Enhanced closure scheme for lattice Boltzmann equation hydrodynamics

Reviving the local second-order boundary approach within the two-relaxation-time lattice Boltzmann modelling

This work addresses the Dirichlet boundary condition for momentum in the lattice Boltzmann method (LBM), with focus on the steady-state Stokes flow modelling inside non-trivial shaped ducts. For this task, we revisit a local and highly accurate boundary scheme, called the local second-order boundary (LSOB) method. This work reformulates the LSOB within the two-relaxation-time (TRT) framework, which achieves a more standardized and easy to use algorithm due to the pivotal parametrization TRT properties. The LSOB explicitly reconstructs the unknown boundary populations in the form of a Chapman-Enskog expansion, where not only first- but also second-order momentum derivatives are locally extracted with the TRT symmetry argument, through a simple local linear algebra procedure, with no need to compute their non-local finite-difference approximations. Here, two LSOB strategies are considered to realize the wall boundary condition, the original one called Lwall and a novel one Lnode, which operate with the wall and node variables, roughly speaking. These two approaches are worked out for both plane and curved walls, including the corners. Their performance is assessed against well-established LBM boundary schemes such as the bounce-back, the local second-order accurate CLI scheme and two different parabolic multi-reflection (MR) schemes. They are all evaluated for 3D duct flows with rectangular, triangular, circular and annular cross-sections, mimicking the geometrical challenges of real porous structures. Numerical tests confirm that LSOB competes with the parabolic MR accuracy in this problem class, requiring only a single node to operate. This article is part of the theme issue 'Fluid dynamics, soft matter and complex systems: recent results and new methods'.

Numerical investigation of non-Newtonian fluids in annular ducts with finite aspect ratio using lattice Boltzmann method

In this work the instability of the Taylor-Couette flow for Newtonian and non-Newtonian fluids (dilatant and pseudoplastic fluids) is investigated for cases of finite aspect ratios. The study is conducted numerically using the lattice Boltzmann method (LBM). In many industrial applications, the apparatuses and installations drift away from the idealized case of an annulus of infinite length, and thus the end caps effect can no longer be ignored. The inner cylinder is rotating while the outer one and the end walls are maintained at rest. The lattice two-dimensional nine-velocity (D2Q9) Boltzmann model developed from the Bhatnagar-Gross-Krook approximation is used to obtain the flow field for fluids obeying the power-law model. The combined effects of the Reynolds number, the radius ratio, and the power-law index n on the flow characteristics are analyzed for an annular space of finite aspect ratio. Two flow modes are obtained: a primary Couette flow (CF) mode and a secondary Taylor vortex flow (TVF) mode. The flow structures so obtained are different from one mode to another. The critical Reynolds number Re(c) for the passage from the primary to the secondary mode exhibits the lowest value for the pseudoplastic fluids and the highest value for the dilatant fluids. The findings are useful for studies of the swirling flow of non-Newtonians fluids in axisymmetric geometries using LBM. The flow changes from the CF to TVF and its structure switches from the two-cells to four-cells regime for both Newtonian and dilatant fluids. Contrariwise for pseudoplastic fluids, the flow exhibits 2-4-2 structure passing from two-cells to four cells and switches again to the two-cells configuration. Furthermore, the critical Reynolds number presents a monotonic increase with the power-law index n of the non-Newtonian fluid, and as the radius ratio grows, the transition flow regimes tend to appear for higher critical Reynolds numbers.

Involving the Navier-Stokes equations in the derivation of boundary conditions for the lattice Boltzmann method

By means of the continuity equation of the incompressible Navier-Stokes equations, additional physical arguments for the derivation of a formulation of the no-slip boundary condition for the lattice Boltzmann method for straight walls at rest are obtained. This leads to a boundary condition that is second-order accurate with respect to the grid spacing and conserves mass. In addition, the boundary condition is stable for relaxation frequencies close to two.

Analysis of the lattice Boltzmann Bhatnagar-Gross-Krook no-slip boundary condition: ways to improve accuracy and stability

An analytical and numerical analysis of the no-slip boundary condition at walls at rest for the lattice Boltzmann Bhatnagar-Gross-Krook method is performed. The main result of this analysis is an alternative formulation for the no-slip boundary condition at walls at rest. Numerical experiments assess the accuracy and stability of this formulation for Poiseuille and Womersley flows, flow over a backward facing step, and unsteady flow around a square cylinder. This no-slip boundary condition is compared analytically and numerically to the boundary conditions of Inamuro [Phys. Fluids 7, 2928 (1995)] and Zou and He [Phys. Fluids 9, 1591 (1997)] and it is found that all three make use of the same mechanism for the off-diagonal element of the stress tensor. Mass conservation, however, is only assured by the present one. In addition, our analysis points out which mechanism lies behind the instabilities also observed by Lätt [Phys. Rev. E 77, 056703 (2008)] for this kind of boundary conditions. We present a way to remove these instabilities, allowing one to reach relaxation frequencies considerably closer to 2.

