Impulse response and spatio-temporal wave-packets: The common feature of rogue waves, tsunami, and transition to turbulence

Abstract

Here, we present the impulse response of the canonical zero pressure gradient boundary layer from the dynamical system approach. The fundamental physical mechanism of the impulse response is in creation of a spatio-temporal wave-front (STWF) by a localized, time-impulsive wall excitation of the boundary layer. The present research is undertaken to explain the unit process of diverse phenomena in geophysical fluid flows and basic hydrodynamics. Creation of a tsunami has been attributed to localized events in the ocean-bed caused by earthquakes, landslides, or volcanic eruptions, whose manifestation is in the run up to the coast by surface waves of massive amplitude but of very finite fetch. Similarly rogue waves have often been noted; a coherent account of the same is yet to appear, although some explanations have been proposed. Our studies in both two- and three-dimensional frameworks in Sengupta and Bhaumik [“Onset of turbulence from the receptivity stage of fluid flows,” Phys. Rev. Lett. 107(15), 154501 (2011)] and Bhaumik and Sengupta [“Precursor of transition to turbulence: Spatiotemporal wave front,” Phys. Rev. E 89(4), 043018 (2014)] have shown that the STWF provides the central role for causing transition to turbulence by reproducing carefully conducted transition experiments. Here, we furthermore relax the condition of time behavior and use a Dirac-delta wall excitation for the impulse response. The present approach is not based on any simplification of the governing Navier-Stokes equation (NSE), which is unlike solving a nonlinear shallow water equation and/or nonlinear Schrödinger equation. The full nonlinear Navier-Stokes equation (NSE) is solved here using high accuracy dispersion relation preserving numerical schemes and using appropriate formulation of the NSE which minimizes error. The adopted numerical methods and formulation have been extensively validated with respect to various external and internal 2D and 3D flow problems. We also present results from the Orr-Sommerfeld equation to show that the origin of the STWF is via a linear mechanism. Nonlinearity and nonparallelism play the central role in causing these phenomena of geophysics and transition to turbulence.