The Phenomenon of Friction Resistance Due to Streamwise Heterogeneous Roughness with Modified Wall-Function RANSE

Surface roughness can reduce the performance of a system of fluid mechanics due to an increase in frictional resistance. The ship hull, which is overgrown by biofouling, experiences a drag penalty which causes energy wastage and increased emission levels. The phenomenon of fluid flow that passes over a rough surface still has many questions, one of which is the phenomenon of frictional resistance on heterogeneous roughness in the streamwise direction. In the ship hull, biofouling generally grows heterogeneous along the hull with many factors. RANSE-based Computational Fluid Dynamics was used to investigate the friction resistance for heterogeneous roughness phenomenon. The modified wall-function method represented equivalent sand grain roughness (ks) and a roughness function were applied together with k-epsilon turbulence model to simulate rough wall turbulent boundary layer flow. As the heterogeneous roughness, three different ks values were denoted as P (ks = 81.25 μm), Q (ks = 325.00 μm) and R (ks = 568.75 μm), and they are arranged by all possible combinations. The combined roughness, whether homogeneous (PPP, QQQ, or RRR) and inhomogeneous (PQR, PRQ, QPR, etc.), results in unique skin friction values. The step-change in the height of the heterogeneous roughness produced a sudden change in the local skin friction coefficient in the form of overshoot or undershoot, followed by a relaxation where the inhomogeneous local skin friction is slowly returning to the homogeneous local one, which was explained in more detail by plotting the distribution of the mean velocity profile near the step-up or step-down. The order of roughness arrangement in a streamwise heterogenous roughness pattern plays a key role in generating overall skin friction with values increasing in the following order: PQR < PRQ < QPR < QRP < RPQ < RQP. Those inhomogeneous cases with three different values of ks can be represented by a single value (being like homogeneous) by the calculations provided in this paper.


