## 1. Introduction

For offshore structures or ships, the surface roughness mostly has a negative effect. In particular, the roughness of the hull surface increases the resistance, thereby causing an increase in fuel consumption and in carbon dioxide emissions. It is difficult to accurately predict the resistance of a ship considering the roughness of the ship surface, even though many studies regarding this have been made over the last half century. The ship surface is smooth after surface painting. As the ship is operating in sea water and over time as shown in Figure 1, surface roughness is increased due to the fouling phenomenon caused by various marine organisms, and partial damage of hull painting surface such as erosion, thereby increasing frictional resistance.

When Computational Fluid Dynamics (CFD) is used to simulate the behavior or phenomena of the ocean structure in consideration of the surface roughness, various effects such as the height of the surface roughness, its distribution, or other factors should be considered. If modeling is performed considering the actual surface roughness, it may be difficult to get converged solutions along with the increase of the computing time due to the excessive number of meshes. Therefore, the surface roughness value can be predicted by considering it as a surface roughness function in a commonly used wall function.

Since the thickness of the boundary layer becomes thinner as the Reynolds number increases in the flat plate boundary layer flow ($\delta \sim L/\sqrt{Re}$), it is necessary to distribute the elements more closely in the adjacent region of the wall surface, thereby increasing the number of elements, simultaneously degrading the mesh quality. The wall function is used as a practical method to solve these problems and to compute efficiently. In the Reynolds-averaged Navier-Stokes (RANS) turbulence model such as the k- model simulated by a wall function, ε it is generally known that the value of the first mesh height from the wall is relatively well predicted when located in the log-law layer.

Many studies concerning the high-Reynolds number turbulent boundary layer have been carried out and substantial results were obtained. Schultz (2004) studied the effect of coating roughness and fouling for ship resistance. Candries (2001) showed the drag and roughness characteristics of marine surface. Candries et al. (2003) measured drag characterics for various coatings. Flack and Schultz (2014) studied roughness effects in the turbulent boundary layer experimentally. Choi and Kim (2010) simulated velocity profile in the boundary layer for various y plus values. Dassler et al. (2010) proposed a model of roughness-induced transition and validated developed model by applying flat plates. Chatzikyriakou et al. (2015) carried out direct numerical simulations (DNS) and large eddy simulations (LES) to study the effect of roughness elements on the wall layer structure. Park et al. (2013) investigated the turbulent boundary layer with wall functions for a ship using open source libraries, and compared the results with a commercial CFD code. However, the correlation between the mesh height and the surface roughness height, in various Reynolds numbers (especially in the high Reynolds number), was not studied.

In this paper, CFD open source libraries (OpenFOAM ver 4.0) were used to simulate the surface roughness of the plate and submerged body (Coder, 1983) depending on various Reynolds numbers, and analyzed the effect for various surface roughnesses in the wall function. For this purpose, simple plate and submerged body (Coder, 1983) were used to analyze the correlation between the velocity law of the velocity distribution in the boundary layer and the height of the first grating and surface roughness in the law of wall logarithm.

## 2. Surface Roughness

### 2.1. Surface roughness function

Schoenherr (1932) proposed the following equation by gathering, summarizing and fitting the results of the friction resistance test of the smooth flat plate:(1)

where *C _{F}* is the friction coefficient, and

*Re*is the Reynolds number. This curve is called the 'Schoenherr curve' or 'ATTC 1947 curve'.

Considering surface roughness compared to smooth flat plate, flow analysis of the plate using the wall law reduces the moment profile and increases the surface frictional resistance by reducing the velocity profile. The general wall function for the mean velocity profile in the interior region of the turbulent boundary layer can be expressed as:(2)

where *U*^{+} is the dimensionless velocity, *κ* is the Von Karman constant and is usually 0.41. *y*^{+} is the dimensionless distance from the wall, B is the correction constant of the wall function at the smooth wall surface, and generally uses a number of about 5.45. The dimensionless velocity (*U*^{+}) is reduced by the influence of the surface roughness function (*ΔU*^{+}), as shown in the following equation:(3)

where the surface roughness function (*ΔU*^{+}) is defined as the surface roughness Reynolds number (${k}^{+}=k{U}_{\tau}/\nu $). Here *U _{τ}* is the friction velocity.

