Estimation of Turbulent Vertical Velocity from Nonlinear Simulations of Aircraft Response

Abstract

The objective of this paper is to estimate the unknown vertical velocity component of atmospheric turbulence on a transport aircraft based on data in the flight data recorder with accuracy consistent with the measurement of the alpha sensor. Such vertical velocity component of low frequency (to be called the wind) is not measurable quantitatively with the existing onboardweather radars. The idea is to estimate the vertical wind speed from aircraft response by integrating the nonlinear flight dynamic equations. The motivation of developing this new approach arises from flight trajectory reconstruction of a transport aircraft in an accident with the aerodynamics being unsteady and nonlinear. Because the conventional forward integration with a Runge–Kutta fourth-order scheme tends to diverge, in particular in the predicted altitude, an innovative approach which can maintain the accuracy of numerical integration over a long time period for a system of implicit nonlinear flight dynamic equations is introduced in this paper. The new approach is to add and subtract a linear part of the nonlinear differential equations, with one term being treated implicitly (i.e., unknown) and the other term, together with the other aerodynamic forces and moments, being known function of time. This implicit linear part serves as the numerical damping that reduces or eliminates the growth of errors in numerical forward integrations. Application to a transport aircraft in severe turbulence encounter is illustrated.