# A Power Based Analysis for a Transonic Transport Aircraft Configuration through 3D RANS Simulations
## 2 Methodology  
### Power-Based Analysis in 3D RANS Simulations  
Computational fluid dynamics simulations obtain the flow field so that the continuity, momentum, and energy equations are satisfied. On the other hand, the mechanical energy equation is different from the energy equation. The former is obtained by multiplying the momentum equation with velocity [28], while the latter is related to the first law of thermodynamics. The diagram in Figure 2 illustrates the relation between the conservations of mechanical energy (mechanical energy/Navier-Stokes momentum equation) and energy (Navier-Stokes energy equation). Power-based analysis utilizes the integral form of the kinetic energy equation. As a result, numerical simulations provide sufficient and necessary conditions for power-based analysis due to the conservation of momentum. In power-based analysis, the integral mechanical energy equation can be simplified as Equation (1). This equation establishes the equilibrium of mechanical power by using three terms, namely power inputs; outputs; and an error, \(e\). The power inputs and outputs are introduced in the following paragraphs. The error, \(e\), is elaborated in detail in the section on power imbalance error. \[\mathrm{P_{in}}=\mathrm{P_{out}}+e \tag{1}\]  
Figure 3 depicts a control volume of a simplified transport aircraft that is bounded by an inlet, an outlet, a cylindrical far-field boundary, and the aircraft surface. A survey plane is introduced as an internal plane perpendicular to the flow direction. The aircraft surface consists of a fuselage, wing, and propulsor. These aircraft components can be categorized into two groups: the fuselage and wing are power-consuming components, while the propulsor is the power-adding component. The power terms associated with the aircraft components and control volume are listed in Table 1. For the entire control volume, the total power input is the summation of the wake energy inflow (rate), \(\mathrm{\dot{E}_{w,in}}\); drag power, \(\mathrm{DV_{\infty}}\); and shaft power (mechanical power addition), \(\mathrm{P_{s}}\), as expressed by Equation (2). The total power output is the summation of the wake energy outflow (rate), \(\mathrm{E_{w,out}}\); thrust power, \(\mathrm{TV_{\infty}}\); and viscous dissipation rate, \(\Phi\), as given in Equation (3). The individual power terms are elaborated in the following paragraphs. The power balance equation of the entire  
Figure 2: The diagram of the conservation of mechanical energy and total energy [23]. control volume can be established by combining these power terms and the error, \(e\), as given in Equation (1). \[\mathrm{P_{in}} =\dot{\mathrm{E}}_{\mathrm{w,in}}+\mathrm{DV}_{\infty+}\mathrm{P_{S}} \tag{2}\] \[\mathrm{P_{out}} =\dot{\mathrm{E}}_{\mathrm{w,out}}+\mathrm{TV}_{\infty}+\Phi \tag{3}\]  
The power term of the wake energy flow rate, \(\dot{\mathrm{E}}_{\mathrm{w}}\), refers to the mechanical energy passing through a plane. For the inlet plane, it corresponds to the mechanical energy inflow, \(\dot{\mathrm{E}}_{\mathrm{w,in}}\). Its value is simply zero for the free stream incoming flow, \(\mathrm{V}_{\infty}\), but it has a finite value for a possible headwind or backwind entering the control volume. For the out plane, \(\dot{\mathrm{E}}_{\mathrm{w,out}}\), it is the mechanical energy of the aircraft wake and jet that flows out of the control volume. For an internal survey plane (SP), the surface integral of \(\dot{\mathrm{E}}_{\mathrm{w}}\) evaluates the mechanical energy of the flow through the plane. On the other hand, depending on the physical property, \(\dot{\mathrm{E}}_{\mathrm{w}}\) can be broken down into the kinetic energy flow rate, \(\dot{\mathrm{K}}\dot{\mathrm{E}}_{\mathrm{w}}\), and the pressure power, \(\dot{\mathrm{E}}_{\mathrm{p}}\), as shown in Equation (4). The former refers to the kinetic energy of the flow, as given in Equation (5). The latter is the pressure work, which might transform isentropically into \(\dot{\mathrm{K}}\dot{\mathrm{E}}_{\mathrm{w}}\), as expressed by Equation (6).