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.

翻译：本文提出了一种数值方法，用于加速Favre平均非线性谐波（FNLH）方法的收敛。该方案提供了一个统一的数学框架，以显式或隐式方式求解由平均流和时域线性化谐波流形成的稀疏线性系统。该方法探索了FNLH稀疏线性系统的相似性，并推导出一种内存高效的过程，使其内存消耗不依赖于待计算的谐波数量。所提出的方法已在工业CFD求解器HYDRA中实现。通过两个测试案例，在收敛性、计算效率和内存消耗方面对显式和隐式方案进行了比较研究。比较表明，隐式方案比显式方案具有更好的收敛性，并且其计算效率比采用4级多网格的显式方案大约高出7至10倍。此外，隐式方案的内存消耗仅为采用4级多网格的显式方案的约$50\%$。与全环非定常雷诺平均Navier-Stokes（URANS）模拟相比，隐式方案在计算时间和内存消耗上均降低了两个数量级，同时获得了与URANS相当的结果。