Unsteady transonic two-dimensional Euler solutions using finite elements

Abstract

A finite element solution of the unsteady Euler equations is presented, and demonstrated for two-dimensional airfoil configurations oscillating in transonic flows. Computations are performed by spatially discretizing the conservation equations using the Galerkin weighted residual method and then employing a multistage RungeKutta scheme to march forward in time. Triangular finite elements are employed in an unstructured O-mesh computational grid surrounding the airfoil. Grid points are fixed in space at the far-field boundary and are constrained to move with the airfoil surface to form the near-field boundary. A mesh deformation scheme has been developed to efficiently move interior points in a smooth fashion as the airfoil undergoes rigid-body pitch and plunge motion. Both steady and unsteady results are presented, and a comparison is made with solutions obtained using finite volume techniques. The effects of using either a lumped or consistent mass matrix were studied and are presented. Results show the finite element method provides an accurate solution for unsteady transonic flows about isolated airfoils. HE accurate determination of the unsteady flowfield surrounding an oscillating airfoil or wing in transonic flow is of paramount importance in the prediction of the flutter characteristics of the body. Transonic flutter remains an active research topic because of the continued interest by both the military and commercial sectors to operate in these flight regimes with vehicles that are ever lighter and, as a result, more flexible. Typically, analysis of fluid-structure interaction in flight vehicles involves describing the vehicle structure using finite elements, whereas finite difference methods are used to model the surrounding fluid. Coupling of the two dissimilar models then presents problems because of the disparity between the two solution techniques. Additional complexities arise because of phase mismatching between the fluid pressures and the displacements of the structure. It seems a natural progression to use finite element methods to describe the behavior of both the fluid and structure, thus bringing more commonality into the analysis of the two media. Early research efforts concentrated on using the transonic small-disturbance equation or the nonlinear full potential equation as the model for the gas dynamic behavior in transonic flows. Unfortunately, this idealization leads to errors when strong shocks are present because of their inability to account for the production of entropy and vorticity. To properly account for these effects within the framework of the conservation laws, the Euler equations must be used. These equations, although they neglect fluid viscosity, have become the standard when unsteady transonic solutions are desired. The absence of a viscosity representation is an important limitation of the present formulation and must be considered