Steady-state two-relaxation-time lattice Boltzmann formulation for transport and flow, closed with the compact multi-reflection boundary and interface-conjugate schemes

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Highlights

  • The steady-state two-relaxation-time LBM is introduced for flow and transport equations.

  • The method is stable for highly discontinuous transport coefficients and sources.

  • The system is closed with the compact multi-reaction MR boundary and interface-conjugate.

  • The parabolic solutions are exact with the developed Dirichlet, Neumann and Robin MR rules.

  • Two-phase Stokes flow and heterogeneous linear ADE exemplify the method in grid-rotated slabs.

Abstract

We introduce the steady-state two-relaxation-time (TRT) Lattice Boltzmann method. Owing to the symmetry argument, the bulk system and the closure equations are all expressed in terms of the equilibrium and non-equilibrium unknowns with the half discrete velocity set. The local mass-conservation solvability condition is adjusted to match the stationary, but also the quasi-stationary, solutions of the standard TRT solver. Additionally, the developed compact, boundary and interface-conjugate, multi-reflection (MR) concept preserves the efficient directional bulk structure and shares its parametrization properties. The method is exemplified in grid-inclined stratified slabs for two-phase Stokes flow and the linear advection-diffusion equation featuring the discontinuous coefficients and sources. The piece-wise parabolic benchmark solutions are matched exactly with the novel Dirichlet, pressure-stress, Neumann flux and Robin MR schemes. The popular, anti-bounce-back and shape-fitted Dirichlet continuity schemes are improved in the presence of both interface-parallel and perpendicular advection velocity fields. The steady-state method brings numerous advantages: it skips transient numerical instability, overpasses severe von Neumann parameter range limitations, tolerates high physical contrasts and arbitrary MR coefficients. The method is promising for faster computation of Stokes/Brinkman/Darcy linear flows in heterogeneous soil, but also heat and mass transfer problems governed by an accurate boundary and interface treatment.

Introduction

The LBM, Lattice Boltzmann method [57], [58], destines for the intensive computations of the single/multi-phase flow and transport in a complex geometry, e.g. [10], [53], [76], [77], [98], [2], [92], [74], [87], [25], [4], [119], due to the extreme simplicity of its explicit-time marching bulk algorithm, local mass-conserving boundary handling and implicit-interface tracking. However, the porous media simulations are often intended for stationary velocity and phase field distributions, which are attainable after a very long number of computational steps. Hence, several strategies have been proposed to improve this issue. On the one side, the iterative momentum relaxation techniques [71], [68], [69], that adjust the driven body forcing until its complete cancellation by friction, enables the time steps reduction by half using the single-relaxation-times BGK model [97]. On the other side, much more radically, Verberg & Ladd [110] reduce the BGK Stokes-flow model with τ=1 to linear algebraic system with respect to the macroscopic variables alone, pressure and momentum, and then gain one to two orders of magnitude in step reduction with the bi-conjugate matrix inversion in a random arrays of spheres, for about 30% longer single-node update. The subsequent LBM “matrix” formulations concentrate their efforts on the efficiency of the linear and non-linear solvers, e.g. [108], [109], [88], [80]; a recent work [84] reports an enhanced efficiency of the non-linear, implicit, finite-volume, steady-state BGK algorithm for high Reynolds number simulations. Alternatively, the “preconditioned” equilibrium formulations [54], [66], [96], [89] aim at introducing of small adjustable prefactors in front of the time-derivatives in the modelled macroscopic equations. An independent scaling of the mass and diffusive-flux variables also accelerates the advection-diffusion ADE schemes [33]; however their gain in time steps is counterbalanced by a reduction in stable velocity amplitude [51]. In flow schemes, the number of steps to steady-state depends on the lattice value of the kinematic viscosity ν but its optimal choice is not obvious and depends upon the convergence criteria [106], [70], irregularity of structure [106], the penetrability of porous inclusions [44] and the image size [87]. Finally, the “squared sound velocity” cs2, relating pressure to density, is free tunable in linear flow computations: the number of steps and the convergence dependency upon ν decreases as cs2 increases [106] and tρ=cs2tP diminishes, in accord with the “preconditioned” techniques.

