Non-unique Numerical Solutions and Their Stability of the Euler Equations for Transonic Flow over Airfoils

Multiple numerical solutions of inviscid transonic flow over several airfoils exist at certain Mach number and angle of attack. Global linear stability analysis of the multiple solutions is conducted in this paper. Linear perturbation equations of the Euler equations around a steady-state solution are formed, an eigenvalue problem is then constructed using the modal analysis approach, then it is discretized using a numerical scheme that is similar to the JST scheme for the Euler equations. Only a small portion of the eigen spectrum is needed and thus can be found efficiently by using implicit restarted Arnoldi’s algorithm. The eigenvalue that has the largest real part determines stability of the steady-state solution, the corresponding eigenmode can also be found. Multiple numerical solutions and their stability characteristics are studied for the Hafez airfoils with flat and wavy surfaces, original and modified NACA 0012 airfoil. For the Hafez airfoils, the numerical solutions and stability are consistent with previous studies in the literature, which suggest the validity of the present method. Analysis of the NACA 0012 airfoil indicates stability of symmetric solutions of the Euler equations at conditions where buffet is found from unsteady Navier-Stokes equations. Euler solutions of the same airfoil but modified to include the displacement thickness of the boundary layer computed from the Navier-Stokes equations, however, exhibit instability based on the present linear stability analysis. The asymmetry J78 airfoil exhibits different type of multiple solutions and stability mechanism.