Linearization of unsteady transonic flows containing shocks

The problem of determining unsteady airloads on a thin, three-dimensional, planar wing oscillating with infinitesimal amplitude in a transonic flow is considered. The flow is assumed to be governed by the transonic small disturbance equation. The unsteady disturbance is taken to be a small perturbation superposed on a given steady mean flowfield. The equations governing the unsteady field, allowing for induced oscillations of any embedded shocks, are obtained. The linearization is shown to fail, locally, at the intersection of a shock with the wing surface, although the failure has little influence on the sectional characteristics of the wing. HE problem of determining unsteady aerodynamic loads on wings executing infinitesimal oscillation is of fundamental importance in aeroelasticity. Methods for evaluating these loads are well developed for thin profiles in subsonic and supersonic flows,' where, to a first ap- proximation, the steady-state disturbance field due to thickness and mean orientation does not interact with the unsteady flow due to the oscillations. At freestream Mach numbers sufficiently close to 1, however, the steady distur- bances can become large enough to influence the unsteady flow directly if the frequency is low and the aspect ratio not too small (at high frequencies or small aspect ratio the in- teraction can be neglected and classical methods apply2). Mathematical tools for predicting loads in this "strong in- teraction" regime have only recently been developed. Much of the work to date has dealt with finite-difference schemes for solving various nonlinear field equations, e.g., Euler's equations,3 the full potential equation,4 and the transonic small disturbance equation.5 The aeroelastician, however, is normally confronted with determining the stability of a configuration with respect to infinitesimal disturbances. For this problem a linear aerodynamic theory is desirable. The linearization of the equations of motion in the tran- sonic regime has been extensively discussed by Landahl,2 who defined the region in the Mach number/frequency/aspect ratio parameter space within which steady disturbances have a first-order effect on the unsteady flow. Landahl's primary interest, however, was in solutions outside this regime, where the interaction can be neglected. Moreover, within the strong interaction region his formulation is incomplete, since it does not account for the presence of shocks, which are normally generated in the steady flow and oscillate in response to the motion of the surface. This shock motion creates a con- centrated load on the surface which can represent a sub- stantial fraction of the total unsteady loads.6 Two recent finite-difference studies7'8 have considered the linearized problem within the strong interaction regime. Neither of these investigations, however, properly accounted for the displacement of the shock, and their results, therefore, must be viewed with some skepticism. In an early paper Eckhaus9 obtained analytical results for a two-dimensional airfoil by replacing the actual steady disturbance field by a simple normal shock discontinuity