Aircraft Drag Polar Estimation Based on a Stochastic Hierarchical Model Junzi Sun, Jacco M. Hoekstra, Joost Ellerbroek Control and Simulation, Faculty of Aerospace Engineering Delft University of Technology, the Netherlands Abstract—The aerodynamic properties of an aircraft determine airspeed, and air density, and the remaining effects of the flow a crucial part of the aircraft performance model. Deriving for both the lift and drag are described with coefficients for accurate aerodynamic coefficients requires detailed knowledge both forces. The most complicated part is to model these lift of the aircraft’s design. These designs and parameters are well protected by aircraft manufacturers. They rarely can be used and drag coefficients. These parameters depend on the Mach in public research. Very detailed aerodynamic models are often number, the angle of attack, the boundary layer and ultimately not necessary in air traffic management related research, as they on the design of the aircraft shape. For fixed-wing aircraft, often use a simplified point-mass aircraft performance model. these coefficients are presented as functions of the angle of In these studies, a simple quadratic relation often assumed to attack, i.e., the angle between the aircraft body axis and the compute the drag of an aircraft based on the required lift. This so-called drag polar describes an approximation of the airspeed vector. In air traffic management (ATM) research, drag coefficient based on the total lift coefficient. The two key however, simplified point-mass aircraft performance models parameters in the drag polar are the zero-lift drag coefficient and are mostly used. These point-mass models consider an aircraft the factor to calculate the lift-induced part of the drag coefficient. as a dimensionless point, where the angle of attack, as well Thanks to this simplification of the flight model together with as the side-slip angle, and the effect of the angular rates are accurate flight data, we are able to estimate these aerodynamic parameters based on flight data. In this paper, we estimate the not explicitly considered. Hence, the step of calculating these drag polar based on a novel stochastic total energy model using is often skipped and the drag polar is used instead. Bayesian computing and Markov chain Monte Carlo sampling. The relationship between drag coefficient and lift coeffi- The method is based on the stochastic hierarchical modeling cient is the main factor determining the aircraft performance. approach. With sufficiently accurate flight data and some basic Knowledge of the drag polar is therefore essential for most knowledge of aircraft and their engines, the drag polar can be estimated. We also analyze the results and compare them to the ATM research such as trajectory prediction, fuel optimization, commonly used Base of Aircraft Data model. The mean absolute parameter estimation. difference among 20 common aircraft for zero-lift drag coefficient Many methods exist to explore the aircraft performance and lift-induced drag factor are 0.005 and 0.003 respectively. At during the preliminary design phase, often with a focus on the end of this paper, the drag polar models in different flight the modeling of the aerodynamics. Hence, one source of phases for these common commercial aircraft types are shared. open information regarding drag polar models comes from Keywords - aircraft performance, drag polar, aerodynamic coef- textbooks [3], [4], [5], [6]. However, only older aircraft models ficient, Bayesian computing, MCMC are available in the literature. In [7], an empirical model for estimating zero-lift drag coefficients was proposed using I. INTRODUCTION existing literature data based on several aircraft models. In Since the invention of aircraft, researchers have been study- general, open data on drag polar model is rare, especially for ing the aerodynamic properties of airfoils and aircraft. Exam- modern commercial aircraft. ples of fundamental studies on aerodynamic drag are given The aircraft manufacturers who design the aircraft do have in [1], [2]. From the start, the goal has been to optimize the accurate aerodynamic data. However, these data are rarely lift over drag ratio for the cruise flight. Much effort has been publicly available due to commercial competition. The most dedicated to creating designs that would reduce drag and thus comprehensive collection of drag polar data is the Base increase the lift efficiency of aircraft. While the zero drag of Aircraft Data (BADA) developed by Eurocontrol [8]. It coefficient contains the parasitic drag of the whole aircraft, contains the drag polars for nearly all common aircraft types. the wing is mainly responsible for the lift-induced drag. Next, BADA is the default “go-to” aircraft performance model for to