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 w...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.展开更多
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 sim...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.展开更多
Fluid flows in microfluidic devices are often characterized by non-Newtonian rheology with non-linear wall slip behavior also observed.This work solves this problem class with the lattice Boltzmann method(LBM),proposi...Fluid flows in microfluidic devices are often characterized by non-Newtonian rheology with non-linear wall slip behavior also observed.This work solves this problem class with the lattice Boltzmann method(LBM),proposing new advanced boundary scheme formulations to model the joint contribution of non-linear rheology and non-linear wall slip laws in application to microchannels of planar and circular cross-section.The non-linear stress-strain-rate relationship of the microflow is described by a generalized Newtonian model where the viscosity function follows the Sisko model.To guarantee that LBM steady-state solutions are not contaminated by numerical errors that depend on the viscosity local value,the two-relaxation-time(TRT)collision is adopted.The fluid-wall accommodation model considers different slip laws,such as the Navier linear,Navier non-linear,empirical asymptotic and Hatzikiriakos slip laws.They are transcribed into the LBM framework by adapting the local second-order boundary(LSOB)scheme strategy to this problem class.Theoretical and numerical analyses developed for a steady and slow viscous fluid within 2D slit and 3D circular pipe channels demonstrate the parabolic level of accuracy of the developed LSOB scheme throughout the considered non-linear slip and non-Newtonian models.展开更多
We investigate a two-relaxation-time(TRT)lattice Boltzmann algorithm with the asymptotic analysis technique.The results are used to analyze invariance properties of the method.In particular,we focus on time dependent ...We investigate a two-relaxation-time(TRT)lattice Boltzmann algorithm with the asymptotic analysis technique.The results are used to analyze invariance properties of the method.In particular,we focus on time dependent Stokes and Navier-Stokes problems.展开更多
文摘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.
基金The author is thankful to D.d’Humi`eres for his parallel work on the Fourier analysis of the TRT AADE model and to anonymous referee for constructive suggestions.
文摘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.
基金Fundacao para a Ciencia e a Tecnologia(FCT)for its financial support via the project LAETA Base Funding(DOI:10.54499/UIDB/50022/2020).
文摘Fluid flows in microfluidic devices are often characterized by non-Newtonian rheology with non-linear wall slip behavior also observed.This work solves this problem class with the lattice Boltzmann method(LBM),proposing new advanced boundary scheme formulations to model the joint contribution of non-linear rheology and non-linear wall slip laws in application to microchannels of planar and circular cross-section.The non-linear stress-strain-rate relationship of the microflow is described by a generalized Newtonian model where the viscosity function follows the Sisko model.To guarantee that LBM steady-state solutions are not contaminated by numerical errors that depend on the viscosity local value,the two-relaxation-time(TRT)collision is adopted.The fluid-wall accommodation model considers different slip laws,such as the Navier linear,Navier non-linear,empirical asymptotic and Hatzikiriakos slip laws.They are transcribed into the LBM framework by adapting the local second-order boundary(LSOB)scheme strategy to this problem class.Theoretical and numerical analyses developed for a steady and slow viscous fluid within 2D slit and 3D circular pipe channels demonstrate the parabolic level of accuracy of the developed LSOB scheme throughout the considered non-linear slip and non-Newtonian models.
文摘We investigate a two-relaxation-time(TRT)lattice Boltzmann algorithm with the asymptotic analysis technique.The results are used to analyze invariance properties of the method.In particular,we focus on time dependent Stokes and Navier-Stokes problems.