Introduction
The issue of using energy more efficiently on ships seems urgent, and how to do this is greatly helped by the existence of CFD. In the last three decades, CFD as a numerical method, which is very sophisticated that can help humans to solve various problems in the fields of science and technology [1][2][3][4][5][6], but at a competitive cost, has played a very important role in advancing transportation technology [7], especially in naval architecture [8]. With this highly sophisticated method, the need for increased energy efficiency on ships is greatly helped. In 2012, the International Maritime Organization (IMO) noted that the total emissions from the shipping sector worldwide were 2.2% compared to all human-made CO 2 emissions [9]. This number was predicted to increase 2-3 times in 2050 if there are no prevention efforts [10]. There are many ways to save on the use of energy on board [11,12], such as improving hull and propeller design for more hydrodynamic performance. With the help of CFD, efforts to improve energy efficiency can be made more accessible.
Caring about the cleanliness of the hull due to biofouling is one of the efforts to maintain the hydrodynamic performance to prevent energy waste. Roughness can increase friction resistance, and then power requirements increase, resulting in losses, which have a significant impact on large vessels such as the VLCC (Very Large Crude Carrier) [13] or ships with low Fr (Froude number) [14]. A roughness, namely tubeworm fouling, can increase the friction resistance of ships by 23-34% [15], while a heavily fouled ship hull can increase the friction resistance by up to 80% [16]. Due to the growth of biofouling on ship hull, fuel consumption can increase over the operational time and can increase significantly just in a year [17]. The total economic losses from biofouling, including fuel additions, cleaning, and repainting, can reach $ 15 million a year [18].
The phenomenon of the effect of roughness on fluid flow was first investigated by Nikuradse [19]. The mean velocity profile of the structure turbulence boundary layer of the smooth case (see Eq. (1)) is exposed to a downward shift in the log law region by a roughness to become a new velocity profile (see Eq. (2)) [20]. Thus, the concept of the roughness function (ΔU þ ) as the downward shift and the roughness Reynold number (k s þ ) were used (see Eqs. 3 and 4). Where: U þ is the nondimensional mean velocity profile equal to U=U τ ; U is the mean velocity at y (the normal from the wall); U τ is the friction velocity defined as ffiffiffiffiffiffiffiffiffiffi τ w =ρ p ; τ w is the shear stress magnitude and ρ is the density of the fluid; y þ is the non-dimensional normal distance from the wall defined as yU τ =ν; ν is the kinematic viscosity; κ is the von Karman constant and B is the smooth wall log-law intercept; k s is equivalent sand roughness height. From the new velocity profile, there is an indication of an increased momentum deficit compared to the smooth case.
The flow over surface roughness phenomenon can be simulated using CFD, which is generally solved by two different methods: modified wall function and geometrically resolved. The modified wall function is a method in which the geometry model (mesh) remains smooth, and the roughness length scale represents the roughness, generally using k s (equivalent sand grain roughness height) as a variable for a roughness function (ΔU þ ) which will shift/modify the mean velocity profile. This method is only supported when using the RANSE (Reynolds-Averaged Navier-Stokes Equations). The modified wall function method is very effective for modeling large objects such as the hull of a ship, where the k s and ΔU þ values have been previously known and inputted, as was done by Demirel et al. [21], Song et al. [22], Andersson et al. [23], and also it can be used for propeller [24] and tidal turbine [25]. The modified wall function method prioritizes seeing results (impacts), such as increased drag, wake, and the like. Meanwhile, the geometrically resolved method used the real roughness geometry that formed from the mesh. These methods generally use to know how the k s value and the ΔU þ characteristic from a roughness. DNS (Direct Numerical Simulation) and LES (Large Eddy Simulation) are well known to be very good at doing this task. The geometrically resolved can also be done with RANSE as done by Atencio & Chernoray [26] with a difference of about 7% with their experimental results.
A reduction in hull performance due to roughness can be simulated using CFD with a modified wall function method with acceptable accuracy. Reynolds-averaged Navier-Stokes Equations (RANSE) simulation to study the friction resistance of flat plates due to the antifouling coating performed by Demirel et al. [27]. They used the roughness function from the experimental result of Schultz [28]. Using the Kriso Container Ship model, Demirel et al. [21] continued the CFD simulation to predict the impact of marine coatings and biofouling. Song et al. [22] also looked at the effect of biofouling on the ship's hydrodynamic characteristics, using a different roughness function. Anderson et al. [23] performed a comprehensive review and comparative analysis of different methods to model hull roughness. The comparison of acceptable CFD and experimental results was carried out by Song et al. [29].
Much of the literature reported assumes that the roughness distribution is homogeneous, but in reality, it is much non-homogeneous. Hull roughness, mainly that arising from biofouling, rarely occurs homogeneously. From a personal review in the field, the authors found that the biofouling growth was thicker in the stern of the vessel than in the bow, which may be influenced by the distribution of shear stress and flow compressive forces, which are more favorable for biofouling to grow better at the stern [30].
Many studies are looking at the roughness problems at an inhomogeneous pattern. The smooth-to-rough and rough-to-smooth patterns in streamwise direction were studied by Antonia & Luxton [31,32]. In a streamwise phenomenon, the flow of fluid through abruptly different roughness conditions produces an internal boundary layer, which limits the near-wall layer, which senses the new surface conditions, from the flow further away from the wall, which keeps a memory of the upstream surface conditions before the surface transition [33]. The internal boundary layer and the local wall shear stress, at the transition of the two difference roughness conditions, exceed the equilibrium value when the roughness is homogeneous and then change to relax according to the homogeneous roughness equilibrium value certain distance [34]. Experimental methods and numerical simulations carried out several studies related to this inhomogeneous roughness. However, the numerical simulations for solving this case are mostly DNS and LES. Few do that through the RANSE, as Suastika et al. [35] on a flat plate, and Song et al. [36] using the Wigley hull model.
The frictional resistance acting on the hull due to inhomogeneous roughness becomes important to be modeled and analyzed, considering that the hull roughness due to biofouling is mostly inhomogeneous. In a simple method where the ship hull is represented by a flat plate, then an inhomogeneous roughness is applied by dividing the plate into three equal parts, namely the fore, middle, and after. The parts are given different roughness values in the form of k s and are arranged according to several combinations. From the simulation results on inhomogeneous roughness in the form of frictional resistance values will be compared with where if the condition is a smooth surface and some surfaces with homogeneous roughness. Then how are the three roughness values in the inhomogeneous condition correlated to become one roughness value (homogenized) which is close to the inhomogeneous roughness value. This CFD simulation uses the basis of Reynolds-averaged Navier-Stokes Equations (RANSE), where the roughness model uses a modified wall function. Research with variations in roughness that is streamwise inhomogeneous, which is then analyzed systematically, according to our knowledge is still a little done.

