# Validation of a Wind Tunnel Propeller Dynamometer for Group 2 Unmanned Aircraft
### Propeller Theory
#### 1.4.2 Blade Element Momentum Theory  
Blade Element Momentum Theory (BEMT) is a common methodology for predicting propeller performance in terms of the coefficients defined in Section 1.4.1. BEMT requires only a few inputs. The code implemented for this research is described succinctly by the flowchart presented as Figure 4, and is similar to examples found in [22; 23; 24; 25]. The first step in the BEMT solution process is to discretize propeller geometry for analysis. Input files catalog propeller twist and the local airfoil profile for \(n\) radial segments, specified by distance from the hub (\(r\)), each of length \(dr\), from the hub to the tip. The measurements describing propeller geometry specification are shown as Figure 5.  
Figure 4: BEMT solution process. In addition to propeller geometry data, the propeller operating condition is input by specifying freestream velocity (\(V_{\infty}\)) and RPM for a given run of the BEMT code. After a run case begins, the code takes on assumed values for the axial and angular inflow factors, \(a_{axial_{i}}\) and \(a_{angular_{i}}\), respectively, for each propeller segment of length \(dr_{i}\). The initial assumed values for \(a_{axial_{i}}\) and \(a_{angular_{i}}\) are 0.1 and 0.01, respectively. These terms are induction factors describing the axial and angular velocity components, \(V_{axial_{i}}\) and \(V_{\Theta_{i}}\), respectively, within an annular streamtube containing \(dr_{i}\). Due to the propeller rotation, the fluid within streamtube \(i\) acquires the velocity components modeled as  
\[V_{axial_{i}}=[a_{axial_{i}}]V_{\infty}\]  
and  
\[V_{\Theta_{i}}=[a_{angular_{i}}]\omega r_{i}\]  
which are accounted for during application of momentum conservation equations. Subsequently, the total downwash angle is computed for each blade segment. The local flow geometry and definitions for force directions for a blade segment are shown in Figure 6.  
Given the freestream velocity, rotational velocity at the radial location under consideration, and the induced velocities due to the propeller motion, the total downwash angle at segment \(i\) is computed as shown in Equation (7). \[\varepsilon_{Tot_{i}}=\tan^{-1}\left(\frac{V_{\infty}+V_{axial_{i}}}{\omega r_ {i}-V_{\Theta_{i}}}\right) \tag{7}\]  
Next, with the total downwash angle defined, the local lift and drag coefficients for the airfoil sections can be determined. For each blade segment, the effective angle of attack  
Figure 5: Propeller discretized into blade segments. Figure 6: Velocities and force directions on propeller blade segment \(i\). \(\alpha_{eff_{i}}\) is the sum of the geometric angle of attack (AOA), \(\alpha_{i}\) and zero-lift AOA, \(\alpha_{L=0_{i}}\). The geometric AOA is defined in Equation (8), where \(\beta_{i}\) is the geometric pitch angle  
\[\alpha_{i}=\beta_{i}-\varepsilon_{Tot_{i}}. \tag{8}\]  
Given the effective angle of attack, \(\alpha_{eff_{i}}\), the \(C_{l}\) and \(C_{d}\) for each section is straightforward to determine from tables of 2-D aerodynamic data. As the APC (Advanced Precision Composites) propellers studied experimentally are predominately made up of National Advisory Committee for Aeronautics (NACA) 4412 airfoils [14], this cross-section was assumed for each propeller segment in the BEMT code. In this work, the 2-D input aerodynamic data are developed from XFOIL [26] analysis at the Reynolds number computed based on the vector sum of the freestream and rotational velocity and chord at the 75% radial location, as is common in propeller aerodynamics [11]. Then, the total thrust and torque the propeller is producing are estimated. For each blade segment, the incremental thrust and torque are shown as Equations (9) and (10), respectively. \[dT_{i}=q_{i}c_{i}[C_{l_{i}}\cos\left(\varepsilon_{Tot_{i}}\right)-C_{d_{i}} \sin\left(\varepsilon_{Tot_{i}}\right)]Adr_{i} \tag{9}\]  
Figure 4:0  
where dynamic pressure at radial location \(i\) is defined as shown in Equation (11).