*5.2.2 Quantification of the overall skin friction between inhomogeneous and homogeneous rough surface*

Apart from looking at the percent increase in overall skin friction e*<sup>r</sup>*; *<sup>s</sup>* 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 *CF* 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).

$$\mathbf{e}\_{i;q} = \frac{\mathbf{C}\_{F;i} - \mathbf{C}\_{F;QQ}}{\mathbf{C}\_{F;QQ}} \times \mathbf{100\%} \tag{25}$$

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 *CF*; *QQQ* is chosen as a reference because the arithmetic average of *ks* 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*<sup>i</sup>*; *<sup>q</sup>* represents the error of assuming a single (average) roughness value for an in-homogeneously rough hull. A negative value of e*<sup>i</sup>*; *<sup>q</sup>* indicates that the *CF* 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*<sup>i</sup>*; *<sup>q</sup>*. **Table 2** shows that the e*<sup>i</sup>*; *<sup>q</sup>* values are negative for all the inhomogeneous cases, indicating that the *CF* 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*<sup>i</sup>*; *<sup>q</sup>* decreases orderly in the sequence of PQR to RQP, indicating that the order of roughness arrangement plays a key role.

**Figure 6.** *Plotting the results of the verification of the predicted ks values for inhomogenous cases with ks*; *<sup>i</sup>*; *<sup>p</sup> and ks*; *<sup>i</sup>*; *a.*
