Fig 1

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The accuracy of the Moment Method for imposing no-slip boundary conditions in the lattice Boltzmann algorithm is investigated numerically using lid-driven cavity flow. Boundary conditions are imposed directly upon the hydrodynamic moments of the lattice Boltzmann equations, rather than the distribution functions, to ensure the constraints are satis...

## Contexts in source publication

**Context 1**

... are several popular alternatives to bounce-back. Maxwell-Broadwell conditions [12,13] have been applied to the LBM for rarefied flow and flow in the slip regime by Ansumali and Karlin [14], but no explicit condition is placed on the tangential velocity at the boundary and the artificial slip remains (at least for the popular D2Q9 lattice, see Fig. 1). Zou and He's non-equilibrium bounce back [15] does not have numerical slip and allows the user to specify the no- slip boundary condition but it uses a rather ad-hoc closure in its derivation and questions remain about its stability and accuracy for complex flows. All of these implementations are formulated in terms of the lattice ...

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## Citations

... Recently, the moment-based boundary condition has received increased interest and attention [40][41][42][43][44][45], which is based on the moments of the LBM [40] and has not yet been introduced to the DUGKS. Numerical simulations show that the moment-based boundary condition converges with second-order accuracy using dipole-wall collisions [40], natural convection in the square cavity [41], and lid-driven cavity flow [43][44][45]. ...

... Recently, the moment-based boundary condition has received increased interest and attention [40][41][42][43][44][45], which is based on the moments of the LBM [40] and has not yet been introduced to the DUGKS. Numerical simulations show that the moment-based boundary condition converges with second-order accuracy using dipole-wall collisions [40], natural convection in the square cavity [41], and lid-driven cavity flow [43][44][45]. A numerical simulation of the Poiseuille flow shows that the moment-based boundary condition can eliminate the spurious oscillations seen in solutions using other boundary conditions, considering the nonzero deviatoric stress [42]. ...

The boundary conditions are crucial for numerical methods. This study aims to contribute to this growing area of research by exploring boundary conditions for the discrete unified gas kinetic scheme (DUGKS). The importance and originality of this study are that it assesses and validates the novel schemes of the bounce back (BB), non-equilibrium bounce back (NEBB), and Moment-based boundary conditions for the DUGKS, which translate boundary conditions into constraints on the transformed distribution functions at a half time step based on the moment constraints. A theoretical assessment shows that both present NEBB and Moment-based schemes for the DUGKS can implement a no-slip condition at the wall boundary without slip error. The present schemes are validated by numerical simulations of Couette flow, Poiseuille flow, Lid-driven cavity flow, dipole–wall collision, and Rayleigh–Taylor instability. The present schemes of second-order accuracy are more accurate than the original schemes. Both present NEBB and Moment-based schemes are more accurate than the present BB scheme in most cases and have higher computational efficiency than the present BB scheme in the simulation of Couette flow at high Re. The present Moment-based scheme is more accurate than the present BB, NEBB schemes, and reference schemes in the simulation of Poiseuille flow and dipole–wall collision, compared to the analytical solution and reference data. Good agreement with reference data in the numerical simulation of Rayleigh–Taylor instability shows that they are also of use to the multiphase flow. The present Moment-based scheme is more competitive in boundary conditions for the DUGKS.

... It has been shown that the LBM-MRT is superior to the LBM-SRT model in simulating higher Reynolds number flows (Perumal and Dass, 2015;Mohammed and Reis, 2017). E. Aslan et al. (2014) have studied the SRT and MRT stability for the lid-driven cavity flow in 2D. ...

The Lattice Boltzmann method (LBM) is a relatively new development in computational fluid dynamics (CFD). Instead of approximating the Navier-Stokes equations, the approach is based on solving a simplified version of the Boltzmann equation on a specific discrete space. It can be shown via a Chapman Enskog expansion that for vanishing Knudsen number the LBM recovers the Navier-Stokes equations. The standard LBM is limited to equally spaced Cartesian grids, making this approach notably expensive for capturing thin boundary layers, and therefore impractical for most relevant problems in aerodynamics. In this work, conventional numerical methods are implemented to solve the two-dimensional discrete-velocity Boltzmann equation in generalised curvilinear coordinates to simulate fluid flows with non-Cartesian grids. Several test cases are used for verification and validation, and the results have been extensively compared with the available numerical and experimental literature with very favourable outcome confirming the feasibility of the implemented methodology. The generalised LBM resolves large gradients in the wall vicinity with fewer mesh elements than the Cartesian LBM; therefore, it substantially reduces the computational effort. Flows over the 2D circular cylinder and the NACA0012 aerofoil are specifically investigated to assess the accuracy and performance of this approach for external aerodynamics. Moreover, the generalised LBM has been implemented into AMROC (Adaptive Mesh Refinement in Object-oriented C++) software infrastructure, a fully parallelised finite-volume framework that uses a block-structured adaptive mesh refinement. This development is a new class of LBM scheme with block-structured adaptive mesh refinement on curvilinear grids. The latest is implemented with the mapped mesh strategy, and various test cases are solved for validation, including heat transfer flows. Overall, this project enhanced the current capabilities of the standard LBM by combining the generalised LBM with AMROC capabilities.

