Christian Heining

Bottom reconstruction in thin-film flow over topography: Steady solution and linear stability

We consider viscous gravity-driven films flowing over undulated substrates. Instead of the widely studied direct problem of finding the free surface for a given bottom topography, we focus on the inverse problem: Given a specific free surface shape, we seek the corresponding bottom topography which causes this free surface profile. As an asymptotic approach for thin films and moderate Reynolds numbers, we apply the weighted-residual integral boundary-layer method which enables us to derive a set of two evolution equations for the film thickness and the flow rate. We prescribe the free surface as a monofrequent periodic function and discuss the influence of inertia, film thickness, and surface tension on the shape of the corresponding substrate. For small free surface undulations, we can solve the bottom contour analytically and study its parametric dependence. The analytical results are then validated with numerical simulations. Furthermore, we consider the stability of the corresponding direct problem, w...

Weighted-residual integral boundary-layer model for the nonlinear dynamics of thin liquid films falling on an undulating vertical wall

A set of first-order weighted-residual integral boundary-layer equations describing the nonlinear dynamics of a thin liquid film falling on a corrugated periodic vertical wall is derived. The spatiotemporal dynamics of the films is analyzed analytically and numerically in the framework of this set. A steady-state flow is found in the case of an asymptotically small wall corrugation and its stability is investigated. It is found that steady flow regimes arise in the case of a relatively small wall wavelength for the Reynolds number below its critical value corresponding to the flat-wall flow and for larger amplitudes of the wall corrugation when the Reynolds number exceeds its critical value. In the case of a larger wall wavelength, the emerging flows are either genuinely nonstationary or time periodic. The temporal period of the time-periodic flows increases with the amplitude of the wall corrugation and decreases with the Reynolds number. A possibility of the emergence of reverse flows is also discussed.

Suppression of eddies in films over topography

We study inertial film flow down undulated inclines. With increasing Reynolds number, eddies are formed in the troughs of the bottom undulation. Further increase leads to a diminution of the eddies until they vanish completely. At even higher Reynolds numbers, they reappear yielding an eddy-free window of Reynolds numbers. Within this window, the free-surface shape changes abruptly. The change comes along with a sudden decrease in the mean film thickness and an abrupt transition of the surface shape type from anharmonic with a strong indentation to harmonic. The anharmonic surface shape shows typical features of a hydraulic jump, which vanishes during the transition. We find that the eddies disappear at Reynolds numbers where the first harmonic of the free-surface contour is sufficiently strong regardless of the exact surface shape. Numerical calculations are in good agreement with the experimental findings.

Pattern formation and mixing in three-dimensional film flow

The effect of inertia on gravity-driven free surface flow over different three-dimensional periodic corrugations is considered analytically, numerically and experimentally. In the case of high bottom amplitudes, compared to the film thickness, the results predict complex free surface structures especially in cases where the topography is not fully flooded by the liquid film. The investigation of the flow field shows a rich variety of pattern formation phenomena depending on the interplay between the geometry of the topography and the inertia of the film. Finally, we show how the complex topographical structure enhances the laminar mixing within the film.

Velocity field reconstruction in gravity-driven flow over unknown topography

A numerical method for reconstructing the velocity field of a viscous liquid flowing over unknown topography is presented. For a given fluid this procedure allows one to determine the velocity field as well as the topographic structure from the free-surface shape only. First, we confirm the results with previous computations in the thin-film limit and then generalize the numerical solution to arbitrary film thicknesses and focus on the velocity field. It is documented that even smoothly corrugated free-surface shapes require strongly undulated topographies to maintain the flow structure. Finally, we discuss details of the implementation in applications, solvability in general, and sensitivity of the solution.

One-dimensional bathymetry based on velocity measurements

Modelling the hydrodynamics of open channel flows requires the prior knowledge of the channel bed topography in order to accurately determine the flow features. As an alternative to measure the bed topography either by direct or airborne optical measurement, a numerical technique which uses the measured flow velocity to infer the channel bed topography is presented. The depth-averaged one-dimensional shallow water equations along with an empirical relationship between the free-surface and the depth-averaged velocities are used for the inverse problem analysis. It is shown that after a series of algebraic manipulation and integration, the equation governing the inverse problem simplifies to a simple integral equation. The proposed method is tested on a range of analytical and experimental benchmark test cases and the results confirm that, within the given assumptions, it is possible to reconstruct the river bed topography from a known velocity distribution (either free-surface velocity or depth-averaged velocity).

Flow domain identification in three-dimensional creeping flows

This study presents a new method to reconstruct the three-dimensional flow domain in thin gravity-driven film flows using an inversion strategy of the lubrication equation. With only the knowledge of the free surface velocity, it is possible to reconstruct the film thickness, the internal pressure field, and the topography shape. For each unknown variable, we derive the corresponding partial differential equation and present numerical algorithms for the solution. The success of the reconstruction is underpinned with examples of flows over trench and bump topographies. It can be shown that the inversion strategy is robust with respect to external perturbations in the form of noisy input data which occur in experimental setups. The proposed method is finally compared to experimental data in the literature and to numerical solutions of the full Navier-Stokes equations.