Copyright @ lFAC System Identification, Copenhagen. Denmarlc, 1994 IDENTIFICATION OF UNSTEADY AERODYNAMIC MODEL OF A DELTA WING AT HIGH ANQLES OF ATTACK M. GOMAN·, A. KHRABROV· and S. USOLTSEV· • Central Aerohydrodynamics Institute (TsAGI), Zhukovsky-3, Moscow region, 140160. Russia Abstract. A model for the lift force of an aircraft operating at high angles of attack is presented. In this model the Wlsteady aerodynamic effect is expressed as a dependence. of the lift on an internal variable for which a differential equation is postulated. The resulting model is applied to a delta wing subjected to forced pitching moment oscillations. The unknown parameters representing the unsteady aerodynamics are estimated by nonlinear and linear least squares procedures from wind tunnel data at three different frequencies and the angle of attack between 30 and 60 deg. The predicted aerodynamic terms in the model agree well with measured data. Key Words. Aerodynamics modeling; nonlinear equations; linearization techniques; least-squares estimation (2) 1. INTRODUCTION The problem of adequate mathematical model- ing of aerodynamic characteristics at high angles of attack when separated and vortex breakdown flow conditions occur is very important for the flight dynamics simulation. In many practical cases the unsteady aerodynamic forces and mo- ments are approximated by linear terms using the unsteady aerodynamic derivatives (Klein, 1993; Etkin, 1972). This method is sufficiently accu- rate for attached flow at low angles of attack. For example, the representation of the lift coefficient has the following form where CI I (0-) is the lift coefficient for steady state conditions, CL q and CL .. are the rotation and un- steady derivatives, T = tVlc - nondimensional . - -IV' do- time, q = qc ,0- = ([T' But in the region of 0- where separated and vortex flow occurs this representation is not correct. The values of unsteady derivatives obtained in forced oscillations wind tunnel experiments strongly de- pend on the amplitude and the frequency of oscil- lations. To overcome this problem one can use the well known approach of the generalized descrip- tion using the indicial response functionals (Schiff and Tobak, 1976). A simple but sufficiently ac- curate approach was proposed by Goman and Khrabrov (1992). To describe the variation of nonlinear aerodynamic characteristics associated with the motion past history is possible by utiliz- ing the ordinary differential equations. The fol- lowing mathematical model, describing simple ex- amples with an airfoil and a delta wing, was pro- posed by Goman and Khrabrov (1992) : CL = Cdo-, x) + Ci.4 +Ci .. o TI x + x = xo (0- - T20) where x is the internal dynamic variable which characterizes for example the position of the vor- tex breakdown on the delta wing or the separa- tion point on the airfoil (0 S x SI), TI is the nondimensional relaxation time constant, T2 is the parameter which defines the total time delay of the vortex breakdown or flow separation appear- ance. Function Xo defines the vortex breakdown or separation point position in steady conditions; the terms with derivatives Ci q , Ci .. outline the unsteady effects at the angles of attack, where =0. The mathematical model (2) permits to describe the main features observed in wind tunnel exper- iments at high angle of attack conditions. Un- known parameters TI, T2 and functions CL(o-,x), xo(o-) have to be defined from steady and un- steady experimental data. The parameters es- timation technique should take into account the structure of the mathematical model (2) and fea- tures of the available experimental data. In this paper results of the parameters estimation of the 1039