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：本文提出了一种加速Favre平均非线性谐波（FNLH）方法收敛的数值计算流程。该方案为显式或隐式求解FNLH方法中由平均流场和时域线性化谐波流场所形成的稀疏线性系统提供了统一的数学框架。该方法通过探究FNLH稀疏线性系统的相似性，形成了一种内存高效的求解流程，其内存消耗不依赖于所需计算的谐波数量。所提出的方法已在工业计算流体力学求解器HYDRA中实现。通过两个测试算例，对显式和隐式方案在收敛性、计算效率和内存消耗方面进行了对比研究。对比结果表明，隐式方案比显式方案具有更好的收敛性，并且在采用4层多重网格时，其计算效率约为显式方案的7至10倍。此外，隐式方案的内存消耗仅为采用4层多重网格的显式方案的大约50%。与全环非定常雷诺平均Navier-Stokes（URANS）模拟相比，隐式方案能够产生与URANS相当的结果，而计算时间和内存消耗则降低了两个数量级。