The front runner in roll waves produced by local disturbances

Roll waves produced by a local disturbance comprise a group of shock waves with steep fronts. We used a robust and accurate numerical scheme to capture the steep fronts in a shallow-water hydraulic model of the waves. Our simulations of the waves in clear water revealed the existence of a front runner with an exceedingly large amplitude – much greater than those of all other shock waves in the wave group. The trailing waves at the back remained periodic. Waves were produced continuously within the group due to nonlinear instability. The celerity depended on the wave amplitude. Over time, the instability produced an increasing number of shock waves in an ever-expanding wave group. We conducted simulations for three types of local disturbances of very different duration over a range of amplitudes. We interpreted the simulation results for the front runner and the trailing waves, guided by an analytical solution and the laboratory data available for the smaller waves in the trailing end of the wave group.

Hydrodynamic instability of downslope flow with respect to two-dimensional perturbations

<p>Hydrodynamic instability of open flows down inclines is an important phenomenon which leads perturbation growth, turbulence, roll waves formation etc. It has been widely studied for flows of Newtonian rheology with respect to longitudinal perturbations (perturbations that spread along the flow velocity vector), for example, see works [1 - 4]. From mathematical point of view, the study of the stability of open flow down an inclined planes with respect to two- or three-dimensional perturbations (i.e., with respect to oblique perturbations, spreading under an arbitrary angle to the flow velocity vector) is quite difficult, especially, if the fluid has non-Newtonian rheological properties, which can be important in the context of geophysical applications. Nonetheless, works exist, where these two factors (non-Newtonian rheology of the moving medium and arbitrary angle of spreading of perturbations) are taken into account, e.g., [5,6]. In more recent work [5], the problem of downslope flow linear stability is solved in complete formulation (continuity and momentum equations are used with no averaging over the depth, stability with respect to 3D perturbations is studied); this significant work uses complex mathematics, and can be difficult for applications.</p><p>This abstract is based on the work [6], where linear stability analysis was first conducted for the downslope flow that is described by hydraulic equations, but 1) the rheology of the flow and flow regime (laminar or turbulent) were arbitrary, 2) oblique perturbations were taken into account. The stability criterion is obtained analytically, it contains basic flow characteristics and can be applied to the flow if it's depth-averaged velocity <strong><em>u</em></strong>, depth <em>h</em>, relation between the bottom friction and <em>h</em>, <em>u</em> (<em>u</em> is the velocity modulus), slope angle are known. It is shown that the flow can be unstable (i.e., small perturbations grow, and this can lead, for example, to roll waves formation, or turbulisation of the flow) to oblique perturbations, even if standard stability criterion for longitudinal 1D perturbations is satisfied. This takes place, e.g., for dilatant fluids with power law index greater than 2).</p><p>The result is important not only for experimentalists, but for researchers who use numerical modeling, because knowledge of the flow behavior (for example, possible roll waves development) plays crucial role when choosing a computational scheme that will allow one to get the correct result.</p><p>[1] Benjamin T.B. Wave formation in laminar flow down an inclined plane. J. Fluid Mech. 1957. V. 2. P. 554 – 574.</p><p>[2] Yih C-S. Stability of liquid flow down an inclined plane. Phys. Fluids. 1963. V. 6(3). P. 321 – 334.</p><p>[3] Trowbridge J.H. Instability of concentrated free surface flows. J. Geophys. Res. 1987. V. 92(C9). P. 9523 – 9530.</p><p>[4] Coussot P. Steady, laminar, flow of concentrated mud suspensions in open channel. J. Hydraul. Res. 1994. V. 32. P. 535 – 559.</p><p>[5] Mogilevskiy E. Stability of a non-Newtonian falling film due to three-dimensional disturbances. Phys. Fluids. 2020. V. 32. 073101.</p><p>[6] Zayko J., Eglit M. Stability of downslope flows to two-dimensional perturbations. Phys. Fluids. 2019. V. 31. No. 8. 086601.</p>