CFD modeling
A Reynolds-averaged Navier-Stokes Equations (RANSE) model, implemented in ANSYS Fluent, was used to solve the governing equations that can model turbulent flow over rough walls using modified wall function. The governing equations consist of averaged continuity and momentum equations, which for an incompressible flow without body forces, are given using tensor notation as described in Eqs. 5 and 6, respectively. Where i, j =1,2,3,U i is the mean velocity component, P is the mean pressure, ρ is the fluid density, ν is the kinematic viscosity, u 0 i is the fluctuating velocity component, and u i 0 u j 0 is the Reynolds stresses [37].
The realizable k-ε with standard wall function turbulence model, which relates the Reynolds stresses to the mean flow properties, was used to close the system of Eqs. (5) and (6). The turbulence model is a two-equation model representing the transports of turbulence kinetic energy k and turbulence dissipation rate ε [38].
The roughness function (ΔU þ ) model used in this study is that proposed by Cebeci and Bradshaw [39], whose model follows Nikuradse's uniform sand-grain roughness data [19]. Therefore, the roughness height utilized in this study is referred to as equivalent sand-grain roughness height k s . The generalized Cebeci and Bradshaw's roughness function model is given in Eq. (7), where A =0,k þ s; smooth = 2.25, k þ s; rough = 90.00 and C s = 0.253. In which the power a is given in Eq. (8).
a ¼ sin The boundary conditions were set for this study, as illustrated in Figure 1a. The inlet was velocity inlet where the free stream velocity prescribes the flow velocity U ∞ . The outlet was the pressure outlet set to be hydrostatic to ensure no upstream propagation of disturbances. The no-slip condition is applied on the plate's surface while the top and side boundaries are modeled as free-slip walls. The boundary conditions, governing, and turbulence modeling equations are discretized using a finite volume second-order method. These sets are then solved using a finite volume solver, utilizing a SIMPLE algorithm in which gradient calculations are carried out using the least-square cell-based method. The residual is set at 10 À5 as a convergence criterion. For all simulations in this study, the plate length, fluid properties, and free stream velocity are kept constant. The plate length L is 30 m. The fluid is seawater with mass density ρ = 1025 kg/m 3 and dynamic viscosity μ = 0.001077 kg/(ms). Finally, the free stream velocity is set at U ∞ = 9.77 m/s (19 knots).

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The Phenomenon of Friction Resistance Due to Streamwise Heterogeneous Roughness with… DOI: http://dx.doi.org /10.5772/intechopen.99137 In CFD modeling, the distance of the frame against the wall is set to decrease exponentially as it moves typically towards the wall., as shown in Figure 1b.A hexahedron-type mesh is chosen, and it is arranged manually with adjustable grid size. It is crucial to determine the mesh size near the wall to obtain an appropriate value for the dimensionless normal coordinate y þ , defined as y þ ¼ uτy=ν, where y is the outward wall-normal coordinate. To model the roughness effects correctly, the y þ value for the first cell center above the wall must be larger than the local roughness Reynolds number k þ s . To ensure this condition is always satisfied, ANSYS Fluent will virtually shift the wall if y þ <k þ s . For the roughness cases considered in this study, a blockage effect of 50% of the roughness height is assumed, and the corrected y þ value for the first cell center above the wall is given as In this way, the singularity issue is avoided, and fine meshes can be handled correctly.