... Verschaeve (2009) proposes a model that although not being a moments-based method, add to the solution the relation between second-order moments and the strain rate tensor and the mass conservation condition for incompressible flows in its differential form, i.e., ∇ ⋅ = 0. Malaspinas, Chopard and Latt (2011) set a system of equations based on density, first and second-order moments, using a minimization procedure to solve the overdetermined system of equations for the moments of the particle distribution function. In the model of Mohammed and Reis (2017), they carefully pick only linearly independent equations for the hydrodynamic moments and solve these equations for the unknown distributions that are going back to the fluid portion of the domain. ...

... In this work we perform a comparative analysis between moments-based models: Latt et al. (2008), Malaspinas et al. (2011), Mohammed and Reis (2017) and Hegele Jr et al. (2018). These models were chosen for this comparative analysis because, although they are all moments-based models, they rely on different methods for solving the resulting system of equations for the boundary conditions. ...

... Some works have been published on regularized or moment-based boundary conditions in recent years. For comparison purposes, we selected the works of Latt et al. (2008), Malaspinas et al. (2011) and Mohammed and Reis (2017) ...

Dealing with boundary conditions (BC) was ever considered a puzzling question in the Lattice–Boltzmann (LB) method. The most popular BC models are based on Ad-Hoc rules and, although these BC models were shown to be suitable for low-order LB equations, their extension to high-order LB was shown to be a very difficult problem and, at authors knowledge, never solved with satisfaction. The main question to be solved is how to deal with a problem when the number of unknowns (the particle populations coming from the outside part of the numerical domain) is greater than the number of equations at our disposal at each boundary site. Recently, BC models based on the regularization of the LB equation, or moments-based models, were proposed. These moments replace the discrete populations as unknowns, independently of the number of discrete velocities that are needed for solving a given problem. The full set of moments-based BC leads, nevertheless, to an overdetermined system of equations, and what distinguishes one model from another is the way this system is solved. In contrast with previous work, we base our approach on second-order moments. Four versions of this model are compared with previous moments-based models considering, in addition to the accuracy, some main model attributes such as global and local mass conservation, rates of convergence, and stability. For this purpose, the complex flow patterns displayed in a two-dimensional lid-driven cavity are investigated.

... As observed in Fig. 3a, the horizontal velocity along the vertical height at the center line of the lid-driven cavity is used for comparison. Air at Re = 1000 is used as working fluid inside the cavity and the results obtained with ANSYS FLUENT are compared against the results of Ghia et al. [36] as well as with Mohammed and Reis [37]. From Fig. 3a, the obtained results are in good match with the results of Ghia et al. [36] and that of with Mohammed and Reis [37]. ...

... Air at Re = 1000 is used as working fluid inside the cavity and the results obtained with ANSYS FLUENT are compared against the results of Ghia et al. [36] as well as with Mohammed and Reis [37]. From Fig. 3a, the obtained results are in good match with the results of Ghia et al. [36] and that of with Mohammed and Reis [37]. ...

Two-dimensional conjugate heat transfer performance of stepped lid-driven cavity was numerically investigated in the present study under forced and mixed convection in laminar regime. Pure water and Aluminium oxide (Al2O3)/water nanofluid with three different nanoparticle volume concentrations were considered. All the numerical simulations were performed in ANSYS FLUENT using homogeneous heat transfer model for Reynolds number, Re = 100 to 500 and Grashof number, Gr = 5000, 13,000 and 20,000. Effective thermal conductivity of the Al2O3/water nanofluid was evaluated by considering the Brownian motion of nanoparticles which results in 20.56% higher value for 3 vol.% Al2O3/water nanofluid in comparison with the lowest thermal conductivity value obtained in the present study. A solid region made up of silicon is present underneath the fluid region of the cavity in three geometrical configurations (forward step, backward step and no step) which results in conjugate heat transfer. For higher Re values (Re = 500), no much difference in the average Nusselt number (Nuavg) is observed between forced and mixed convection. Whereas, for Re = 100 and Gr = 20,000, Nuavg value of mixed convection is 24% higher than that of forced convection. Out of all the three configurations, at Re = 100, forward step with mixed convection results in higher heat transfer performance as the obtained interface temperature is lower than all other cases. Moreover, at Re = 500, 3 vol.% Al2O3/water nanofluid enhances the heat transfer performance by 23.63% in comparison with pure water for mixed convection with Gr = 20,000 in forward step.

