Two-dimensional instability of the bottom boundary layer under a solitary wave

Abstract

The objective of this paper is to establish a detailed map for the temporal instability of the bottom boundary layer (BBL) flow driven by a soliton-like wave induced pressure gradient in a U-tube-shaped tunnel, which serves as an approximation to the BBL under surface solitary waves. Both linear stability analysis and fully nonlinear two-dimensional simulations using high-order numerical methods have been carried out. The process of delineation of the stability regions as a function of boundary layer thickness-based Reynolds number of the temporally evolving base flow, Reδ, consists of two parts. In the first part, we assess the lower limit of the Reδ range within which the standard, quasi-steady, linear stability analysis is applicable when considering individual base flow profiles sampled during its transient evolution. Below this limit, transient linear stability analysis serves as a more accurate predictor of the stability properties of the base flow. In the second step, above the Reδ limit where the BBL is determined to be linearly unstable, the base flow is further classified as unconditionally stable, conditionally unstable, or unconditionally unstable in terms of its sensitivity to the amplitude and the insertion time of perturbations. Two distinct modes of instability exist in this case: post- and pre-flow-reversal modes. At a moderate value of Reδ, both modes are first observed in the wave deceleration phase. The post flow reversal mode dominates for relatively low Re and it is the one observed in Sumer et al. [“Coherent structures in wave boundary layers. Part 2. Solitary motion,” J. Fluid Mech. 646, 207 (2010)]. For Re above a threshold value of the base flow in the unconditionally unstable regime, the pre-flow reversal mode, which is longer in wavelength than its post-reversal counterpart, becomes dominant. In the same regime, the threshold Reδ value above which instability is observed in the acceleration phase of the wave is also identified. In this case, the base flow velocity profiles lack any inflection point, suggesting that the origin of such an instability is viscous. Finally, the lower Reδ limit above which quasi-steady linear stability analysis is valid may be independently obtained by adapting to the surface solitary wave BBL, an instability criterion which links the average growth rate and wave event timescale, as previously proposed in studies of the instability of the interior of progressive and solitary internal waves.