Various Collision Models for LBE

Lattice Boltzmann method with multiple relaxation times require a transformation of the particle distribution function into an equivalent set of moments such that each non-conserved moment can be assigned its own relaxation rate. The choice of the moment set is not unique in serval aspects. For example, moments can be taken in the laboratory frame of reference or in the frame co-moving with the fluid. They can be grouped in different ways, e.g. based on physical association (e.g. heat flux) or according to their symmetry and transformation properties.

Most importantly, moments have to be independent of each other if they evolve at different time scales. Unfortunately, this independence is often only vaguely defined in the LBM literature. Pioneering lattice Boltzmann methods with multiple relaxation rates [Phil. Trans. R. Soc. A 360, 1792 pp. 437-451] use unweighted orthogonalization of the moment base in an attempt to enforce mutual independence. In [JCP 190, 2 pp. 351-370] Dellar presented a weighted orthogonal moment basis but did not address the mathematical consequences of this decision. More recently also non-orthogonal moments, Hermite moments and cumulants have been proposed as candidates.

The choice of the moment basis and in particular the type of orthogonality (if any) has profound and often surprising consequences on stability and accuracy. However, the topic is underrepresented in the scientific literature and educational material on the lattice Boltzmann method. In this course I will demonstrate that unweighted orthogonality is incompatible with certain lattices, in particular in three dimensions. Hermite moments and cumulants lead to favorable types of orthogonality.

Equivalent partial differential equations and applications

Lattice Boltzmann schemes replace Boltzmann's binary collision operator with a model using linear relaxation of the distribution functions towards their equilibrium values. Examples of these models include linearisations of lattice gas collision operators, the single-relaxation-time or BGK collision operator, and various multiple-relaxation-time collision operators designed to optimise different measures of accuracy or stability. The two-relaxation-time collision operator groups the discrete particles velocities into anti-parallel pairs, called "dumb-bells" in the theory of lattice gas collision operators, and assigns different relaxation times to the odd and even combinations of anti-parallel velocities. This is equivalent to assigning different relaxation times to the odd and even moments of the distribution functions. For example, momentum flux is an even moment while heat or energy flux is an odd moment. A particular "magic" combination of the two relaxation rates has good properties for computing Poiseuille flow parallel to one of the coordinate axes. The point of zero velocity is then located precisely half-way between lattice points.

We will describe a different interpretation of the two-relaxation-time collision operator that assigns different relaxation times to the forward-propagating and backward-propagating parts of each anti-parallel pair of discrete velocities. The "magic" combination sets the forward-propagating distribution function to equilibrium in this interpretation. The distribution function at any lattice point and time level thus depends only on the distribution function propagating in the reverse direction at the previous time level, and on the equilibrium distributions that are known functions of the fluid density and velocity. By considering two successive lattice Boltzmann timesteps, and hence two reversals of direction, we can extend this to show that the distribution function at any lattice point and time level depends on the same distribution function at the same lattice point two time levels earlier, and on the equilibrium distributions. This allows us to construct closed finite difference schemes for evolving the fluid density and velocity alone across three time levels.

The discrete evolution across three time levels can be thought of as a discrete approximation to partial differential equations with second derivatives with respect to time. However, we prefer to intepret them as discrete approximation to a first order system, the expected conservation laws for mass and momentum, and separate evolution equations for the mass and momentum fluxes. The latter depend only on the fluid density and velocity, so we retain just enough kinetic behaviour to simulate a Maxwell fluid with a finite stress relaxation time. Our approach also extends to include bounce-back boundary conditions with a one timestep delay, the natural implementation based on a halo of "ghost" cells outside the computational domain. We will show numerical experiments including a vortex dipole colliding with a rigid boundary, and the flow driven by an oscillating boundary.

In this work a numerical method for fully three dimensional simulations of a capsule in the electrohydrodynamic regime is proposed. A quasi-steady dual time-stepping scheme allows for iterative computation of the capsule’s fluid velocity using the multigrid lattice Boltzmann method at each time step. The capsule’s elasticity is computed using a linear finite element method and the membrane’s bending resistance is computed from the Helfrich bending energy. The immersed interface method (IIM) is used to compute the electric field arising due to the electrical properties of the interior fluid, exterior fluid, and the membrane. The fluid structure interaction is facilitated through the immersed boundary method (IBM), which is coupled to the IIM through least squares interpolation between the control points of the IIM and the Lagrangian nodes of the IBM. The method is validated by comparing the numerical results to analytical solutions and previously published studies. The method is then used to study the deformation of a capsule in a combined shear flow and DC electric field for various membrane conductances, membrane capacitances, and conductivity ratios. For nonconducting membranes the interaction between the distribution of the electric forces and the capsule inclination angle due to the shear flow result in complex equilibrium dynamics not reported for neutral capsules.

