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1. Hyperbolic Numerical Models for Unsteady Incompressible, Surcharged Stormwater Flows

   Hodges, Ben R. ; Vasconcelos, Jose G. ; Sharior, Sazzad ; Geller, Vitor G. ( June 2022 , Proceedings of the 39th IAHR World Congress)

   The transition from open-channel to surcharged flow creates problems for numerical modeling of stormwater systems. Mathematically, problems arise through a discrete shock at the boundary between the hyperbolic Saint-Venant equations and the elliptic incompressible flow equations at the surcharge transition. Physically, problems arise through trapping of air pockets, creation of bubbly flows, and cavitation in rapid emptying and filling that are difficult to correctly capture in one-dimensional (1D) models. Discussed herein are three approaches for modeling surcharged flow with hyperbolic 1D equations: (i) Preissmann Slot (PS), (ii) Two- component Pressure Approach (TPA) and (iii) Artificial Compressibility (AC). Each provides approximating terms that are controlled by model coefficients to alter the pressure wave celerity through the surcharged system. Commonly, the implementation of these models involve slowing the pressure celerity below physical values, which allows the numerical solution to dissipate the transition shock between the free surface and surcharged flows without resorting to extraordinarily small time-steps. The different methods provide different capabilities and numerical implementations that affect their behavior and suitability for different problems. https://doi.org/10.3850/IAHR-39WC2521711920221370 

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2. Pressure-driven gas ow in viscously deformable porous media: application to lava domes

   https://doi.org/10.1017/jfm2019-211  

   Hyman, D. ; Bursik, M. ; Pitman, E.B ( June 2019 , Journal of fluid mechanics)

   The behaviour of low-viscosity, pressure-driven compressible pore fluid flows in viscously deformable porous media is studied here with specific application to gas flow in lava domes. The combined flow of gas and lava is shown to be governed by a two-equation set of nonlinear mixed hyperbolic–parabolic type partial differential equations describing the evolution of gas pore pressure and lava porosity. Steady state solution of this system is achieved when the gas pore pressure is magmastatic and the porosity profile accommodates the magmastatic pressure condition by increased compaction of the medium with depth. A one-dimensional (vertical) numerical linear stability analysis (LSA) is presented here. As a consequence of the pore-fluid compressibility and the presence of gravitation compaction, the gradients present in the steady-state solution cause variable coefficients in the linearized equations which generate instability in the LSA despite the diffusion-like and dissipative terms in the original system. The onset of this instability is shown to be strongly controlled by the thickness of the flow and the maximum porosity, itself a function of the mass flow rate of gas. Numerical solutions of the fully nonlinear system are also presented and exhibit nonlinear wave propagation features such as shock formation. As applied to gas flow within lava domes, the details of this dynamics help explain observations of cyclic lava dome extrusion and explosion episodes. Because the instability is stronger in thicker flows, the continued extrusion and thickening of a lava dome constitutes an increasing likelihood of instability onset, pressure wave growth and ultimately explosion. 

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3. Oblique interactions between solitons and mean flows in the Kadomtsev–Petviashvili equation

   https://doi.org/10.1088/1361-6544/abef74  

   Ryskamp, S ; Hoefer, M A ; Biondini, G ( June 2021 , Nonlinearity)

   Abstract The interaction of an oblique line soliton with a one-dimensional dynamic mean flow is analyzed using the Kadomtsev–Petviashvili II (KPII) equation. Building upon previous studies that examined the transmission or trapping of a soliton by a slowly varying rarefaction or oscillatory dispersive shock wave (DSW) in one space and one time dimension, this paper allows for the incident soliton to approach the changing mean flow at a nonzero oblique angle. By deriving invariant quantities of the soliton–mean flow modulation equations—a system of three (1 + 1)-dimensional quasilinear, hyperbolic equations for the soliton and mean flow parameters—and positing the initial configuration as a Riemann problem in the modulation variables, it is possible to derive quantitative predictions regarding the evolution of the line soliton within the mean flow. It is found that the interaction between an oblique soliton and a changing mean flow leads to several novel features not observed in the (1 + 1)-dimensional reduced problem. Many of these interesting dynamics arise from the unique structure of the modulation equations that are nonstrictly hyperbolic, including a well-defined multivalued solution interpreted as a solution of the (2 + 1)-dimensional soliton–mean modulation equations, in which the soliton interacts with the mean flow and then wraps around to interact with it again. Finally, it is shown that the oblique interactions between solitons and DSW solutions for the mean flow give rise to all three possible types of two-soliton solutions of the KPII equation. The analytical findings are quantitatively supported by direct numerical simulations. 

