title: The evolutionary steps from nonlinear to turbulent waves on gravity-driven films over topography
Creators: name: Dauth, Markus
Publisher: name: University of Bayreuth
Year: 2019
Subjects: subject: Nonlinear waves
subject: turbulence
subject: transition to turbulence
subject: particle-tracking
subject: wavebreaking
subject: gravity-driven films
subject: films over topographies
subject: 500
subject: 530
subject: 600
subject: 620

description: A gravity-driven viscous film flow over an inclination is a fundamental problem in the field of fluid mechanics. It serves as a model to understand the physics behind a wide range of environmental and technical processes, which silently influence everyday life. For an idealized setup, which presumes that the viscous fluid flows over an inclined or vertical substrate of infinite extent which is perfectly flat, hydrodynamic effects have been studied extensively and are well-understood. In particular, the Navier-Stokes equations can be solved analytically. The well-known Nusselt solution with its quadratic velocity profile is at hand under these idealized constraints. Going beyond the idealized world reveals many fascinating effects, which are also essential for practical applications. The price to be paid, however, is a sharp increase in complexity. A prominent role plays, for example, the roughness of the underlying substrate. This roughness is either induced by accuracy limitations in preparation or intentionally. The influence of the substrate's undulations on the flow characteristics is tremendous and the interplay between the undulations and the flow gives rise to new phenomena, e.g., resonant standing waves and eddies. In general the flow over such inclined topographies cannot be calculated analytically in closed form from the Navier-Stokes equations. Since already small impurities on the substrate have a serious effect on the film flow and a predictable flow is required for various technical applications, e.g., curtain coating, falling film reactors, and heat exchangers, the understanding of the physical phenomena is vital. Also, most systems in the environment and technical applications have a finite extent and are bounded by, e.g., side walls which prohibit the assumption of a two-dimensional flow. In the respected film flows the interface between the fluid and the surrounding gas is considered as a shear free, deformable boundary. Hence, waves can emerge spontaneously at the free surface if a critical volume flux is exceeded. The waves appear as the flow reacts to disturbances, like ambient noise or external forcing. On their way downstream these waves can decrease or increase their amplitude depending on the flow parameters, the underlying substrate, and the wave parameters. Similar to the substrate's undulation, the waves at the free surface drastically impact the application of film flow systems in technical facilities. Therefore, extensive experimental and computational research is inevitable to understand the flow mechanisms. This dissertation deals with the influence of inertia and various substrate shapes on the evolution of linear, nonlinear, and turbulent free surface waves on a gravity-driven viscous film flow. The aim of this study is to follow all evolutionary steps of waves from the linear over the nonlinear to a turbulent regime and to assign accompanying phenomena to interactions of the waves and the flow. Combining the results of a wide range of experiments made it possible to reveal novel flow phenomena. In particular, the shape of the traveling free surface wave, the mechanism of wave breaking, the transition of regular wave fronts to irregular wave fronts, and the transition to a comprehensive turbulent flow will be discussed in detail and assigned to the interaction of the traveling free surface waves with the underlying steady flow. In the first step, linear and nonlinear waves were analyzed in detail. To do so, different geometrical shapes and excitation frequencies and amplitudes were considered in the experiments. Although sinusoidal waves were impressed in the experiments, two distinct wave shapes were found, namely convex and concave. By analyzing these wave shapes, an energy transfer from the excitation frequency to higher harmonics was inspected. The importance of the influence of the interaction between the waves and the substrate can be shown by the emergence of beat frequencies in the Fourier spectrum for undulated substrates. Additionally, particle tracking measurements of the steady state and the perturbed state unveiled major differences in the magnitude of the interaction between the traveling wave and the steady flow depending on the substrate's shape. The exact shape of the substrate is not determining the wave's shape but the shape of the steady state free surface is decisive for the evolution of the traveling free surface wave. The shape of the steady state free surface is a reaction of the flow to the underlying boundaries. Thereby, e.g., eddies have a strong impact on the flow and cause an effective substrate shape leading to the shape of the free surface. Convectively unstable nonlinear waves are in the focus of the next study, which uncovered a completely new phenomena on gravity-driven thin film flows, namely, the wave breaking. Wave breaking in different flow configurations, e.g., at a beach, is well-known textbook knowledge. However, the physical effects provoking the wave breaking at the beach are not present in the system considered in this work. Up to now, neither experimental nor theoretical work has shown this novel phenomena for film flows and hence, there is no explanation of the physical mechanism behind it. The evolutionary behavior of nonlinear waves can be categorized in three different domains: saturation, single wave breaking, and multiple wave breaking. In the experiments, a wave frequency selection for the wave breaking was found to differ for the substrates. Along the event of wave breaking, air is encapsulated underneath the flow over the whole channel width via plunging of the wave into a trough of the steady state free surface. Due to an energetically favorable state the air encapsulation contracts to an orderly bubble. As the decisive parameter for the wave breaking mechanism the global or mean curvature of the steady state free surface was identified, due to the fact that wave breaking needs obstacles with protruding, prominent, and sufficiently high peaks. These obstacles provoke corresponding peaks on the steady state free surface which act as stumbling stones for the waves. The subsequent study on nonlinear waves went one step further to unveil the phenomenon of the transition of regular wave fronts to irregular wave fronts. Thereby, a subdivision of the wave into a chaotic wave front and regular wave tail was found. This means that a coexistence of a regular and turbulent regime can be achieved on film flows by varying the inertia in the system. Based on the finding of the irregular wave front, an extensive study on the transition was performed. That way, the necessity of wave breaking as a precursor is revealed and a transition to turbulence in line with the Landau theory for the transition to turbulence is at hand. In this theory the state of turbulence is reached via multiple bifurcations which repeat themselves ``indefinitely''. The wave breaking acts as the bifurcation and new stable and unstable waves emerge from the collapse. Since wave breaking needs protruding, prominent, sufficiently high peaks as obstacles, this transition of regular to irregular wave fronts was found on strongly undulated, anharmonic, rectangular substrates. The necessity of strongly undulated anharmonic substrates for the transition of regular to irregular wave fronts leads to the question: Can harmonic undulated substrates also show this kind of turbulence? This question was challenged with new measurements concerning a flow over a sinusoidal substrate. Within the framework of this dissertation the transition of regular to irregular wave fronts on a sinusoidal topography was not observed. Nevertheless, a transition to turbulence of the whole flow was recorded. In the experiments the transition to turbulence was observed for a sufficiently high inertia and irrespective of an external excitation.