Modelling powder snow avalanches using a depth-averaged turbulent shear model

<p>Two different mathematical models of fluid mechanics are now being investigated  at the WSL Institute for Snow and Avalanche Research in Davos to model powder-snow avalanches. The first approach is to solve the full three-dimensional multiphase (ice-dust, air) incompressible Navier-Stokes equations; the second approach is to apply depth-averaged models to simulate both the formation and independent propagation of the powder cloud. The final goal of both models is to predict the dynamics of powder avalanches in three-dimensional terrain and specifically cloud impact pressures. Both models are driven by the same set of terrain dependent mass and momentum exchanges defined by the flow state (speed, density, height) of the avalanche core. The great advantage of the depth-average approach is computational speed, allowing the investigation of different hazard scenarios involving variable release locations, snow temperature and entrainment depths. This fact has allowed the widespread application of the depth-average model to many historical case studies, including the avalanches measured at the Vallée de la Sionne (VdlS) test site. However, a central modelling problem needs to be resolved: both air-entrainment (cloud height and density) and drag (cloud speed) are intimately linked to the turbulence created during the cloud formation phase.</p><p>In this presentation, we present a depth-averaged turbulence model proposed by V. M. Teshukov [1] and extended by Richard and Gavrilyuk [2] and Gavrilyuk et al. [3], Ivanova et al. [5, 6]. The mathematical model is a 2D hyperbolic non-conservative system of equations that is mathematically equivalent to the Reynolds-averaged model of barotropic turbulent flows. The system is non-conservative, extending the classical shallow water equations to contain three independent components of the symmetric Reynolds stress tensor. We simulate the measured powder cloud heights of two VdlS avalanches using both the incompressible Navier-Stokes and turbulent shallow-water models, capturing the unsteady formation of billow height and width measured by ground based photogrammetry [4]. This can only be achieved by making air-entrainment dependent on the vorticity predicted by the turbulence model. We conclude by summarizing why we believe shallow-water type models can be applied for practical hazard engineering problems.</p><p>References:</p><p>[1] V. M. Teshukov in "Gas-dynamics analogy for vortex free-boundary flows.", J. Appl. Mech. Tech. Phys., 2007.</p><p>[2] G. L. Richard, S. L. Gavrilyuk in "A new model of roll waves: comparison with Brock’s experiments", Journal of Fluid Mechanics, 2012.</p><p>[3] S.L. Gavrilyuk, K.A. Ivanova, N. Favrie in "Multi-dimensional shear shallow water flows : problems and solutions", Journal of Computational Physics, 2018.</p><p>[4] Dreier, L., Bühler, Y., Ginzler, C., and Bartelt, P.: Comparison of simulated powder snow avalanches with photogrammetric measurements, Annals of Glaciology, 57, 371 - 381, 10.3189/2016AoG71A532, 2016.]</p><p>[5] K.A. Ivanova, S.L. Gavrilyuk, ”Structure of the hydraulic jump in convergent radial flows”,Journal of Fluid Mechanics, Volume 860, 10 February2019 , pp. 441-464.</p><p>[6] K.A. Ivanova, S.L. Gavrilyuk, B. Nkonga, G.L. Richard, ”Formation and coarsening of roll-waves in shear shallow water flows down an inclinedrectangular channel”,Computers& Fluids, 159, pp 189203, 2017</p>

Using High Sample Rate Lidar to Measure Debris-Flow Velocity and Surface Geometry

ABSTRACT Debris flows evolve in both time and space in complex ways, commonly starting as coherent failures but then quickly developing structures such as roll waves and surges. These processes are readily observed but difficult to study or quantify because of the speed at which they evolve. Many methods for studying debris flows consist of point measurements (e.g., flow height or basal stresses), which are inherently limited in spatial coverage and cannot fully characterize the spatiotemporal evolution of a flow. In this study, we use terrestrial lidar to measure debris-flow profiles at high sampling rates to examine debris-flow movement with high temporal and spatial precision and accuracy. We acquired measurements during gate-release experiments at the U.S. Geological Survey debris-flow flume, a unique experimental facility where debris flows can be artificially generated at a large scale. A lidar scanner was used to record repeat topographic profiles of the moving debris flows along the length of the flume with a narrow swath width (∼1 mm) at a rate of 60 Hz. The high-resolution lidar profiles enabled us to quantify flow front velocity of the debris flows and provided an unprecedented record of the development and evolution of the flow structure with a sub-second time resolution. The findings of this study demonstrate how to obtain quantitative measurements of debris-flow movement. In addition, the data help us to quantitatively define the development of a saltating debris-flow front and roll waves behind the debris-flow front. Such measurements may help constrain future modeling efforts.

Roll waves and their generation criteria

ABSTRACT Pulsating waves (also known as roll waves) might occur on the free surface of extreme events like mud and debris flows, among others, usually intensifying the caused damage. This technical note aims to inform about the roll wave phenomenon developing in a free-surface laminar flow, and analyze its generation criteria, centered on the concepts of Froude number and disturbance frequency. The complete linear stability analysis of the new depth-averaged model was proven a useful theoretical tool in determining new generation criteria for roll waves developing in non-Newtonian fluids. The results showed that the roll wave generation depends on two criteria: the first is associated to the minimum Froude number, and the second is related to the cut-off frequency. In addition, we have confirmed that the new generation criteria can be verified via numerical simulation based on a second model with full equations (Fluent software). Globally, the emergence of roll waves is favored by the non-Newtonian properties of the flowing fluid and the fact that the cut-off frequency decreases along with the minimum Froude number. Lastly, both generation criteria were tested in order to examine a case study involving the occurrence of roll waves in a watershed.