Prediction of the moments in advection-diffusion lattice Boltzmann method. Ⅰ. Truncation dispersion, skewness, and kurtosis

摘要

The effect of the heterogeneity in the soil structure or the nonuniformity of the velocity field on the modeledresident time distribution (RTD) and breakthrough curves is quantified by their moments. While the first momentprovides the effective velocity, the second moment is related to the longitudinal dispersion coefficient (k_T ) in thedeveloped Taylor regime; the third and fourth moments are characterized by their normalized values skewness (Sk)and kurtosis (Ku), respectively. The purpose of this investigation is to examine the role of the truncation correctionsof the numerical scheme in k_T , Sk, andKu because of their interferencewith the second moment, in the form of thenumerical dispersion, and in the higher-order moments, by their definition. Our symbolic procedure is based on therecently proposed extended method of moments (EMM). Originally, the EMM restores any-order physical momentsof the RTD or averaged distributions assuming that the solute concentration obeys the advection-diffusionequation in multidimensional steady-state velocity field, in streamwise-periodic heterogeneous structure. In ourwork, the EMM is generalized to the fourth-order-accurate apparent mass-conservation equation in two- andthree-dimensional duct flows. The method looks for the solution of the transport equation as the product of a longharmonic wave and a spatially periodic oscillating component; the moments of the given numerical scheme arederived from a chain of the steady-state fourth-order equations at a single cell. This mathematical technique isexemplified for the truncation terms of the two-relaxation-time latticeBoltzmann scheme, using plug and parabolicflow in straight channel and cylindrical capillary with the d2Q9 and d3Q15 discrete velocity sets as simple butillustrative examples. The derived symbolic dependencies can be readily extended for advection by another,Newtonian or non-Newtonian, flow profile in any-shape open-tabular conduits. It is established that the truncationerrors in the three transport coefficients k_T , Sk, and Ku decay with the second-order accuracy.While the physicalvalues of the three transport coefficients are set by P´eclet number, their truncation corrections additionally dependon the two adjustable relaxation rates and the two adjustable equilibrium weight families which independentlydetermine the convective and diffusion discretization stencils. We identify flow- and dimension-independentoptimal strategies for adjustable parameters and confront them to stability requirements. Through specific choicesof two relaxation rates and weights, we expect our results be directly applicable to forward-time central differencesand leap-frog central-convective Du Fort–Frankel–diffusion schemes. In straight channel, a quasi-exact validationof the truncation predictions through the numerical moments becomes possible thanks to the specular-forwardno-flux boundary rule. In the staircase description of a cylindrical capillary, we account for the spurious boundarylayerdiffusion and dispersion because of the tangential constraint of the bounce-back no-flux boundary rule.