**4. Verification and validation of modeling results**

#### **4.1 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 *CF* 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 *CF* is defined in Eq. (9), where *D* is the drag per unit width, *τ<sup>w</sup>* is the wall shear stress, *ρ* is the fluid density, *U*<sup>∞</sup> 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 *en*þ1,*<sup>n</sup>* between the lower and higher cell numbers is defined in Eq. (10) [40].

*CF* <sup>¼</sup> *<sup>D</sup> ρU*∞<sup>2</sup> *L=*2 ¼ Ð *L* <sup>0</sup> *τwdx ρU*∞<sup>2</sup> *<sup>L</sup>=*<sup>2</sup> (9)

$$\varepsilon\_{n+1,n} = \frac{C\_F(n+1) - C\_F(n)}{C\_F(n)} \times 100\,\% \tag{10}$$

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 *CF* 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 *en*þ1,*<sup>n</sup>* 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).

#### **4.2 Verification and validation**

In addition to the grid independence tests, further analyses are carried out for varied viscous-scaled wall-normal distance *y*<sup>þ</sup> 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 *CF* 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


#### **Table 1.**

*Friction coefficients CF calculated using increasing number of cells in the simulations for the smooth plate (SSS).*


#### **Table 2.**

*Overall friction coefficients for the plate segments CF*; 1, *CF*; 2, *CF*; 3 *and the entire plate CF, and the percent errors* e*<sup>r</sup>*; *<sup>s</sup>*, e*<sup>i</sup>*; *<sup>q</sup> and* e*<sup>i</sup>*,*h.*


#### **Table 3.**

*Overall friction coefficient CF calculated using different ranges of y*<sup>þ</sup> *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.*

between the CFD result and the 1947 ATTC line and, in a similar manner, between the CFD result and the 1957 ITTC ship-model correlation line to quantify the accuracy of the CFD results. The percent error *e* in the latter case is calculated using Eq. (13).

$$\frac{0.242}{\sqrt{\mathbf{C}\_F}} = \log\_{10}(\text{Re}\,\mathbf{C}\_F) \tag{11}$$

$$C\_F = \frac{0.075}{\left[\log\_{10}(\text{Re}) - 2\right]^2} \tag{12}$$

$$e = \frac{C\_{F; \text{CFD}} - C\_{F; \text{ITTC 1957}}}{C\_{F; \text{ITTC 1957}}} \times 100\text{\%} \tag{13}$$

The results are summarized in **Table 3**, showing *CF* values calculated using different *y*<sup>þ</sup> ranges for the first cell center above the wall targeted in the simulations. **Table 3** shows that for the SSS case, using a *y*<sup>þ</sup> range between 155 and 254 would result in the closest *CF* value to the 1947 ATTC and 1957 ITTC lines with percent errors of *e* = 0.620% and *e* = 0.814%, respectively. A smaller *y*<sup>þ</sup> 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*<sup>þ</sup> range between 64 and 254 will result in an acceptable *CF* value with a maximum magnitude of percent errors of 1.57%

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

when compared with the 1947 ATTC line and 1.76% when compared with the 1957 ITTC line. This result is following the recommended *y*<sup>þ</sup> range in the literature for smooth flat plate CFD simulations between 50 and 300 [42].

