Numerical visualization of nappe oscillation
Abstract The unsteady global dynamics of a gravitational liquid sheet interacting with a onesided adjacent air enclosure, typically referred to as nappe oscillation, is addressed, under the assumptions of potential flow and presence of surface tension effects. Preliminary physical insights of the sheet centerline sinuous modes show that the nappe dynamics features the propagation of two wave fronts both directed downstream or one downstream and the other one upstream depending on whether the flow Weber number is greater or less than unity, respectively; moreover, the overall system behaves as a driven damped spring-mass oscillator, where the liquid sheet acts as the mass and the air enclosure as the spring. The findings of the eigenvalues spectral analysis closely agree with direct numerical simulations of the partial differential equation governing in the space-time domain the global evolution of the disturbances, starting from an initial gaussian-like shape.
On the stability of a falling liquid curtain
The stability of a falling liquid curtain is investigated. The sheet of liquid is assumed two-dimensional, driven by gravity and influenced by a compressible cushion of air enclosed on one side of the curtain. The linear stability problem is formulated in the form of an integro-differential eigenvalue problem. Although experimental efforts have consistently reported a peak in the low-frequency range of the spectrum, the linear stability results do not show instabilities at these frequencies. However, a multi-modal approach combined with a projection onto low-frequency modes reveals a dominant and robust instability feature that is in good agreement with experimental measurements. This instability manifests itself as a wave packet, consisting of a linear superposition of linear global modes, that travels down the curtain and causes a strong pressure signal in the enclosed air cushion.
The mechanics of edge-tones
By consideration of the frequency spectrum of the sound field of an oscillating linear jet, it is found that one may expect a continuous spectrum of frequencies from 0 to 0.08U0/d, where U0 = mean velocity of efflux of the jet, d = width of slit, the maximum amplitude occurring at the frequency 0.055U0/d which is found experimentally. To explain the tones produced when an edge is placed in such a flat jet, it is shown that the hydrodynamic effect of the edge is to amplify those oscillations of the jet which have a certain particular frequency in the band of frequencies to which the jet is unstable. This is due to the reciprocal influence of the maximum transverse velocity in the jet upon the shedding of vortices at the edge, and of the sudden production of free vorticity at the edge in deflecting the jet and so causing a slight extra production of vorticity on one side of the slit, which is in turn amplified as it moves downstream. From this one deduces that vortices of opposite circulation are produced simultaneously at the upper edges of the slit and the wedge, and similarly at the lower edges, and also that the jet wave-length λ and the edge to slit distance h are related by an equation h = (i + 1/4) λ, where i is an integer. It is suggested that the value of i is such that λ is near the jet-tone wave-length which gives maximum amplitude. Independently of this, an expression, partly theoretical and partly empirical, is obtained for the velocity with which the vortices move along. A formula for the edge-tone frequency is then deduced in the form f = 1/2 Uo {i + 1/4 / h – 1/30d}, where d = slit-width. This formula applies only when h/d > 10, i.e. when the edge to slit distance is at least 10 slit-widths. The predictions of the theory as regards the numerical values of the jumps in frequency, dependence of frequency upon slit-width, and absolute values of wave-length and frequency, are examined in the light of previous experimental results, and are found to be satisfactory.