Transition to turbulence at the bottom of a solitary wave

Abstract

A linear stability analysis of the laminar flow in the boundary layer at the bottom of a solitary wave is made to determine the conditions leading to transition and the appearance of turbulence. The Reynolds number of the phenomenon is assumed to be large and a 'momentary' criterion of instability is used. The results show that the laminar regime becomes unstable during the decelerating phase, when the height of the solitary wave exceeds a threshold value which depends on the ratio between the boundary layer thickness and the local water depth. A comparison of the theoretical results with the experimental measurements of Sumer et al. (J. Fluid Mech., vol. 646, 2010, pp. 207-231) supports the analysis

Linear stability of boundary layers under solitary waves

“…Special care has been taken to ensure that the numerical results are valid. Linear stability predicts that the boundary layer flow is unstable for the entire parameter range considered, confirming qualitatively the results by Blondeaux et al (2012). As a result of this analysis the stability of this flow cannot be described by a critical Reynolds number unlike what is atempted in previous publications.…”

“…Investigation of the bottom boundary layer under a solitary wave has been initiated by Liu et al (2007) when they published theoretical and experimental results concerning the shape of the boundary layer profile. This work has led to subsequent publications by Sumer et al (2010), Blondeaux (2008, 2011) and Blondeaux et al (2012) investigating the transitions in the boundary layer. Sumer et al (2010) investigated experimentally the stability of the boundary layer flow under a solitary wave, Blondeaux (2008, 2011) performed direct numerical simulations to this end and Blondeaux et al (2012) were the first to perform a linear stability analysis on this type of flow.…”

“…This work has led to subsequent publications by Sumer et al (2010), Blondeaux (2008, 2011) and Blondeaux et al (2012) investigating the transitions in the boundary layer. Sumer et al (2010) investigated experimentally the stability of the boundary layer flow under a solitary wave, Blondeaux (2008, 2011) performed direct numerical simulations to this end and Blondeaux et al (2012) were the first to perform a linear stability analysis on this type of flow. A result of the works by Sumer et al (2010) and Vittori and Blondeaux (2011) is that the regimes of the boundary layer flow can be categorized into three to four regimes Sumer et al (2010): laminar, laminar with regular vortex tubes, transitional and fully turbulent.…”

“…Vittori and Blondeaux (2011) proposed that circumstantial laboratory conditions, such as wall roughness or vibrations, perturbed the system and led to a lowering of the critical Reynolds number. As Blondeaux et al (2012) concluded later, the flow under a solitary wave is always unstable in the sense of linear stability and they suggested to use the growth of the kinetic energy attached to the perturbations as a measure for the appearance of transitions in the flow. However, a linear stability analysis as presented in Blondeaux et al (2012) cannot predict whether the flow after transition is turbulent or not, contrary to what the title of Blondeaux et al (2012) suggests.…”

“…As Blondeaux et al (2012) concluded later, the flow under a solitary wave is always unstable in the sense of linear stability and they suggested to use the growth of the kinetic energy attached to the perturbations as a measure for the appearance of transitions in the flow. However, a linear stability analysis as presented in Blondeaux et al (2012) cannot predict whether the flow after transition is turbulent or not, contrary to what the title of Blondeaux et al (2012) suggests. In a recent study Pedersen et al (2013) measured boundary layers during runup of solitary waves on a beach.…”