Surface roughness modeling
In this study, we will investigate just single parameter variations, namely the roughness height k s . This section will explain details of the k s set up and their possible combinations. Four surface roughness with different k s values are considered in this study, namely, smooth wall (S), small roughness height (P), medium roughness height (Q), and high roughness height (R). All three combinations of roughness P, Q , and R are considered to form either homogeneous or inhomogeneous rough walled turbulent boundary layer flow. For example, a three-surface combination of PPP, QQQ , or RRR forms a homogeneous roughness, while a combination of PQR, PRQ , QPR, etc., forms an inhomogeneous roughness, where those are described in Figure 2. These k s values are specifically chosen so that the average height of three different k s of P = 81.25 μm, Q = 325.00 μm, and R = 568.75 μm will give an average height equal of Q , i.e., (81.25 μm + 325.00 μm + 568.75 μm)/ 3 = 325.00 μm. The selected k s values are also designed to simulate the various stages of ship-hull biofouling growth, ranging from light slime [16] to about small calcareous fouling [15].

Grid independence study
To ensure optimum numbers of cells are used in the final simulations, several grid independence tests using flat plate smooth wall (SSS) base cases are conducted. For each case, the overall friction coefficient C F is calculated using the increasing number of cells in the simulation. The number of cells in the latter simulation is approximately twice than that in the former. The overall friction coefficient C F is defined in Eq. (9), where D is the drag per unit width, τ w is the wall shear stress, ρ is the fluid density, U ∞ is the free stream velocity, L is the plate length, and x is the distance downstream from the leading edge of the plate. Furthermore, a percent error e nþ1,n between the lower and higher cell numbers is defined in Eq. (10) [40].
The results of the grid independence study are summarized in Table 1. The number of cells is varied from 750,000 to 6,127,550. Table 1 shows that the C F value increases monotonically with the increasing number of cells, which is expected to reach an asymptotic value at a very large number of cells. The value of e nþ1,n as listed in Table 1 decreases with the increasing number of cells used in the simulation. The results show that the error is very low (in the range 0.0206-0.131%), well below the recommended 2% from the literature [41]. Based on this grid independence test, N = 3,000,000 is chosen as an optimum number of cells for all the cases (including homogeneous and inhomogeneous roughness).