*U*is expressed using the free stream velocity (

_{τ}*U*) and the Reynolds number (Walderhaug, 1986).(4)(5)

_{e}

In order to calculate the resistance considering the boundary layer thickness depending on the Reynolds number during the operation of the ship in which the ship attachment occurs, this equation is used in the present paper. The definition of the Reynolds number used is given:(6)

where x is the length of the plate and longitudinal length of the submerged body (Coder, 1983).

### 2.2. Characteristics of surface roughness function

To consider the surface roughness, the surface roughness function is applied according to various Reynolds numbers to fit the surface scale. The surface roughness can be considered as the roughness Reynolds number (*k*^{+}). The surface roughness has the following characteristics according to the size of the roughness Reynolds number.

That is, in the case of small surface roughness Reynolds number (*k*^{+}<5), it can be defined that the interference due to roughness is considerably small and slowed by the viscosity of the fluid and is called “smooth”. On the other hand, as the surface roughness Reynolds number (*k*^{+}) increases, the viscous force is no longer reduced by the formation of eddy. In this case, the shape resistance and the viscous resistance are related to the total surface resistance. This transition region of the surface roughness is known to 5<*k*^{+}<70. If the surface roughness Reynolds number increases further, the surface friction due to the shape resistance of the surface roughness and the Reynolds number becomes an independent mechanism and the shape resistance of the surface roughness plays a much larger role.

### 2.3. Wall function

Pope (2000) formulated the production of turbulent kinetic energy in the turbulent kinetic energy equation, as follows(7)

where *τ _{w}* is the wall shear stress, ρ is the density, k is the turbulent kinetic energy, and

*y*is the distance from the wall surface to cell center.

_{p}*C*is the empirical constant of 0.09. This production of turbulent kinetic energy is formulated in OpenFOAM as the name of “kqRWallFunction.” Park et al. (2013) was proposed the modified wall function to simulate rapidly changed flow around the bulbous bow of a ship. Therefore, this study selected the wall function proposed by Park et al. (2013) and Park (2014) to simulated the 3D submerged body. The wall function (Park et al., 2013) is expressed as(8)

_{μ}

## 3. Numerical Methods

### 3.1. Roughness description

The height of the surface roughness (*k _{o}*) considered is 12 (

*k*

_{1}), 500 (

*k*

_{2}), 5000 (

*k*

_{3}), 10000 (

*k*

_{4}) and 100000 (

*k*

_{5}) as the surface roughness height, and the CFD analysis results were compared and analyzed with the smooth case ignoring the surface roughness.

### 3.2. Numerical methods

OpenFOAM is a tensorial approach to computational continuum mechanics. It contains various tensor fields and tensorial derivatives required in the FVM technique. The code syntax can be written very similar to the partial differential equations one wish to solve. This paper used 'simpleFoam' solver, which is an incompressible flow solver of 'OpenFOAM 4.0'. The computations are performed to solve mass-conservation, momentum-conservation, and turbulence model equations. The velocity and pressure are coupled using the ‘SIMPLE’ algorithm. The realizable k- model (ε Shih et al., 1995) is selected for the turbulence model. The convection term is discretized using the Van Leer (Van Leer, 1979) limiter in the Total Variation Diminishing (TVD) scheme, and the diffusion term is discretized using the central difference scheme. The Algebraic Multi-Grid (AMG) method (Weiss et al., 1999) is used to increase the convergence of the algebraic equations, and the algebraic equations are solved using Gauss-Seidel iteration.

