Weakly nonlinear instability of planar viscous sheets

A second-order instability analysis has been performed for sinuous disturbances on two-dimensional planar viscous sheets moving in a stationary gas medium using a perturbation technique. The solutions of second-order interface disturbances have been derived for both temporal instability and spatial instability. It has been found that the second-order interface deformation of the fundamental sinuous wave is varicose or dilational, causing disintegration and resulting in ligaments which are interspaced by half a wavelength. The interface deformation has been presented; the breakup time for temporal instability and breakup length for spatial instability have been calculated. An increase in Weber number and gas-to-liquid density ratio extensively increases both the temporal or spatial growth rate and the second-order initial disturbance amplitude, resulting in a shorter breakup time or length, and a more distorted surface deformation. Under normal conditions, viscosity has a stabilizing effect on the first-order temporal or spatial growth rate, but it plays a dual role in the second-order disturbance amplitude. The overall effect of viscosity is minor and complicated. In the typical condition, in which the Weber number is 400 and the gas-to-liquid density ratio is 0.001, viscosity has a weak stabilizing effect when the Reynolds number is larger than 150 or smaller than 10; when the Reynolds number is between 150 and 10, viscosity has a weak destabilizing effect.

Numerical simulations of dense clouds on steep slopes: application to powder-snow avalanches

In this paper, two-dimensional direct numerical simulations (DNS) of dense clouds moving down steep slopes are presented for the first time. The results obtained are in good agreement with the overall characteristics, i.e. the spatial growth rate and velocity variations, of clouds studied in the laboratory. In addition to the overall flow structure, DNS provide local density and velocity variations inside the cloud, not easily accessible in experiments. The validity of two-dimensional simulations as a first approach is confirmed by the dynamics of the flow and by comparison with experimental results. The interest of the results for powder-snow avalanches is discussed; it is concluded that two-dimensionality is acceptable and that large density differences need to be taken into account in future simulations.

Long-wavelength asymptotics of unstable crossflow modes, including the effect of surface curvature

Stationary vortex instabilities with wavelengths significantly larger than the thickness of the underlying three-dimensional boundary layer are studied with asymptotic methods. The long-wavelength Rayleigh modes are locally neutral and aligned with the direction of the local inviscid streamline. For a spanwise wave number β ≪ 1, the spatial growth rate of these vortices is O ( β 3/2 ). When β becomes O ( R -1/7 ), the viscous correction associated with a thin sublayer near the surface modifies the inviscid growth rate to the leading order. As β is further decreased through this regime, viscous effects assume greater significance and dominate the growth-rate behaviour. The spatial growth rate becomes comparable to the real part of the wave number when β = O ( R -¼ ). At this stage, the disturbance structure becomes fully viscous-inviscid interactive and is described by the triple-deck theory. For even smaller values of β , the vortex modes become nearly neutral again and align themselves with the direction of the wall-shear stress. Thus the study explains the progression of the crossflow-vortex structure from the inflectional upper branch mode to nearly neutral long-wavelength modes that are aligned with the wall-shear direction.