Convergence Acceleration of Favre-Averaged Non-Linear Harmonic Method

This paper develops a numerical procedure to accelerate the convergence of the Favre-averaged Non-Linear Harmonic (FNLH) method. The scheme provides a unified mathematical framework for solving the sparse linear systems formed by the mean flow and the time-linearized harmonic flows of FNLH in an explicit or implicit fashion. The approach explores the similarity of the sparse linear systems of FNLH and leads to a memory efficient procedure, so that its memory consumption does not depend on the number of harmonics to compute. The proposed method has been implemented in the industrial CFD solver HYDRA. Two test cases are used to conduct a comparative study of explicit and implicit schemes in terms of convergence, computational efficiency, and memory consumption. Comparisons show that the implicit scheme yields better convergence than the explicit scheme and is also roughly 7 to 10 times more computationally efficient than the explicit scheme with 4 levels of multigrid. Furthermore, the implicit scheme consumes only approximately $50\%$ of the explicit scheme with four levels of multigrid. Compared with the full annulus unsteady Reynolds averaged Navier-Stokes (URANS) simulations, the implicit scheme produces comparable results to URANS with computational time and memory consumption that are two orders of magnitude smaller.