Various external forcing formulations of the lattice Boltzmann method (LBM) are analyzed by deriving the analytic solutions of the fully developed Poiseuille flows with and without the porous wall.

For uniform driving force, all the forcing formulations recover the second-order accurate discretized Navier-Stokes equation. However, the analytic solutions show that extra force gradients arise due to variable force, and this form differs from the analysis using Chapman-Enskog expansion.

It is possible to remove these extra terms of single-relaxation-time (SRT) LBM using specific relaxation time depending on the force formulation adopted. However, this limits the broader applicability of the SRT LBM. Moreover, the multiple-relaxation-time (MRT) LBM may provide an option to remove the variable-force gradient term benefiting from separating relaxation parameters for each moment.

Off-lattice Boltzmann methods generally offer more flexibility than the customary lattice Boltzmann method, but at the price of higher computational costs. Nevertheless, features such as changeable time step sizes, body-matched meshes, or special velocity sets can be reasons for using off-lattice Boltzmann methods. As an alternative to finite difference or finite volume lattice Boltzmann methods, the semi-Lagrangian lattice Boltzmann method (SLLBM) has established itself as a competitive off-lattice Boltzmann method.

Semi-Lagrangian lattice Boltzmann methods follow the characteristics of the particle distribution functions back in time, similar to the original lattice Boltzmann method. However, when using time steps of a size other than one, when using non-standard velocity sets, or when using a non-Cartesian mesh, the advection step will not be from node to node. In this case, an interpolation step is required, and its implementation is crucial for the accuracy of the simulation. In our SLLBM approach, the simulation domain is discretized into cells, similar to the finite element method. Then, high-order shape functions are used to determine the particle distribution functions during the advection step. This procedure reduces mass losses of the interpolation and increases the overall accuracy.

In recent years, we have extended the SLLBM to simulate three-dimensional compressible viscous flows. Our ansatz was as follows: density, momentum, and temperature of the flow are simulated with one distribution function requiring a high-order expansion of the equilibrium up to fourth order. This also requires large velocity sets with high degree of quadrature, which usually makes the simulation too expensive. However, by using cubature rules for the velocity discretization, we have successfully reduced the size of velocity sets to only 45 discrete velocities in three dimensions. This paved the way for affordable 3D compressible simulations with the SLLBM.

In this contribution, we recapitulate the method, discuss advantages and disadvantages, and show recent results, e.g., viscous transonic and supersonic flows around 2D NACA airfoils and 3D spheres, both using body-fitted meshes.

Microfluidic multiphase flow is a topic of increasing interest because of its applications and the possibilities it offers in various fields. The major advantages of operating in the microscale include the large surface-volume ratios, control of fluid flow and lower costs. Thus, it is imperative to study the flow patterns in such flows and the parameters which subsequently influence them. This research aims to examine the efficacy of LBM in simulating the flow of two immiscible fluids in a Y-Y channel by validating it with

experimental results and investigating the effect of dimensionless flow parameters and geometry on the flow patterns. The LBM can be easily extended to multiphase flow, and the interface need not be tracked in the case of multiphase LBM, thus retaining the simplicity of the single-phase model. This research uses the model proposed by Rothman and Keller to simulate multiphase flow (Gunstensen et al. [1991]). Particle

distributions are defined for each fluid in this method, and the interfacial tension is applied as a body force to ensure fluid separation. The case of two-phase flows in Y-Y channels is a little more complex than many earlier studies on multiphase LBM, primarily because of the range of velocities involved in the study. Previous studies dealing with multiphase channel flows have rarely taken the contact angle into account, and

when they have, it is for the simpler case of a rectangular or T-T channel. The problem is compounded by the inclination of the inlets and outlets, especially the corners located at the intersection of the inlets. This necessitates a different method of applying the contact angle. The method used in this research for implementing the contact angle was proposed by Xu et al (Xu et al. [2017]), where the contact angle is applied by correcting the position of the interface normal to match the contact angle. The complexity of this case, however, necessitates some modifications near the intersection of the inlets in the determination of the surface normal. Traditionally, an 8th order discretization is used to estimate the surface normal, but for certain boundaries, this introduces ambiguities as the boundary nodes might be located on either side of two fluid nodes. A lower-order discretization for these nodes is, therefore, necessary to avoid such ambiguities, which

is especially important when dealing with low Capillary numbers. At these values, traditionally slug flow is observed and at higher Capillary numbers, parallel flow is observed. The model is validated by comparing it to experimental results obtained by Liu (Liu [2022]) for various Capillary numbers. Simulations performed using the model were observed to accurately predict the flow regimes seen in the experiments. Additionally,

the length of the slugs obtained from the simulations was comparable to those from experiments. The phenomenon of leakage during parallel flow was also captured. Finally, the geometry of the channel is varied to understand the influence of geometry on the flow regimes and the capability of the method itself to model different geometries.

Bibliography:

Bastien Chopard, Alexandre Dupuis, Alexandre Masselot, and Pascal Luthi. Cellular automata and lattice boltzmann techniques: An approach to model and simulate complex systems. Advances in complex systems, 5(02n03):103{246, 2002.

Andrew K. Gunstensen, Daniel H. Rothman, Stephane Zaleski, and Gianluigi Zanetti. Lattice Boltzmann model of immiscible fluids. Physical Review A, 43(8):4320{4327, 1991. ISSN 10502947. doi: 10.1103/PhysRevA.43.4320.

Zheng Liu. Purifying radionuclides with microfluidic technology for medical purpose. PhD Thesis, TU Delft Repository, 2022. doi: https://doi.org/10.4233/uuid:e1bebcdd-185a-4515-b352-76d68f65ace8.

Zhiyuan Xu, Haihu Liu, and Albert J Valocchi. Lattice boltzmann simulation of immiscible two-phase flow with capillary valve effect in porous media. Water Resources Research, 53(5):3770{3790, 2017.

The main objective of this contribution is to investigate Multiple Relaxation Time Lattice Boltzmann schemes for Convection-Diffusion Equations. In particular, we discuss the issue of obtaining second order exact schemes when the advection velocity is not constant.

Our study is based on previous results by the authors [1] that allow to identify the extra terms, coming from the non-constant advection velocity, that prevent the exactness of such shemes. We show how these terms can be cancelled, first by a suitable choice of the momentum equilibrium, and then by adding an ancillary distribution that acts as a forcing term applied to the non-conserved momentum. Numerical experiments are performed on Poiseuille and Taylor-Green test cases. They show that the proposed approaches are relevant regarding the stated problem.

References

[1] J. Michelet, M. M. Tekitek, and M. Berthier, “Multiple relaxation time lattice Boltzmann model for advection-diffusion equations with application to radar image processing,” Submitted in Journal of Computational Physics, dec 2021.