New insights on impulse wave formation from a Newtonian collapse in water

Impulse waves have been the focus of numerous studies due to their relationship with natural disasters like tsunamis. Because they can be generated from many different geographical configurations, from submarine earthquakes to subaerial landslides, these numerous studies consider different triggering mechanisms, resulting in a wealth of analyses which remain somewhat ununified. Similarly, and just in the case of subaerial landslides, to unravel the dynamics of these waves, physicists have adopted many different approaches, from solid blocks to granular media penetrating through a body of water.

As scientists were faced with a huge number of slide and terrain properties influencing the waves dynamics, these quantities have been grouped as dimensionless numbers to develop semi-analytical models that correlate these numbers with meaningful wave properties like the maximum wave amplitude. Unfortunately, such models only correlate the slide and wave properties after the slide has penetrated through water and the wave has started propagating – hence too late to anticipate the wave formation!

“Owing to the societal stakes behind these studies, it is paramount to predict the wave formation from initial conditions prior to the slide motion” argue PhD candidate Quentin Kriaa, Dr. Sylvain Viroulet and Dr. Laurent Lacaze from Université de Toulouse in a joint statement to Advances in Engineering. In their recent work currently published in the journal, Physical Review Fluids, the authors point out that common features stand out from the eclectic approaches in the literature: for them, this calls for a canonical modeling that gets rid of secondary effects to focus only on the common ingredients between all these situations. “We have found beneficial and vital to synthetize the physics to the core ingredients. For example, compared to solid blocks, granular media bring in the essential ingredient of the slide deformability. But unfortunately, it goes with other ingredients which can hardly be dissociated, like the slide porosity and dilatancy… This can be simplified” they remark.

In their study, the group of researchers from Toulouse proposed a synthetic model of impulse waves produced by subaerial landslides. Essentially, such waves are produced by a transfer of energy from a falling slide to a body of water which is displaced during the fall. The authors expected the energy transfer to be nourished by the slide fall and deformation, hence by its inertia and dissipation. Thus, to investigate the formation of an impulse wave, they adopted the collapse of a column of Newtonian slide from air to water. Performing 2D three-phase simulations, the authors explored the influence of slide inertia (slide-to-water density ratio) and dissipation (slide-to-water viscosity ratio) beyond laboratory and geophysical values to gain understanding and, thus, provide a predictive model of wave formation.

The research team showed that the column collapse of the Newtonian slide succeeded in capturing the physics of wave formation. From rollers to solitary-like waves, the different well-known types of waves were recovered with this physics, and so was the long-established correlation between the maximum wave amplitude η_0,max and the maximum slide front velocity – in dimensionless form, the maximum Froude number Fr_max.

Formation of a specific wave among the three observed is determined by the interplay between viscous dissipation and slide inertia, “because this is what controls the slide dynamics” insist the researchers. “The wave maximum amplitude is usually correlated to the maximum slide front velocity through a relationship η_0,max(Fr_max) because indeed, the wave growth is nourished by the kinematics of the slide-water interface. Yet, this very kinematics is controlled by the slide dynamics, quantified by a Reynolds number which can be predicted!”. In fact, with a simple analytical model and with their numerical simulations, the researchers described how the slumping slide transitions from an accelerating fall to an inertial or viscous regime, and they managed to compute the maximum Reynolds number solely from initial conditions. The maximum Froude number was then deduced, offering the possibility to predict the wave type and its maximum amplitude.

“We didn’t stop there. To get an in-depth understanding and to predict the wave evolution without using correlations, we wanted to analytically describe how the slide motion nourishes the transient wave growth,” add the researchers. Their models capture the wave growth just with volume conservation: during a short amount of time, a volume of water is displaced by the slide, so an equal amount has to rise above the still water level as a perturbation. The faster the slide motion, the larger the volume displaced per unit time, the taller the water perturbation and the faster it propagates. As long as the perturbations do not run away from the slide, the slide fall keeps feeding the wave as an accumulation of such perturbations. When perturbations move faster than the slide tries to feed them, the wave propagates away: this is the end of the wave generation. “Despite its simplicity, this reasoning captures the correlation η_0,max(Fr_max), so combining all the elements together, it is now possible to estimate the wave maximum amplitude directly from initial conditions!” remarks the research group.