... Previous applications of the moment method determine the tangential momentum flux at a boundary by imposing ∂ x u x = 0. Moreover, it was always implicitly assumed that the deviatoric stress, T, is given by Newtonian constitutive equation [32][33][34][35][37][38][39]. A closer inspection shows that the stress obtained from the D2Q9 discrete velocity Boltzmann equation includes a non-zero Burnett contribution at second-order in relaxation time that reassembles a non-objective viscoelastic constitutive equation [12]. ...

... Gorban and Packwood [42] and Brownlee et al. [43] used the classical lid-driven cavity flow to study the stability of lattice Boltzmann algorithms and concluded that multiple relaxation time models and filtering techniques are superior to BGK and entropic collision models. Mohammed and Reis [39] used the cavity flow to assess the moment-based method with simple boundary conditions and although they reported very accurate results in agreement with the most sophisticated numerical methods when using an MRT collision operator they also found that the method becomes unstable at higher Reynolds numbers on coarse meshes when used with the BGK collision operator (as is this case with other on-node methods -they are less stable than bounce back when used with BGK). The classic lid-driven cavity flow has stress singularities in the upper corners where the moving horizontal lid meets stationary vertical walls. ...

... The classic lid-driven cavity flow has stress singularities in the upper corners where the moving horizontal lid meets stationary vertical walls. In was suggested in [39] that this may be the cause of instabilities of the moment-based boundary conditions at high Re on coarse meshes. To circumvent the singularity we simulate here the so-called "regularized" cavity [44], where the velocity of the lid (north horizontal boundary) in box 0 ≤ x, y ≤ 1 is a smooth function of the horizontal coordinate x: The velocity condition (1.1) influences the stress field since it has non-vanishing gradients. ...

Stress boundary conditions for the lattice Boltzmann equation that are consistent to Burnett order are proposed and imposed using a moment-based method. The accuracy of the method with complicated spatially-dependent boundary conditions for stress and velocity is investigated using the regularized lid-driven cavity flow. The complete set of boundary conditions, which involve gradients evaluated at the boundaries, are implemented locally. A recently-derived collision operator with modified equilibria and velocity-dependent collision rates to reduce the defect in Galilean invariance is also investigated. Numerical results are in excellent agreement with existing benchmark data and exhibit second-order convergence. The lattice Boltzmann stress field is studied and shown to depart significantly from the Newtonian viscous stress when the ratio of Mach to Reynolds numbers is not negligibly small.

... The lattice Boltzmann method with moment-based based boundary conditions has been successfully applied to various physical systems, where exact hydrodynamic boundary conditions must be employed [39][40][41][42] . Furthermore, its stability and accuracy has also been commented upon briefly 43 , and some theoretical analysis has been performed 37 , but there is still room for an indepth stability analysis. Studying stability of the present 3D method is left for future research. ...

... It is noted that the LSOB method 28 solves the rectangular matrix formulation and seems to avoid any symmetry breaking. Some preliminary stability analysis for the moment-based boundary method in 2D has been conducted 37,43 , however, an in-depth stability analysis of the current 3D method requires further studies and is a topic for future. At present, a shortcoming of the moment-based method for boundary constraints is its limitation to boundaries aligned with grid points. ...

In this paper, moment‐based boundary conditions for the lattice Boltzmann method are extended to three dimensions. Boundary conditions for velocity and pressure are explicitly derived for straight on‐grid boundaries for the D3Q19 lattice. The method is compared against the bounce‐back scheme using both single and two relaxation time collision schemes. The method is verified using classical benchmark test cases. The results show very good agreement with the data found in the literature. It is confirmed from the results that the derived moment‐based boundary scheme is of second order accuracy in grid spacing and does not produce numerical slip, and therefore offers a transparent way of accurately prescribing velocity and pressure boundaries that are aligned with grid points in 3D.

... If we take the first three moments of equation (1), apply a Chapman-Enskog expansion, and consider only terms up to leading order in relaxation times, we can show that embedded within the discrete velocity Boltzmann equation are the weakly compressible Navier-Stokes equations 32,35,42 ...