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4. Plume-SPH 1.0: a three-dimensional, dusty-gas volcanic plume model based on smoothed particle hydrodynamics

   https://doi.org/10.5194/gmd-11-2691-2018  

   Cao, Zhixuan ; Patra, Abani ; Bursik, Marcus ; Pitman, E. Bruce ; Jones, Matthew ( January 2018 , Geoscientific Model Development)

   Abstract. Plume-SPH provides the first particle-based simulation ofvolcanic plumes. Smoothed particle hydrodynamics (SPH) has several advantagesover currently used mesh-based methods in modeling of multiphase freeboundary flows like volcanic plumes. This tool will provide more accurateeruption source terms to users of volcanic ash transport anddispersion models (VATDs), greatly improving volcanic ash forecasts. The accuracy ofthese terms is crucial for forecasts from VATDs, and the 3-D SPH modelpresented here will provide better numerical accuracy. As an initial effortto exploit the feasibility and advantages of SPH in volcanic plume modeling,we adopt a relatively simple physics model (3-D dusty-gas dynamic modelassuming well-mixed eruption material, dynamic equilibrium and thermodynamicequilibrium between erupted material and air that entrained into the plume,and minimal effect of winds) targeted at capturing the salient features of avolcanic plume. The documented open-source code is easily obtained andextended to incorporate other models of physics of interest to the largecommunity of researchers investigating multiphase free boundary flows ofvolcanic or other origins.

   The Plume-SPH code (https://doi.org/10.5281/zenodo.572819) also incorporates several newly developed techniques inSPH needed to address numerical challenges in simulating multiphasecompressible turbulent flow. The code should thus be also of general interestto the much larger community of researchers using and developing SPH-basedtools. In particular, the SPH−ε turbulence model is used to capturemixing at unresolved scales. Heat exchange due to turbulence is calculated bya Reynolds analogy, and a corrected SPH is used to handle tensile instabilityand deficiency of particle distribution near the boundaries. We alsodeveloped methodology to impose velocity inlet and pressure outlet boundaryconditions, both of which are scarce in traditional implementations of SPH.

   The core solver of our model is parallelized with the message passinginterface (MPI) obtaining good weak and strong scalability using novel techniquesfor data management using space-filling curves (SFCs), object creationtime-based indexing and hash-table-based storage schemes. These techniques areof interest to researchers engaged in developing particles in cell-typemethods. The code is first verified by 1-D shock tube tests, then bycomparing velocity and concentration distribution along the central axis andon the transverse cross with experimental results of JPUE (jet or plume thatis ejected from a nozzle into a uniform environment). Profiles of severalintegrated variables are compared with those calculated by existing 3-D plumemodels for an eruption with the same mass eruption rate (MER) estimated forthe Mt. Pinatubo eruption of 15 June 1991. Our results are consistent withexisting 3-D plume models. Analysis of the plume evolution processdemonstrates that this model is able to reproduce the physics of plumedevelopment.

    

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5. Constitutive relations for compressible granular flow in the inertial regime

   https://doi.org/10.1017/jfm.2019.476  

   Schaeffer, D. G. ; Barker, T. ; Tsuji, D. ; Gremaud, P. ; Shearer, M. ; Gray, J. M. ( September 2019 , Journal of Fluid Mechanics)

   Granular flows occur in a wide range of situations of practical interest to industry, in our natural environment and in our everyday lives. This paper focuses on granular flow in the so-called inertial regime, when the rheology is independent of the very large particle stiffness. Such flows have been modelled with the $\unicode[STIX]{x1D707}(I),\unicode[STIX]{x1D6F7}(I)$ -rheology, which postulates that the bulk friction coefficient $\unicode[STIX]{x1D707}$ (i.e. the ratio of the shear stress to the pressure) and the solids volume fraction $\unicode[STIX]{x1D719}$ are functions of the inertial number $I$ only. Although the $\unicode[STIX]{x1D707}(I),\unicode[STIX]{x1D6F7}(I)$ -rheology has been validated in steady state against both experiments and discrete particle simulations in several different geometries, it has recently been shown that this theory is mathematically ill-posed in time-dependent problems. As a direct result, computations using this rheology may blow up exponentially, with a growth rate that tends to infinity as the discretization length tends to zero, as explicitly demonstrated in this paper for the first time. Such catastrophic instability due to ill-posedness is a common issue when developing new mathematical models and implies that either some important physics is missing or the model has not been properly formulated. In this paper an alternative to the $\unicode[STIX]{x1D707}(I),\unicode[STIX]{x1D6F7}(I)$ -rheology that does not suffer from such defects is proposed. In the framework of compressible $I$ -dependent rheology (CIDR), new constitutive laws for the inertial regime are introduced; these match the well-established $\unicode[STIX]{x1D707}(I)$ and $\unicode[STIX]{x1D6F7}(I)$ relations in the steady-state limit and at the same time are well-posed for all deformations and all packing densities. Time-dependent numerical solutions of the resultant equations are performed to demonstrate that the new inertial CIDR model leads to numerical convergence towards physically realistic solutions that are supported by discrete element method simulations. 

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