Characteristic-based formulation of boundary conditions for preconditioned or non-preconditioned flow equations

A numerical implementation of characteristic boundary conditions for Euler and Navier-Stokes equations is presented. The method combines the specified boundary conditions and the outgoing characteristic variables according to the wave propagation directions. In the general case, the proposed boundary conditions update the primitive variables by solving a small system of linear equations (4×4 in 2D, 5×5 in 3D) at each boundary point/cell. The method can be used for both preconditioned and non-preconditioned equations. For a perfect gas without preconditioning, a closed analytical solution is provided. Two possible methods of extrapolating the outgoing characteristic variables are discussed. Finally, the numerical approach is validated for the 2D internal flow in a channel with a bump.