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Efficient Fourier-based algorithms for time-periodic unsteady problems.

机译：基于高效傅立叶的算法，用于求解时间周期非定常问题。

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This dissertation work proposes two algorithms for the simulation of time-periodic unsteady problems via the solution of Unsteady Reynolds-Averaged Navier-Stokes (URANS) equations. These algorithms use a Fourier representation in time and hence solve for the periodic state directly without resolving transients (which consume most of the resources in a time-accurate scheme). In contrast to conventional Fourier-based techniques which solve the governing equations in frequency space, the new algorithms perform all the calculations in the time domain, and hence require minimal modifications to an existing solver. The complete space-time solution is obtained by iterating in a fifth pseudo-time dimension.;Various time-periodic problems such as helicopter rotors, wind turbines, turbomachinery and flapping-wings can be simulated using the Time Spectral method. The algorithm is first validated using pitching airfoil/wing test cases. The method is further extended to turbomachinery problems, and computational results verified by comparison with a time-accurate calculation. The technique can be very memory intensive for large problems, since the solution is computed (and hence stored) simultaneously at all time levels. Often, the blade counts of a turbomachine are rescaled such that a periodic fraction of the annulus can be solved. This approximation enables the solution to be obtained at a fraction of the cost of a full-scale time-accurate solution. For a viscous computation over a three-dimensional single-stage rescaled compressor, an order of magnitude savings is achieved.;The second algorithm, the reduced-order Harmonic Balance method is applicable only to turbomachinery flows, and offers even larger computational savings than the Time Spectral method. It simulates the true geometry of the turbomachine using only one blade passage per blade row as the computational domain. In each blade row of the turbomachine, only the dominant frequencies are resolved, namely, combinations of neighbor's blade passing. An appropriate set of frequencies can be chosen by the analyst/designer based on a trade-off between accuracy and computational resources available. A cost comparison with a time-accurate computation for an Euler calculation on a two-dimensional multi-stage compressor obtained an order of magnitude savings, and a RANS calculation on a three-dimensional single-stage compressor achieved two orders of magnitude savings, with comparable accuracy.
机译：本文提出了两种算法，通过求解非稳态雷诺平均纳维-斯托克斯方程（URANS）来求解时间周期非稳态问题。这些算法在时间上使用傅立叶表示，因此可以直接解决周期状态而无需解决瞬态问题（瞬态方案消耗了大部分资源）。与传统的基于傅立叶的技术求解频率空间中的控制方程式相反，新算法在时域中执行所有计算，因此需要对现有求解器进行最少的修改。通过对第五个伪时间维进行迭代，可以获得完整的时空解决方案。可以使用时间谱方法模拟各种时间周期问题，例如直升机旋翼，风力涡轮机，涡轮机械和襟翼。首先使用俯仰翼型/机翼测试案例对算法进行验证。该方法进一步扩展到涡轮机械问题，并且通过与时间精确计算进行比较来验证计算结果。由于解决方案是在所有时间级别同时计算（并因此存储的），因此对于大问题，该技术可能会占用大量内存。通常，涡轮机的叶片数会重新调整比例，以便可以解决环空的周期性部分。这种近似使解决方案的获得成本仅为全面时间精确解决方案的一小部分。对于在三维单级重定标的压缩机上进行粘性计算，可以节省一个数量级。第二种算法，降阶谐波平衡方法仅适用于涡轮机械流量，并且比时间频谱方法。它使用每叶片行仅一个叶片通道作为计算域来模拟涡轮机的真实几何形状。在涡轮机的每个叶片排中，只有主频率被分解，即相邻叶片通过的组合。分析人员/设计人员可以根据精度和可用计算资源之间的折衷选择一组合适的频率。在二维多级压缩机上进行欧拉计算时，采用时间精确计算进行成本比较可以节省一个数量级，而在三维单级压缩机上进行RANS计算则可以节省两个数量级，其中相当的精度。

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