In accord with the analysis by Emanuel [J Eng Math, 117:79–105, 2019 ], the upstream and downstream vorticities are provided for an unsteady, three-dimensional shock, where the upstream flow may be non-uniform. The transformation for the upstream velocity Taylor series coefficients, $$e_{ij}^{*}\rightarrow e_{ij}$$ e ij ∗ → e ij , is also provided. The analytic approach is illustrated for the vorticity with a simplified flow model. Additionally, the intrinsic coordinate basis, on the downstream side of the shock, is obtained. The analytical representation of the vorticities is in a shock-based, flow-plane coordinate system, whereas the intrinsic coordinate basis is associated with a shock-based, osculating plane. In all cases, results are exact, explicit, and simplified as much as possible.

中文翻译：

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与不稳定的三维冲击相关的涡度和其他特性

根据 Emanuel [J Eng Math, 117:79–105, 2019] 的分析，上游和下游涡流是为不稳定的三维激波提供的，其中上游流动可能是不均匀的。还提供了上游速度泰勒级数系数 $$e_{ij}^{*}\rightarrow e_{ij}$$ e ij ∗ → e ij 的变换。用简化的流动模型说明了涡度的解析方法。此外，还获得了冲击下游侧的固有坐标基础。涡量的解析表示在基于激波的流动平面坐标系中，而固有坐标基础与基于激波的密切平面相关联。在所有情况下，结果都尽可能准确、明确和简化。