The permeability and quality of velocity field in a square array of solid and permeable cylindrical obstacles with the TRT–LBM and FEM Brinkman schemes

Silva, Gonçalo; Ginzburg, Irina

doi:10.1016/j.crme.2015.05.003

Cited by 13 publications

(10 citation statements)

References 33 publications

(114 reference statements)

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“…[56], confirms that the IBF effectively suppresses strong interface fluctuations typical in BF and FEM. The random and bimodal two-dimensional benchmarks are of the special interest in this context, since they provide nontrivial validation to scheme (54), which has been derived for the straight channel.…”

Section: Discussionsupporting

confidence: 64%

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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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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Section: Discussionsupporting

confidence: 64%

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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“…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). The most evident difference between them is that the FEM explicitly prescribes exact continuity conditions for velocity and stress components (with effective viscosity) on the vertex-centred grid, while the LBM imposes them implicitly and approximately in-between the centres of grid cells.…”

Section: Numerical Evaluationmentioning

confidence: 99%

“…This section makes use of numerical simulations validated at sufficiently fine grids for the parameter range studied Silva and Ginzburg 2015), which allows one to evaluate the applicability/limitations of the cell model and lubrication theory solutions.…”

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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“…(8) , even when Error (II) is absent due to A = 0 , the removal of Error (I) will require = 3 / 8 for this setup. Worse, the studies [32][33][34] have identified this problem and showed that Error (I), which is intrinsic to the LBM discretization of non-uniform body forces, gets transferred from bulk to boundaries; for example, in the BB case it leads to body force -dependent artefacts that affect the precision of the no-slip wall location on a much larger extent, at both O(δx ) and O(δx 2 ) (i.e. they affect the BB accommodation of both the flow gradient and the flow curvature).…”

Section: A Note On the Effect Of The Lbm Force Errors On Boundariesmentioning

confidence: 99%

“…they affect the BB accommodation of both the flow gradient and the flow curvature). This implies that, in this case, there is no single value capable of fixing the BB wall location at the desired place; for a detailed discussion on the pros and cons of using = 3 / 16 vs. = 3 / 8 in the BB wall modeling of this problem class we refer to [32][33][34] .…”

Section: A Note On the Effect Of The Lbm Force Errors On Boundariesmentioning

confidence: 99%

Discrete effects on the forcing term for the lattice Boltzmann modeling of steady hydrodynamics

Silva

2020

Computers & Fluids

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This work concerns with the clean inclusion of the forcing term in the lattice Boltzmann method (LBM) for the modeling of non-uniform body forces in steady hydrodynamics. The study is conducted for the two-relaxation-time (TRT) scheme. Here, we consider a simple, but yet sufficiently generic, flow configuration driven by a spatially varying body force based on which we derive the analytical solution of the forced LBM-TRT and compare two force strategies set by first-and second-order expansions. This procedure exactly establishes the macroscopic system satisfied by each force formulation at discrete level. The obtained theoretical results are further verified in two distinct benchmark channel flow problems. Overall, this study shows that the spatial discrete effects posed by the LBM modeling of the force term may come in through two sources. The first one is a defect inherent to LBM, arising from the non-local spatial discretization of the forcing term, and given by the discrete Laplacian of the body force. While it corrupts the discrete momentum balance with a non-linear viscosity dependent term in the single-relaxation-time schemes, this inconsistency is avoided with the TRT scheme, through its free-tunable relaxation degree of freedom . The other error source is unique to the use of second-order force expansions, where its nonzero second-order velocity moment interferes with the discrete momentum balance through a spurious first-order derivative term, leading to several forms of inconsistency in the LBM steady solution. For simulations under the convective scaling it makes the LBM scheme a numerical representation of a differential system distinct from the physical one, whereas under the diffusive scaling it leads to viscosity-dependent numerical errors, which corrupt the otherwise consistent structure of TRT steady-state solutions. In contrast, the defects reported herein are absent with the first-order force expansion scheme operated within the scope of the LBM-TRT model for steady hydrodynamics.

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The permeability and quality of velocity field in a square array of solid and permeable cylindrical obstacles with the TRT–LBM and FEM Brinkman schemes

Cited by 13 publications

References 33 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

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

Discrete effects on the forcing term for the lattice Boltzmann modeling of steady hydrodynamics

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