Two-Relaxation-Time Lattice Boltzmann Scheme:About Parametrization,Velocity,Pressure and Mixed Boundary Conditions

We develop a two-relaxation-time (TRT) Lattice Boltzmann model for hydrodynamicequations with variable source terms based on equivalent equilibriumfunctions. A special parametrization of the free relaxation parameter is derived. Itcontrols, in addition to the non-dimensional hydrodynamic numbers, any TRT macroscopicsteady solution and governs the spatial discretization of transient flows. Inthis framework, the multi-reflection approach [16, 18] is generalized and extended forDirichlet velocity, pressure and mixed (pressure/tangential velocity) boundary conditions.We propose second and third-order accurate boundary schemes and adapt themfor corners. The boundary schemes are analyzed for exactness of the parametrization,uniqueness of their steady solutions, support of staggered invariants and for the effectiveaccuracy in case of time dependent boundary conditions and transient flow.When the boundary scheme obeys the parametrization properly, the derived permeabilityvalues become independent of the selected viscosity for any porous structureand can be computed efficiently. The linear interpolations [5, 46] are improved withrespect to this property.

Study of Simple Hydrodynamic Solutions with the Two-Relaxation-Times Lattice Boltzmann Scheme

For simple hydrodynamic solutions, where the pressure and the velocity arepolynomial functions of the coordinates, exact microscopic solutions are constructedfor the two-relaxation-time (TRT) Lattice Boltzmann model with variable forcing andsupported by exact boundary schemes. We show how simple numerical and analyticalsolutions can be interrelated for Dirichlet velocity, pressure and mixed (pressure/tangential velocity) multi-reflection (MR) type schemes. Special care is taken toadapt themfor corners, to examine the uniqueness of the obtained steady solutions andstaggered invariants, to validate their exact parametrization by the non-dimensionalhydrodynamic and a “kinetic” (collision) number. We also present an inlet/outlet“constant mass flux” condition. We show, both analytically and numerically, that thekinetic boundary schemes may result in the appearance of Knudsen layers which arebeyond the methodology of the Chapman-Enskog analysis. Time dependent Dirichletboundary conditions are investigated for pulsatile flow driven by an oscillating pressuredrop or forcing. Analytical approximations are constructed in order to extend thepulsatile solution for compressible regimes.

Truncation Errors,Exact and Heuristic Stability Analysis of Two-Relaxation-Times Lattice Boltzmann Schemes for Anisotropic Advection-Diffusion Equation

This paper establishes relations between the stability and the high-order truncated corrections for modeling of the mass conservation equation with the tworelaxation-times(TRT)collision operator.First we propose a simple method to derive the truncation errors from the exact,central-difference type,recurrence equations of the TRT scheme.They also supply its equivalent three-time-level discretization form.Two different relationships of the two relaxation rates nullify the third(advection)and fourth(pure diffusion)truncation errors,for any linear equilibrium and any velocity set.However,the two relaxation times alone cannot remove the leading-order advection-diffusion error,because of the intrinsic fourth-order numerical diffusion.The truncation analysis is carefully verified for the evolution of concentration waves with the anisotropic diffusion tensors.The anisotropic equilibrium functions are presented in a simple but general form,suitable for the minimal velocity sets and the d2Q9,d3Q13,d3Q15 and d3Q19 velocity sets.All anisotropic schemes are complemented by their exact necessary von Neumann stability conditions and equivalent finite-difference stencils.The sufficient stability conditions are proposed for the most stable(OTRT)family,which enables modeling at any Peclet numbers with the same velocity amplitude.The heuristic stability analysis of the fourth-order truncated corrections extends the optimal stability to larger relationships of the two relaxation rates,in agreementwith the exact(one-dimensional)and numerical(multi-dimensional)stability analysis.A special attention is put on the choice of the equilibrium weights.By combining accuracy and stability predictions,several strategies for selecting the relaxation and free-tunable equilibrium parameters are suggested and applied to the evolution of the Gaussian hill.