## 4. Results and Discussion

For a submerged body, Series 58 body is selected due to axisymmetric shape. The submerged body has a central parallel part and a sharp tail shape as shown in Fig. 2. Due to the axisymmetric shape, axi-symmetric computations are performed. The length of the body (L) is 6.283 m and the free stream velocity is 15.9 m/s, thus the Reynolds number is 1.0 × 10^{8} . The domain extent is set to 2L in the nose, 5L in the tail and 3L in the lateral direction, respectively (Song and Park, 2017). For the case of the y+ value of 10, about 720,000 meshes are used in the computation. the grid number with y+ values from 10 to 5000 is represented in Table 1.

The simulations with the Reynolds number of 10^{8} for various y+ values are carried out. As the Reynolds number increases, y+, which satisfies the log layer, increase. Fig. 3 shows the wall shear stress distribution along the body with existing experimental data (Coder, 1983). The y+ values from about 30 to 500 are included in the log law layer, thus the wall function (Park et al., 2013) is used. A somewhat different result is obtained at the larger y+ values above 5000. On the other hand, ordinarily, the y+ uses the wall function above the value of 13. Therefore, in this simulation, when the value of y+ is 10, the result is ambiguous as shown in Fig. 3. However, the result of this computation is not significantly different from other y+ values.

Fig. 4. Friction resistance coefficient for various y+ values. The global friction resistances with the Reynolds number of 10^{8} coefficients are almost the same. However, at a y+ value of 5000, the result is large, indicating that care must be taken when determining the y+ value. As the Reynolds number increases, the thickness of the boundary layer decreases and the coefficient of friction resistance decreases.

Fig. 5 shows the x-velocity and turbulent kinetic energy contours when y+ are 100 and 5,000. For the case with large y+ value, the boundary layer is depicted thick and the turbulence energy are also excessively generated. Thus, it is known that the wall shear stress is excessively predicted.

The simulations with the Reynolds number of 10^{8} for various roughness heights are carried out. Fig. 6 shows the wall shear stress distribution along the body. The roughness heights are varied from 12 to 1,000,000 *μm*. As the surface roughness increases, the wall shear stress increases. In the plate, the results of 12 *μm* and smooth surfaces are the same, but in the submerged body, the result with 12 *μm* is calculated to be large. This is presumed to be a form factor by 3D shape effect. It can be seen that the 3D effect appears largely in the nose where the 3D shape starts. Also, when the roughness is more than 5000 *μm*, it can be confirmed that the wall shear stress distribution converged. In the completely rough region, the roughness is not important.

Fig. 7 shows the friction resistance coefficient for various surface roughnesses. As the surface roughness with the Reynolds number of 10^{8} increases, the frictional resistance increases. The friction resistance coefficient shows the same tendency as the wall shear stress. As the Reynolds number increases, the thickness of the boundary layer decreases and thus the coefficient of friction resistance decreases.

Fig. 8 shows the x-velocity and turbulent kinetic energy contours for the smooth and rough (*k*_{2}= 500 *μm*) surfaces. In the case of large roughness, the x-velocity at the wall decreases and thus, the boundary layer becomes thicker. As the result, the turbulence energy at the surface of the body is predicted to be large.

## 5. Concluding Remarks

In this paper, CFD simulations for simple plate model and submerged body are performed by setting the height of the surface roughness inside the wall function model provided by the open source libraries (OpenFOAM 4.0).

The 3D submerged body, Series 58 body, is simulated. Considering the influence of the y+ value, it is confirmed that the friction resistance coefficient is not computed well when y+ value is 5000. However, it is confirmed that there is no significant difference in the overall frictional resistance values. When y+ value is constant, it is confirmed that the coefficient of the friction decreases as the Reynolds number increases. For the case with large y+ value, the boundary layer is depicted thick and the turbulence energy is also excessively generated. Thus the wall shear stress is excessively predicted. As the surface roughness increases, the frictional resistance of the surface increases and then converges to a constant value. The friction resistance with the small roughness is different from the result with the smooth surface, which is presumed by the influence of the form factor. As the roughness increases, the boundary layer becomes thicker and the turbulence kinetic energy on the surface increases. The change of resistance according to the surface roughness is considered to be the basic data that can be applied to real scale ships and offshore structures in the future.