It should be understood that the free tuning of the lattice transport coefficients, like kinematic viscosity, is only admissible when the numerical scheme respects the dimensionless group of the considered problem, basically formed from the Darcy, Péclet, Reynolds, Froude numbers, the transport coefficient contrasts and aspect ratios. The two-relaxation-times TRT operator [34], [37], [38] accomplishes this requirement by keeping the specific freely-adjustable combination Λ of its two collision rates at a fixed value, when the transport coefficient varies linearly with one of the two, [31], [65]. The Λ control has been proven [65] through the generic form of the steady-state TRT recurrence solutions, also there extended to more complicated operators. Moreover, the multiple-relaxation-times MRT collisions [64] may achieve the same objective by keeping fixed all Λ combinations of the symmetric and anti-symmetric mode rates, [38], [65], [70]; this property is not available either with the BGK or regularized MRT operators with fixed kinetic “ghost” rates [58], [32], [16], [86], where Λ depends, respectively, quadratically and linearly on the transport coefficient. On the other side, it has been shown [37], [51] that the steady-state implicit-interface tracking can be identically reformulated through the anti-bounce back [34], [72] and the bounce-back, ABB and BB interface-conjugate directional TRT relations. The steady-state solutions of these boundary rules share the TRT bulk parametrization [65]; consequently, the same Λ values locate exactly the mid-grid straight/diagonal boundary and interface on the piece-wise parabolic, velocity and diffusion scalar, profiles, [28], [29], [37], [18], [49]. In this context, several numerical studies have performed consistent measurements of the transport coefficients at fixed Λ, and examined their Λ impact, including the prediction of the permeability and drag in homogeneous [106], [70], [8] or heterogeneous [44], [100] structures, effective permeability in non-Newtonian fluids [107], effective diffusivity [26], [11] and thermal conductivity of the composite materials [19]. On the theoretical perspective, Λ impacts the truncation [42], [24], the numerical and Taylor dispersion [46], the high-order moments [48], the asymptotic convergence rate: quite uniformly in regular and random sphere packings [70], and stability [41], [75], [42].

The von Neumann TRT stability analysis [41] reveals one singular choice: Λ=14, called the “optimal” OTRT subclass, where the non-equilibrium becomes fully expressed through the directional central-difference operators of the equilibrium alone. The advanced OTRT stability is confirmed numerically through the Taylor-dispersion ADE modelling in a broad Péclet range [46], but also in high Reynolds number dipole-wall collisions [90], [91]. The OTRT also distinguishes itself for an accurate modeling of the Taylor dispersion [46], effective diffusivity [11] and liquid/solid flux exchange [63], [120], [85]. The TRT steady-state recurrence equations [65], [40] also allow to find exact solutions of the scheme in grid-aligned geometry, e.g. discrete-exponential, heterogeneous Brinkman channel flow, [44], [47]. The initial suspicion that the BB exhibits spurious Knudsen-type accommodation [17] in the grid-inclined channels [30] was later algebraically expressed [65] and quantified (i) with the Navier–Stokes steady-state channel flow [39]; and (ii), for tangential bounce-back moments impact in the transient Gaussian evolution along the flat wall [45], [49]. The underlying discrete-exponential mechanism is beyond the scope of the Chapman-Enskog or perturbative analysis, but its presence is not negligible, as it degrades the BB accuracy to the first order in inclined [30], [99], [102] and spherical [70] geometries. The steady-state linear ADE model excites the whole, equilibrium and non-equilibrium, perturbation even on diagonally-oriented implicit interface, when the advection and/or diffusion equilibrium weights are freely rotated with respect to the lattice, and the scalar field evolves along all interface-cut lattice directions, [52].

