Non-modal stability analysis in viscous fluid flows with slippery walls

A study of optimal temporal and spatial disturbance growths is carried out for three-dimensional viscous incompressible fluid flows with slippery walls. The non-modal temporal stability analysis is performed under the framework of normal velocity and normal vorticity formulations. A Chebyshev spectral collocation method is used to solve the governing equations numerically. For a free surface flow over a slippery inclined plane, the maximum temporal energy amplification intensifies with the effect of wall slip for the spanwise perturbation, but it attenuates with the wall slip when perturbation considers both streamwise and spanwise wavenumbers. It is found that the boundary for the regime of transient growth appears far ahead of the boundary for the regime of exponential growth, which raises a question on the critical Reynolds number for the shear mode predicted from the eigenvalue analysis. Furthermore, the eigenvalue analysis or the modal stability analysis reveals that the unstable region for the shear mode decays rapidly in the presence of wall slip, which is followed by the successive amplification of the critical Reynolds number for the shear mode and ensures the stabilizing effect of slip length on the shear mode. On the other hand, for a channel flow with slippery bounding walls, the maximum spatial energy amplification intensifies with the effect of wall slip in the absence of angular frequency, but it reduces with the wall slip if the angular frequency is present in the disturbance. Furthermore, the maximum spatial energy disturbance growth can be achieved if the disturbance excludes the angular frequency. Furthermore, it is observed that the angular frequency plays an essential role in the pattern formation of optimal response. In addition, the pseudo-resonance phenomenon occurs due to external temporal and spatially harmonic forcings, where the pseudo-resonance peak is much higher than the resonance peak.