## 3 Validation
### NACA 0012 Airfoil  
Testing progresses to consider the two-dimensional flow around a NACA 0012 airfoil; this test case remains highly simplified, but brings in pressure drag, and thus lends itself to studying the effect of riblets on this non-frictional drag component. Unfortunately, little information is available for validation. Once again, the airfoil is assumed to be fully covered by riblets of locally optimal size, i.e., ensuring \(l_{g}^{+}=10.5\) everywhere. Figure 4 shows the mean velocity profile on the upper surface of the airfoil at \(x/c=0.5\), for two different angles of attack, namely 0 and 4 degrees. The expected unitary upward shift \(\Delta U^{+}\) due to the grooves is consistently observed. At \(\alpha=4^{\circ}\), experimental data are available from ref. [34], and the present results appear to agree with them. However, the agreement is less satisfactory at other incidences. It must be mentioned that experiments were carried out with riblets of constant physical size, with a size that is about 10 viscous lengths. The percentage total drag reduction at \(\alpha=0^{\circ}\) is measured to be 7%, which is in agreement with the experimental data from ref. [34,35], as well as with CFD results obtained by [PERSON] and [PERSON] in [32]. Our data indicate only a mild variation of drag reduction with the angle of attack, while ref. [35] mentions an increase beyond 16% at \(\alpha=0^{\circ}\) and a sudden drop to zero at \(\alpha=10^{\circ}\). The lower part of Figure 4 compares the evolution of the skin-friction coefficient along the airfoil, and shows very good agreement with the same quantity taken from ref. [32] (except for the precise location of the transition region). The pressure coefficient \(C_{p}\) and the difference \(\Delta C_{p}=C_{p,0}-C_{p}\) at \(\alpha=4^{\circ}\) are shown in Figure 5. Changes are visible, to attest once again the effect of riblets on the pressure  
Figure 3: Mean velocity profile with/without riblets over the flat plate, at \(Re_{x}=5\times 10^{5}\) (**left**) and \(Re_{x}=9\times 10^{5}\) (**right**). The riblets profile consistently shows the upward shift of \(\Delta U^{+}=1\) on the logarithmic region. Figure 2: Zero-pressure-gradient boundary layer over a flat plate. **Left**: evolution of the skin-friction coefficient with/without riblets, and comparison with data from ref. [15]; **right**: change along the plate of the slip length (**left**\(y\) axis) and riblets height (**right**\(y\) axis), in dimensional units. distribution along the airfoil. Changes in the expansion peak at the leading edge and on the pressure recovery at the trailing edge due to riblets provide a significant additional contribution to drag reduction; form drag is reduced by 7.7%, adding to the friction reduction of 6.7%. These results agree with several findings by [PERSON] and [PERSON] (2018), who interpreted the reduced form drag by observing that riblets change the flow field by making it more similar to the inviscid solution, where the slip length is infinite, and form drag is exactly zero. Moreover, the altered pressure distribution leads to a larger lift in all tested incidences; this effect, which will be discussed later when discussing the full aircraft, is important for the reduction of the total drag: since the aircraft in cruise always needs the same lift, increased aerodynamic efficiency implies a reduced angle of attack in cruise, thus bringing in an additional contribution to drag reduction. Figure 4: Mean velocity profiles (**top**) and skin-friction coefficient (**bottom**) for the NACA0012 airfoil, at an incidence of \(\alpha=0^{\circ}\) (**left**) and \(\alpha=4^{\circ}\) (**right**). The **top row** shows the mean profile over the suction side in law-of-the-wall form at \(x/c=0.5\), and compares with data from ref. ([PERSON] et al., 2019). The **bottom row** plots the evolution of the friction coefficient along the chord, and compares with data from ref. ([PERSON] et al., 2019), represented with symbols.