One sometimes wants to run lattice Boltzmann simulations with more precision than is available in commonly supported floating point arithmetic. This is particularly true for confirming qualitative properties such as conservation laws. It is also true for GPUs where one may be restricted to 32b-bit floating point numbers. This talk presents a rescaling of the Skordos (1993) variable transformation to formulate a lattice Boltzmann scheme that can be implemented using integer arithmetic. In particular, 128-bit integer arithmetic is efficient on common hardware such as x86-64. Standard lattice Boltzmann schemes require division, while variants that target the Chorin artificial compressibility equations do not. The extra division comes at significant computational cost, even in floating point arithmetic, and is implemented using Newton's method in integer arithmetic.

Sharp and diffuse interface models are usually considered to be two fundamentally different approaches to the modeling of multiphase flows. In this contribution we discuss a conservative lattice Boltzmann phase field model as a phase separator and a momentum balance boundary condition forming a sharp interface. In this approach, the interaction between the two phases is limited to the nodes directly adjacent to the interface. This allows us to directly apply the super convergent cumulant lattice Boltzmann method in both domains and benefit from its excellent stability properties.

A convergence study of LB diffusion-interface method

The simulation of fluid flow around wind turbines is key to improve the performance and reliability of wind turbines in varying wind conditions. To quantify the impact of complex wind patterns on wind turbine components, high-fidelity simulations are essential. Thus, we investigate a method for simulating wind dynamics induced by a three-blade turbine with the Lattice Boltzmann Method (LBM) implemented for GPUs. Our approach uses a rotating overset lattice, where the wind turbine rotor is embedded in a lattice that rotates synchronously with the rotor. The rotating lattice is coupled with a static lattice that represents the surrounding fluid domain. An efficient and accurate interpolation technique for the coupling of the lattices is pivotal for this approach. We use a compact quadratic interpolation technique tailored to the cumulant LBM. The interpolation process involves rotating the distributions by manipulating their moments, while also accounting for Coriolis and centrifugal forces on the rotating mesh. In addition, the compact quadratic interpolation is optimized for massively parallel execution on a graphics processing unit (GPU) to reduce simulation times. This improves the scalability of our approach and provides a promising solution for advancing high-fidelity simulations of rotating objects.

The modelling of fluid-filled vesicles \textemdash such as red blood cells (RBCs) \textemdash has obvious importance in the field of biomedical science. Given a healthy human RBC has dimensions of approximately 8$\mu$m in length and 2$\mu$m in width, the modelling of RBCs often utilises mesoscale modelling approaches such as single-component lattice Boltzmann method (SCLBM).

Presented here is an alternative approach to simulating vesicles in three-dimensions using chromodynamic multi-component lattice Boltzmann method (cMCLBM).

By capitalising on the natural disparity in length scales—typified by the RBC membrane's width of approximately 40nm—and operating within the continuum approximation of fluids, the RBC's interior is conceptualised as one immiscible fluid, its exterior as another, and the quasi-two-dimensional interface between them as the \textit{de-facto} membrane. The primary advantage of this approach lies in its unified methodology, requiring only the cMCLBM framework to model both fluids and the deformable body. This presentation outlines the model, showing: (i) the foundational methodology, (ii) two-component simulations (a single vesicle), and (iii) the expansion of the methodology to simulate multiple vesicles, with attention being drawn to the advantages of such an approach.

In the last decades, Micro-channels and micro-cavities becomes a subject of interest of many researchers due to their higher efficiency. The contemporary study aims to numerically analyze the convective heat transfer and entropy generation analysis for the case of a micro-channel filled with Cu/water under the effect of inclination angle in the slip flow regime. a series of numerical simulations based on the Lattice Boltzmann method are carried out in a micro medium filled with nanoliquid. To resolve the governing equations, a special treatment of boundary conditions precisely at the side walls is done. The obtained results are presented in terms of heat transfer rate and surface distribution of entropy generation.

Estimation of permeability of porous media dates back to Henry Darcy (1856). The literature data on permeability are scattered, and this scatter not always can be attributed to the accuracy of experiment or simulation or to the sample variability. In this work, we prepare porous samples and determine their permeability experimentally. Hereafter, the samples are scanned in 3D using X-ray computed tomography, and the images are used for Stokes flow simulations using the lattice Boltzmann method with the simple bounce-back boundary condition. Careful execution of all steps in this combined experiment-simulation study resulted in an excellent agreement (<1%) between laboratory results and simulations. Here, experimental results are extensive and stable, while flow is simulated directly on three-dimensional images and without fitting parameters. Analyzing the conditions when experiments and simulations agree reveals a flaw affecting many experimental measurements with the out-of-sample placement of pressure ports, including industry standards. The flaw originates from (1) incorrect calculation of the applied pressure gradient, (2) omitting virtual part of the measured system, and (3) pressure loss at the sample–tube contact. Contrary to common wisdom, the relative magnitude of (3) is defined by the sample–tube diameter ratio and is independent of the size of sample pores. Removing or taking the flaw into account advances the understanding and control of flow-related processes in complex geometries.