The symbolic solution of the recurrence equations becomes tedious in grid-inclined slabs and motivates our interest for a robust steady-state analysis tool, which is able to handle any boundary and interface modifications. This seemingly difficult task turns out to be relatively simple and perhaps of much broader, numerical interest. The task of this work is hence threefold: (i) to formulate the system of the TRT steady-state bulk equations; (ii) to extend and recast the multi-reflection MR boundary rules, and (iii), to adapt them with the directional interface-conjugate equations. Every couple of bulk equations are simply the aforementioned ABB and BB, linear directional combinations of the equilibrium (macroscopic) and non-equilibrium variables, expressed with the half discrete velocity set owing to the symmetry argument; their form is the same for any lattice; it allows for non-uniform or discontinuous external sources and collision rates. The directional equations are complemented with the local mass and, in flow problem, momentum conservation steady-state solvability conditions. The implicit-interface tracking is matched automatically; otherwise, a couple of two directional equations are to be replaced by two interface-conjugate conditions, formulated in terms of the bulk unknowns. We use this opportunity to systematize and further generalize the MR Dirichlet approach for velocity [31], [38] and scalar fields [35], [38], [81], but also to extend the MR for particular pressure-stress, advection-diffusive flux, diffusive flux and Robin conditions.

By construction, the MR rule is equivalent to directional Taylor expansion along the cut-link, from the grid boundary node to an arbitrarily-shaped surface. Hereafter, we call “linear” those boundary techniques which only account for the first-order gradients in the reconstruction of the wall-incoming populations, e.g. with finite-differences [105], [20], interpolations [21], [9], [116], [16], three-population “linear” multi-reflection [35], [38], [81], or locally, through the non-equilibrium projections [32], [67], [86], [117]. Although the linear rules neglect the “kinetic” non-equilibrium component, their effective accuracy depends on the “ghost” collision rates, very noticeably in slow (porous) flow [31], [39] and diffusion-dominate problems, at least. The parabolic MR rules [31], [35], [38] account for the second-order derivatives through the third-order accurate Chapman-Enskog population expansion [28], [31], [35] or the recurrence solution form [38]. They attain then a quasi-analytical quality of velocity field and notably smooth pressure fluctuations on modest grids, e.g. in low Reynolds number flow around circular and spherical surface [31], [94], [70]; the parabolic rules also surpasses the partially-saturated or immersed boundary approaches in porous media simulations [94], [16], [95], [14]. The MR is further adapted for moving bodies [31], free-interface pressure-stress condition [7], spatial resistance variation in Brinkman porous models [102] and the finite Knudsen number slip-flows [103], [104]. Our particular emphasis will be put on the parabolic MR rules because they are able to locate exactly grid-rotated parabolic profiles, unreachable with the BB, ABB and linear rules. These solutions were originally achieved with the local but parabolic, boundary concept LSOB [30]; its linear counterpart [117] is recently revived, and the parabolic LSOB is now extended [101] for smooth shaped-surface and acute angles. Finally, in this topic, it is worth mentioning that the LSOB and the moment-based boundary approach [93], [6] developed [90], [73], [91] for on-grid straight boundaries, are more suitable than the MR for independent prescription of the normal and tangential stress conditions [32].

Beyond of the formal accuracy, our main concern is the exact steady-state parametrization of the boundary rules by the governing physical numbers as, otherwise, the numerical dimensionless solutions and their relative errors will depend on the lattice values assigned for physical coefficients, such as the kinematic viscosity or molecular diffusion. The sufficient parametrization conditions [38] are expressed through the MR coefficients, and we extend them for all type MR schemes developed in the present work. In particular, we show that the linear Dirichlet scalar MR family [81], [82], [83], referred to as LMK hereafter, and its local equivalent [118], are not parametrized by the Péclet number, except for their particular coefficient sets. In the past, linear Dirichlet flow schemes [9], [116] have been improved for parametrization within the infinite linear MGLI family [38], [39]. In the same fashion, we extend the LMK to two infinite parametrized families, called MPLI and PLI [simultaneously extending the MR pressure-linear scheme [38], [39]], and examine their respective accuracy for diffusion-and advection-dominant regimes. The compact MR steady-state formulation is recasted through the bulk unknowns alone; it is “one-node” for the linear schemes and “two-node” for the parabolic ones; hence, the computationally-efficient TRT directional structure [87], [4] remains preserved. We also postulate and satisfy the “portability” conditions on the parabolic MR coefficients, from the TRT to the principal, flow and ADE, MRT collisions, and outline their heuristic, adjustable stability bounds.

