## 3 Data Reduction Methodologies
### Application of Response Surface Methodology (RSM) to Flight Data  
As mentioned above, such flow features as boundary layer profiles around the aircraft airframe cannot always be the same at a targeted flight condition due to the difference in remaining fuel, the resultant difference at the angle of attack, and atmospheric air conditions for each flight. This situation leads to difficulty in evaluating such testing devices as the riblets aimed at reducing skin friction drag. For example, in the case of evaluating the reduction of skin friction drag by using the riblets, a comparison can be made between a case not using riblets and a case using riblets. The datasets of both cases would be obtained at different flights. In order to properly compare a dataset (e.g., boundary layer profile) obtained at different flights by eliminating the influence of the flight conditional difference, an approach was taken to interpolate the data by using response surface methodology. RSM consists of a group of mathematical and statistical techniques for empirical model building. The objective of RSM is to optimize a response or an output variable to some influential independent variables (input variables). RSM was originally developed to model experimental responses [40; 41], but its application field was later expanded to the modelling of numerical experiments. An inaccuracy of RSM can be defined as measurement errors. The following describes the sequence of the interpolation approach conducted in this study based on RSM. 1. First, flight experimental data obtained with a smooth wall surface on the aircraft airframe without involving any flow control device such as riblets were considered as reference data to be compared. The dataset for the reference case (as well as other flight cases) contains pitot pressure data and flight condition data such as Mach number, Reynolds number, etc. (hereinafter, flight data) measured by ADS onboard. The pressure coefficient based on pitot pressure divided by flight dynamic pressure was considered the primary parameter of interest and is expressed as follows: \[C_{p}=\frac{P_{pitot_{ref}}-P_{\infty_{ref}}}{\frac{1}{2}\cdot\gamma P_{\infty_ {ref}}\cdot M_{\infty_{ref}}^{2}}\] (4)  
Figure 5: Sensitivities of flight conditions on \(s^{+}\): (a) angle of attack; (b) yaw angle, (c) pressure; (d) temperature. From Equation (4), the pitot pressure at each pitot-probe is obtained as follows:  
\[p_{pitot_{ref}}=C_{p_{ref}}\cdot\left(\frac{1}{2}\gamma p_{\infty_{ref}}M_{ \infty_{ref}}^{2}\right)+p_{\infty_{ref}} \tag{5}\]  
Based on Equation (5) and assuming the isentropic flow process, the Mach number at each pitot-probe is obtained by Equation (6) by further assuming that the pitot pressure at each pitot-probe is considered to be the local total pressure at each probe tip. Here, an assumption was made that no total pressure loss was involved in the shock wave formation in front of each pitot-probe, since the entire flowfield is considered to be subsonic in accounting for flight speed conditions. \[M_{pitot_{ref}}=\sqrt{\left(\frac{2}{\gamma-1}\right)\left\{\left(\frac{p_{ pitot_{ref}}}{p_{\infty_{ref}}}\right)^{\frac{\gamma-1}{\gamma}}-1\right\}} \tag{6}\]  
Rewriting Equation (6) in terms of \(U\)-velocity yields the \(U\)-velocity (\(U_{pitot}\)) at each pitot-probe as Equation (7). \[U_{pitot_{ref}}=M_{pitot_{ref}}\cdot\sqrt{\gamma\cdot R\cdot T_{0\infty}\cdot \left(1+\frac{\gamma-1}{2}M_{pitot_{ref}}^{2}\right)^{-1}} \tag{7}\]  
Similar to the reference case, the pressure coefficient for the compared case, \(C_{p_{com}}\), is created by pitot pressures and flight data for the compared case (Equation (8)). \[C_{p_{com}}=\frac{p_{pitot_{com}}-p_{\infty_{com}}}{\frac{1}{2}\gamma p_{ \infty_{com}}M_{\infty_{com}}^{2}} \tag{8}\]  
Assuming that the pressure coefficient at each pitot-probe can be expressed by using some measured variables yields an expression of \(C_{p_{com}}\) shown as Equation (9).