Computations with finite element methods for the Brinkman problem

This work focuses on the numerical solution of the Stokes-Brinkman equation for a voxel-type porous-media grid, resolved by one to eight spacings per permeability contrast of 1 to 10 orders in magnitude. It is first analytically demonstrated that the lattice Boltzmann method (LBM) and the linear-finite-element method (FEM) both suffer from the viscosity correction induced by the linear variation of the resistance with the velocity. This numerical artefact may lead to an apparent negative viscosity in low-permeable blocks, inducing spurious velocity oscillations. The two-relaxation-times (TRT) LBM may control this effect thanks to free-tunable two-rates combination Λ. Moreover, the Brinkman-force-based BF-TRT schemes may maintain the nondimensional Darcy group and produce viscosity-independent permeability provided that the spatial distribution of Λ is fixed independently of the kinematic viscosity. Such a property is lost not only in the BF-BGK scheme but also by "partial bounce-back" TRT gray models, as shown in this work. Further, we propose a consistent and improved IBF-TRT model which vanishes viscosity correction via simple specific adjusting of the viscous-mode relaxation rate to local permeability value. This prevents the model from velocity fluctuations and, in parallel, improves for effective permeability measurements, from porous channel to multidimensions. The framework of our exact analysis employs a symbolic approach developed for both LBM and FEM in single and stratified, unconfined, and bounded channels. It shows that even with similar bulk discretization, BF, IBF, and FEM may manifest quite different velocity profiles on the coarse grids due to their intrinsic contrasts in the setting of interface continuity and no-slip conditions. While FEM enforces them on the grid vertexes, the LBM prescribes them implicitly. We derive effective LBM continuity conditions and show that the heterogeneous viscosity correction impacts them, a property also shared by FEM for shear stress. But, in contrast with FEM, effective velocity conditions in LBM give rise to slip velocity jumps which depend on (i) neighbor permeability values, (ii) resolution, and (iii) control parameter Λ, ranging its reliable values from Poiseuille bounce-back solution in open flow to zero in Darcy's limit. We suggest an "upscaling" algorithm for Λ, from multilayers to multidimensions in random extremely dispersive samples. Finally, on the positive side for LBM besides its overall versatility, the implicit boundary layers allow for smooth accommodation of the flat discontinuous Darcy profiles, quite deficient in FEM.

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“…[21][22][23][24] and references herein) and the lattice Boltzmann method (LBM). The LBM can be further subdivided into two main groups.…”

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confidence: 99%

Analysis and improvement of Brinkman lattice Boltzmann schemes: Bulk, boundary, interface. Similarity and distinctness with finite elements in heterogeneous porous media

Ginzburg¹,

Silva²,

Talon

2015

Phys. Rev. E

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“…The Taylor-Hood velocity/pressure pair is one of the most popular finite elements for generalized Stokes flow, but it is known not to be robust in the Darcy limit. Specifically, the accuracy of the Q 2 /Q 1 Taylor-Hood velocity field reduces to first-order in the Darcy limit [33]. This is due to a lack of strong coercivity (in the kernel) with respect to the H(div; Ω)-norm.…”

Section: Darcy-dominated Flow With Sharp Boundary Layersmentioning

confidence: 99%

Isogeometric Divergence-conforming B-splines for the Darcy-Stokes-Brinkman Equations