the chosen airfoil, the aspect ratio of the wing plays an current ATM researchers. However, it imposes a strict license important role. The wing can be seen as a drag to lift converter, in terms of sharing and redistribution of the model data. The of which the already high efficiency can be increased further. project-based license for newer versions makes it even harder This is an on-going effort: some examples of current research for the same researcher to reuse the model, which will also are boundary layer suction, morphing wings, plasma control, apply for new users of older versions. and blended wing-body aircraft shapes. The goal of this paper is to propose an alternative path to Aerodynamic lift and drag forces of an aircraft are com- estimate the drag polar models for modern fixed-wing com- plicated and computationally intensive to compute. Lift and mercial aircraft, as well as share the drag polar models that we drag are considered as functions of the wing area, dynamic have obtained. We approach this estimation problem using a Eighth SESAR Innovation Days, 3rd – 7th December 2018 2 novel stochastic total energy (STE) model. The STE approach where A is the aspect ratio of the wing (span divided by the treats the parameters of the standard total energy model as average chord) and where e is the Oswald factor, which lies random variables. Then, we try to solve the parameter esti- typically in the range 0.70-0.90. This equation is often written mation using Bayesian computing, specifically, Markov chain as: Monte Carlo (MCMC) approximations. Finally, a database of 2 drag polar models for different common commercial aircraft, CD = CD0 + kCL (4) which were produced using this method, is provided. with: The structure of this paper is as follows. In section two, the fundamentals of the point-mass drag polar model are 1 k = (5) introduced. In section three, we focus on the hierarchical πAe model approach. In section four, experiments are conducted to These two parameters, CD0 and k are the zero-lift drag examine and obtain drag polar data of multiple aircraft types coefficient and lift-induced drag coefficient factor respectively. based on this method. Section five and six are dedicated to The values of both parameters are considered as constants the discussions and conclusions of this research. under a specific aerodynamic configuration of the aircraft. Fig. 1 illustrates an example of drag polar by using a computational II. THEORY OF AERODYNAMIC MODELING fluid dynamics (CFD) simulation for an aircraft with a clean A. Drag polar in point-mass models (no flaps or extended landing gear) configuration. While an aircraft flies, the drag force is produced by the airflow interacting with the aircraft body. The lift force is 0.09 produced due to the pressure difference between the upper 0.08 and lower surface of the lifting devices (wings). With the same 0.07 airspeed and altitude conditions, control of lift is performed 0.06 by re-configuring the aircraft angle of attack and/or modifying D 0.05 C the surface shape of lifting devices. By changing the elevator 0.04 settings, the pitch angle and the angle of attack can be 0.03 controlled. On the other hand, a change of the lifting device 0.02 CD0 = 0.0184 surface is primarily performed by re-configuring flaps. K = 0.0484 0.01 In general, the lift and drag forces of an aircraft that is 0.2 0.0 0.2 0.4 0.6 0.8 1.0 1.2 traveling in the free stream can be computed as: CL Figure 1. Drag polar (B747-400 simulated, clean configuration, α < 10◦) 1 2 L = CL ρV S 2 (1) 1 2 B. Aircraft aerodynamic configurations D = CD ρV S 2 Besides the angle of attack that affects the values of the lift where CL and CD are lift and drag coefficients, respectively. ρ, and drag coefficients, the change in the shape of the aircraft V , and S are air density, true airspeed, and the lifting surface can alter these values. The most notable change in aircraft is area of the aircraft. In practice, CL and CD can be modeled flaps (and slats), speed brakes and landing gear. Each structural as functions of the angle of attack (α), Mach number(M) and setting also has its own corresponding drag polar model. flap deflection(δf ): Flaps are common aircraft surfaces deployed in order to provide an increase in the maximum lift coefficient. They are deployed to be able to fly at lower speeds, typically at low CL = fcl(α; M; δf ) (2) altitudes (for example, during takeoff, initial climb, and ap- CD = fcd(α; M; δf ) proach). Different aircraft types have different configurations of flaps and flap settings. An increase in flap angle leads to an In aerodynamic models, multi-dimensional table interpolation increase in the lift coefficient under the same angle of attack, at or higher order polynomials are used. However, in many ATM the expense of a higher drag.