The interface-conjugate is built for two linear problems exemplified in arbitrarily-rotated slabs: the well known benchmark of the two-phase Stokes flow with the flat interface, e.g. [29], [37], [56], [122], [79], and the ADE with the discontinuous mass-source and diffusion coefficient due to heterogeneous porosity [111], [50], [51], which is typical boundary problem found in the heat and mass transfer applications, e.g. [112], [115], [82], [55], [19], [78], [61], [60]. Our transient interface-conjugate update [TIC] is inspired by the “decoupled interface treatment” [82], [55]; the steady-state interface-conjugate equations [SIC] then replace every couple of bulk equations on interface cut-links. The SIC is able to model exactly the piece-wise parabolic rotated velocity profile thanks to parabolic, Dirichlet velocity and pressure-stress, MR rules. In turn, the exact piece-wise parabolic ADE solutions are closed by a broad panel of the interface-jump and boundary conditions, in the presence of grid-rotated interface-parallel advective velocity. A particular emphasis is placed on the “opposite” problem [50], [51] of a constant, interface-perpendicular Darcy velocity through the heterogeneous blocks in series; this tough benchmark has not been yet examined with the interface-conjugate, or sometimes is forbidden with it, e.g. [59].

Finally, we tackle another challenging problem. Basically, the explicit interface-conjugate algorithms maintain the flux balance only on the straight midway interface, e.g. [36], [82], while the grid-shifted boundary and interface, but also inlet/outlet conditions, e.g. [87], most often do not guarantee it exactly. Although the mass deficit can be recompensed, e.g. locally through the immobile population, it has been demonstrated [30] that exact boundary schemes for grid-inclined parabolic flow conserve the outgoing mass over the whole periodic segment; a harmful effect of the artificial mass conservation has been numerically demonstrated for flow field [1], [113] and scalar ADE solutions with the Neumann flux schemes [23]. Typically, when the MR Dirichlet velocity condition does not preserve the global mass, the stationary momentum solution copes with the constant-rate mass flux over the computational domain; in other words, the immobile population is not at equilibrium and hence, an additive pressure constant will continuously vary in time [31]. We will propose and approve a simple, uniform modification of the local mass-conservation solvability condition able to mimics “quasi-stationary” solutions.

The paper is organized as follows. Section 2 summarizes the transient TRT bulk update with external sources and introduces its steady-state counterpart. Section 3 analyses the BB and ABB, defines the “linear” and “parabolic” closure relations, recalls the multi-reflection MR concept, recasts it in steady-state compact form and builds the sufficient parametrization conditions for Dirichlet, flux and shear-stress MR schemes. Section 4 extends the Dirichlet velocity class MRq(u), constructs and applies the interface-conjugate TIC and SIC schemes in a two-phase stratified grid-rotated flow. Section 5 extends the “linear” and “parabolic” Dirichlet-scalar MRq(p) families and builds the particular, directional advective-diffusive flux, diffusive-flux and Robin boundary schemes. Section 6 extends the TIC and SIC for the ADE problem with discontinuous collision rates, mass-sources and interface jumps. Section 7 examines the existence and accuracy of the stationary ADE solutions in the interface-parallel and interface-perpendicular flow. Section 8 concludes the paper. Appendix A provides details on the construction of the MR schemes and tabulates their coefficients. Appendix B outlines the steady-state numerical algorithm with the MR boundary and interface-conjugate closure.

