# Estimation of Stability Parameters for Wide Body Aircraft Using Computational Techniques
## 1 Introduction  
The Boeing 747-200 is a long-range, wide-body, large aircraft designed and developed by Boeing, a commercial airplane company in the USA. It is powered by a high-bypass turbofan engine developed by Pratt and Whitney. Some different variants also have GE-CF6 engines, and original variants have Rolls-Royce RB211 engines. It has a pronounced wing sweep, allowing a cruise Mach number of 0.90. The 747-200B was the basic passenger version of its series with increased fuel capacity and more powerful engines. General specifications of the Boeing 747-200 are given in Table 1.  
Flow field characteristics can be obtained by applying various aerodynamic flow models. Navier-Stokes (NS) equations are the most accurate flow models that describe viscous, rotational, and compressible flows. These are based on the Eulerian approach and primarily describe the momentum equation. The mass and energy equations are defined using additional relations to solve NS equations [2]. Based on the accuracy and vast applications, NS equations are used in different forms, such as direct numerical solution (DNS), large eddy simulation (LES), unsteady Reynolds-averaged Navier-Stokes (URANS), Reynolds-averaged Navier-Stokes (RANS), thin-layer Navier-Stokes (TLNS), and turbulence modeling (TM) [3]. Boundary-layer equations are based on the concept that at large Reynolds numbers, the viscous effects are limited to a thin region of flow adjacent to the body surface. BL equations are found by dropping the viscous diffusion terms from Navier-Stokes equations because they have little contribution to the solution. The pressure is assumed to be constant along the length in BL models [4]. Euler equations are the conservation laws assuming inviscid flow. The continuity equations do not change; however, energy relations may vary according to the flow characteristics. The assumption of inviscid flow does not limit the applicability of these equations. Moreover, outside the boundary layer, this assumption is also feasible for the flow properties. Modeling flow can be further simplified if the flow is assumed to be irrotational (zero vorticity) [5]. The full potential model combines all the conservation laws (continuity, momentum, and energy equations) [6]. Linearized potential equations are developed by decomposing the velocity into an unperturbed component and a perturbed component, which helps one to solve the non-linearity. Prandtl-Gluert's relation applies to the flows outside the transonic and hypersonic regimes [7]. For an incompressible flow, the Laplace equation in full potential form is reduced to partial differential equations. The Laplace equation has linearized behavior and has an exact solution. The discretization form of this equation is not required, and hence numerical errors do not exist [8]. Empirical methods do not model the flow; however, these can be used for the estimation of aerodynamic performance [9]. Estimations of lift, drag, and aerodynamic efficiency can be performed by using empirical methods. These methods are further used for stability and control derivatives of the complete aircraft, for instance, in US DATCOM (United States Data Compendium). The evolution of various aerodynamic flow models is described in the flow chart shown in Figure 1. \begin{table}
\begin{tabular}{c c c} \hline
**S No.** & **Specification** & **Value** \\ \hline
1 & Maximum takeoff weight & 833,000 lbs \\
2 & Total length & 231 ft 10 in \\
3 & Cabin width & 239.5 in \\
4 & Vertical tail height & 63 ft 5 in \\
5 & Fuel capacity & 53,985 US gal \\
6 & Thrust & 54,750 lbf \\
7 & Range & 6560 nm \\
8 & Take off & 10,900 ft \\
9 & Type of engine & JT9D-7 \\ \hline \end{tabular}
\end{table} 
Table 1: General specifications of the BOEING 747-200 aircraft [1]. Aerodynamic coefficients can be estimated using experimental and computational techniques. There are various computational techniques available, e.g., DATCOM, XFLR, TORNADO, AVL, PANAIR, and ANSYS. Each technique has its limitations that restrict its areas of application.