Calculation of Airfoil Flutter by an Euler Method with Approximate Boundary Conditions

Abstract

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.

Prediction of Flutter and LCO by an Euler Method on Non-moving Cartesian Grids with Boundary-Layer Corrections

“…As we mentioned earlier that the first mode is symmetrical and the second mode antisymmetric, it is no wonder that the pitching and plunging displacements are roughly in phase for the first flutter point while there is obvious phase difference for the second flutter point and the phase lag is almost equal to π for the third flutter point. Gao et al 16 predicted the flutter boundary for the Isogai wind model using method (1) earlier and their result is compared with our current result in Fig. 4.…”

“…4,16,[21][22][23] For convenience, here we briefly describe the basics including unsteady Euler method, the approximate boundary condition approach on non-moving cartesian grids, the integral boundary layer method and its coupling with the Euler solver, the full Navier-Stokes method using k − ω turbulence model, and the structural solver as well as the strong coupling CFD-CSD scheme.…”

Flutter Prediction by a Cartesian Mesh Euler Method with Small Perturbation Gridless Boundary Conditions

“…Namely, one can write where V n is the surface normal velocity based on the motion of the surface. In Equations (19), the first two relations are reflection conditions, and the last equation…”

“…Writing the derivative shape function for one of these surrounding nodes in the abbreviated notation n φ′ , one can consider a perturbation of the derivative shape function given by represents a balance between the pressure in the fluid and the centrifugal force associated with the fluid motion along a curved path defined by the local surface radius of curvature, R. The tildes indicate that the velocity components are in a local (surface normal) coordinate system. The surface boundary conditions of Equations (19) are written in terms of gridless shape functions at any surface node p using where ∆θ represents a small angular displacement of the airfoil (and associated surface normal at p), and is the perturbed value of shape function associated with the displacement. Thus, for small displacements, Equation (21) assumes that there is a linear variation of derivative shape function with angular separation between p and n. It is interesting to note that the least squares shape functions exhibit such near linear behavior for small deflections.…”

“…Gao et al recently proposed another small perturbation Cartesian approach. 18,19 In their method, a single gridline of the Cartesian was used to represent the airfoil, and the finite thickness of the airfoil is included in the perturbation boundary condition.…”