Verification and validation
In addition to the grid independence tests, further analyses are carried out for varied viscous-scaled wall-normal distance y þ ranges of the first cell center above the wall. For the smooth plate SSS case, the calculation result is verified using the well-known Schoenherr's friction coefficient and the 1957 ITTC (International Towing Tank Conference) ship-model correlation line. Schoenherr's friction coefficient C F is given in Eq. (11). It was adopted by ATTC (American Towing Tank Conference) as a standard for the clean hull skin friction resistance in 1947, and it is often referred to as the 1947 ATTC line. The second correlation is the 1957 ITTC ship-model correlation line, which is given in Eq. (12). A percent error is defined The results are summarized in Table 3, showing C F values calculated using different y þ ranges for the first cell center above the wall targeted in the simulations. Table 3 shows that for the SSS case, using a y þ range between 155 and 254 would result in the closest C F value to the 1947 ATTC and 1957 ITTC lines with percent errors of e = 0.620% and e = 0.814%, respectively. A smaller y þ range will result in larger differences, but the percentage differences do not exceed 1.76%, which is relatively small. Despite these anomalies, overall, the CFD and the 1947 ATTC or the 1957 ATTC line are small. Despite some discrepancies, for the smooth surface cases considered in this study, any y þ range between 64 and 254 will result in an acceptable C F value with a maximum magnitude of percent errors of 1.57%  Table 3.
Overall friction coefficient C F calculated using different ranges of y þ value for the first cell center above the wall compared with 1947 ATTC (Eq. (11)) and 1957 ITTC (Eq. (12)) lines for the smooth SSS case.
when compared with the 1947 ATTC line and 1.76% when compared with the 1957 ITTC line. This result is following the recommended y þ range in the literature for smooth flat plate CFD simulations between 50 and 300 [42].
To model the roughness effects correctly, the y þ value for the first cell center above the wall denoted as (Δy þ ) 1 , must be larger than the local equivalent sand grain roughness Reynolds number k þ s , i.e., (Δy þ ) 1 > k þ s . However, when one employs a fine mesh near the wall, the (Δy þ ) 1 value may have a smaller value than the k þ s value, i.e., (Δy þ ) 1 < k þ s . If such a case happens, ANSYS Fluent applies a virtual shift of the wall by increasing the value of (Δy þ ) 1 with an amount k þ s /2, such that (Δy þ ) 1 > k þ s . To gain more insight into the CFD results for rough conditions, they are verified using the empirical calculation, Granville's similarity law scaling method [43]. The simplified Granville similarity scaling can be calculated using Eqs. 14, 15, and 16. Where: C F R is the coefficient of frictional resistance for rough condition, where the empirical formula as the foundation is taken from the approximated Kármán-Schoenherr formula [44]; Re r is the Reynolds number for calculate the C F R using the empirical formula, which equal of the Reynolds number for smooth condition ( Re s ) that is shifted as described in Eq. (15). Then, κ is the von Kármán constant; k s þ is roughness Reynolds number; ν is kinematic viscosity; U τ is friction velocity defined as ffiffiffiffiffiffiffiffiffiffi τ w =ρ p or approached by U ∞ C F =2 ðÞ 1=2 ; τ w is the shear stress magnitude, where to get it is necessary to do iterative calculations against C F R . The roughness function ΔU þ is from Cebeci and Bradshaw [39] in Eq. (7), according to this study.
The verification result using the similarity scaling from Granville [43] can be seen in Table 4. The calculation uses e C; G , as described in Eq. (17). From the results of the calculation of e C; G , it can be concluded that CFD modeling for homogeneous roughness can be accepted with the difference in error against the empirical is not exceed 1.8%.
Log 10 Re r À 1:729 ÀÁ 2 (14) The homogeneous roughness simulation results were also verified using other literature from Granville [45], as written in Eqs. 18 and 19. This method can predict the characteristic roughness function, ΔU þ ¼ fk s þ ÀÁ , by plotting the predicted value of ΔU þ (Eq. (19)) against k s þ (Eq. (18)) with the difference in the overall drag results from the rough conditions (C F R ) to the smooth conditions (C F S ). Where, L is the plate length, Re is the Reynolds number, k s is the roughness height, and ΔU þ 0 is the roughness function slope, which is the slope of ΔU þ as a function of ln k s þ ÀÁ . The verification results are plotted in Figure 3 and the simulation results successfully approached the planned roughness function model, namely from Cebeci and Bradshaw [39], with C s = 0.253. Verification of the simulation results using Granville [45] method described in Figure 3. The results collapse on the roughness function used (Cebeci and Bradshaw [39]).

Results and discussion
A systematic analysis of the results from the inhomogeneous rough surface cases is given in this section. To study the roughness effects, the local (c f ) and overall (C F ) skin friction coefficients are calculated for both the homogeneous and inhomogeneous roughness cases. The effects from the roughness height and the roughness sequence in the streamwise direction are studied by analyzing the plots of local skin friction coefficient as a function of the length and plotting the mean velocity profile for the step up and step-down phenomenon and by comparing its integral values (C F ) for the different cases. We also calculate the skin friction coefficient percentage differences between rough surfaces (both homogeneous and inhomogeneous) and the smooth wall reference case and between inhomogeneous roughness cases (combination of PQR) and the homogeneous roughness reference case (i.e., QQQ). Lastly, we carried out the prediction of how the single k s value of the homogeneous case that equal to the three different k s that composed the inhomogeneous case. ΔU þ and k þ s for homogeneous cases that calculated using Granville [45] compared with the used roughness function, and other roughness functions, Colebrook-type roughness function [46], and Schultz and Flack [47].