Lattice Boltzmann model for exterior flows with an annealing preconditioning method

In this paper we propose a highly efficient and stable lattice Boltzmann method for solving low Reynolds number exterior flows using a preconditioning technique. The present method is based on replacing the constant preconditioning parameter (gamma) within uniform grids [Guo, Phys. Rev. E 70, 066706 (2004)] by a space- and time-dependent one in a nested mesh-refined domain. To do this, for the transition from a fine to the neighboring coarser grid, gamma has been divided by a factor K , which is large initially and anneals stepwise to a small value after some iterations. With this technique, more than one order of magnitude larger convergence rate can be achieved, and several orders of magnitude larger system size can be treated.

Acceleration of steady-state lattice Boltzmann simulations for exterior flows

The simulation of a stationary fluid flow past an obstacle by the lattice Boltzmann method (LBM) in two dimensions is discussed. The combination of second-order expressions for far-field boundary conditions and a suitable treatment of the no-slip boundary condition at the obstacle surface with the nested grid-refinement technique can be applied to the LBM, resulting in a highly efficient method for the treatment of exterior flows. Furthermore, via replacing the nested time stepping by local time stepping, the resolution process can be substantially accelerated. A highly accurate drag coefficient was used to make an error assessment for various no-slip boundary conditions imposed on the obstacle surface. The analysis showed that the equilibrium method for treating the no-slip conditions is superior to halfway bounce-back and full-way bounce-back no-slip conditions when the relaxation time tau=1 . Also a tau -dependence test was made to evaluate the performance of different methods in the treatment of the no-slip boundary conditions.

Straight velocity boundaries in the lattice Boltzmann method

Various ways of implementing boundary conditions for the numerical solution of the Navier-Stokes equations by a lattice Boltzmann method are discussed. Five commonly adopted approaches are reviewed, analyzed, and compared, including local and nonlocal methods. The discussion is restricted to velocity Dirichlet boundary conditions, and to straight on-lattice boundaries which are aligned with the horizontal and vertical lattice directions. The boundary conditions are first inspected analytically by applying systematically the results of a multiscale analysis to boundary nodes. This procedure makes it possible to compare boundary conditions on an equal footing, although they were originally derived from very different principles. It is concluded that all five boundary conditions exhibit second-order accuracy, consistent with the accuracy of the lattice Boltzmann method. The five methods are then compared numerically for accuracy and stability through benchmarks of two-dimensional and three-dimensional flows. None of the methods is found to be throughout superior to the others. Instead, the choice of a best boundary condition depends on the flow geometry, and on the desired trade-off between accuracy and stability. From the findings of the benchmarks, the boundary conditions can be classified into two major groups. The first group comprehends boundary conditions that preserve the information streaming from the bulk into boundary nodes and complete the missing information through closure relations. Boundary conditions in this group are found to be exceptionally accurate at low Reynolds number. Boundary conditions of the second group replace all variables on boundary nodes by new values. They exhibit generally much better numerical stability and are therefore dedicated for use in high Reynolds number flows.

Improved simulation of drop dynamics in a shear flow at low Reynolds and capillary number

The simulation of multicomponent fluids at low Reynolds number and low capillary number is of interest in a variety of applications such as the modeling of venule scale blood flow and microfluidics; however, such simulations are computationally demanding. An improved multicomponent lattice Boltzmann scheme, designed to represent interfaces in the continuum approximation, is presented and shown (i) significantly to reduce common algorithmic artifacts and (ii) to recover full Galilean invariance. The method is used to model drop dynamics in shear flow in two dimensions where it recovers correct results over a range of Reynolds and capillary number greater than that which may be addressed with previous methods.

A many-component lattice Boltzmann equation simulation for transport of deformable particles

We review the analysis of single and N-component lattice Boltzmann methods for fluid flow simulation. Results are presented for the emergent pressure field of a single phase incompressible liquid flowing over a backward-facing step, at moderate Reynolds Number, which is compared with the experimental data of Denham & Patrick (1974 Trans. IChE 52, 361-367). We then access the potential of the N-component method for transport of high volume fraction suspensions of deformable particles in pressure-driven flow. The latter are modelled as incompressible, closely packed liquid drops. We demonstrate the technique by investigating the particles' transverse migration in a uniform shear ('lift'), and profile blunting and chaining.