Andrea Vacca

A two-phase model for fast geomorphic shallow flows

The paper introduces a 2D shallow water model based on a two-phase formulation for the analysis of fast geomorphic transients occurring in the context of river morphodynamics. Mass and momentum conservation principles are separately imposed for both phases. The model naturally accounts for non-equilibrium solid transport, since neither instantaneous adaptation hypothesis nor any lag equation is employed to represent sediment dynamics. The hyperbolic character of the proposed model is shown to be preserved independently on the flow conditions. Results from numerical simulations of both 1D and 2D test-cases are compared with literature experimental data and with available numerical solutions.

On the convective nature of roll waves instability

A theoretical analysis of the Saint-Venant one-dimensional flow model is performed in order to define the nature of its instability. Following the Brigg criterion, the investigation is carried out by examining the branch points singularities of dispersion relation in the complex ω and k planes, where ω and k are the complex pulsation and wave number of the disturbance, respectively. The nature of the linearly unstable conditions of flow is shown to be of convective type, independently of the Froude number value. Starting from this result a linear spatial stability analysis of the one-dimensional flow model is performed, in terms of time asymptotic response to a pointwise time periodic disturbance. The study reveals an influence of the disturbance frequency on the perturbation spatial growth rate, which constitutes the theoretical foundation of semiempirical criteria commonly employed for predicting roll waves occurrence.

Minimum channel length for roll-wave generation

The initial phase of roll-wave development is investigated by means of spatial linear stability analysis using the St. Venant equations, subject to a pointwise time-varying oscillating disturbance. The predicted spatial growth is compared with both Vedernikovs results and those computed with a fully non-linear model. It is shown that for large values of the channel slope Vedernikovs theory systematically overpredicts the roll waves spatial growth rate, whereas the present analysis yields significant improvements. A modification of Montuoris criterion for the minimum channel length prediction is finally proposed, which agrees with available experimental data independently of the channel slope.

Green's Function of the Linearized Saint-Venant Equations in Laminar and Turbulent Flows

In the present paper, an analytical expression of the Green’s function of linearized Saint-Venant equations (LSVEs) for shallow water waves is provided and applied to analyse the propagation of a perturbation superposed to a uniform flow. Independently of the kinematic character of the base flow, i.e., subcritical or supercritical uniform flow, the effects of a non-uniform vertical velocity profile and a non-constant resistance coefficient are accounted for. The use of the Darcy-Weisbach friction law allows a unified treatment of both laminar and turbulent conditions. The influence on the wave evolution of the wall roughness and the fluid viscosity are finally discussed, showing that in turbulent regime the assumption of constant friction coefficient may lead to an underestimation of both amplification and damping factors on the wave fronts, especially at low Reynolds numbers. This conclusion has to be accounted for, particularly in describing hyper-concentrated suspensions or other kinds of Newtonian mixtures, for which the high values of the kinematic viscosity may lead to relatively low Reynolds numbers.

Influence of Relative Roughness and Reynolds Number on the Roll-Waves Spatial Evolution

The paper investigates the influence of the resistance coefficient variability onto the spatial development of roll-waves. Two models, based on time-asymptotic solutions of the linearized St. Venant equations, subject to either impulsive or oscillating perturbation, have been modified by including the dependence of the resistance coefficient on flow conditions, wall roughness, and fluid viscosity. Independently of the perturbation type, it has been shown that the hypothesis of constant resistance coefficient leads to underestimate the disturbance spatial growth. Theoretical predictions are finally compared with results of a fully nonlinear model and with literature experimental data for several combinations of Froude and Reynolds numbers and relative roughness values. The representation of variability of the resistance coefficient fundamentally improves the performance of minimum channel length criteria, whereas its neglect may lead to noncautious channel design.