Local skin friction c f
The local skin friction coefficient c f is defined in Eq. (20). Where τ w is the wall shear stress (obtained from CFD simulation), ρ is the fluid density and U ∞ is the free stream velocity. The streamwise length x and lateral position y for the inhomogeneous RPQ case are plotted in Figure 4. Generally, c f is plotted against Re x , but in this case, Re x is represented by x (streamwise distance) because we want to study the step-up and step-down phenomena. The factors of streamwise length (L) and freestream velocity (U ∞ ), which are components of the Re x value, have different effects on the increase in friction resistance [48]. Figure 4 shows that the rough-wall homogeneous cases have a higher c f than that of the smooth wall case at the same position on the streamwise. This value indicates that a rough wall surface indeed deviates from the smooth wall case and increases skin friction drag [19]. Within the homogeneous rough wall cases, the plots show that the highest k s case (RRR) has a higher c f value than those of the lower k s cases (Q and P respectively) at equal place. Such behavior shows that a rougher surface will experience an elevated wall drag compared with less rough surfaces. The four homogeneous cases (including the smooth wall case) show a similar monotonic decrease in c f with increasing x. This classical result of exponential decrease of c f with x or Re x illustrates large friction near the leading edge of the plate (low Reynolds numbers), which decreases exponentially towards the trailing edge. Similar behavior has been reported in various experimental and numerical studies [15,49].

Homogeneous and inhomogeneous roughness
For the inhomogeneous cases with step changes in the equivalent sand grain roughness height k s (PQR, PRQ , QPR, QRP, RPQ and RQP), the c f values show step responses following the step-change in k s . For example, Figure 4a with PQR step changes case show an increase in c f every time there is an increase from P to Q and from Q to R height. Figure 4a shows that in the first one-third part of the inhomogeneous rough plate (0 m < x < 10 m) with k s value of 81.25 mm (P), the inhomogeneous case line (solid red line) collapses with the homogeneous PPP case (yellow dotted line) well. However, as the inhomogeneous case arrives at the start of the second one-third part of the plate (10 m < x < 20 m), where it has k s value of 325.00 μm (Q), the red line slightly jumps over (overshoots) the homogeneous QQQ case represented with a green dashed-dotted line, and then the red line gradually falls onto the homogeneous QQQ case. Finally, the last one-third part of the inhomogeneous rough plate (20 m < x < 30 m) has k s value of 568.75 μm (R), and the plot clearly shows that the red line slightly overshoots the homogeneous RRR case (dashed blue line) in the first few x and then gradually collapses to the homogeneous RRR case. Similar behavior is observed for all of the other five inhomogeneous roughness combinations (Figure 4b-f).
Such a jump in c f , Andreopoulos and Wood [50] have reported values between one surface profile to another surface profile. They measured the response of a smooth wall boundary layer to a perturbation/disturbance caused by a short sandpaper strip. The measured τ w behind the strip was around three times the undisturbed value (fully smooth case). The sudden jump in c f is followed by a 11 The Phenomenon of Friction Resistance Due to Streamwise Heterogeneous Roughness with… DOI: http://dx.doi.org/10.5772/intechopen.99137 relaxation, where the c f is slowly returning to the smooth wall value. The relaxation rate was found to be very slow and Andreopoulos and Wood [50] were unable to record any full recovery, even at the last measuring point. As observed by Andreopoulos and Wood [50], our CFD results of local c f shown in Figure 9 also exhibit a slow relaxation rate. However, RANSE cannot pick up the small-scale turbulence structures near the wall that occur at the border between the two roughness zones, influencing the flow downstream. The wall model cannot fully capture the flow physics, but it provides us with some indications of the effect. Figure 4 indicate that the c f value of the inhomogeneous rough surface will recover the underlying homogeneous rough wall c f further downstream if the distance is sufficiently long. To quantify this, an averaged overshoot/undershoot will be defined and calculated in the following sub-subsection.
To see what happens to the overshoot and undershoot phenomena, the mean velocity profile of the difference c f values are plotted in Figure 5. The mean velocity profile plot was taken at a distance of 10.25 m from the leading edge. The step-up case where the c f value overshoot was taken in the PRQ (see Figure 4c) and RRR cases, while the step-down case, where the c f value was undershot, was taken in the RPQ (see Figure 4f) and PPP cases. The outer scaling method is used to compare the mean velocity of the profile, where y is the vertical distance from the wall, δ is the thickness of the boundary layer taken 0.99 U ∞ , U ∞ is the free stream velocity, and U is the velocity at each y. The plot results show that in the roughness step-up where overshoot occurs, the velocity profile is shifted upward (see Figure 5a). Conversely, in the roughness step-down, where there is an undershoot of the cf. value, the velocity profile is shifted downwards (see Figure 5b).

