Roll waves on laminar sheet flow of Newtonian fluid with negligible surface tension

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We conduct direct numerical simulations (DNS) to study the temporal and spatial developments of the roll waves on a laminar sheet flow of Newtonian fluid. The DNS unveil the physics of the wavefront and show the limitation of the widely used shallow-layer approximations. The most prominent wave, the front runner, is determined by the DNS for the first time in studying the spatial development of the laminar sheet flow with negligible surface tension. Depending on the Froude and Reynolds numbers, the front runner can be a multi-peaked undular bore or a single-peaked non-breaking or breaking wave. The simulation has uncovered an extended region behind the wavefront, where the bed-friction stress is much higher than the corresponding friction in the undisturbed uniform flow. It also produces an uplift velocity needed in the description of wave breaking. For comparison, we also examine the nonlinear development of the instability using two-equation and four-equation shallow-layer models. The two-equation shallow-layer model has produced the bulk of the wave profile but is deficient because it fails to predict the uplift velocity and the substantial increase in bed friction in the frontal region. The four-equation shallow-layer model correctly predicts the bed friction but cannot produce the breaking wave. The simulations also determine the celerity and amplitude of the front runner to follow a linear relationship, qualitatively similar to the roll waves in a turbulent flow.