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Investigation of various solution strategies for the time-spectral method for incompressible, viscous flows

机译：不可压缩粘性流时谱方法的各种解决方案研究

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Time-spectral methods show a huge potential for decreasing computation time of time-periodic flows. While time-spectral methods are often used for compressible flows, applications to incompressible flows are rare. This paper presents an extension of the time-spectral method (TSM) to incompressible, viscous fluid flows using a pressure-correction algorithm in a finite volume flow solver. Several algorithmic treatments of the time-spectral operator in a pressure-correction algorithm have been investigated. Initially the single time instances were solved using the Jacobi method as preconditioner. While the existing fluid code is easily adapted, the solver shows a fast degradation in stability. Thus the solution matrix was reordered with respect to time and a block Gauss-Seidel preconditioner was applied. The single time blocks were directly solved using the Cholesky algorithm. The solver is more robust, but the current implementation is inefficient. To alleviate this problem an approach, coupling all time instances and control volumes, was developed. For the complete time and spatial system two different treatments in the preconditioner were researched. To outline the advantages and disadvantages of the proposed solution strategies the laminar flow around the pitching NACA0012 airfoil was investigated. Moreover, unsteady simulations using first and second order time-stepping techniques were used and the time-spectral results were compared to regular time-stepping approaches. It is shown that the time-spectral implementations solving the whole temporal-spatial system are faster than the regular time-stepping schemes. The efficiency of the time-spectral solver decreases with increasing number of harmonics. Furthermore, with a small number of harmonics the lift coefficient over time is not accurately predicted.
机译：时间谱方法显示出减少时间周期流计算时间的巨大潜力。虽然时间谱方法通常用于可压缩流，但不可压缩流的应用很少。本文介绍了在有限体积流量求解器中使用压力校正算法将时间谱方法（TSM）扩展到不可压缩的粘性流体流的方法。已经研究了压力校正算法中时间谱算子的几种算法处理。最初，使用Jacobi方法作为前置条件求解单个时间实例。尽管可以轻松地修改现有的流体代码，但求解器显示出快速的稳定性下降。因此，解矩阵相对于时间进行了重新排序，并应用了块高斯-赛德尔预调节器。使用Cholesky算法直接求解单个时间块。求解器更健壮，但是当前的实现效率低下。为了减轻这个问题，开发了一种将所有时间实例和控制量耦合的方法。对于完整的时间和空间系统，研究了预处理器中的两种不同处理方法。为了概述提出的解决方案的优点和缺点，研究了俯仰NACA0012机翼周围的层流。此外，使用了使用一阶和二阶时间步长技术的不稳定模拟，并将时间谱结果与常规时间步长方法进行了比较。结果表明，解决整个时空系统的时间谱实现比常规的时间步长方案要快。时间频谱求解器的效率随谐波数量的增加而降低。此外，在谐波数量较少的情况下，无法准确预测随时间的升力系数。

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