**2. 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, *Ui* is the mean velocity component, *P* is the mean pressure, *ρ* is the fluid density, *ν* is the kinematic viscosity, *u*<sup>0</sup> *<sup>i</sup>* is the fluctuating velocity component, and *ui* 0*uj* <sup>0</sup> is the Reynolds stresses [37].

$$\frac{\partial U\_i}{\partial \mathbf{x}\_i} = \mathbf{0} \tag{5}$$

$$\frac{\partial U\_i}{\partial t} + \frac{\partial \left(U\_i U\_j\right)}{\partial \mathbf{x}\_j} = -\frac{1}{\rho} \frac{\partial P}{\partial \mathbf{x}\_i} + \frac{\partial}{\partial \mathbf{x}\_j} \left[\nu \left(\frac{\partial U\_i}{\partial \mathbf{x}\_j} + \frac{\partial U\_j}{\partial \mathbf{x}\_i}\right)\right] - \frac{\partial \left(\overline{u\_i u\_j}\right)}{\partial \mathbf{x}\_j} \tag{6}$$

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 *ks*. The generalized Cebeci and Bradshaw's roughness function model is given in Eq. (7), where *A* = 0, *k*<sup>þ</sup> *<sup>s</sup>*; *smooth* = 2.25, *k*<sup>þ</sup> *<sup>s</sup>*;*rough* = 90.00 and *Cs* = 0.253. In which the power *a* is given in Eq. (8).

$$
\Delta U^{+} = \begin{cases} 0 & \to k\_{s}^{+} \le k\_{s;smooth}^{+} \\\\ \frac{1}{\kappa} \ln \left[ A \left( \frac{k\_{s}^{+} / k\_{s;smooth}^{+}}{k\_{s;rough}^{+} - k\_{s;smooth}^{+}} \right) + C\_{s} k\_{s}^{+} \right]^{a} & \to k\_{s;smooth}^{+} < k\_{s}^{+} \le k\_{s;rough}^{+} \\\\ \frac{1}{\kappa} \ln \left( A + C\_{s} k\_{s}^{+} \right) & \to k\_{s}^{+} > k\_{s;rough}^{+} \end{cases} \tag{7}
$$

*The Phenomenon of Friction Resistance Due to Streamwise Heterogeneous Roughness with… DOI: http://dx.doi.org/10.5772/intechopen.99137*

$$a = \sin\left[\frac{\pi}{2} \frac{\log\left(k\_s^+/k\_{s;smooth}^+\right)}{\log\left(k\_{s;rough}^+/k\_{s;smooth}^+\right)}\right] \tag{8}$$

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�<sup>5</sup> 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/m3 and dynamic viscosity μ = 0.001077 kg/(ms). Finally, the free stream velocity is set at *U*<sup>∞</sup> = 9.77 m/s (19 knots).

*The flow domain used in the simulations with three plate segments at the bottom boundary (a), and the hexahedron mesh with an exponential cell height gradation near the wall bottom boundary (b).*

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*<sup>þ</sup> ¼ *uτy=ν*, where *y* is the outward wall-normal coordinate. To model the roughness effects correctly, the *y*<sup>þ</sup> value for the first cell center above the wall must be larger than the local roughness Reynolds number *k*<sup>þ</sup> *<sup>s</sup>* . To ensure this condition is always satisfied, ANSYS Fluent will virtually shift the wall if *y*<sup>þ</sup> <*k*<sup>þ</sup> *<sup>s</sup>* . For the roughness cases considered in this study, a blockage effect of 50% of the roughness height is assumed, and the corrected *y*<sup>þ</sup> value for the first cell center above the wall is given as *y*<sup>þ</sup> ¼ *y*<sup>þ</sup> þ *k*<sup>þ</sup> *<sup>s</sup> =*2. In this way, the singularity issue is avoided, and fine meshes can be handled correctly.

### **3. Surface roughness modeling**

In this study, we will investigate just single parameter variations, namely the roughness height *ks*. This section will explain details of the *ks* set up and their possible combinations. Four surface roughness with different *ks* 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 *ks* values are specifically chosen so that the average height of three different *ks* 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 *ks* 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].

#### **Figure 2.**

*Combinations of three plate segments resulting inhomogeneous (including fully smooth) and inhomogeneous rough surface conditions.*

*The Phenomenon of Friction Resistance Due to Streamwise Heterogeneous Roughness with… DOI: http://dx.doi.org/10.5772/intechopen.99137*
