A multilayer shallow water approach for the approximate description of polydisperse sedimentation in a viscous fluid is presented. The fluid is assumed to carry finely dispersed solid particles that belong to a finite number of species that differ in density and size. These species segregate and form areas of different composition. In addition, the settling of particles influences the motion of the ambient fluid. A distinct feature of the new approach is the particular definition of the average velocity of the mixture. It takes into account the densities of the solid particles and the fluid and allows us to recover the global mass conservation and linear momentum balance laws of the mixture. This definition motivates a modification of the Masliyah–Lockett–Bassoon (MLB) settling velocities of each species. The multilayer shallow water model allows one to determine the spatial distribution of the solid particles, the velocity field, and the evolution of the free surface of the mixture. The final model can be written as a multilayer model with variable density where the unknowns are the average velocities and concentrations in each layer, the transfer terms across each interface, and the total mass. An explicit formula of the transfer terms leads to a reduced form of the system. Finally, an explicit bound of the minimum and maximum eigenvalues of the transport matrix of the system is utilized to design a Harten–Lax–van Leer (HLL)-type path-conservative numerical method. Numerical simulations illustrate the coupled polydisperse sedimentation and flow fields in various scenarios, including sedimentation in a type of basin that is used in practice in mining industry and in a basin whose bottom topography gives rise to recirculations of the fluid and high solids concentrations.
Mots-clés : Multilayer shallow water model, polydisperse sedimentation, path-conservative method, viscous flow, recirculation
@article{M2AN_2019__53_4_1391_0,
author = {B\"urger, Raimund and Fern\'andez-Nieto, Enrique D. and Osores, V{\'\i}ctor},
title = {A dynamic multilayer shallow water model for polydisperse sedimentation},
journal = {ESAIM: Mathematical Modelling and Numerical Analysis },
pages = {1391--1432},
publisher = {EDP-Sciences},
volume = {53},
number = {4},
year = {2019},
doi = {10.1051/m2an/2019032},
zbl = {1477.35159},
mrnumber = {3980062},
language = {en},
url = {http://www.numdam.org/articles/10.1051/m2an/2019032/}
}
TY - JOUR AU - Bürger, Raimund AU - Fernández-Nieto, Enrique D. AU - Osores, Víctor TI - A dynamic multilayer shallow water model for polydisperse sedimentation JO - ESAIM: Mathematical Modelling and Numerical Analysis PY - 2019 SP - 1391 EP - 1432 VL - 53 IS - 4 PB - EDP-Sciences UR - http://www.numdam.org/articles/10.1051/m2an/2019032/ DO - 10.1051/m2an/2019032 LA - en ID - M2AN_2019__53_4_1391_0 ER -
%0 Journal Article %A Bürger, Raimund %A Fernández-Nieto, Enrique D. %A Osores, Víctor %T A dynamic multilayer shallow water model for polydisperse sedimentation %J ESAIM: Mathematical Modelling and Numerical Analysis %D 2019 %P 1391-1432 %V 53 %N 4 %I EDP-Sciences %U http://www.numdam.org/articles/10.1051/m2an/2019032/ %R 10.1051/m2an/2019032 %G en %F M2AN_2019__53_4_1391_0
Bürger, Raimund; Fernández-Nieto, Enrique D.; Osores, Víctor. A dynamic multilayer shallow water model for polydisperse sedimentation. ESAIM: Mathematical Modelling and Numerical Analysis , Tome 53 (2019) no. 4, pp. 1391-1432. doi : 10.1051/m2an/2019032. http://www.numdam.org/articles/10.1051/m2an/2019032/
, A multilayer Saint-Venant model: derivation and numerical validation. Discrete Contin. Dyn. Syst. Ser. B 5 (2005) 189–214. | DOI | MR | Zbl
and , Finite-volume solvers for a multilayer Saint-Venant system. Int. J. Appl. Math. Comput. Sci. 17 (2007) 311–319. | DOI | MR | Zbl
, and , Numerical simulations of 3D free surface flows by a multilayer Saint-Venant model. Int. J. Numer. Methods Fluids 56 (2008) 331–350. | DOI | MR | Zbl
, , and , A multilayer Saint-Venant system with mass exchanges for shallow water flows. Derivation and numerical validation. ESAIM: M2AN 45 (2011) 169–200. | DOI | Numdam | MR | Zbl
, , and , Approximation of the hydrostatic Navier-Stokes system for density stratified flows by a multilayer model: kinetic interpretation and numerical solution. J. Comput. Phys. 230 (2011) 3453–3478. | DOI | MR | Zbl
, Critical regimes of two-phase flows with a polydisperse solid phase. In: Fluid Mechanics and Its Applications, Springer, Dordrecht (2010).
