An efficient implicit Fourier time-spectral method is developed for the unsteady Reynolds-averaged Navier–Stokes (URANS) computations of turbulent flows. In addition to the time-spectral URANS equations, the time-spectral Spalart–Allmaras turbulence model equation is also solved by the space–time lower-upper symmetric Gauss–Seidel (LU-SGS) implicit scheme. The efficient LU-SGS sweeps are directly extended to the time domain without introducing additional approximation error, delayed implicit temporal coupling, and additional algorithmic steps. The ability of the algorithm is verified through the test cases of the pitching NACA0012 airfoil and pitching ONERA M6 wing. Dominant Fourier modes of the drag coefficient and its components are identified. Special attention is paid to the nonlinear behavior of the surface pressure and skin friction coefficients in the shock-free area, especially the region downstream the shock where the influence of turbulence is significant. For the test case of a pitching ONERA M6 wing, difference in local flow unsteadiness for the two branches of the $\lambda$-type shock is observed. The phenomenon is further discussed through study of the shock motion range and the upstream Mach number for each shock branch.

一种有效的隐式傅立叶时间谱方法被开发用于湍流的非稳态雷诺平均Navier-Stokes（URANS）计算。除了时谱的URANS方程外，时谱的Spalart-Allmaras湍流模型方程还可以通过时空上下对称高斯-赛德尔（LU-SGS）隐式方案求解。高效的LU-SGS扫描直接扩展到时域，而不会引入额外的近似误差，延迟的隐式时间耦合和额外的算法步骤。通过俯仰NACA0012机翼和俯仰ONERA M6机翼的测试案例，验证了算法的能力。确定了阻力系数及其分量的主要傅里叶模式。在无冲击区域，特别是在冲击下游的湍流影响很大的区域，特别要注意表面压力和皮肤摩擦系数的非线性行为。对于俯仰的ONERA M6机翼的测试案例，该飞机的两个分支的局部流动不稳定性差异$\lambda$观察到类型的冲击。通过研究冲击运动范围和每个冲击分支的上游马赫数，进一步讨论了该现象。