CO2 sequestration in porous media: experimental data and LB simulations

This study investigates conjugate natural convection in a 2D enclosure with multiple solid blocks, incorporating volumetric thermal radiation. The enclosure features a sinusoidal temperature profile on one side and cooling on the other, with N periodically arranged, thermally conductive blocks. The enclosure has a sinusoidal temperature profile on one side and cooling on the other, with N thermally conductive blocks. The non-orthogonal multi-relaxation time lattice Boltzmann method (NMRT-LBM) is used, employing D2Q9 and D2Q5 schemes for velocity and temperature, respectively. Radiative transfer is modeled using a D2Q8 BGK LBM scheme. The influence of various parameters on heat transfer is explored, including thermal properties, radiation effects (Planck number, emissivity, extinction coefficient), and geometrical factors (solid-to-fluid volume fraction, number of blocks). The accuracy of our model is validated against benchmark problems. Results show that solid fraction and block number significantly impact heat transfer. Wall emissivity enhances heat transfer at low Planck numbers. Conversely, reducing solid fraction, Planck number, and extinction coefficient all improve heat transfer. Interestingly, subdividing the initial block can reduce heat transfer, especially for a large number of blocks and low Planck numbers. These findings highlight the potential of NMRT-LBM as a valuable tool for simulating complex heat transfer phenomena involving thermal radiation, with applications in building design optimization for energy efficiency.

This study introduces a numerical tool that utilizes the non-orthogonal multi-relaxation time lattice Boltzmann method (MRT-LBM) to simulate the coupled thermal and mass diffusion occurring within porous media exposed to naturally convecting moist air flow.The D2Q9 velocity distribution function and separate D2Q5 schemes for temperature and concentration capture double-diffusive convection. The validity of model is established through comparison with a differentially heated porous cavity under varying conditions. Our numerical investigations delve into the influence of key parameters—porosity (ε), Lewis number (Le), Rayleigh number (Ra), buoyancy ratio (Br), and Darcy number (Da)—on flow, temperature, and solute distributions. The resulting isotherms, iso-concentrations, streamlines, Nusselt numbers, and Sherwood numbers reveal an important impact on flow structures and thermal-mass transport mechanisms. Notably, an increased Darcy number enhances both heat and mass transfer. This validated model provides a comprehensive understanding of porous media behavior, potentially informing future building comfort strategies related to thermal management and species transport.

To improve thermal efficiency of such concentrating solar power system (CSP), thermal heat losses from the receiver must be minimized. In this manuscript, convective and radiative heat losses from solar cavity receiver are numerically assessed by using Lattice Boltzmann Method (LBM). Combined natural convection-surface radiation heat transfer mode in open rectangular solar cavity receiver is presented. The two parallel walls are insulated while the wall facing the opening is subjected to a constant temperature with parabolic profile. The open boundary is assumed to be a black surface at ambient temperature while the other walls are diffuse, gray and opaque. LBM-BGK model with double distribution functions (D2Q9-D2Q4) is adopted here to predict dynamic and thermal fields. Effects of heating temperature, inclination angle, radiative proprieties and geometric aspect ratio on heat losses inside the cavity are analyzed and discussed.

It was found that an increasing of the inclination angle induces a large increasing on convective heat loss. Also, by increasing the heating temperature from 200°C to 600°C an amplification of 88% is observed for the total heat loss inside the cavity. On the other hand, the doubling of the aspect ratio of the cavity induces a reduction in thermal losses of about 10%.

