Free surface water waves generated by instability of an exponential shear flow in arbitrary depth

Abstract

The stability of an exponential current in water to infinitesimal perturbations in the presence of gravity and capillarity is revisited and reformulated using the Weber and Froude numbers. Some new results on the generation of gravity-capillary waves are presented, which supplement the previous works of Morland et al. [“Waves generated by shear layer instabilities,” Proc. Math. Phys. Sci. 433, 441–450 (1991)] and Young and Wolfe [“Generation of surface waves by shear-flow instability,” J. Fluid Mech. 739, 276–307 (2014)] on finite depth. To consider perturbations at much larger scales, special attention is given to the stability of exponential currents only in the presence of gravity. More precisely, the present investigation reveals significant insights into the stability of exponential shear currents under different environmental conditions. Notably, we have identified that the dimensionless growth rate increases with the Froude number, providing a deeper understanding of the interplay between shear layer thickness and surface velocity. Furthermore, our analysis elucidates the dimensional wavelength of the most unstable mode, emphasizing its relevance to the characteristic shear layer thickness. Additionally, within the realm of gravity-capillary instabilities, we have established a sufficient condition for the stability of exponential currents based on the Weber number. Our findings are supported by stability diagrams at finite depth, showing how the size of stable domains correlates with the characteristic thickness of the shear layer. Moreover, we have explored the stability of a thin film of liquid in an exponential shearing flow, further enriching our understanding of the complex dynamics involved in such systems.