To model the roughness effects correctly, the *y*<sup>þ</sup> value for the first cell center above the wall denoted as (Δ*y*þ)1, must be larger than the local equivalent sand grain roughness Reynolds number *k*<sup>þ</sup> *<sup>s</sup>* , i.e., (Δ*y*þ)1> *k*<sup>þ</sup> *<sup>s</sup>* . However, when one employs a fine mesh near the wall, the (Δ*y*þ)1value may have a smaller value than the *k*<sup>þ</sup> *<sup>s</sup>* value, i.e., (Δ*y*þ)1< *k*<sup>þ</sup> *<sup>s</sup>* . If such a case happens, ANSYS Fluent applies a virtual shift of the wall by increasing the value of (Δ*y*þ)1 with an amount *k*<sup>þ</sup> *<sup>s</sup>* /2, such that (Δ*y*þ)1> *k*<sup>þ</sup> *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: *CFR* 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 <sup>r</sup>* is the Reynolds number for calculate the *CFR* using the empirical formula, which equal of the Reynolds number for smooth condition ( *Res*) that is shifted as described in Eq. (15). Then, *κ* is the von Kármán constant; *ks* <sup>þ</sup> is roughness Reynolds number; *ν* is kinematic viscosity; *U<sup>τ</sup>* is friction velocity defined as ffiffiffiffiffiffiffiffiffiffi *<sup>τ</sup>w=<sup>ρ</sup>* <sup>p</sup> or approached by *<sup>U</sup>*∞ð Þ *CF=*<sup>2</sup> <sup>1</sup>*=*<sup>2</sup> ; *τ<sup>w</sup>* is the shear stress magnitude, where to get it is necessary to do iterative calculations against *CFR* . The roughness function *ΔU*<sup>þ</sup> 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 *eC*; *<sup>G</sup>*, as described in Eq. (17). From the results of the calculation of *eC*; *<sup>G</sup>*, 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%.

$$C\_{F\_R} = \frac{0.0795}{\left(\log\_{10}{Re\_r} - 1.729\right)^2} \tag{14}$$

$$Re\_r = Re\_t - \mathbf{10}^{\left(\frac{4U+\kappa}{\ln\left(10\right)}\right)} \tag{15}$$

$$
\Delta U^{+} = f\left(k\_{\varepsilon}^{+}\right) = f\left(\frac{k\_{\varepsilon}U\_{\tau}}{\nu}\right) \tag{16}
$$

$$e\_{C;G} = \frac{C\_{F;GFD} - C\_{F;Gannville}}{C\_{F;Gannville}} \times 100\text{\%} \tag{17}$$

$$k\_s^+ = \left(\frac{k\_s}{L}\right) \left(\frac{\text{Re C}\_{FS}}{2}\right) \left(\sqrt{\frac{2}{C\_{FR}}}\right) \left[1 - \frac{1}{\kappa} \left(\sqrt{\frac{2}{C\_{FR}}}\right) + \frac{1}{\kappa} \left(\frac{3}{2\kappa} - \Delta U^{+'}\right) \left(\frac{C\_{FR}}{2}\right)\right] \tag{18}$$


**Table 4.**

*Overall friction coefficient CF of homogeneous rough condition calculated compared with the similarity law scaling procedure from Granville [43].*

**Figure 3.**

Δ*U*<sup>þ</sup> *and k*<sup>þ</sup> *<sup>s</sup> 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].*

$$
\Delta U^{+} = \left(\sqrt{\frac{2}{\mathbf{C}\_{F}}}\right) - \left(\sqrt{\frac{2}{\mathbf{C}\_{FR}}}\right) - \mathbf{19.7} \left[\left(\sqrt{\frac{\mathbf{C}\_{FS}}{2}}\right) - \left(\sqrt{\frac{\mathbf{C}\_{FR}}{2}}\right)\right] - \frac{\mathbf{1}}{\kappa} \Delta U^{+\prime} \left(\sqrt{\frac{\mathbf{C}\_{FR}}{2}}\right) \tag{19}
$$

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*<sup>þ</sup> ¼ *f ks* <sup>þ</sup> � �, by plotting the predicted value of *ΔU*<sup>þ</sup> (Eq. (19)) against *ks* <sup>þ</sup> (Eq. (18)) with the difference in the overall drag results from the rough conditions (*CFR*) to the smooth conditions (*CFS*). Where, *L* is the plate length, *Re* is the Reynolds number, *ks* is the roughness height, and Δ*U*þ0 is the roughness function slope, which is the slope of Δ*U*<sup>þ</sup> as a function of ln *ks* <sup>þ</sup> � �. 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 *Cs* = 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]).

### **5. 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 (*cf* ) and overall (*CF*) 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 (*CF*) 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 *ks* value of the homogeneous case that equal to the three different *ks* that composed the inhomogeneous case.

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