“Cardiovascular disease is the world’s biggest killer, as well as the second biggest cause of death in the UK with over three million people suffering from atherosclerotic cardiovascular disease” (UK Government, 2020). Atherosclerosis, the narrowing of arteries due to a build-up of plaque, can be treated through medical interventions such as balloon angioplasty and stent implantation to dilate the stenosed vessel and to restore and maintain physiological blood flow. However, these procedures can result in adverse complications like stent thrombosis and in-stent restenosis. Factors such as stent morphology, stent spacing and stent implantation position alter the local haemodynamic environment, which is known to regulate stent thrombosis. The Lattice Boltzmann method is used to numerically model two-dimensional blood flow around a stent strut to explore how stent morphology affects the local haemodynamics. This method is a niche CFD approach with possible addition of mesoscopic physics, examination of real stent profiles and the coupling of an advection-diffusion equation to obtain relative residence time (RRT) values. Three different stent strut profiles (circular, square and trapezium) from the market are analysed in this work. As these struts had different strut heights, additional simulations that studied the same profiles but with equal strut height were required to highlight the effects of strut profile on local haemodynamics. Metrics of haemodynamics including wall shear stress (WSS), shear rate, RRT and re-circulation region length post strut were measured for each profile. These metrics were then used to assess the thrombogenicity of each stent strut design. The results showed that the circular profile was the least thrombotic profile, with the smallest: (i) max WSS, (ii) max shear rate, (iii) max RRT and (iv) re-circulation region post strut of the three struts. The trapezium shaped profile was the most thrombotic, with the largest of the previously mentioned metrics. Therefore, the simulation results from different stent profiles with equal strut height show that the more streamlined a stent strut is, the less likely stent thrombosis is to occur.

References

UK Government. (2020). UK Government tackles heart disease with new partnership [Press release]. Retrieved from https://www.gov.uk/government/news/uk-government-tackles-heart-disease-with-new-partnership#:~:text=Cardiovascular%20disease%20is%20the%20world%27s

,suffering%20from%20atherosclerotic%20cardiovascular%20disease

This study introduces a predictive hydrodynamic model to investigate the distribution characteristics of synthetic glass particles injected from a time-dependent source in a turbulent free surface flow, while also examining the impact of pulsation on the spatial and temporal distributions of particles. The simulations are conducted using a Computational Fluid Dynamics (CFD) code based on the finite volume approach. The Discrete Phase Model (DPM) is selected to capture the movement of particles. The k-ε turbulence closure model is employed to simulate turbulence generation, and the Volume of Fluid (VOF) method is used to accurately capture the time-varying free surface. The trajectory and deposition of particles are analyzed in detail.

The numerical results demonstrate that the pulsation plays a predominant role at the early stages of glass particles distribution, and that particles transportation can be enhanced due to the synchronization of particle movement with the oscillating potential. It was also observed that the pulsation affects the distribution of the injected material, particularly near the front, and that a significant swirling action is developed compared to constant-rate-injection case.

The findings presented in this study can guide watershed managers in implementing effective and cost-efficient conservation measures; by providing insights into the dynamics of water flow and pollutant dispersion, these results can improve the design of water treatment facilities, optimize sediment transport systems, and enhance pollutant dispersion models.

Realistic pore-scale simulations of flow through porous media frequently use discrete images (pixels in 2D or voxels in 3D) of real-life samples as inputs. Today's commonly held belief is that high-accuracy simulations require high-resolution images, which often result in lengthy scanning and/or simulation times. Yet, decreasing the resolution destroys the simulation accuracy when the features of the sample (e.g., pores) are unresolved. Here, we report the discovery of superstructures in discrete images, which emerge from the sample's features and discrete mesh. These superstructures - and not the original features of the sample - control flow in low-resolution simulations. Consequently, decreases in resolution change the topology (flow “pathways”) and morphology (pore “shapes”) of the discrete image of the sample. Using permeability as an example, we present a new methodology to significantly improve the flow simulation accuracy for both low resolution CT-imaged and computer-generated samples. This methodology is based on the novel concept of “null point”, P0, and voxel-based resolution parameter, \chi. The presented methodology improves extraction of quantitative information from discrete images. Our findings are not limited by image dimensionality, imaging technique, or simulated processes.

Efficiently accounting for uncertainties in computational fluid dynamics (CFD) models remains a crucial challenge. For computing statistical solutions of partial differential equations in fluid flow models, it is promising to combine scalable deterministic solvers based on lattice Boltzmann methods (LBMs), such as OpenLB, with uncertainty quantification (UQ) techniques. By sampling uncertain parameters and conducting many simulations, the uncertainty can be accurately quantified using statistical analysis. However, non-intrusive Monte Carlo (MC) methods are often computationally expensive when applied to CFD problems. In this talk, we propose to employ a generalized polynomial chaos (gPC) expansion method within a stochastic Galerkin (SG) framework on LBM. Our novel SG LBM offers a more efficient alternative to MC for estimating uncertainty in LBM simulations of incompressible fluid flows. Numerical results of a Taylor--Green vortex flow problem validate that the SG LBM achieves comparable accuracy to the MC LBM, while significantly reducing the computational costs. By combining the strengths of both the LBM and gPC-based SG methods, we thus provide a robust and intrinsically efficient framework for UQ in CFD simulations.

We propose an adaptation of regularized schemes initially introduced by Latt and Chopard to the MRT lattice Boltzmann scheme. We treat the D2Q9 case and establish the associated asymptotic partial differential equations. We present first numerical test cases.