An adjoint approach is developed to compute the receptivity of the rotating disc boundary layer to surface roughness. The adjoint linearised Navier–Stokes equations, in cylindrical coordinates, are derived and receptivity characteristics are computed for a broad range of azimuthal mode numbers using a fully equivalent velocity–vorticity formulation. For each set of flow conditions (i.e. azimuthal mode number), the adjoint method only requires that the linear and adjoint solutions be computed once. Thus, the adjoint approach offers significant computational and time advantages over alternative receptivity schemes (i.e. direct linearised Navier–Stokes) as they can be used to instantaneously compute the receptivity of boundary layer disturbances to many environmental mechanisms. Stationary cross-flow disturbances are established by randomly distributed surface roughness that is periodic in the azimuthal direction and modelled via a linearisation of the no-slip condition on the disc surface. Each roughness distribution is scaled on its respective root-mean-square. A Monte-Carlo type uncertainty quantification analysis is performed, whereby mean receptivity amplitudes are computed by averaging over many thousands of roughness realisations with variable length and wavelength filters. The amplitude of the cross-flow instability is significantly larger for roughness distributions near the conditions for neutral linear instability, while roughness elements radially outboard have a negligible effect on the receptivity process. Furthermore, receptivity increases sharply for roughness distributions that encompass wavelength scales equivalent to that associated with the cross-flow instability. Finally, mean receptivity characteristics are used to predict the radial range that stationary cross-flow vortices achieve amplitudes sufficient to invalidate the linear stability assumptions.

中文翻译：

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一种计算旋转圆盘边界层对表面粗糙度接受度的辅助方法

开发了一种辅助方法来计算旋转圆盘边界层对表面粗糙度的接受度。导出了柱坐标中伴随的线性化 Navier-Stokes 方程，并使用完全等效的速度-涡度公式计算了广泛的方位角模式数的接收特性。对于每组流动条件（即方位模数），伴随方法只需要计算一次线性和伴随解。因此，伴随方法提供了显着的计算和时间优势，而不是替代的感受性方案（即直接线性化 Navier-Stokes），因为它们可以用于瞬时计算边界层扰动对许多环境机制的感受性。静态横流扰动是通过随机分布的表面粗糙度建立的，该粗糙度在方位角方向上是周期性的，并通过圆盘表面上的无滑移条件的线性化来建模。每个粗糙度分布在其各自的均方根上进行缩放。执行蒙特卡罗类型的不确定性量化分析，从而通过使用可变长度和波长滤波器对数千个粗糙度实现进行平均来计算平均接收幅度。对于中性线性不稳定性条件附近的粗糙度分布，横流不稳定性的幅度明显更大，而径向外侧的粗糙度元素对接受度过程的影响可以忽略不计。此外，对于包含与横流不稳定性相关的波长尺度的粗糙度分布，接受度急剧增加。最后，平均接收特性用于预测静态横流涡流达到足以使线性稳定性假设无效的幅度的径向范围。