Partitioned coupling of advection–diffusion–reaction systems and Brinkman flows

Abstract

We present a partitioned algorithm aimed at extending the capabilities of existing solvers for the simulation of coupled advection-di↵usion-reaction systems and incompressible, viscous flow. The space discretization of the governing equations is based on mixed finite element methods defined on unstructured meshes, whereas the time integration hinges on an operator splitting strategy that exploits the di↵er-ences in scales between the reaction, advection, and di↵usion processes, considering the global system as a number of sequentially linked sets of partial di↵erential, and algebraic equations. The flow solver presents the advantage that all unknowns in the system (here vorticity, velocity, and pressure) can be fully decoupled and thus turn the overall scheme very attractive from the computational perspective. The robustness of the proposed method is illustrated with a series of numerical tests in 2D and 3D, relevant in the modelling of bacterial bioconvection and Boussinesq systems.

On a vorticity-based formulation for reaction-diffusion-Brinkman systems

“…strategies and thorough comparisons between dedicated solvers have been reported in [24]. The model parameters are set as µ = 1, D c = 0.01, D s = 0.5, g = (0, −1) T , and K = I; the reaction kinetics are specified by G j (c, s) = 6π sin(t) exp(−4D j π 2 t)x 2 (x − 1) 2 y 2 (y − 1) 2…”

“…On top of that, the proposed finite element method provides divergence-free velocities, thus preserving an essential constraint of the underlying physical system. Based on the present formulation, the stability properties of different staggered coupling techniques have been recently analysed in [24]. Other recent contributions regarding the numerical discretisation for related models for reactive solute transport in porous media (but using Darcy descriptions of flow), include the formulation and convergence studies for mixed and conformal methods as presented in [1,8,20,27,30,31].…”

Kinetic‐fluid derivation and mathematical analysis of the cross‐diffusion–Brinkman system

“…Lemma 1. Assume that the hypotheses (8), (9), and (10) hold. Then, the following properties of operators  1 and  2 hold true.…”

“…In the works of Anaya et al, 6,7 the authors proposed the analysis of this model using the mixed finite element method for standard and nonstandard boundary conditions, respectively. Later, in the work of Lenarda et al, 8 the authors studied numerically an advection-diffusion-reaction system coupled with an incompressible viscous flow. When the fluid is at rest (u = 0), several works have been proposed in the literature to investigate the theoretical and numerical analysis of the cross-diffusion model.…”

A fully-mixed finite element method for the steady state Oberbeck–Boussinesq system

“…Motivated by the vast applications and the challenging mathematical structure of such nonlinearly coupled system, the interest in analyzing it and in developing efficient numerical techniques to simulate related phenomena has significantly increased, see, e.g., [2,4,5,7,9,11,14,16,21,22,25,28,29,32,34,36,39,42,41,44,46,48] and the references therein. Those works include numerical algorithms based on finite volume approaches, standard finite element techniques, parallel and projection-based stabilization methods, spectral collocation, and mixed finite element methods; and they concentrate on heat-driven flows and double-diffusion convection, including cases in which the phenomena occur in porous enclosures, with either constant or variable physical parameters.…”