... That is because at higher Reynolds numbers, the dissipation of the small secondary vortex is slower. Moreover, increasing the slip length reduces the rolling-mill effect (i.e. the continual generation of new dipoles) observed for no-slip conditions 35 especially for high Reynolds numbers since the space between the two primary monopoles is increased. ...

We study the physics of flow due to the interaction between a viscous dipole and boundaries that permit slip. This includes partial and free slip, and interactions near corners. The problem is investigated by using a two relaxation time lattice Boltzmann equation with moment-based boundary conditions. Navier-slip conditions, which involve gradients of the velocity, are formulated and applied locally. The implementation of free-slip conditions with the moment-based approach is discussed. Collision angles of 0°, 30°, and 45° are investigated. Stable simulations are shown for Reynolds numbers between 625 and 10 000 and various slip lengths. Vorticity generation on the wall is shown to be affected by slip length, angle of incidence, and Reynolds number. An increase in wall slippage causes a reduction in the number of higher-order dipoles created. This leads to a decrease in the magnitude of the enstrophy peaks and reduces the dissipation of energy. The dissipation of the energy and its relation to the enstrophy are also investigated theoretically, confirming quantitatively how the presence of slip modifies this relation.

... The momentbased method is similar in spirit to Noble et al.'s approach but is far more general in terms of the types of boundary conditions that can be implemented and the lattice stencils they can be applied to. Moment based boundary conditions have been applied to several flows and problems already [21,44,[46][47][48] . However, a detailed study of moment-based boundary conditions for no-slip flows has not been conducted. ...

The accuracy of moment-based boundary conditions for no slip walls in lattice Boltzmann simulations is examined numerically by using the dipole-wall collision benchmark test for both normal and oblique cases. In the normal case the dipole hits the wall perpendicularly while in the oblique case the dipole hits the wall at an angle of 30° to the horizontal. Boundary conditions are specified precisely at grid points by imposing constraints upon hydrodynamic moments only. These constraints are then translated into conditions for the unknown lattice Boltzmann distribution functions at boundaries. The two relaxation time (TRT) model is used with a judiciously chosen product of the two relaxation times. Stable results are achieved for higher Reynolds number up to 10,000 for the normal collision and up to 7500 for the oblique case. Excellent agreement with benchmark data is observed and the local boundary condition implementation is shown to be second order accurate.

... Moment-based conditions eliminate the viscosity-dependent error associated with bounce-back and allows the user to impose a variety of hydrodynamic constraints at grid points. This method has already been applied to several flows such as rarefied flow with first order Navier-Maxwell slip boundary conditions [11], and a diffusive slip [3] and no-slip such as [9] and wetting conditions for multiphase flow (Hantsch and Reis) [6], and adiabatic and heat source conditions (Allen and Reis) [1]. In all cases, second order accuracy has been confirmed numerically. ...

A Lattice Boltzmann Method (LBM) with moment based boundary conditions is used to numerically simulate the two-dimensional flow between parallel plates, driven by a pulsating pressure gradient. The flow is simulated by using a single relaxation time model under both non-slip and Navier-slip boundary conditions. Convergence is investigated by using two distinct approaches. The first approach uses acoustic scaling in which we fix Mach, Reynolds and Womersley numbers whilst varying the LBM relaxation time. Diffusive scaling is used in the second approach-here we fix the Reynolds and Womersley numbers and the relaxation time whilst the Mach number decreases with increasing grid size. For no-slip conditions using acoustic scaling, the numerical method converges, but not always to the appropriate analytical result. However, the diffusive scaling approach performs as expected in this case, showing second-order convergence to the correct analytical result. Convergence to the analytical solution (though not always second-order) is also observed for the simulations with Navier-slip using diffusive scaling.

We revisit force evaluation methodologies on rigid solid particles suspended in a viscous fluid that is simulated via the lattice Boltzmann method (LBM). We point out the noncommutativity of streaming and collision operators in the force evaluation procedure due to the presence of a solid boundary, and provide a theoretical explanation for this observation. Based on this analysis, we propose a discrete force calculation scheme with enhanced accuracy. The proposed scheme is essentially a discrete version of the Reynolds transport theorem (RTT) in the context of a lattice Boltzmann formulation. Besides maintaining satisfactory levels of reliability and accuracy, the method also handles force evaluation on complex geometries in a simple and transparent way. We run benchmark simulations for flow past cylinder and NACA0012 airfoil (for Reynolds numbers ranging from 102 to 0.5×106) and show that the current approach significantly reduces the grid size requirement for accurate force evaluation.