We perform a resolvent analysis to examine the perturbation dynamics over the laminar separation bubble (LSB) that forms near the leading edge of a NACA 0012 airfoil at a chord-based Reynolds number (Re) of 500 000 and an angle of attack of 8 degrees. While we focus on the LSB residing over $6\%$ of the chord length, the resolvent operator is constructed about the global mean flow over the airfoil, avoiding numerical issues arising from domain truncation. Moreover, randomized singular value decomposition is adopted in the present analysis to relieve the computational cost associated with the high-Re global base flow. To examine the local physics over the LSB, we consider the use of exponential discounting to limit the time horizon that allows for the instability to develop with respect to the base flow. With discounting, the gain distribution over frequency accurately captures the spectral content over the LSB obtained from flow simulation. The peak-gain frequency also agrees with previous flow control results on suppressing dynamic stall over a pitching airfoil. According to the gain distribution and the modal structures, we conclude that the dominant energy-amplification mechanism is the Kelvin-Helmholtz instability. In addition to discounting, we also examine the use of spatial windows for both the forcing and response. From the response-windowed analysis, we find that the LSB serves the main role of energy amplifier, with the amplification saturating at the reattachment point. The input window imposes the constraint of surface forcing, and the results show that the optimal actuator location is slightly upstream of the separation point. The surface-forcing mode also suggests the optimal momentum forcing in the surface-tangent direction, with strong unidirectionality that is ideal for synthetic-jet-type actuators. This study demonstrates the strength of randomized resolvent analysis in tackling high-Reynolds-number base flows, and it calls attention to the care needed for base-flow instabilities. The physical insights provided by the resolvent analysis can also support flow control studies that target the LSB for suppressing flow separation or dynamic stall.

中文翻译：

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Re = 500000时翼型层流分离气泡的溶剂分析

我们执行分解分析，以检查在NACA 0012机翼前缘附近形成的层流分离气泡（LSB）上的扰动动力学，其基于弦的雷诺数（Re）为500 000，迎角为8度。虽然我们专注于居住在$6％$在弦长的基础上，解析​​器算子围绕机翼上的全局平均流构造，避免了因域截断而引起的数值问题。此外，在本分析中采用随机奇异值分解来减轻与高Re全局基流相关的计算成本。为了检查LSB上的局部物理，我们考虑使用指数折现来限制时间范围，该时间范围允许相对于基本流的不稳定性发展。通过打折，频率上的增益分布可以准确地捕获从流量模拟获得的LSB上的频谱内容。峰值增益频率也与先前的流量控制结果一致，该结果可抑制俯仰翼型上的动态失速。根据增益分布和模态结构，我们得出结论，主要的能量放大机制是开尔文-亥姆霍兹不稳定性。除了打折，我们还研究了将空间窗口用于强迫和响应的情况。从响应窗口分析中，我们发现LSB充当了能量放大器的主要角色，放大在重新连接点达到饱和。输入窗口施加了表面力的约束，结果表明，最佳执行器位置在分离点的稍上游。表面强制模式还建议在表面切线方向上具有最佳动量强迫，具有很强的单向性，非常适合合成射流式执行器。这项研究证明了解决高雷诺数基流的随机分解剂分析的优势，它引起人们对基础流不稳定性所需要的注意。分解物分析提供的物理见解还可以支持针对LSB的流动控制研究，以抑制流动分离或动态失速。