Low-order modelling of three-dimensional surface waves in liquid film flow on a rotating disk

Using low-dimensional numerical simulations, we investigate the characteristics of complex and three-dimensional surface waves in a liquid film flowing over a rotating disk, focusing on large flow rates from a nozzle. Existing integral boundary layer (IBL) models, which are based on spatially averaged variables along the direction normal to the disk surface, have primarily focused on the formation of axisymmetric waves under relatively small flow rates. In this study, an extended IBL model that accounts for both laminar and turbulent regimes is developed by considering the non-uniformity of the local flow rate in the spreading film flow and incorporating closure models dependent on the local Reynolds number. Our numerical results successfully capture the generation of concentric waves by an impinging circular liquid jet and their transition into three-dimensional solitary waves. These findings are in good agreement with visualization images and time-series data of free-surface fluctuations from a displacement sensor. The backscattering of small-scale three-dimensional turbulence into large-scale horizontal turbulence inside the film plays a critical role in determining the transition of wave modes and the nonlinear dynamics of the waves in the turbulent regime. Furthermore, the behaviour of three-dimensional waves in the downstream region, including frequent wave coalescence in the transition region and the breakup of small-scale solitons, is distinct from that of gravity-driven falling film flows. The amplitude of the three-dimensional waves is inversely related to the generalized Reynolds number defined for rotating films.