Unsteady transonic airfoil computation using implicit Euler scheme on body-fixed grid

Abstract

The unsteady Euler equations have been derived for the flow relative motion with respect to a frame of reference that is rigidly attached to the moving airfoil. The resulting equations preserve the conservation form. The grid is generated once by an elliptic solver without any need for dynamic grid computation. An implicit approximately factored finite-volume scheme has been developed and implemented through a fully vectorized computer program. The scheme is based on the spatial approximate factorization of Beam and Warming. Implicit second-order and explicit second- and fourth-order dissipations are added to the scheme. The boundary conditions are explicitly satisfied. The scheme is applied to steady and unsteady transonic rigid-airfoil flows. For forced harmonic airfoil motions, periodic solutions are achieved within the third cycle of oscillation. The results are in good agreement with the experimental data. sonic flows past a NACA 0015 airfoil at a constant pitch rate.18 The latter case is the same application considered in Ref. 16. The numerical solution has been obtained by using the flux-vector splitting and the flux-difference splitting meth- ods extended for dynamic meshes.19 In Ref. 18, it is noted that different turbulence models give different normal-force and pitching-moment coefficients. Numerical solutions of the unsteady Euler equations are less expensive than those of the unsteady Navier-Stokes equations. For the unsteady transonic flow, the unsteady Euler equations adequately model most of the real flow features, with the exception of viscous effects whenever they are substantial. The Euler equations model shock waves and their motion, entropy increase across shocks and entropy gradient, and vorticity production and convection behind shocks, as can be seen from Crocco's theorem and the inviscid vorticity transport equa- tion. Recently, successful time-accurate solutions of the un- steady Euler equations have been presented for pitching air- foils19 and wings in transonic flows,20 and for the rolling oscillation of delta wings—a vortex-dominated flow problem in a locally conical supersonic flows.21 To the best of bur knowledge, Ref. 21 is the first work done for unsteady vortex- dominated flows with shock waves, which is directly applica- ble to maneuvering delta wings. In this paper, we present an implicit approximately-factored finite-volume scheme for the time-accurate numerical solution of the unsteady Euler equations of the flow relative motion with respect to an airfoil-fixed frame of reference. The scheme is applied to steady and unsteady transonic flows around a NACA 0012 airfoil. For the unsteady flows, the airfoil is in pitching oscillation about a small and moderate mean angle of attack at a large amplitude. The computational results are compared with the experimental data.