## 5 Adaptive Wing Design and Manufacturing
### Dimensional Analysis with Buckingham's Theorem
The non-dimensional input or output variables are excluded in the analysis. For the lift force analysis, there are only four input variables and one output variable. The removal of the physical dimensions from the variables gives a minimum of two \(\pi\) terms for the lift force, as shown in Equation (3):  
\[\text{4 input variables + 1 output variable - 3 physical dimensions = 2 \pi terms} \tag{3}\]  
Next, we separate the dimensional independent input variables and the input variables containing all three physical dimensions (M,L,T), as shown in Table 4.  
The independent variables of the physical dimensions are matched to reduce the lift force (F) physical dimensions; therefore, the first term fo \begin{table}
\begin{tabular}{c c c c} \hline \hline \multicolumn{2}{c}{**Independent Variable**} & \multicolumn{2}{c}{**Dependent Variable Dimensions**} \\
**Dimensions** & & **and Output Variable of Lift** \\ \hline Density (\(\rho\)) & \(\text{ML}^{-3}\) & Dynamics viscosity (\(\mu\)) & \(\text{ML}^{-1}\ \text{T}^{-1}\) \\ Fluid velocity (\(U\)) & \(\text{LT}^{-1}\) & Lift force (F) & \(\text{MLT}^{-2}\) \\ Wing chord (c) & L & & \\ \hline \hline \end{tabular}
\end{table} 
Table 4: Independent, dependent, and output variables of lift. \begin{table}
\begin{tabular}{c c c} \hline \hline \multicolumn{2}{c}{**Variables Specified for the Analysis of the Adaptive Wing Prototype**} & **Dimensions** \\ \hline (I) & Input variables representing the fluid in the wind tunnel & \\  & (1) & Density (\(\rho\)) & \(\text{ML}^{-3}\) \\  & (2) & Fluid velocity (\(U\)) & \(\text{LT}^{-1}\) \\  & (3) & Dynamics viscosity of the fluid (\(\mu\)) & \(\text{ML}^{-1}\ \text{T}^{-1}\) \\  & (4) & Wing chord (\(c\)) & L \\  & (5) & Angular displacement of the leading edge (LeadingEdge) & non-dimensional \\  & (6) & Angular displacement of the trailing edge (TrailingEdge) & non-dimensional \\  & (7) & Angle Of Attack of the wing (AOA) & non-dimensional \\ \hline (II) & Output variables representing the wind tunnel results \\  & (1) & Lift force & \(\text{MLT}^{-2}\) \\  & (2) & Drag force & \(\text{MLT}^{-2}\) \\ \hline \hline \end{tabular}
\end{table} 
Table 3: Variables for the lift and drag analysis. Only using the independent variables of the physical dimensions to match and reduce the dynamic viscosity (\(\upmu\)) physical dimensions, the second term is found and shown in Table 6.  
The first term \(\pi_{1}\) is associated with the force produced by the wing surface versus the dynamic characteristics of the fluid. This dimensional term is associated with the lift coefficient \(C_{l}\). The second term \(\pi_{2}\) is the ratio between the fluid inertial forces and the viscous forces inside the fluid. This term is associated with the Reynolds number (\(Re\)), and it is essential in wind tunnel testing because it allows the comparison of the fluid flow conditions between the full-scaled wings and the scaled-down models. The Buckingham theorem allows the finding of the relationship between the two \(\pi\) terms, which are thus proportional between the coefficients of the forces of lift and drag and the Reynolds number. The relationship is shown in Equation (4), where \(\varnothing\) is the angle of the leading edge and the trailing edge. \[\frac{F}{\rho\;v^{2}\;l^{2}}=\varnothing\;\bigg{(}\frac{\rho\;v\;l}{\mu}\bigg{)} \tag{4}\]