Direct Numerical Investigation of Turbulence of Capillary Waves

Abstract

We consider the inertial range spectrum of capillary wave turbulence. Under the assumptions of weak turbulence, the theoretical surface elevation spectrum scales with wave number k as I η ∼ k α , where α ¼ α 0 ¼ −19=4, energy (density) flux P as P 1=2 . The proportional factor C, known as the Kolmogorov constant, has a theoretical value of C ¼ C 0 ¼ 9.85 (we show that this value holds only after a formulation in the original derivation is corrected). The k −19=4 scaling has been extensively, but not conclusively, tested; the P 1=2 scaling has been investigated experimentally, but until recently remains controversial, while direct confirmation of the value of C 0 remains elusive. We conduct a direct numerical investigation implementing the primitive Euler equations. For sufficiently high nonlinearity, the theoretical k −19=4 and P 1=2 scalings as well as value of C 0 are well recovered by our numerical results. For a given number of numerical modes N, as nonlinearity decreases, the long-time spectra deviate from theoretical predictions with respect to scaling with P, with calculated values of α < α 0 and C > C 0 , all due to finite box effect. [5]. In special cases of weak (or wave) turbulence, mathematical formulations are more accessible and the cascade spectrum can be obtained as an exact stationary solution of the kinetic equation, which governs the evolution of wave spectrum due to nonlinear resonant interactions.For capillary waves, the framework of weak turbulence theory (WTT) is developed by Zakharov and Filonenko [3] (and reformulated in [6,7]). The isotropic spectrum of surface elevation yields a powerlike solution in the inertial range which is expressed in closed form,

Growth and spectra of gravity–capillary waves in countercurrent air/water turbulent flow

“…Similar power laws were observed in experiments by Wright, Budakian & Putterman (1996), Falcon et al (2007) and Cobelli et al (2011), and more recently in simulations by Deike et al (2014) and Pan & Yue (2014). Although WTT is developed under a number of restrictions (weak nonlinearity, homogeneity, isotropy and separation of scales between forcing and dissipation), it has already been observed ) that some theoretical predictions can hold even if such hypotheses are not fully met.…”

Role of the basin boundary conditions in gravity wave turbulence

“…Indeed, a similar phenomenon has previously been reported both experimentally, in studies of wave turbulence on a metallic plate (Humbert et al 2013;Miquel et al 2014), and of capillary wave turbulence (Deike et al 2014a) when increasing dissipation (e.g. adding dampers on the plate, or working with high enough viscosity fluids), and numerically when reducing the nonlinear interactions (Pan & Yue 2014). Here also, the ratio between dissipation and nonlinearity has to be small enough at all scales to reach a wave turbulence regime.…”

“…by wave breaking) with respect to nonlinear interactions may also explain the steepening of the gravity spectrum at low nonlinearity. In capillary wave turbulence, a similar phenomenon of steepening of the spectrum at low nonlinearity has been reported experimentally when working with fluids of sufficiently high viscosity (Deike et al 2014a) and numerically when reducing the nonlinear interactions (Pan & Yue 2014).…”

Wave turbulence: the case of capillary waves

“…Few works, however, have been done in comparison to the many experimental studies. These simulations were first done by Pushkarev and Zakharov [51,65], then more recently by Deike et al [66] and Pan and Yue [67]. Surface waves are, however, subject to numerical instabilities which make it difficult to obtain an extended inertial zone (less than one decade).…”