Overshoot and undershoot percentage differences
The overshoot and undershoot height of the flow seems to be based on the k s of the following roughness. For example, when we look into the cases PQR and PRQ in the first row of Figure 5a and 5b, the jump from P to Q is lower than that from P to R, resulting in a lower overshoot from P to Q than that from P to R. This also leads to a faster settling time for the P to Q jump than that for P to R case. Such behavior happens because R corresponds to a much higher k s value than that corresponds to the Q case. Such undershoot and overshoot raise a question regarding how much is

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The Phenomenon of Friction Resistance Due to Streamwise Heterogeneous Roughness with… DOI: http://dx.doi.org /10.5772/intechopen.99137 the difference in c f between the homogeneous and inhomogeneous cases. To answer such a question, a percent error e i,h is defined between the areas under the c f curves for the inhomogeneous and homogeneous cases described in Eq. (21), where the subscripts h and i refer to homogeneous and inhomogeneous cases, respectively. The integral boundaries and the surface roughness for the homogeneous and inhomogeneous cases correspond to each other. For example, e i,h for the inhomogeneous case QPR is calculated as described in Eq. (22).
Eq. (21) shows that if the boundary layer responded to the step-change instantly and there is no overshoot/undershoot from the homogeneous roughness curve, e i,h would be zero. A positive value of e i,h means that on average, there is an overshoot while a negative value of e i,h means that there is an undershoot relative to the corresponding homogeneous curves. The values of e i,h are tabulated in Table 2. Table 2 shows cases with decreasing magnitude of overshoot in the following order: PQR > PRQ > QPR and cases with decreasing magnitude of undershoot in the following order: RQP > RPQ > QRP. A consistent trend is observed in all the cases with different plate lengths. The most significant averaged overshoot (1.87%) and undershoot (1.14%) are observed (PQR and RQP, respectively).

Overall skin friction C F
Following the local skin friction analysis from the previous subsection, it is desirable to estimate the overall skin friction coefficient C F of the plates. This allows us to see the influence of individual roughness height k s or the combination of it in a more global way. The overall skin friction coefficient C F is given in Eq. (9). It is related to the local skin friction coefficient c f (x) by the relation described in Eq. (23).
The corresponding overall skin friction coefficients C F s for the plate segments 1, 2, and 3 are given, respectively, as described in Eq. (24). Where x is the distance in the streamwise direction with the origin at the leading edge of the plate. The lengths of plate segments 1, 2, and 3 are denoted as L 1 , L 2 and L 3 , respectively. Table 2 summarizes the overall C F and those for each plate segments (C F;1 , C F;2 , C F;3 ). Table 2 shows that for the homogeneous cases, both the smooth SSS and the three rough cases (PPP, QQQ , and RRR), the overall friction coefficient C F decreases as the flow move from the upstream to downstream (C F;1 > C F;2 > C F;3 ). The RRR case has the largest C F among the three homogeneous roughness cases due to its highest k s value.