Section snippets

Transient (standard) and steady-state algorithms

The LBM is operated on the cuboid computational grid within the penetrable d-dimensional domain r{Vp} composed from Np grid points; a regular mesh-size and time step are set equal to one lattice unit and one iteration, respectively; the characteristic size is L. The neighboring nodes are interconnected by the discrete velocity set dDqQ: it consists of zero-vector c0 and Qm=Q1 vectors cq; the one-half of the non-zero velocity vectors is numbered with the positive numbers qQ12, their

Transient and steady-state closure relations

The population update in Eq. (3) is closed by the boundary rules for the populations incoming from the outside of the computational domain Vp.

Definition I Any node rbVp with at least one wall-cut velocity cqb such that r+cqbVp, cqbQ12Q12, is referred to as a boundary node; the set of the wall-cut velocities is denoted by Qb(rb).

Consider a boundary node rb and a wall-cut velocity cq, qQb(rb). The transient update with Eq. (3) computes fq(rb,t+1) with the boundary rule. In turn,

Flow schemes

The “linear” and “parabolic”, Stokes/Navier–Stokes flow MRq(u) families for the Dirichlet velocity condition have been introduced [31], generalized [38], [39] and extended for the Brinkman flow [102]. Namely, an infinite three-population “linear” family MGLI is correctly parametrized [38], [39], [70], and it includes the BFL [9] and YLI [116] schemes improved for this property with the help of the suitable post-collision correction Fˆq. The MGLI member with Fˆq=0 is called CLI; CLI is most

The Dirichlet scalar and Neumann flux schemes

We first extend and parametrize the Dirichlet scalar MRq(p) families [34], [38], [81]: the “linear” and “parabolic” MRq(p) is constructed in Sections 5.1 and 5.2, respectively. Section 5.3 re-builds the Neumann flux FLI scheme [81] and its novel extension FMR. Section 5.4 introduces the diffusion-flux Neumann families MRq(d)={SFLI,AFLI}. Section 5.5 combines the MRq(p) and MRq(d) for mixed (Robin) MRq(m) family.

Interface-conjugate ADE schemes

We construct the transient [TIC] and steady-state [SIC] interface-conjugate schemes for the heterogeneous advection-diffusion equation with the piece-wise continuous diffusion coefficients and sources. The interface-continuity or jump conditions, prescribed for the scalar and flux variables, are respectively based on the MRq(p) and MRq(f)/MRq(d) schemes. They extend the “decoupled interface treatment” [82], [55] under specific conditions imposed on the tangential advection-diffusive flux. In

Heterogeneous ADE in the interface-parallel and interface-normal flow

We consider steady-state solutions to Eq. (93) in the inclined stratified channel y[-h1,h2], h=h1+h2, using the rotated frame from Eq. (69). The geometry is set periodic along x, the interface is parallel with the x-axis and placed arbitrarily with respect to the grid at y=0. Eq. (93) is modelled in the form:·(u(y)P)M(y)=·(Dϕ(y)P),P=P(y),Dϕ(y)=ϕkD0,withϕk=ϕ1wheny[h1,0];andϕk=ϕ2wheny[0,h2],rϕ=ϕ1ϕ2.

The mass source is set velocity-dependent:M(y)=w1ϕ(y)u

Concluding remarks

We proposed the steady-state TRTLBM formulation sustained with the compact boundary and interface closure equations; the method is expressed through the equilibrium and post-collision non-equilibrium components in the half discrete velocity space. The underlying matrix structure is mainly directional; in-node variables are only inter-connected through the conservation solvability constraints. When the stationary solution exists, it is the same with the transient and steady-state approaches,

Conflict of interest

The authors declare no conflict of interest.

Declaration of Competing Interest

The authors report no declarations of interest.

Acknowledgments

The author thanks Dr. Goncalo Silva for his critical reading of the manuscript.

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