## 5 Adaptive Wing Design and Manufacturing
### Dimensional Analysis with Buckingham's Theorem  
The \"dimensional analysis method\" solved with the aid of \"Buckingham's theorem\" can be a very powerful tool for understanding complex wind tunnel tests [32; 33]. Buckingham's theorem is used to analyse the relationships and interactions between the fluid flow and the wing prototype during wind tunnel tests. The physical unit of mass (Kg), the time (s), the length (m), and the temperature (K) are represented by the independent variables M, T, L, and \(\Theta\), respectively. The forces (lift and drag) are determined by the relationships between the 11 input variables, 7 of them defining the fluid, and 4 of them defining the adaptive wing model, as shown in Table 2.  
The Bernoulli Equation (1) allows the expressing of the subsonic fluid velocities (\(U\)) with three pressure variables (static, dynamic, and total pressure) and one fluid density (\(\rho\)) variable. The three pressure variables can be reduced and their effects expressed by the variable \(U\) magnitude. \[U=\sqrt{\frac{2\Big{(}P_{dym}\Big{)}}{\rho}}\text{ where }P_{dym}=(P_{\rm T}-P_{s}) \tag{1}\] \begin{table}
\begin{tabular}{c c c} \hline \hline  & **Variables for the Analysis on the Adaptive Wing Prototype** & **Dimensions** \\ \hline (I) & Input variables representing the fluid in the wind tunnel & \\  & (1) & Density (\(\rho\)) & \(\rm ML^{-3}\) \\  & (2) & Fluid velocity (\(U\)) & \(\rm LT^{-1}\) \\  & (3) & Dynamics viscosity of the fluid (\(\mu\)) & \(\rm ML^{-1}T^{-1}\) \\  & (4) & Temperature of the fluid (\(t\)) & \(\Theta\) \\  & (5) & Static pressure of the fluid (\(P_{s}\)) & \(\rm M\,L^{-1}\,T^{-2}\) \\  & (6) & Dynamic pressure of the fluid (\(P_{d}\)) & \(\rm M\,L^{-1}\,T^{-2}\) \\  & (7) & Total pressure of the fluid (\(P_{t}\)) & \(\rm M\,L^{-1}\,T^{-2}\) \\ \hline (II) & Input variables representing the adaptive wing prototype & \\  & (1) & Wing chord (\(c\)) & \(\rm L\) \\  & (2) & Angular displacement of the leading edge (LeadingEdge) & non-dimensional \\  & (3) & Angular displacement of the trailing edge (TrailingEdge) & non-dimensional \\  & (4) & Angle of attack of the wing (AOA) & non-dimensional \\ \hline (III) & Output variables representing the wind tunnel results & \\  & (1) & Lift force & \(\rm MLT^{-2}\) \\  & (2) & Drag force & \(\rm MLT^{-2}\) \\ \hline \hline \end{tabular}
\end{table} 
Table 2: Variables for the analysis of the adaptive wing. Figure 19: Adaptive wing prototype experimental tests: (**a**) side view and (**b**) back view. The fluid temperature (T) affects the dynamic viscosity (\(\mu\)) and the density (\(\rho\)) values of the air during wind tunnel tests. The value of the temperature variable (t) is used to define the density and the dynamic viscosity of the air at which the tests will be conducted; therefore, the variable (T) can be reduced in this analysis. The four variables (\(P_{T},\ P_{dyn},\ P_{s},\ T\)) can be excluded from the list of 11 input variables, as their effects are expressed as independent variables (\(U,\rho\)). The aerodynamic forces (drag and lift) can be expressed as a function of seven independent input variables, as shown in Equation (2). \[\text{Force}\ (\text{Lift},\ \text{Drag})=f\ \{\rho\,\ U\,\ \mu\,\ c,\ \text{LeadingEdge},\ \text{TrailingEdge},\ \text{AOA}\} \tag{2}\]  
To find the relationships between these seven variables and their effects on the drag and lift forces considered as output (Table 3), we have to use the Buckingham theorem in Equation (2). We can observe in Table 3 that all seven input variables and the two output variables can be defined by these three M, L, T physical dimensions. The theorem makes it possible to find the number of \(\pi\) terms (symbolized by the Greek letter \(\pi_{1}\,\ \pi_{2}\,\ \text{etc. }\)) needed to define the relationships between the input and output variables, which in this case are the drag and lift forces. The first step of the theorem is to remove the number of dimensional parameters from the input and output variables.