*5.2.3 Prediction of a representative roughness height for an inhomogeneous rough surface*

Having discussed the local and overall skin frictions, a question may arise: "Can we predict a representative roughness height *ks* for an inhomogeneous rough surface?" We do this prediction with the help of the method from Granville [45], namely Eqs. 18 and 19, as well as the minimum error optimization help iterated by the solver to calculate the portion of each segment, *L*1, *L*2, and *L*3. The result of this work creates Eq. (26), where it is found that the segment *L*<sup>1</sup> has a more significant portion, namely 29.7%. The *L*<sup>2</sup> segment has a portion of 24.3% and L3 ≈ 23.1%, where if all the portion of segments are totaled, the value will be smaller than the average value of the inhomogeneous *ks* (see Eq. (27)), *ks*; *<sup>i</sup>*; *<sup>p</sup>* <*ks*; *<sup>i</sup>*; *<sup>a</sup>*.

$$k\_{\rm r;i;p} \approx 29.7\% \cdot k\_{\rm r;1} + 24.9\% \cdot k\_{\rm r;2} + 23.1\% \cdot k\_{\rm r;3} \tag{26}$$

$$k\_{s;i;a} = \frac{k\_{s;1} + k\_{s;2} + k\_{s;3}}{3} \tag{27}$$

The verification of the equation for predicting the *ks* value for the inhomogeneous case is done with the aid of the Granville method [45], which plots the Δ*U*<sup>þ</sup> and *k*<sup>þ</sup> *<sup>s</sup>* , as explained in Eqs. (18 and 19), respectively. The results of the verification are plotted in **Figure 6**, where not only the results of the calculation *ks*; *<sup>i</sup>*; *<sup>p</sup>*, but also the results of the calculation of *ks*; *<sup>i</sup>*; *<sup>a</sup>* are plotted. From the plot results, it can be seen that, prediction using *ks*; *<sup>i</sup>*; *<sup>p</sup>* got a very good match to the roughness function used in this simulation, namely from Cebeci and Bradshaw [39].