Applicability of Kinematic, Diffusion, and Quasi-Steady Dynamic Wave Models to Shallow Mud Flows

Unsteady shallow-layer flows may be described through full dynamic models or using simplified momentum equations, based on kinematic, diffusion, and quasi-steady approximations, which guarantee a reduction of the computational effort. This paper aims to investigate through linear analysis the applicability range of simplified shallow-wave models with special concern to unsteady flows of mud. Considering a three-equation depth-integrated Herschel-Bulkley model, the applicability of the approximated wave models is discussed comparing the propagation characteristics of a small perturbation of an initial steady uniform flow as predicted by the simplified models with those of the full dynamic model. Based on this comparison, applicability criteria for the different wave approximations for mud flows of Herschel-Bulkley fluids, which account for the effects of the rheological parameters, are derived. The results show that accounting for the fluid rheology is mandatory for the choice of an appropriate simplified model. DOI: 10.1061/(ASCE)HE.1943-5584.0000881.

Linear stability analysis of a 1-D model with dynamical description of bed-load transport

In the paper a linear stability analysis of a one–dimensional morphodynamical model is presented. The local equilibrium assumption on the solid discharge is removed and a dynamical description of the bed–load transport is employed. Both unsteady and quasi–steady flow models are analysed and discussed. The results reveal the existence, at moderate Froude number values, of perturbations related to bed elevation instability, migrating in the upstream direction both in quasi–steady and full unsteady models.

Simplified wave models applicability to shallow mud flows modeled as power-law fluids

Simplified wave models — such as kinematic, diffusion and quasi-steady — are widely employed as a convenient replacement of the full dynamic one in the analysis of unsteady open-channel flows, and especially for flood routing. While their use may guarantee a significant reduction of the computational effort, it is mandatory to define the conditions in which they may be confidently applied. The present paper investigates the applicability conditions of the kinematic, diffusion and quasisteady dynamic shallow wave models for mud flows of power-law fluids. The power-law model describes in an adequate and convenient way fluids that at low shear rates fluids do not posses yield stress, such as clay or kaolin suspensions, which are frequently encountered in Chinese rivers. In the framework of a linear analysis, the propagation characteristics of a periodic perturbation of an initial steady uniform flow predicted by the simplified models are compared with those of the full dynamic one. Based on this comparison, applicability criteria for the different wave approximations for mud flood of power-law fluids are derived. The presented results provide guidelines for selecting the appropriate approximation for a given flow problem, and therefore they may represent a useful tool for engineering predictions.

Boundary conditions effect on linearized mud-flow shallow model

The occurrence of roll-waves in mud-flows is investigated based on the formulation of the marginal stability threshold of a linearized onedimensional viscoplastic (shear-thinning) flow model. Since for this kind of non-Newtonian rheological models this threshold may occur in a hypocritical flow, the downstream boundary condition may have a nonnegligible effect on the spatial growth/decay of the perturbation. The paper presents the solution of the 1D linearized flow of a Herschel and Bulkley fluid in a channel of finite length, in the neighbourhood of a hypocritical base uniform flow. Both linearly stable and unstable conditions are considered. The analytical solution is found applying the Laplace transform method and obtaining the first-order analytical expressions of the upstream and downstream channel response functions in the time domain. The effects of both the yield stress and the rheological law exponent are discussed, recovering as particular cases both power-law and Bingham fluids. The theoretical achievements may be used to extend semi-empirical criteria commonly employed for predicting roll waves occurrence in clear water even to mud-flows.

On the applicability of minimum channel length criterion for roll-waves in mud-flows

Abstract The paper addresses the prediction of roll-waves occurrence in mud-flows. The spatial growth of a point-wise disturbance is analytically described, based on the linearized flow model of a Herschel and Bulkley fluid, in the neighborhood of an initial uniform base condition. The theoretical achievements allow to generalize to mud-flows the minimum channel criterion commonly used for the prediction of roll-waves in clear-water. The applicability of the criterion is discussed through the comparison with literature laboratory data concerning unstable flows without rollwaves.