, and , On models of polydisperse sedimentation with particle-size-specific hindered-settling factors. Appl. Math. Model. 33 (2009) 1815–1835. | DOI | MR | Zbl
, , and , Strongly degenerate parabolic-hyperbolic systems modeling polydisperse sedimentation with compression. SIAM J. Appl. Math. 64 (2003) 41–80. | DOI | MR | Zbl
, and , Patterns of sedimentation from polydispersed turbidity currents. Proc. Roy. Soc. Lond. A 452 (1996) 2247–2261. | DOI
, , , and , Linearly implicit IMEX Runge–Kutta methods for a class of degenerate convection-diffusion problems. SIAM J. Sci. Comput. 37 (2015) B305–B331. | DOI | MR | Zbl
, and , Model equations for gravitational sedimentation-consolidation processes. ZAMM Z. Angew. Math. Mech. 80 (2000) 79–92. | DOI | MR | Zbl
, , and , A family of numerical schemes for kinematic flows with discontinuous flux. J. Eng. Math. 60 (2009) 387–425. | DOI | MR | Zbl
, , and , Hyperbolicity analysis of polydisperse sedimentation models via a secular equation for the flux Jacobian. SIAM J. Appl. Math. 70 (2010) 2186–2213. | DOI | MR | Zbl
, , and , On the implementation of WENO schemes for a class of polydisperse sedimentation models. J. Comput. Phys. 230 (2011) 2322–2344. | DOI | MR | Zbl
, and , Antidiffusive Lagrangian-remap schemes for models of polydisperse sedimentation. Numer. Methods Partial Differ. Equ. 32 (2016) 1109–1136. | DOI | MR | Zbl
, , , , , and , Numerical solution of a multi-class model for batch settling in water resource recovery facilities. Appl. Math. Model. 49 (2017) 415–436. | DOI | MR | Zbl
, and , High order finite volume schemes based on reconstruction of states for solving hyperbolic systems with nonconservative products. applications to shallow-water systems. Math. Comput. 75 (2006) 1103–1134. | DOI | MR | Zbl
, , and , Two-dimensional sediment transport models in shallow water equations. A second order finite volume approach on unstructured meshes. Comput. Methods Appl. Mech. Eng. 198 (2009) 2520–2538. | DOI | MR | Zbl
and , A class of computationally fast first order finite volume solvers: PVM methods. SIAM J. Sci. Comput. 34 (2012) A2173–A2196. | DOI | MR | Zbl
, , , and , A HLLC scheme for nonconservative hyperbolic problems. Application to turbidity currents with sediment transport. ESAIM M2AN 47 (2013) 1–32. | DOI | Numdam | MR | Zbl
and , Analogue study of particle segregation in pyroclastic density currents, with implications for the emplacement mechanisms of large ignimbrites. Sedimentology 49 (2002) 907–928. | DOI
, and , Definition and weak stability of nonconservative products. J. Math. Pures Appl. 74 (1995) 483–548. | MR | Zbl
and , A secular equation for the Jacobian matrix of certain multispecies kinematic flow models. Numer. Methods Partial Differ. Equ. 26 (2010) 159–175. | DOI | MR | Zbl
and , Sedimentation of bidisperse suspensions. Int. J. Multiphase Flow 36 (2010) 481–490. | DOI
, , and , The structure of the deposit produced by sedimentation of polydisperse suspensions. J. Geophys. Res. 116 (2011) F01024.
, Modelling and numerical simulation of submarine sediment shallow flows: transport and avalanches. Bol. Soc. Esp. Mat. Apl. SeMA 49 (2009) 83–103. | MR | Zbl
and , Extension of WAF type methods to non-homogeneous shallow water equations with pollutant. J. Sci. Comput. 36 (2008) 193–217. | DOI | MR | Zbl
, , and , A multilayer shallow water system for polydisperse sedimentation. J. Comput. Phys. 238 (2013) 281–314. | DOI | MR | Zbl
, and , A multilayer method for the hydrostatic Navier–Stokes equations: a particular weak solution. J. Sci. Comput. 60 (2014) 408–437. | DOI | MR | Zbl
, and , Polydisperse particle-driven gravity currents. J. Fluid Mech. 472 (2002) 333–371. | DOI | Zbl
, , and , Numerical modeling of sediment transport applied to coastal morphodynamics. Appl. Numer. Math. 104 (2016) 30–46. | DOI | MR | Zbl
and , Sedimentation of binary particle mixtures. Powder Technol. 24 (1979) 1–7. | DOI
, , Computational models for polydisperse particulate and multiphase systems. Cambridge Series in Chemical Engineering. Cambridge University Press, Cambridge (2013). | MR
, Hindered settling in a multiple-species particle system. Chem. Eng. Sci. 34 (1979) 1166–1168. | DOI
, , and , On a shallow water model for the simulation of turbidity currents. Commun. Comput. Phys. 6 (2009) 848–882. | DOI | MR | Zbl
, A new bound for polynomials when all the roots are real. Math. Gazette 95 (2011) 520–526. | DOI
, Numerical methods for nonconservative hyperbolic systems: a theoretical framework. SIAM J. Numer. Anal. 44 (2006) 300–321 (electronic). | DOI | MR | Zbl
and , On the well-balance property of Roe’s method for nonconservative hyperbolic systems. Applications to shallow-water systems. ESAIM: M2AN 38 (2004) 821–852. | DOI | Numdam | MR | Zbl
and , Sedimentation and fluidisation: Part I. Trans. Inst. Chem. Eng. (London) 32 (1954) 34–53.
, Vertically averaged models for the free surface non-hydrostatic Euler system: derivation and kinetic interpretation. Math. Models Methods Appl. Sci. 21 (2011) 459–490. | DOI | MR | Zbl
, and , Sediment composition due to settling of particles of different sizes. Int. J. Multiphase Flow 11 (1985) 419–423. | DOI
, A weak formulation of Roe’s approximate Riemann solver. J. Comput. Phys. 102 (1992) 360–373. | DOI | MR | Zbl
, An Introduction to Gravity Currents and Intrusions. CRC Press, Boca Raton, FL (2009). | DOI | MR | Zbl
Cité par Sources :
