# Advances in Riblets Design
## 7 Pressure Gradients in Riblet Surfaces  
Measuring the performance of riblets under the influence of different pressure gradients in flat plates [16; 55; 59; 95; 96; 101; 132] and curved surfaces such as those in aircraft [70; 96] has been previously studied. The Clauser parameter (\(\beta\)) [94] has been defined to evaluate the effect of varying pressure gradients on the surfaces, as defined as follows:  
\[\beta=\frac{\delta^{*}}{\tau_{w}}\frac{dp}{dx} \tag{2}\]  
In which \(\delta^{*}\), \(\tau_{w}\) and dp/dx represent boundary layer displacement thickness, wall shear stress, and streamwise pressure gradient, respectively. Zero amount of Clauser represents ZPG, and negative and positive values represent Favorable Pressure Gradient (FPG) and Adverse Pressure Gradient (APG), respectively. [PERSON] et al. [101] showed the positive effects of riblets for both FPGs and APGs by testing flat plates in a wind tunnel in which riblets were applied to both sides of the plate. The results showed a reduction in momentum thickness in the grooved surface under positive values of \(\beta\), but it was reduced by increasing the Reynolds number toward \(5\times 10^{5}\) and \(\beta\) values toward nine. [PERSON] et al. [132] conducted tests with Clauser parameters above 5 with trapezoidal riblets. Their results did not show sensible change compared with a smooth surface. In the meantime, [PERSON] et al. [55] indicated the effectiveness of APG on the trapezoidal riblets by concluding that there is almost always a reduction in drag for almost all pressure gradients, except for higher values of the Clauser parameter.  
The effect of semi-circular riblets in ZPG and mild APG flows was numerically studied over an airfoil with a high-resolution LES method by [PERSON] et al. [59]. Simulations were carried out by changing \(s^{+}\) and then comparing the results with [PERSON] et al. [23] and [PERSON] et al. [35]. Their simulations indicated only a small improvement in \(\beta=0.5\) (see Figure 10), showing a moderate reduction in drag. This result was confirmed by [PERSON] et al. [62], which concluded that at mild APGs, the drag reduction increased by about 7%.  
[PERSON] et al. [96] investigated the relation of riblet drag reduction with \(s^{+}\), and different pressure gradients by use of the Clauser parameter in flat plates and the TRA2022 conceptual aircraft. The study indicated the effect of pressure gradient based on the non-dimensional diameter of vortices in the different Clauser values as a function of normalized wall units and the impeding mechanism of longitudinal vortices. The drag reduction effect decreased for higher values of \(\beta\) and \(s^{+}\), respectively, while it increased for lower values of \(\beta\) and higher values of \(s^{+}\). Numerical simulations on the surface of the TRA2022 aircraft showed a small portion of the aircraft surface with \(\beta\approx 2.0\) in the trailing edge position of the wing (roughly 15% of the total suction surface), as shown in Figure 11, while almost all of the aircraft surfaces had near zero pressure gradient. By comparing the original with the riblet surface, they estimated a total drag reduction of about 2%. As the friction drag was about 50% of the total drag, a 4% reduction in friction drag is roughly the same as the effectiveness of riblets in the flat plate with a ZPG. The study concluded that APG has less effect on drag reduction when all of the aircraft surface is covered by riblets.  
Figure 11: Distribution of Clauser parameter on TRA2022 aircraft [96].  
Figure 10: Simulation of an airfoil in ZPG and APG with different \(s^{+}\)[59].