title: 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 Iη∼k(α), where α=α0=-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=C0=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 C0 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 C0 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<α0 and C>C0, all due to finite box effect.
authors.affiliation: Department of Mechanical Engineering, Massachusetts Institute of Technology, Cambridge, Massachusetts 02139, USA.
authors.forename: Yulin
authors.initials: Y
authors.lastname: Pan
authors.affiliation: Department of Mechanical Engineering, Massachusetts Institute of Technology, Cambridge, Massachusetts 02139, USA.
authors.forename: Dick K P
authors.initials: DK
authors.lastname: Yue