Evaluation of the Stability Derivatives Using the Sensitivity Equations

Abstract

In this paper, we present a numerical method to predict the static and dynamic stability derivatives of the aircraft by solving the flow governing equations and the associated sensitivity equations. The aerodynamic model and the derivation of the sensitivity equations are discussed in detail. The key features of the present method are that the unsteady effects are taken into consideration and the complete set of stability derivatives, both static and dynamic, can be obtained simultaneously. The flow and sensitivity equations are solved using a computational fluid dynamics technique. The stability derivatives of a NACA 0012airfoil undergoing a pitching oscillation are computed based on the solutions of the sensitivity equations. The present method is verified by comparing the aerodynamic forces obtained from the unsteady flow simulation with those predicted by the stability derivatives. I. Introduction T HEdesignofatmosphericvehiclesandtheircontrolsystemsisa challenging area of ongoing research. The quality of a design is evaluated by the behavior of the aircraft when performing required tasks and maneuvers. For this purpose, one must be able to calculate the aerodynamic forces (and moments) acting on the aircraft at any instant of the flight. Because the aerodynamic forces depend on the aircraft motion history and there are infinitely many aircraft maneuvers, it is not viable to analyze the behavior of an aircraft unless a mathematical model for the representation of the aerodynamic forces is introduced. Up to now, the most widely used mathematical model is to express the aerodynamic forces as functions of the stability derivatives, which can be traced back to the work of Bryan [1] at the beginning of the 20th century. Even though the mathematical models using the concept of the stability derivatives are usually simple, it is far from trivial to determinethesestabilityderivativesbecauseofthecomplexityofthe flowfields and the lack of technique for “separating” the stability derivativesfromtheaerodynamicforcesorfromthecombinationsof the stability derivatives. Traditionally, the stability derivatives are evaluatedthroughwind-tunnelexperiments, flighttests,orempirical methods. The wind-tunnel experiments and flight tests are very expensive and time-consuming. The empirical methods are limited to certain stability derivatives and are sometimes not reliable. In recentyears,therehavebeenpersistenteffortstowarddevelopingthe computational fluid dynamics (CFD) technique for evaluating the stability derivatives numerically [2–4]. These methods are usually based on the solution of the sensitivity equations [5] that have been widely used in many fields such as aerodynamic optimizations. In this paper, a numerical method for evaluating the stability derivatives using the equations of sensitivity is presented. This method is similar in spirit to that of Limache and Cliff [3] and Limache [4]. However, the present method allows the computation of dynamic stability derivatives by solving the unsteady flow governing equationsandtheunsteadysensitivity equations,whereas the method of Limache and Cliff [3] and Limache [4] only computes the static stability derivatives by considering the aerodynamically steady flows.Afterpresentingtheaerodynamicmodelandsensitivity equations in the next section of the present paper, we will apply this method to compute the stability derivatives of a NACA 0012 airfoil undergoing the pitching oscillations. The present method is verified by comparing the aerodynamic forces obtained from the unsteady flow simulation and predicted by the stability derivatives.