Time-dependent coupling of Navier–Stokes and Darcy flows

Çeşmelioǧlu, Ayçıl; Girault, Vivette; Rivière, Béatrice

doi:10.1051/m2an/2012034

Cited by 57 publications

(39 citation statements)

References 31 publications

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“…Now we bound the interface terms. To bound I 13 , we first use Lemma 3.1, then (11), (13), (12), (17) and (18) and obtain…”

Section: Error Boundsmentioning

confidence: 99%

“…One of them is the Stokes-Darcy coupling where the effects of elasticity property of the structure is neglected. There has been an extensive amount of work in this topic a few of which can be found in [2][3][4][5][6][7][8][9][10][11][12][13][14]. The other subproblem is the fluid-structure interaction (FSI) problem where the porous property of the structure is neglected.…”

Section: Introductionmentioning

confidence: 99%

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Numerical analysis of the coupling of free fluid with a poroelastic material

Çeşmelioǧlu

Chidyagwai

2019

Numerical Methods Partial

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This paper presents a numerical solution of the coupled system of the time‐dependent Stokes and fully dynamic Biot equations. The numerical scheme is based on standard inf‐sup stable finite elements in space and the Backward Euler scheme in time. We establish stability of the scheme and derive error estimates for the fully discrete coupled scheme. To handle realistic parameters which may cause nonphysical oscillations in the pore fluid pressure, a heuristic stabilization technique is considered. Numerical errors and convergence rates for smooth problems as well as tests on realistic material parameters are presented.

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“…Now we bound the interface terms. To bound I 13 , we first use Lemma 3.1, then (11), (13), (12), (17) and (18) and obtain…”

Section: Error Boundsmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Numerical analysis of the coupling of free fluid with a poroelastic material

Çeşmelioǧlu

Chidyagwai

2019

Numerical Methods Partial

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“…The discrete velocity u h pois is a suitable approximation of u pois in the finite element space X fh . (The mathematical analysis of the time-dependent NSD problem has been recently carried out in [14]. )…”

Section: Space Discretizationmentioning

confidence: 99%

“…Under those conditions, the analysis of a coupled Stokes/Darcy problem has been studied in [10,17,19,20,28,29,26,27,30,40,48] in the steady case, and in [12,53] in the time-dependent case. Moreover, the case of the Navier-Stokes equations has been considered in [3,14,19,31].…”

Section: Introductionmentioning

confidence: 99%

Navier–Stokes/Forchheimer models for filtration through porous media

Cimolin

Discacciati

2013

Applied Numerical Mathematics

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a b s t r a c tModeling the filtration of incompressible fluids through porous media requires dealing with different types of partial differential equations in the fluid and porous subregions of the computational domain. Such equations must be coupled through physically significant continuity conditions at the interface separating the two subdomains. To avoid the difficulties of this heterogeneous approach, a widely used strategy is to consider the Navier-Stokes equations in the whole domain and to correct them introducing suitable terms that mimic the presence of the porous medium. In this paper we discuss these two different methodologies and we compare them numerically on a sample test case after proposing an iterative algorithm to solve a Navier-Stokes/Forchheimer problem. Finally, we apply these strategies to a problem of internal ventilation of motorbike helmets.

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“…This is the so called added-mass effect. Furthermore, accounting for a poroelastic material model requires addressing issues typical of partitioned methods for flows [13,14,15] arising in particular for the coupling of Stokes-Darcy models, which represents the paradigm for studying the interaction of free flows with subsurface filtration [9,10,12,16,17,31,30,41]. An additional difficulty is combining the Eulerian description of the moving fluid domain with the typical Lagrangian parametrization of the structure [18,28].…”

Section: Introductionmentioning

confidence: 99%

Partitioning strategies for the interaction of a fluid with a poroelastic material based on a Nitsche’s coupling approach

Bukač

Yotov

Zakerzadeh

et al. 2015

Computer Methods in Applied Mechanics and Engineering

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We develop a computational model to study the interaction of a fluid with a poroelastic material. The coupling of Stokes and Biot equations represents a prototype problem for these phenomena, which feature multiple facets. On one hand, it shares common traits with fluid-structure interaction. On the other hand it resembles the Stokes-Darcy coupling. For these reasons, the numerical simulation of the Stokes-Biot coupled system is a challenging task. The need of large memory storage and the difficulty to characterize appropriate solvers and related preconditioners for the equations at hand are typical shortcomings of classical discretization methods applied to this problem, such as the finite element method for spatial discretization and finite differences for time stepping. The application of loosely coupled time advancing schemes mitigates these issues, because it allows to solve each equation of the system independently with respect to the others, at each time step. In this work, we develop and thoroughly analyze a loosely coupled scheme for Stokes-Biot equations. The scheme is based on Nitsche's method for enforcing interface conditions. Once the interface operators corresponding to the interface conditions have been defined, time lagging allows us to build up a loosely coupled scheme with good stability properties. The stability of the scheme is guaranteed provided that appropriate stabilization operators are introduced into the variational formulation of each subproblem. The error of the resulting method is also analyzed, showing that splitting the equations pollutes the optimal approximation properties of the underlying discretization schemes. In order to restore good approximation properties, while maintaining the computational efficiency of the loosely coupled approach, we consider the application of the loosely coupled scheme as a preconditioner for the monolithic approach. Both theoretical insight and numerical results confirm that this is a promising way to develop efficient solvers for the problem at hand.

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Time-dependent coupling of Navier–Stokes and Darcy flows

Cited by 57 publications

References 31 publications

Numerical analysis of the coupling of free fluid with a poroelastic material

Numerical analysis of the coupling of free fluid with a poroelastic material

Navier–Stokes/Forchheimer models for filtration through porous media

Partitioning strategies for the interaction of a fluid with a poroelastic material based on a Nitsche’s coupling approach

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