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.
Instability of a spatially developing liquid sheet
The linear stability of an inviscid two-dimensional liquid sheet falling under gravity in a still gas is studied by analysing the asymptotic behaviour of a localized perturbation (wave-packet solution to the initial value problem). Unlike previous papers the effect of gravity is fully taken into account by introducing a slow length scale which allows the flow to be considered slightly non-parallel. A multiple-scale approach is developed and the dispersion relations for both the sinuous and varicose disturbances are obtained to the zeroth-order approximation. These exhibit a local character as they involve a local Weber number Weη. For sinuous disturbances a critical Weη equal to unity is found below which the sheet is locally absolutely unstable (with algebraic growth of disturbances) and above which it is locally convectively unstable. The transition from absolute to convective instability occurs at a critical location along the vertical direction where the flow Weber number equals the dimensionless sheet thickness. This critical distance, as measured from the nozzle exit section, increases with decreasing the flow Weber number, and hence, for instance, the liquid flow rate per unit length. If the region of absolute instability is relatively small it may be argued that the system behaves as a globally stable one. Beyond a critical size the flow receptivity is enhanced and self-sustained unstable global modes should arise. This agrees with the experimental evidence that the sheet breaks up as the flow rate is reduced. It is conjectured that liquid viscosity may act to remove the algebraic growth, but the time after which this occurs could be not sufficient to avoid possible nonlinear phenomena appearing and breaking up the sheet.
Secoiridoids from Olea europaea
Abstract Three new secoiridoids have been isolated from the leaves of Olea europaea along with the previously reported. The compounds were identified by spectral means.