# A Power Based Analysis for a Transonic Transport Aircraft Configuration through 3D RANS Simulations
## 2 Methodology
### Power-Based Analysis in 3D RANS Simulations
It is noted that \(\dot{\mathrm{E}}_{\mathrm{w}}\) is zero for the free stream, and the magnitude of \(\dot{\mathrm{E}}_{\mathrm{w}}\) is based on the state of the free stream flow. For the entire control volume, the inflow and outflow terms (\(\dot{\mathrm{E}}_{\mathrm{w,in}}\) and \(\dot{\mathrm{E}}_{\mathrm{w,out}}\)) are listed in Table 1.  
\[\dot{\mathrm{E}}_{\mathrm{w}}=\dot{\mathrm{E}}\dot{\mathrm{E}}_{\mathrm{w}}+ \dot{\mathrm{E}}_{\mathrm{p}} \tag{4}\]  
\[\dot{\mathrm{K}}\dot{\mathrm{E}}_{\mathrm{w}}=\dot{\mathrm{E}}_{\mathrm{a}}+ \dot{\mathrm{E}}_{\mathrm{v}}=\iint\limits_{\mathrm{SP}}\rho\mathrm{u}\frac{1} {2}(\mathrm{u}-\mathrm{V}_{\infty})^{2}\mathrm{d}\mathrm{S}+\iint\limits_{ \mathrm{SP}}\rho\mathrm{u}\frac{1}{2}\Big{(}\mathrm{v}^{2}+\mathrm{w}^{2} \Big{)}\mathrm{d}\mathrm{S} \tag{5}\] \begin{table}
\begin{tabular}{c c c c} \hline \hline  & **Control Volume** & **Wing/Body** & **Propulsor** \\ \hline Input term & \(\dot{\mathrm{E}}_{\mathrm{w,in}}\) & \(\mathrm{DV}_{\infty}\) & PS \\ Output term & \(\dot{\mathrm{E}}_{\mathrm{w,out}},\Phi\) & - & \(\mathrm{TV}_{\infty}\) \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Power terms in the control volume of an aircraft. Figure 3: Control volume of a simplified aircraft with a BLI propulsor. \[\dot{E}_{P}=\iint\limits_{SP}(p-p_{\infty})(u-V_{\infty})dS \tag{6}\]  
The drag force, \(D\), is exerted on the surface of the aircraft components of the body, wing, empennage, etc. Regarding the conversion of mechanical power, the drag power, \(DV_{\infty}\), is the input power of an aircraft. It corresponds to the power of an ideal force dragging the aircraft in a flow field. This power is obtained by simply multiplying \(D\) by \(V_{\infty}\). It is straightforward that the propulsive system converts shaft power, \(P_{s}\), into thrust power, \(TV_{\infty}\). Therefore, \(P_{s}\) is an input term, while thrust power, \(TV_{\infty}\), is an output term. In this study, \(P_{s}\) is the mechanical power addition that enters the control volume through the boundary of the propulsor surface. For a simplified propulsor of actuator disc model only changing the axial flow, the expression of \(P_{s}\) is given in Equation (7). The output power term thrust power, \(TV_{\infty}\), is obtained by multiplying thrust, \(T\), by \(V_{\infty}\). \[P_{s}=\iint\limits_{prop}(p-p_{\infty})\ udS=\iint\limits_{prop}\Delta p\ udS \tag{7}\]  
Viscous dissipation, \(\Phi\), is an output term of the control volume. \(\Phi\) exists in the viscous region (boundary layer and wake) where the velocity gradient exists. This term uses the volume integral of a domain. The integrand of \(\Phi\) contains nine components. According to the physical property, it can be broken down into laminar, turbulent, and bulk viscous dissipation, as expressed by Equation (8). The expressions of the three components are given in Equations (9)-(11). The turbulent viscosity, \(\mu_{tur}\), is introduced to access the turbulent dissipation, \(\Phi_{tur}\), by applying the Boussinesq assumption. It is noted that \(\Phi\) is a sink term in the mechanical energy equation. \[\Phi=\Phi_{lam}+\Phi_{tur}+\Phi_{bulk} \tag{8}\]  
\[\Phi_{lam}=\iiint\limits_{CV}\mu_{lam}\Bigg{[}\bigg{(}\frac{\partial v}{ \partial x}+\frac{\partial u}{\partial y}\bigg{)}^{2}+\bigg{(}\frac{\partial w }{\partial y}+\frac{\partial v}{\partial z}\bigg{)}^{2}+\bigg{(}\frac{ \partial u}{\partial z}+\frac{\partial w}{\partial x}\bigg{)}^{2}\Bigg{]}dV \tag{9}\]  
\[\Phi_{tur}=\iiint\limits_{CV}\mu_{tur}\Bigg{[}\bigg{(}\frac{\partial v}{ \partial x}+\frac{\partial u}{\partial y}\bigg{)}^{2}+\bigg{(}\frac{\partial w }{\partial y}+\frac{\partial v}{\partial z}\bigg{)}^{2}+\bigg{(}\frac{ \partial u}{\partial z}+\frac{\partial w}{\partial x}\bigg{)}^{2}\Bigg{]}dV \tag{10}\]  
\[\Phi_{bulk}=\iiint\limits_{CV}(\mu_{lam}+\mu_{tur})\Bigg{[}2\bigg{(}\frac{ \partial u}{\partial x}\bigg{)}^{2}+2\bigg{(}\frac{\partial v}{\partial y} \bigg{)}^{2}+2\bigg{(}\frac{\partial w}{\partial z}\bigg{)}^{2}\Bigg{]}dV \tag{11}\]