Falling film flow along steep two-dimensional topography: The effect of inertia

Steady film flow along a vertical wall with isolated step changes is studied numerically for Reynolds numbers Re ∼ O(10⁻³–10²) and capillary numbers Ca ∼ O(10⁻²–10¹). The lengthscale of free surface capillary features upstream of a step-in or step-out decreases uniformly with Re and switches from a −1/3 to a −1/2 power-law dependence on Ca. The height of the capillary features first grows with Re, but eventually diminishes when inertia forces overpower capillary forces. Simultaneously, the key dynamics move from upstream to downstream of the step, and switch from capillary arrest to inertial re-directioning of the falling liquid. The latter mechanism involves a low-pressure region originating from the edge of the step. At a step-out, a new free surface feature appears with increasing Re, which is caused by liquid overshoot in the horizontal direction and is restrained initially by capillary and subsequently by inertial forces. Simple scaling arguments are shown to predict many of the above characteristics.