Session A14: Free-Surface Flows: General

High order finite volume method for solving porous shallow water equations over discontinuous geometry

The present study presents the Riemann solvers for solving porous shallow water equations over the discontinuous porosity and bottom topography. Firstly, extending previous works for solving non-strictly hyperbolic systems, we construct the exact Riemann solver. In which, the stationary wave is considered as the elementary wave reflecting the effects of the standing discontinuity in geometry. In particular, in order to guarantee the uniqueness of the stationary wave, we prove that there are geometric functions to which the monotonic criterion can be applied despite the co-existence of variations in porosity and bottom. From the intensive investigation of the Riemann problem of porous shallow water equations, 17 types of solution structures, 10 wet and 7 dry cases, are confirmed. Secondly, reflecting the structure of the exact Riemann solver, we implement approximate Riemann solvers based on the path-conservative finite volume method. Utilizing the geometric functions which satisfy the monotonic criterion for regularization paths, the non-conservative products are measured in order to reflect the effect of the geometric discontinuity at the cell interface. Furthermore, a Well-balanced WENO reconstruction is implemented to achieve high-order accuracy and exact C-property. Positivity-preserving property is considered to prevent depth from becoming negative during simulation. Stationary wave reconstruction is formulated to reflect the structure of the exact solution. Extensive numerical tests are performed to examine the well-balanced property, high order accuracy, and convergence to Riemann solutions for all types of wave configurations. The numerical results show good agreements with analytical solutions.   

Marangoni flow induced in a waterbody by the impact of a raindrop

The particle image velocimetry (PIV) technique is used to measure the transient flow induced near the surface of a waterbody due to the impact of a freshwater drop. The focus of this work is on the influence of surfactant and salt concentrations of the waterbody on the induced Marangoni flow. The latter arises due to the mixing of the drop's water with the surface water, which reduces the concentration of salt and surfactant on the surface, changing the surface tension of the water in the impact area. Since the surface tension of water increases with increasing salt concentration and decreases with increasing surfactant concentration, the net Marangoni flow depends on the salt and surfactant concentrations in the waterbody. Experiments show that when the surfactant-induced Marangoni flow dominates, the radially outward flow on the surface due to the drop's impact quickly reverses direction and becomes directed towards the point of impact. However, when the salt-induced Marangoni flow dominates, the radially outward flow on the surface due to the impact of the drop is enhanced and persists for a longer time interval.   

Corner universality in polygonal hydraulic jumps

Steady polygonal hydraulic jumps are formed when a circular jump loses stability through an increase in the downstream liquid height beyond a critical value. We report the experimental observation of a universal corner shape in polygonal hydraulic jumps over a wide range of experimental conditions that include the flow rate, weir geometry, and flow history, defining the different universality classes through the tip radius of curvature and corner angle. The tip radius of curvature is nearly constant over all experimental conditions, whereas the corner angle weakly depends on gravitational effects. Knowledge of the corner angle allows one to determine the global jump shape, as defined by a dimensionless geometry number related to the isoperimetric inequality, thus giving a complete description of the jump shape through patching.

    

Modulated hexagons in Marangoni convection with deformable surface covered by surfactant

Hexagons are typical patterns generated by the Marangoni instability in a liquid layer. In the vicinity of the instability threshold, the pattern evolution can be described by a system of Ginzburg-Landau-type amplitude equations. Due to the asymmetry of the boundary conditions at the top and the bottom of the layer, those equations contain additional non-gradient quadratic terms with spatial derivatives. These terms are especially significant for non-equilateral hexagons based on the resonance of wave vectors with slightly different lengths.

We consider the onset of Marangoni convection created by thermocapillary and solutocapillary stresses in a layer with deformable free interface covered by surfactant. In that case, the description has to include two additional variables, a large-scale deformation of the boundary and disturbance of the surfactant, which evolve slowly because of the conservation of the

liquid volume and of the surfactant amount. The interaction of longwave disturbances near the bifurcation point can create a modulational instability of periodic patterns.

We study the joint action of the deformability of the liquid free surface and the effect of insoluble surfactant spread over the surface on the regular (equilateral) and deformed (non-equilateral) hexagons and their modulational instability. The governing parameters of the problem are Biot number characterizing the heat transfer resistance of the surface and Galileo number indicating the role of gravity. Stability maps are plotted for different concentration of insoluble surfactant.

    

Wave transformation and air entrainment by a forced plunging liquid jet

Plunging liquid jets are ubiquitous in real-world flows, ranging from breaking waves to foundries. These jets are known to be important to mass exchange between the liquid and air, through their ability to entrain air within the liquid bulk. In the case of circular plunging jets, these entrainment events historically have been understood to occur with either very high velocities, or very large jet disturbances. Here, harmonic disturbances are imposed upon a circular plunging jet that is otherwise undisturbed, to determine the influence of such disturbances on air entrainment. This jet is found to entrain air at a low plunge velocity with small disturbances, the mechanism for entrainment being an interaction between subharmonic surface waves and the submerged jet flow. These surface waves appear due to harmonic forcing of the jet, a physical mechanism previously observed with surface-piercing bodies that lacked the submerged flow driven by the plunging jet. Experimental results concerning the onset of these waves, the inception characteristics and volume of their air entrainment, and their interactions with the jet flow will be presented, as well as comparisons to observations and measurements from the literature.   

Snail feeding: Thin film flow with an undulating surface

Apple snails (Pomacea canaliculata) collect food particles on the free surface by deforming their feet and generating periodic undulations underneath the air-water interface. Inspired by this peculiar feeding behavior, we develop a 2D thin-film mathematical model to explore the physical mechanism of free surface flows driven by periodic undulations. We find the surface undulations cause normal stresses that deform the air-fluid interface and drive a net flow, which is directly controlled by the fluid properties and the undulation wave speed. Our model also demonstrates the effects of inertia that can significantly alter thin-film flows, in good agreement with the experimental observations. In this talk, we will discuss some preliminary model results along with the experimental findings and offer plausible physical mechanisms driving the flow system.