**5.1 Local skin friction c***<sup>f</sup>*

The local skin friction coefficient c *<sup>f</sup>* is defined in Eq. (20). Where *τ<sup>w</sup>* is the wall shear stress (obtained from CFD simulation), *ρ* is the fluid density and *U*<sup>∞</sup> is the free stream velocity. The streamwise length *x* and lateral position *y* for the inhomogeneous RPQ case are plotted in **Figure 4**. Generally, c *<sup>f</sup>* is plotted against *Re <sup>x</sup>*, but in this case, *Re <sup>x</sup>* is represented by *x* (streamwise distance) because we want to study the step-up and step-down phenomena. The factors of streamwise length (*L*) and freestream velocity (*U*∞), which are components of the *Re <sup>x</sup>* value, have different effects on the increase in friction resistance [48].

$$\mathbf{c}\_f = \frac{\mathbf{r}\_w}{\rho \mathbf{U}\_\infty^2 / 2} \tag{20}$$

#### *5.1.1 Homogeneous and inhomogeneous roughness*

**Figure 4** shows that the rough-wall homogeneous cases have a higher c *<sup>f</sup>* than that of the smooth wall case at the same position on the streamwise. This value indicates that a rough wall surface indeed deviates from the smooth wall case and increases skin friction drag [19]. Within the homogeneous rough wall cases, the plots show that the highest *ks* case (RRR) has a higher c *<sup>f</sup>* value than those of the lower *ks* cases (Q and P respectively) at equal place. Such behavior shows that a rougher surface will experience an elevated wall drag compared with less rough surfaces. The four homogeneous cases (including the smooth wall case) show a similar monotonic decrease in c *<sup>f</sup>* with increasing *x*. This classical result of exponential decrease of c *<sup>f</sup>* with *x* or *Re <sup>x</sup>* illustrates large friction near the leading edge of the plate (low Reynolds numbers), which decreases exponentially towards the trailing edge. Similar behavior has been reported in various experimental and numerical studies [15, 49].

For the inhomogeneous cases with step changes in the equivalent sand grain roughness height *ks* (PQR, PRQ, QPR, QRP, RPQ and RQP), the c *<sup>f</sup>* values show step responses following the step-change in *ks*. For example, **Figure 4a** with PQR step changes case show an increase in c *<sup>f</sup>* every time there is an increase from P to Q and from Q to R height. **Figure 4a** shows that in the first one-third part of the inhomogeneous rough plate (0 m < *x* < 10 m) with *ks* value of 81.25 mm (P), the inhomogeneous case line (solid red line) collapses with the homogeneous PPP case (yellow dotted line) well. However, as the inhomogeneous case arrives at the start of the second one-third part of the plate (10 m < *x* < 20 m), where it has *ks* value of 325.00 *μ*m (Q), the red line slightly jumps over (overshoots) the homogeneous QQQ case represented with a green dashed-dotted line, and then the red line gradually falls onto the homogeneous QQQ case. Finally, the last one-third part of the inhomogeneous rough plate (20 m < *x* < 30 m) has *ks* value of 568.75 *μ*m (R), and the plot clearly shows that the red line slightly overshoots the homogeneous RRR case (dashed blue line) in the first few *x* and then gradually collapses to the homogeneous RRR case. Similar behavior is observed for all of the other five inhomogeneous roughness combinations (**Figure 4b**–**f**).

Such a jump in c *<sup>f</sup>* , Andreopoulos and Wood [50] have reported values between one surface profile to another surface profile. They measured the response of a smooth wall boundary layer to a perturbation/disturbance caused by a short sandpaper strip. The measured *τ<sup>w</sup>* behind the strip was around three times the undisturbed value (fully smooth case). The sudden jump in c *<sup>f</sup>* is followed by a

#### **Figure 4.**

*The distribution of local skin friction coefficient cf along the plate length for the inhomogeneous PQR (a), PRQ (c), QPR (b), QRP (d), RPQ (f), and RQP (e) cases compared with the homogeneous SSS, PPP, QQQ, and RRR cases.*

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

relaxation, where the c *<sup>f</sup>* is slowly returning to the smooth wall value. The relaxation rate was found to be very slow and Andreopoulos and Wood [50] were unable to record any full recovery, even at the last measuring point. As observed by Andreopoulos and Wood [50], our CFD results of local c *<sup>f</sup>* shown in Figure 9 also exhibit a slow relaxation rate. However, RANSE cannot pick up the small-scale turbulence structures near the wall that occur at the border between the two roughness zones, influencing the flow downstream. The wall model cannot fully capture the flow physics, but it provides us with some indications of the effect. **Figure 4** indicate that the c *<sup>f</sup>* value of the inhomogeneous rough surface will recover the underlying homogeneous rough wall c *<sup>f</sup>* further downstream if the distance is sufficiently long. To quantify this, an averaged overshoot/undershoot will be defined and calculated in the following sub-subsection.

