime (Sand 220) and medium slime (Sand 60-80) surface roughness characteristics as the test surfaces would experience a maximum increase in effective power requirements of 27.01% and 38.90%, respectively. Finally, it is of note that the Granville method is limited to the assumption that the velocity is constant for the length of the plate (i.e., ship).

Further investigation could also be conducted on the prediction of resistance of the fouling control coatings (FCCs) at different speeds, on different hulls, and using heterogeneous patch distribution of the roughness [47]. It would also be beneficial to investigate the hydrodynamic performance of the same FCC under the effect of biofouling growth. Exposing surfaces to dynamically grown biofouling would give shipowners and operators a better indication of what powering penalty they should expect from these coatings after a certain time in active service. It is of note that such real biofouling could soon be simulated in the biofouling farm under development at the University of Strathclyde. Applying different mimicked biofouling to the panels before or after the coating application could also serve as a better method to predict the resistance behaviour of the as-applied condition to an existing rough ship hull.

Above all, the present study has provided several important findings, including the procedure to conduct pressure drop measurements with an FTFC, the application of Granville’s method for pipes to develop roughness functions, as well as the introduction of roughness functions for a novel or widely adopted marine surfaces and mimicked biofouling. The findings presented can help predict the required power, fuel consumption and greenhouse gas emissions of ships with hulls coated with certain fouling control coatings and/or in the fouled condition. As a final remark, the authors would like to emphasise that there is an enormous opportunity for growth in the area of research on FTFCs. Indeed, the present study only represents an infinitesimal fraction of the number of coating products and surface roughness conditions that can be tested.