In revolving or flapping wings, radial planetary vorticity tilting (PVTr) is a mechanism that contributes to the removal of radial (spanwise) vorticity within the leading-edge vortex (LEV), while vorticity advection increases its strength. Dimensional analysis predicts that the PVTr and advection should scale with the wing aspect-ratio (AR) in identical fashion, assuming a uniform characteristic length is used. However, the authors’ previous work suggests that the vorticity advection decreases more rapidly than the PVTr as AR increases, indicating that separate normalizations should be applied. Here, we aim to develop a comprehensive scaling for the PVTr and vorticity advection based on simulation results using computational fluid dynamics. Two sets of simulations of revolving rectangular wings at an angle of attack of 45° were performed, the first set with the wing-tip velocity maintained constant, so that the Reynolds number (Re) defined at the radius of gyration equals 110, and the second set with the wing angular velocity maintained constant, so that Re defined at one chord length equals 63.5. We proposed two independent length scales based on LEV geometry, i.e., wing-span for the radial and tangential directions and wing chord for the vertical direction. The LEV size in the radial and tangential directions was limited by the wing-span, while the vertical depth remained invariant. The use of two length scales successfully predicted not only the scaling for the PVTr and the vorticity advection but also the relative magnitude of advection in three directions, i.e., tangential advection was strongest, followed by the vertical (downwash) and then the radial that was negligible.

中文翻译：

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缩放具有两个方向长度尺度的旋转机翼前沿涡旋中的涡旋动力学

在旋转或拍动的机翼中，径向行星涡旋倾斜（PVTr）是一种机制，有助于消除前缘涡旋（LEV）内的径向（跨度）涡旋，而涡旋对流可提高强度。尺寸分析预测，假设使用统一的特征长度，PVTr和对流应以相同的方式与机翼长宽比（AR）成比例。但是，作者先前的工作表明，涡流对流的下降速度比PVTr作为AR下降得更快增加，表示应应用单独的归一化。在这里，我们的目标是基于使用计算流体动力学的模拟结果，为PVTr和涡旋对流开发一个全面的标度。进行了两组以45°迎角旋转的矩形机翼的模拟，第一组模拟的翼尖速度保持恒定，因此在回转半径处定义的雷诺数（Re）等于110，而第二套与机翼角速度保持恒定，从而使Re定义为一个和弦长度等于63.5。我们基于LEV几何提出了两个独立的长度刻度，即径向和切线方向的翼展，垂直方向的翼弦。径向和切线方向上的LEV大小受机翼跨度的限制，而垂直深度则保持不变。使用两个长度尺度不仅可以成功预测PVTr和涡度对流的尺度，而且还可以预测三个方向的对流相对大小，即切向对流最强，其次是垂直（向下冲洗），然后是径向微不足道。