Fourth-order analysis of a diffusive lattice Boltzmann method for barrier coatings

Abstract

We examine the applicability of diffusive lattice Boltzmann methods to simulate the fluid transport through barrier coatings, finding excellent agreement between simulations and analytical predictions for standard parameter choices. To examine more interesting non-Fickian behavior and multiple layers of different coatings, it becomes necessary to explore a wider range of parameters. However, such a range of parameters exposes deficiencies in such an implementation. To investigate these discrepancies, we examine the form of higher-order terms in the hydrodynamic limit of our lattice Boltzmann method. We identify these corrections to fourth order and validate these predictions with high accuracy. However, it is observed that the validated correction terms do not fully explain the bulk of observed error. This error was instead caused by the standard finite boundary conditions for the contact of the coating with the imposed environment. We identify a self-consistent form of these boundary conditions for which these errors are dramatically reduced. The instantaneous switching used as a boundary condition for the barrier problem proves demanding enough that any higher-order corrections meaningfully contribute for a small range of parameters. There is a large parameter space where the agreement between simulations and analytical predictions even in the second-order form are below 0.1%, making further improvements to the algorithm unnecessary for such an application.

Optimizing coating performance for diffusion under cyclic moisture exposure

“…It has previously been shown [19] that this setup results in excellent numerical accuracy for moisture exposure over long times, typically on the order of 0.01% of the predicted theory value, several orders of magnitude better than typical experimental measurements. To verify that cycling the reservoir, which introduces large concentration gradients near the reservoir, retains the desired numerical accuracy to theory, we run a lattice Boltzmann simulation with the given parameters, cycling the reservoir on and off every T timesteps for a total cycle period of 2T timesteps.…”

Moment Independent Expansion for Fourth-Order Corrections in Lattice Boltzmann Methods

“…This is not a diffusion equation, but it has a similar form to the Telegrapher's equation. In the previous analysis in [11], Eqns. (38-39) were used to give a diffusion equation which takes the form…”

“…(20)(21)(22)(23)(24) are all set to zero. Using these definitions, we can re-derive the results from previous work by Strand et al [11].…”

“…Section 4 then shows applications of the proposed expansion to various simple one dimensional system. We re-derive results previously derived by Strand et al [11] using this new moment independent expansion and reanalyze the way the hydrodynamic limit treats temporal derivatives in the diffusive case. We then present two separate derivations for fourth order phase separating systems.…”

“…Since its initial development in the late 1980s, lattice Boltzmann methods have been growing as a powerful tool in computational physics. Originally introduced for simulating hydrodynamic systems [1][2][3], the lattice Boltzmann methods have been an active area of research which consistently is being improved and finding new applications in other fields of physics, namely phase separation [4,5], electrostatics [6], quantum mechanics [7], diffusion [8][9][10], and moisture transport through barrier coatings [11,12]. As a consequence of the many developments in lattice Boltzmann methods, deeper analysis on the fun-Email address: kyle.t.strand@ndsu.edu (Kyle Strand) damental methods have been required for better understanding as to how the method treats any given system [13].…”