To see what happens to the overshoot and undershoot phenomena, the mean velocity profile of the difference *cf* values are plotted in **Figure 5**. The mean velocity profile plot was taken at a distance of 10.25 m from the leading edge. The step-up case where the *cf* value overshoot was taken in the PRQ (see **Figure 4c**) and RRR cases, while the step-down case, where the *cf* value was undershot, was taken in the RPQ (see **Figure 4f**) and PPP cases. The outer scaling method is used to compare the mean velocity of the profile, where *y* is the vertical distance from the wall, *δ* is the thickness of the boundary layer taken 0.99 *U*∞, *U*<sup>∞</sup> is the free stream velocity, and *U* is the velocity at each *y*. The plot results show that in the roughness step-up where overshoot occurs, the velocity profile is shifted upward (see **Figure 5a**). Conversely, in the roughness step-down, where there is an undershoot of the cf. value, the velocity profile is shifted downwards (see **Figure 5b**).

#### *5.1.2 Overshoot and undershoot percentage differences*

The overshoot and undershoot height of the flow seems to be based on the *ks* of the following roughness. For example, when we look into the cases PQR and PRQ in the first row of **Figure 5a** and **5b**, the jump from P to Q is lower than that from P to R, resulting in a lower overshoot from P to Q than that from P to R. This also leads to a faster settling time for the P to Q jump than that for P to R case. Such behavior happens because R corresponds to a much higher *ks* value than that corresponds to the Q case. Such undershoot and overshoot raise a question regarding how much is

#### **Figure 5.**

*Comparison of the mean velocity profile plot at x = 10.25 m from the leading edge with the outer scaling method to see the overshoot phenomenon for the step-up roughness (a), and undershoot for the step-down roughness (b).*

#### *Applications of Computational Fluid Dynamics Simulation and Modeling*

the difference in c *<sup>f</sup>* between the homogeneous and inhomogeneous cases. To answer such a question, a percent error e*i*,*<sup>h</sup>* is defined between the areas under the c *<sup>f</sup>* curves for the inhomogeneous and homogeneous cases described in Eq. (21), where the subscripts *h* and *i* refer to homogeneous and inhomogeneous cases, respectively. The integral boundaries and the surface roughness for the homogeneous and inhomogeneous cases correspond to each other. For example, e*i*,*<sup>h</sup>* for the inhomogeneous case QPR is calculated as described in Eq. (22).

$$\mathbf{e}\_{i,h} = \frac{\int (c\_{f;i} - c\_{f;h}) \,\mathrm{d}\,\mathrm{Re}\_{\mathrm{x}}}{\int (c\_{f;h}) \,\mathrm{d}\,\mathrm{Re}\_{\mathrm{x}}} \times \mathbf{100\%}\tag{21}$$

$$\mathbf{e}\_{i,h} = \frac{\int\_{0}^{L\_{1}} \left(c\_{f;QPR} - c\_{f;\text{QQQ}}\right) \mathrm{d}\,\mathrm{Re}\_{\mathbf{x}} + \int\_{L\_{1}}^{L\_{1}+L\_{2}} \left(c\_{f;\text{QPR}} - c\_{f;\text{PPP}}\right) \mathrm{d}\,\mathrm{Re}\_{\mathbf{x}} + \int\_{L\_{1}+L\_{2}}^{L\_{2}+L\_{3}} \left(c\_{f;\text{QPR}} - c\_{f;\text{RRP}}\right) \mathrm{d}\,\mathrm{Re}\_{\mathbf{x}}}{\int\_{0}^{L\_{1}} c\_{f;\text{QQQ}} \mathrm{d}\,\mathrm{Re}\_{\mathbf{x}} + \int\_{L\_{1}}^{L\_{2}+L\_{3}} c\_{f;\text{PPP}} \mathrm{d}\,\mathrm{Re}\_{\mathbf{x}}} \times 100\,\mathrm{\mathcal{H}} \,\mathrm{d}\,\mathrm{s} \,\tag{22}$$

Eq. (21) shows that if the boundary layer responded to the step-change instantly and there is no overshoot/undershoot from the homogeneous roughness curve, e*<sup>i</sup>*,*<sup>h</sup>* would be zero. A positive value of e*<sup>i</sup>*,*<sup>h</sup>* means that on average, there is an overshoot while a negative value of e*<sup>i</sup>*,*<sup>h</sup>* means that there is an undershoot relative to the corresponding homogeneous curves. The values of e*<sup>i</sup>*,*<sup>h</sup>* are tabulated in **Table 2**.

**Table 2** shows cases with decreasing magnitude of overshoot in the following order: PQR *>* PRQ *>* QPR and cases with decreasing magnitude of undershoot in the following order: RQP *>* RPQ *>* QRP. A consistent trend is observed in all the cases with different plate lengths. The most significant averaged overshoot (1.87%) and undershoot (1.14%) are observed (PQR and RQP, respectively).
