A linear stability analysis of a three‐dimensional shear‐imposed fluid flowing down an inclined plane is studied based on the evolution equations for normal velocity and normal vorticity components. Both modal and nonmodal stability analyses are carried out. The modal stability analysis demonstrates that shear‐imposed flow is more unstable to solo streamwise perturbation. On the contrary, the nonmodal stability analysis demonstrates that shear‐imposed flow is more unstable to solo spanwise perturbation. There is evidence of existing transient energy growth in the wavenumber space that intensifies in the presence of imposed shear stress. Further, the boundary for the zone of transient growth appears far ahead of the boundary for the zone of exponential growth. This fact indicates that the onset of instability for the shear mode may occur before than that predicted from the eigenvalue analysis. A new critical surface parameter involving imposed shear stress is determined, which in fact reveals the existence criterion of instability for the surface mode. Moreover, we have found three different regimes of surface mode instability, shear mode instability, and transient growth phenomenon in the wavenumber space. The unstable region for the surface mode reduces while the unstable region for the shear mode enhances in the wavenumber space with the increasing value of surface parameter, or equivalently, with the increasing value of imposed shear stress. Finally, the unstable region for the surface mode disappears from the wavenumber space as soon as the surface parameter exceeds its critical value; instead, the wavenumber space is occupied by the regime of transient energy growth. In addition, the incident of pseudoresonance takes place on the response curve to external harmonic forcing.

中文翻译：

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施加剪切力的降膜的最佳扰动增长

基于法向速度和法向涡度分量的演化方程，研究了在倾斜平面上流动的三维剪力流的线性稳定性分析。进行了模态和非模态稳定性分析。模态稳定性分析表明，施加剪切流对单向流扰动更为不稳定。相反，非模态稳定性分析表明，施加剪切力的流动对于单独的翼展方向扰动更加不稳定。有证据表明，在存在剪切应力的情况下，波数空间中现有的瞬态能量增长会加剧。此外，瞬时增长区的边界似乎远远超过指数增长区的边界。这一事实表明，剪切模态的不稳定性可能比特征值分析所预测的更早。确定了涉及施加的切应力的新的临界表面参数，这实际上揭示了表面模式不稳定性的存在准则。此外，我们在波数空间中发现了三种不同的表面模态不稳定性，剪切模态不稳定性和瞬态增长现象。随着表面参数值的增加，或者等效地，随着施加的剪应力的值的增加，波模式空间中表面模式的不稳定区域减小，而剪切模式的不稳定区域增大。最后，一旦表面参数超过其临界值，表面模式的不稳定区域就会从波数空间中消失。反而，波数空间被瞬态能量增长机制所占据。此外，伪谐振事件发生在对外部谐波强迫的响应曲线上。