AIAA 94–0056 Fully-Implicit Time-Marching Aeroelastic Solutions

A new fully-implicit approach for computing transonic aeroelastic solutions is presented. The unsteady Euler equations are coupled with a typical section swept wing model and integrated forward in time. The implicit Euler equations are integrated in pseudo-time using multigrid methods, and are coupled with a first-order system decomposition of the structural modal equations. Full convergence of the simultaneous governing equations is achieved at every time step with considerable computational savings over previous approaches. Transient responses for a NACA 64A010 are calculated in different flow regimes, and flutter boundaries are computed and compared to pre-existing numerical data. Nomenclature a non-dimensional location of the elastic axis, positive aft of midchord b airfoil semichord c airfoil chord C l coefficient of lift Cm coefficient of moment about the elastic axis, positive nose up E total energy (internal plus kinetic) f , g Euler flux vectors h plunging displacement of the elastic axis, positive down H total enthalpy Iα section moment of inertia about the elastic axis, Iα = mb 2 r 2 α J jacobian of the transformation from cartesian to body fitted coordinates kc reduced frequency, kc = ωc U∞ K h plunging spring constant Kα pitching spring constant L airfoil section lift (normal to free stream), positive up m airfoil mass per unit span Mea total moment about the elastic axis, positive nose up M∞ free stream Mach number n surface normal vector p static pressure Qi generalized force for ith mode R(wij) flux residual for cell i,j R * modified residual R * ij modified residuals for structural equations Sα static unbalance, positive for c.g. aft of mid-chord, Sα = mbxα t real time t * pseudo-time u, v cartesian velocity components u cartesian velocity vector U∞ free stream velocity U f flutter speed Vij volume of i,j cell V f flutter speed index, V f = U f bωα √ µ w vector of flow variables xt, yt mesh cartesian velocity components xij, zij first order system decomposition intermediate variables α angle of attack, in radians ∆α pitching motion forcing amplitude ∆t implicit real time step γ ratio of specific heats, γ = 1.4 ηi ith normal coordinate µ airfoil mass ratio, µ = m πρb 2 ρ air density ω f frequency of the forced oscillations ω h , ωα uncoupled natural frequencies of typical section in plunge and pitch respectively ωi coupled natural frequency of the ith …