AIAA JOURNAL Vol. 43, No. 2, February 2005 Calculation of Airfoil Flutter by an Euler Method with Approximate Boundary Conditions Chao Gao, ∗ Shuchi Yang, † Shijun Luo, ‡ and Feng Liu § University of California, Irvine, Irvine, California 92697-3975 and David M. Schuster ¶ NASA Langley Research Center, Hampton, Virginia 23681 A numerical method is demonstrated for solving the steady and unsteady Euler equations on stationary Cartersian grids for the purpose of time-domain simulation of aeroelastic problems. Wall boundary conditions are implemented on nonmoving mean chord positions by assuming the airfoil being thin and undergoing small deformation, whereas the full nonlinear Euler equations are used in the flowfield for accurate resolution of shock waves and vorticity. The method does not require the generation of moving body-fitted grids and thus can be easily deployed in any fluid-structure interaction problem involving relatively small deformation of a thin body. The first-order wall boundary conditions are used in solving the full Euler equations, and the results are compared with the Euler solutions using the exact boundary conditions and known experimental data. It is shown that the first-order boundary conditions are adequate to represent airfoils of typical thicknesses with small deformation for both steady and unsteady calculations. Flutter boundaries are accurately predicted by this method for the Isogai wing model test case. Nomenclature b = airfoil half-chord, = c/2 C dw = wave drag coefficient C l = lift coefficient C m = moment coefficient around quarter-chord, positive nose up C p = pressure coefficient c = airfoil chord E = total specific energy (e x , e y ) = unit vectors in (x , y ) directions F (t , x ) = instantaneous upper surface of airfoil f (x ) = mean upper surface of airfoil G = Euler flux vector G(t , x ) = instantaneous lower surface of airfoil g(x ) = mean lower surface of airfoil H = total specific enthalpy h = vertical displacement of elastic axis (i , j ) = grid point index M ∞ = freestream Mach number n = real time level n = outer normal vector to cell surface p = pressure q = velocity vector of fluid particle q b = velocity vector of the surface of control volume R = flux residual Received 10 October 2003; revision received 24 May 2004; accepted for publication 20 August 2004. Copyright c  2004 by the authors. Published by the American Institute of Aeronautics and Astronautics, Inc., with per- mission. Copies of this paper may be made for personal or internal use, on condition that the copier pay the $10.00 per-copy fee to the Copyright Clearance Center, Inc., 222 Rosewood Drive, Danvers, MA 01923; include the code 0001-1452/05 $10.00 in correspondence with the CCC. ∗ Visiting Associate Researcher, Department of Mechanical and Aerospace Engineering; currently Professor, Center for Aerodynamics Re- search and Development, Northwestern Polytechnical University, 710072 Xi’an, People’s Republic of China. † Postdoctoral Researcher, Department of Mechanical and Aerospace Engineering. ‡ Researcher, Department of Mechanical and Aerospace Engineering. § Professor, Department of Mechanical and Aerospace Engineering. Associate Fellow AIAA. ¶ Senior Research Engineer, Aeroelasticity Branch. Associate Fellow AIAA. R ∗ = modified residual Re = Reynolds number based on airfoil chord S = surface of control volume t = real time t ∗ = pseudotime U ∞ = freestream velocity u, v = x and y velocity components of fluid particle u b , v b = x and y components of grid velocity V = control volume V f = flutter speed index W = Euler conservative flow variable x , y = Cartesian coordinates x 0 = x coordinate of the pivot point of airfoil pitching α = instantaneous angle of attack, deg α m = mean angle of attack, deg α 1 = instantaneous increment of angle of attack, deg α 0 = amplitude of the pitching oscillation γ = ratio of specific heats t = real-time step κ = reduced frequency µ = mass ratio ρ = air density τ = structural dimensionless time, = t ω ω = angular frequency I. Introduction C OMPUTATIONAL fluid dynamics (CFD) has proven to be a useful tool for the simulation and prediction of buffet, flutter, and limit-cycle-oscillation (LCO) phenomena of aeroelastic sys- tems. Methods ranging from the linear doublet-lattice method 1 to methods that solve the Euler and the Navier–Stokes equations have been developed. 2−13 Several recent review papers on computational aeroelasticity can be found in Refs. 14–17. Despite its limit in han- dling transonic and other nonlinear flows, the linear doublet-lattice method has been and is still the workhorse for actual design analy- sis in industry because of its efficiency in computer time and, per- haps equally important, the ease in setting up the computational problem. The Reynolds-averaged Navier–Stokes (RANS) methods encompass the most complete flow model short of large-eddy sim- ulations or direct numerical simulations. However, time-domain RANS simulations for aeroelasticity problems at present demand 295