Evans¹,

Hughes²

2012

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Public reporting burden for the collection of information is estimated to average 1 hour per response, including the time for reviewing instructions, searching existing data sources, gathering and maintaining the data needed, and completing and reviewing the collection of information. Send comments regarding this burden estimate or any other aspect of this collection of information, including suggestions for reducing this burden, to Washington Headquarters Services, Directorate for Information Operations and Reports, 1215 Jefferson Davis Highway, Suite 1204, Arlington VA 22202-4302. Respondents should be aware that notwithstanding any other provision of law, no person shall be subject to a penalty for failing to comply with a collection of information if it does not display a currently valid OMB control number. We develop divergence-conforming B-spline discretizations for the numerical solution of the Darcy-Stokes-Brinkman equations. These discretizations are motivated by the recent theory of isogeometric discrete di erential forms and may be interpreted as smooth generalizations of Raviart-Thomas elements. The new discretizations are (at least) patch-wise C0 and can be directly utilized in the Galerkin solution of Darcy-Stokes-Brinkman ow for single-patch con gurations. When applied to incompressible ows, these discretizations produce pointwise divergence-free velocity elds and hence exactly satisfy mass conservation. In the presence of no-slip boundary conditions and multi-patch geometries, the discontinuous Galerkin framework is invoked to enforce tangential continuity without upsetting the conservation or stability properties of the method across patch boundaries. Furthermore, as no-slip boundary conditions are enforced weakly, the method automatically defaults to a compatible discretization of Darcy ow in the limit of vanishing viscosity. The proposed discretizations are extended to general mapped geometries using divergence-preserving transformations. For su ciently regular single-patch solutions, we prove a priori error estimates which are optimal for the discrete velocity eld and suboptimal, by one order, for the discrete pressure eld. Our estimates are additionally robust with respect to the parameters of the Darcy-Stokes-Brinkman problem. We present a comprehensive suite of numerical experiments which indicate optimal convergence rates for both the discrete velocity and pressure elds for general con gurations, suggesting that our a priori estimates may be conservative. The focus of the current paper is strictly on incompressible ows, but our theoretical results naturally extend to ows characterized by mass sources and sinks. AbstractWe develop divergence-conforming B-spline discretizations for the numerical solution of the Darcy-Stokes-Brinkman equations. These discretizations are motivated by the recent theory of isogeometric discrete differential forms and may be interpreted as smooth generalizations of Raviart-Thomas elements. The new discretizations are (at least) patch-wise C 0 and can be directl...

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“…Their consistent formulation, using the TRT-LBM framework, was established in a series of works Ginzburg 2008Ginzburg , 2014. At the same time, the FEM simulation of Brinkman flows has also found vast application (Krotkiewski et al 2011;Hannukainen et al 2011). A detailed comparison on the specificities of both numerical approaches in this class of problems was performed in the recent works Silva and Ginzburg 2015).…”

Section: Numerical Evaluationmentioning

confidence: 99%

Stokes–Brinkman–Darcy Solutions of Bimodal Porous Flow Across Periodic Array of Permeable Cylindrical Inclusions: Cell Model, Lubrication Theory and LBM/FEM Numerical Simulations

Silva

Ginzburg

2016

Transp Porous Med

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An analytical study is devised for the problem of bimodal porous flow across a periodic array of permeable cylindrical inclusions. Such a configuration is particularly relevant for porous media systems of dual granulometry, an idealization often taken, e.g. in the modelling of membranes and fibrous applications. The double-porosity system is governed by the Stokes-Brinkman-Darcy equations, the most general description in this class of flow problems characterized by the permeabilities of the surrounding matrix and inclusions, their porosities and the relative volume fraction. We solve this problem with the Kuwabara cell model and lubrication approach, providing analytical solutions for the system effective permeability in closed analytical form. The ensemble of results demonstrates the self-consistency of the bimodal solutions in eight possible limit configurations and supports the validity of the Beavers-Joseph interface stress jump condition for transmission from the open Stokes flow to low-permeable Darcy region. At the same time, these solutions bring further insight on the relative significance of the governing parameters on the effective permeability, with a focus on the role of the effective viscosity (porosity) distribution. Furthermore, although the cell model is restricted to relatively small volume fractions in open flow, its validity extends in lesspermeable background flow inside Brinkman/Brinkman description. In turn, the lubrication approximation remains more adequate in the opposite limit of the dense impermeable inclusions. These conclusions are drawn from comparisons with the numerical solutions obtained with the developed lattice Boltzmann model and the standard finite element method. The two methods principally differ in the treatment of the interface conditions: implicit and explicit, respectively. The purpose of this task is therefore twofold. While the numerical schemes help quantifying the validity limits of the theoretical approach, the analytical solutions offer a non-trivial benchmark for numerical schemes in highly heterogeneous soil.

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Computations with finite element methods for the Brinkman problem

Cited by 34 publications

References 26 publications

Analysis and improvement of Brinkman lattice Boltzmann schemes: Bulk, boundary, interface. Similarity and distinctness with finite elements in heterogeneous porous media

Analysis and improvement of Brinkman lattice Boltzmann schemes: Bulk, boundary, interface. Similarity and distinctness with finite elements in heterogeneous porous media

Isogeometric Divergence-conforming B-splines for the Darcy-Stokes-Brinkman Equations

Stokes–Brinkman–Darcy Solutions of Bimodal Porous Flow Across Periodic Array of Permeable Cylindrical Inclusions: Cell Model, Lubrication Theory and LBM/FEM Numerical Simulations

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