Quantification of the overall skin friction between rough surface and smooth surface
Having obtained the overall skin friction C F from individual plates, quantification the change in drag penalty between one case to another is made in a more simplified way. The first analysis we are interested in quantifies the roughness wall effects (both homogenous and inhomogeneous cases) on the overall skin friction relative to the smooth wall case. A percent increase in overall skin friction e r; s due to roughness effects is defined as described in Eq. (24). The subscripts r and s refer to rough and smooth, respectively. The results are tabulated in Table 2. Table 2 shows that the RRR case results in the highest e r; s due to the highest k s with a value of 75.23%. For the same reasons, the smallest e r; s resulted from the H240_PPP case, with a value of 18.65%. It is interesting to note that the homogeneous H120_QQQ case with k s = 325 μm experienced an increase in drag penalty of 52.62% compared to the smooth wall case. Such a value of roughness height represents heavy slime [16] or fouled with light calcareous tube-worm fouling [51].
A similar rough and smooth wall e r; s analysis is also conducted for the inhomogeneous cases. The results show that the homogenous QQQ case (with k s = 325 μm has a higher percent increase in overall skin friction e r; s than the inhomogeneous cases. Although the averaged roughness heights for the inhomogeneous cases are the same as the QQQ roughness height, their representative roughness heights are smaller than the QQQ roughness height and depend on the sequence roughness heights in the streamwise direction. It is observed that the values of e r; s for the inhomogeneous cases increase monotonically in the following order: PQR < PRQ < QPR < QRP < RPQ < RQP.

Quantification of the overall skin friction between inhomogeneous and homogeneous rough surface
Apart from looking at the percent increase in overall skin friction e r; s between the rough wall and smooth wall, it is also desirable to quantify the effects of roughness inhomogeneity (combination of PQR) on the overall skin friction C F with respect to the homogeneous QQQ baseline case. For that purpose, a percent decrease between an inhomogeneous roughness case and the homogeneous QQQ case is defined in Eq. (25). e i; q ¼ C F; i À C F; QQQ C F; QQQ Â 100% The subscript i refers to inhomogeneous roughness (variation of PQR) while the subscript QQQ refers to the homogenous rough wall base case. As has been noted above, the friction coefficient C F; QQQ is chosen as a reference because the arithmetic average of k s for the inhomogeneous cases is equal to that of the homogeneous QQQ case. Calculating skin friction from measured surface roughness would normally use a single roughness value which generally comes from the average of measurements over the hull. Thus, e i; q represents the error of assuming a single (average) roughness value for an in-homogeneously rough hull. A negative value of e i; q indicates that the C F values of the inhomogeneous cases (combination of PQR) are lower than the homogeneous base case (QQQ). The opposite is valid for a positive value of e i; q . Table 2 shows that the e i; q values are negative for all the inhomogeneous cases, indicating that the C F values of all the inhomogeneous cases (combination of PQR) are lower than the homogeneous base case (QQQ). For the same plate length, the magnitude of e i; q decreases orderly in the sequence of PQR to RQP, indicating that the order of roughness arrangement plays a key role. the inhomogeneous roughness pattern in a simplified manner using step changes in the equivalent sand grain roughness heights k s , denoted as P, Q , and R.
Some of the findings from this study include: • Such a step-change in the roughness height results in overshoot/undershoot of the local skin friction coefficient c f , followed by a relaxation where the c f value is slowly returning to the underlying rough wall homogeneous c f value. A step up in k s results in an overshoot, while a step down in k s results in an undershoot in the c f values. In some cases where the jump happens over two significantly different k s values (i.e., P to R), the relaxation rate is very slow and unable to recover over the given streamwise distance fully. Cases with decreasing magnitude of overshoot are found in the following order: PQR > PRQ > QPR, and cases with decreasing magnitude of undershooting are in the following order: RQP > RPQ > QRP.
• The sequence of roughness arrangement in a streamwise inhomogeneous roughness pattern plays a key role in the resulting overall skin friction coefficient C F . It is found that C F increases in the following order: PQR < PRQ < QPR < QRP < RPQ < RQP. This result is further reflected in the predicted k s values, showing k s increases in the same order: PQR < PRQ < QPR < QRP < RPQ < RQP. A change of roughness near the leading edge of the plate has a much more significant effect on the overall skin friction coefficient C F than a change of this near the trailing edge. In practical terms, limiting (cleaning) the biofouling from the bow of a ship is of greatest benefit and should be prioritized.
• The overall skin friction (C F ) in inhomogeneous cases (e.g. PQR) is smaller than the k s value with the same mean. Thus, the mean value of k s for the inhomogeneous case is less suitable for predicting the C F value. A new way to predict the k s value for the inhomogeneous case has been proposed in this paper (See Eq. (26)), where the value is very close to the simulation results that have been carried out.