Truncation effect on Taylor–Aris dispersion in lattice Boltzmann schemes: Accuracy towards stability

[Departement_IRSTEA]Eaux [TR1_IRSTEA]ARCEAU

International audience

English. The Taylor dispersion in parabolic velocity field provides a well-known benchmark for advection–diffusion (ADE) schemes and serves as a first step towards accurate modeling of the high-order non-Gaussian effects in heterogeneous flow. While applying the Lattice Boltzmann ADE two-relaxation-times (TRT) scheme for a transport with given Péclet number ( Pe ) one should select six free-tunable parameters, namely, (i) molecular-diffusion- scale, equilibrium parameter; (ii) three families of equilibrium weights, assigned to the terms of mass, velocity and numerical-diffusion-correction, and (iii) two relaxation rates. We analytically and numerically investigate the respective roles of all these degrees of freedom in the accuracy and stability in the evolution of a Gaussian plume. For this purpose, the third- and fourth-order transient multi-dimensional analysis of the recurrence equations of the TRT ADE scheme is extended for a spatially-variable velocity field. The key point is in the coupling of the truncation and Taylor dispersion analysis which allows us to identify the second-order numerical correction delta k T to Taylor dispersivity coefficient k T . The procedure is exemplified for a straight Poiseuille flow where delta k T is given in a closed analytical form in equilibrium and relaxation parameter spaces. The predicted longitudinal dispersivity is in excellent agreement with the numerical experiments over a wide parameter range. In relatively small Pe -range, the relative dispersion error increases with Péclet number. This deficiency reduces in the intermediate and high Pe -range where it becomes Pe -independent and velocity-amplitude independent. Eliminating delta k T by a proper parameter choice and employing specular reflection for zero flux condition on solid boundaries, the d2Q9 TRT ADE scheme may reproduce the Taylor–Aris result quasi-exactly, from very coarse to fine grids, and from very small to arbitrarily high Péclet numbers. Since free-tunable product of two eigenfunctions also controls stability of the model, the validity of the analytically established von Neumann stability diagram is examined in Poiseuille profile. The simplest coordinate-stencil subclass, which is the d2Q5 TRT bounce- back scheme, demonstrates the best performance and achieves the maximum accuracy for most stable relaxation parameters.