Nonlinear coupling of interfacial instabilities with resonant wave interactions in horizontal two-fluid plane Couette–Poiseuille flows: numerical and physical observations

Abstract

We investigate mechanisms governing the initial growth and nonlinear evolution of interfacial waves in horizontal two-fluid plane Couette–Poiseuille flows. Nonlinear coupling of the Kelvin–Helmholtz interfacial instability with resonant wave interactions has been shown to be capable of rapidly generating long waves through the transfer of energy from linearly unstable short waves to stable long-wave components within the context of potential flow theory. The objective of this work is to determine whether that coupled mechanism persists in laminar and turbulent viscous flows. Utilizing both theoretical and computational methods, we analyse the initial Orr–Sommerfeld instability to quantify the frequencies and growth/decay rates of each wave mode for two-fluid laminar and turbulent channel flows. The obtained dispersion relation allows for the identification of resonant and/or near-resonant triads among (unstable and damped) wave components in an interfacial wave spectrum. We perform direct numerical simulations (DNS) of the two-phase Navier–Stokes equations with a fully nonlinear interface to formally establish the validity of our theoretical predictions for viscous flows. DNS results show the existence of a nonlinear energy cascade from unstable short- to damped long-wavelength waves due to resonant subharmonic and/or triadic interactions in both laminar Couette and turbulent Poiseuille flows. Spectral analysis of the interfacial evolution confirms that the combined instability–resonance mechanism persists in the presence of viscosity despite being derived under the assumption of potential flow theory. Finally, we perform a detailed examination of experimentally measured wave power spectra from Jurman et al. (J. Fluid Mech., vol. 238, 1992, pp. 187–219) and carry out a numerical sensitivity study of the flow conditions to demonstrate and verify the existence of the coupled instability–resonance mechanism in physical systems. Our analysis accurately predicts the initial instability and the resulting nonlinear energy cascade through subharmonic and triadic interfacial wave resonances.

The principal stage in wind-wave generation

“…2015; Kurose et al. 2016; Campbell, Hendrickson & Liu 2016). Compared with LES, the Reynolds number in DNS is relatively low when resolving the Kolmogorov scale (Moin & Mahesh 1998).…”

“…Coupled air-water simulations have also been performed, usually focusing on the dynamics of irregular waves or scalar transfer (e.g. Lin et al 2008;Liu et al 2009;Komori et al 2010;Zonta et al 2015;Kurose et al 2016;Campbell, Hendrickson & Liu 2016). Compared with LES, the Reynolds number in DNS is relatively low when resolving the Kolmogorov scale (Moin & Mahesh 1998).…”

Modelling entrainment volume due to surface-parallel vortex interactions with an air–water interface

“…Finally, surface tension effects are implemented through the continuous surface force (CSF) method (Brackbill, Kothe & Zemach 1992) along with a height-function method for the interface curvature (Popinet 2009, 2018). Details of the formulation and validations of the numerical method are available in Campbell, Hendrickson & Liu (2016) and Yu et al. (2019 a ).…”

A Moving-Wave Implementation in WRF to Study the Impact of Surface Water Waves on the Atmospheric Boundary Layer

“…In two-way coupled models, the wind and the wave are coupled dynamically and evolve together. Existing two-way coupled models use either a curvilinear coordinate (e.g., Yang and Shen 2011a,b;Li and Shen 2022b) or the volume-of-fluid method (e.g., Campbell et al 2016;Cimarelli et al 2023;Wu and Deike 2021) to resolve the twophase flow. The majority of the two-way coupling studies employ DNS and focus on the small-scale dynamics of nonlinear wave-wave interaction, turbulence on the waterside, scalar transfer across the interface or wind-wave generation (Lin et al 2008;Liu et al 2009;Komori et al 2010;Campbell et al 2016;Wu and Deike 2021;Li and Shen 2022a,b).…”

“…Existing two-way coupled models use either a curvilinear coordinate (e.g., Yang and Shen 2011a,b;Li and Shen 2022b) or the volume-of-fluid method (e.g., Campbell et al 2016;Cimarelli et al 2023;Wu and Deike 2021) to resolve the twophase flow. The majority of the two-way coupling studies employ DNS and focus on the small-scale dynamics of nonlinear wave-wave interaction, turbulence on the waterside, scalar transfer across the interface or wind-wave generation (Lin et al 2008;Liu et al 2009;Komori et al 2010;Campbell et al 2016;Wu and Deike 2021;Li and Shen 2022a,b). Although two-way coupled models resolve the influence of the wind on waves, they are computationally expensive and limited to small-scale dynamics at relatively low Reynolds numbers.…”