Optimal disturbance growth in shear‐imposed falling film

Abstract

AbstractA 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.