Time-Dependent Response of a Two-Dimensional Airfoil in Transonic Flow

Abstract

A procedure is developed for the aeroelastic analysis of a two-dimensional airfoil in transonic flow. The fluid is assumed to be described by the unsteady low-frequency small-disturbance transonic potential equation for which a fully time-implicit integration scheme exists. Structural equations of motion are integrated in time simultaneously with the potential equation in order to predict the unsteady airfoil motion. As a computational example, a three-degree-o f-freedom NACA 64A010 airfoil is considered using representative values of the structural parameters. The method is shown to be both stable and accurate, and the time response for several choices of initial conditions and reduced freest ream density is presented. Oscillations with either growing or decaying amplitudes are indicated depending upon the prescribed initial conditions. OR the case of flow over an airfoil in a freestream at Mach numbers near 1, small amplitude motions of the body surface can produce large variations in the aerodynamic forces and moments acting on the structure. In addition, phase differences between the flow variables and the resultant forces may be great. These characteristics tend to enhance the probability of encountering aeroelastic instabilities in the transonic flow regime, and thus evidence a need for tech- niques of analyzing the coupled unsteady flowfield and resultant structural response in such situations. In the subsonic and supersonic cases, the governing flow equations may be linearized such that the aerodynamic forces depend upon the body motion in a linear fashion. Moreover, the resultant forces acting on the airfoil may be obtained through superposition by summing the contributions due to each of the various types of body motion permitted. This allows the linear structural equations of motion to be solved independent of the governing aerodynamic equations which provide only the force coefficients. Uncoupling of the fluid and structural equations is not, in general, possible for the transonic regime due to its inherent nonlinear nature. Recent advances in computational methods have made several approaches available for computing unsteady tran- sonic flows. While a number of different techniques have evolved and various physical problems have been con- sidered, M6 the unsteady body motion was generally prescribed as a known function of time, thereby precluding the simulation of true aeroelastic behavior. Only more recently have these procedures been applied to actual aeroelastic problems.17'18 It is the intent here to describe a method for obtaining the time-dependent response of a two- dimensional airfoil in transonic flow and to provide a computational example by applying this technique to a physical situation of practical interest. The governing aerodynamic equation of motion is assumed to be the unsteady low-frequency small-disturbance transonic equation for the velocity potential function which is capable of simulating nonlinear flow phenomena including irregular shock wave motions. Solutions to this equation have corn-