**5.2 Overall skin friction** *CF*

Following the local skin friction analysis from the previous subsection, it is desirable to estimate the overall skin friction coefficient *CF* of the plates. This allows us to see the influence of individual roughness height *ks* or the combination of it in a more global way. The overall skin friction coefficient *CF* is given in Eq. (9). It is related to the local skin friction coefficient *cf* (*x*) by the relation described in Eq. (23).

$$\mathbf{C}\_{F} = \frac{\int\_{0}^{L} c\_{f}(\mathbf{x})d\mathbf{x}}{L} \tag{23}$$

The corresponding overall skin friction coefficients *CFs* 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*<sup>2</sup> and *L*3, respectively.

**Table 2** summarizes the overall *CF* and those for each plate segments (*CF*; 1, *CF*; 2, *CF*; 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 *CF* decreases as the flow move from the upstream to downstream (*CF*; 1 > *CF*; 2 > *CF*; 3). The RRR case has the largest *CF* among the three homogeneous roughness cases due to its highest *ks* value.

#### *5.2.1 Quantification of the overall skin friction between rough surface and smooth surface*

Having obtained the overall skin friction *CF* from individual plates, quantification the change in drag penalty between one case to another is made in a more

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

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*; *<sup>s</sup>* 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**.

$$\mathbf{e}\_{r,s} = \frac{\mathbf{C}\_{F\_r} - \mathbf{C}\_{F\_\sharp}}{\mathbf{C}\_{F\_\sharp}} \times \mathbf{100\%} \tag{24}$$

**Table 2** shows that the RRR case results in the highest e*r*; *<sup>s</sup>* due to the highest *ks* with a value of 75.23%. For the same reasons, the smallest e*r*; *<sup>s</sup>* resulted from the H240\_PPP case, with a value of 18.65%. It is interesting to note that the homogeneous H120\_QQQ case with *ks* = 325 *μ*m 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*<sup>r</sup>*; *<sup>s</sup>* analysis is also conducted for the inhomogeneous cases. The results show that the homogenous QQQ case (with *ks* = 325 *μ*m has a higher percent increase in overall skin friction e*<sup>r</sup>*; *<sup>s</sup>* 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*<sup>r</sup>*; *<sup>s</sup>* for the inhomogeneous cases increase monotonically in the following order: PQR *<* PRQ *<* QPR *<* QRP *<* RPQ *<* RQP.
