Multiple Relaxation Time Lattice Boltzmann Research Articles

Numerical simulation of combined effects of a vertical magnetic field and thermal radiation on natural convection of non-Newtonian fluids confined between circular and square concentric cylinders

BackgroundThis study addresses magnetohydrodynamic (MHD) natural convection involving non-Newtonian fluids. Focusing on the annular region between concentric circular and square cylinders, the research examines the impact of a vertical magnetic field and thermal radiation. MethodsThe Multiple Relaxation Time Lattice Boltzmann Method (MRT-LBM) is employed to simulate momentum and energy interactions in non-Newtonian fluid. Control parameters like Rayleigh number, Hartmann number, Radiation parameter, aspect ratio, square cylinder subdivision, and power-law index are systematically varied, while maintaining a constant Prandtl number. Significant findingsUsing shear-thinning fluid (n < 1) significantly improves cooling, particularly in conjunction with radiation effect. Correlations for the mean Nusselt number are established, offering insights. Results emphasize the substantial influence of parameters like Ra, Rd, and AR. High values of the latter promote the flow intensity and heat evacuation. For n < 1, increasing Ra from 103 to 5 × 104 results in a substantial increase in the heat transfer rate reaching 65 %. For shear-thickening fluid, the heat transfer rate is reduced by 25.31 % at Ra = 5 × 104 when Ha is increased from 0 to 50 in the absence of radiation. In addition, radiation enhances the heat transfer rate by 71.41 % by incrementing Rd from 0 to 0.8 for Ha = 0. The most pronounced effect of the radiation parameter is observed in the cases of Newtonian and shear-thickening fluids. Conversely, Ha and n act by reduction effects on the outputs of the problem. The attenuating effect of the magnetic field is more pronounced in the case of shear-thinning fluids. The plot of mean Nusselt number vs. AR shows an hysteresis phenomenon due to the existence of multiple steady solutions in a range of this parameter. In the latter range of AR, heat transfer induced by multicellular flow is higher than that induced by bicellular one. Heat transfer is somewhat enhanced by the RC configuration, compared to SC one.

A phase-field-based multiple-relaxation-time lattice Boltzmann method for incompressible multiphase flows with density and viscosity contrasts

In this work, we present a unified framework of phase-field-based multiple-relaxation-time lattice Boltzmann (MRT-LB) method for incompressible multiphase flows with density and viscosity contrasts where a block-lower-triangular relaxation matrix is introduced. The present framework includes the classic MRT-LB model and central-moments-based LB model (CLBM, a particular version of the MRT-LB model). The governing equations of incompressible multiphase flows can be reproduced precisely through the direct Taylor expansion method at the second-order of expansion parameters. In this work, the pressure p is re-estimated, and we prove that the present pressure expression is suitable for classic MRT model, CLBM and various lattice structures with an accuracy of O(Δt2+ΔtMa2). In addition, the model construction and analysis are all carried out in the velocity space. As a result, the present model has a similar structure to the traditional Bhatnagar-Gross-Krook (BGK) model. Therefore the process of their analysis and implementation is also similar. We then test the present model by five classic physical problems. The results show that the present model is found to be accurate and has better numerical stability over the BGK model, and has the ability to simulate multiphase flow problems with large density ratio (1000).

Rectangular multiple-relaxation-time lattice Boltzmann method for the Navier-Stokes and nonlinear convection-diffusion equations: General equilibrium and some important issues.

In this paper, we develop a general rectangular multiple-relaxation-time lattice Boltzmann (RMRT-LB) method for the Navier-Stokes equations(NSEs) and nonlinear convection-diffusion equation(NCDE) by extending our recent unified framework of the multiple-relaxation-time lattice Boltzmann (MRT-LB) method [Chai and Shi, Phys. Rev. E 102, 023306 (2020)10.1103/PhysRevE.102.023306], where an equilibrium distribution function (EDF) [Lu et al., Philos. Trans. R. Soc. A 369, 2311 (2011)10.1098/rsta.2011.0022] on a rectangular lattice is utilized. The anisotropy of the lattice tensor on a rectangular lattice leads to anisotropy of the third-order moment of the EDF, which is inconsistent with the isotropy of the viscous stress tensor of the NSEs. To eliminate this inconsistency, we extend the relaxation matrix related to the dynamic and bulk viscosities. As a result, the macroscopic NSEs can be recovered from the RMRT-LB method through the direct Taylor expansion method. Whereas the rectangular lattice does not lead to the change of the zero-, first- and second-order moments of the EDF, the unified framework of the MRT-LB method can be directly applied to the NCDE. It should be noted that the RMRT-LB model for NSEs can be derived on the rDdQq (q discrete velocities in d-dimensional space, d≥1) lattice, including rD2Q9, rD3Q19, and rD3Q27 lattices, while there are no rectangular D3Q13 and D3Q15 lattices within this framework of the RMRT-LB method. Thanks to the block-lower triangular relaxation matrix introduced in the unified framework, the RMRT-LB versions (if existing) of the previous MRT-LB models can be obtained, including those based on raw (natural) moment, central moment, Hermite moment, and central Hermite moment. It is also found that when the parameter c_{s} is an adjustable parameter in the standard or rectangular lattice, the present RMRT-LB method becomes a kind of MRT-LB method for the NSEs and NCDE, and the commonly used MRT-LB models on the DdQq lattice are only its special cases. We also perform some numerical simulations, and the results show that the present RMRT-LB method can give accurate results and also have a good numerical stability.

Fourth-order multiple-relaxation-time lattice Boltzmann model and equivalent finite-difference scheme for one-dimensional convection-diffusion equations.

In this paper, we first develop a fourth-order multiple-relaxation-time lattice Boltzmann (MRT-LB) model for the one-dimensional convection-diffusion equation(CDE) with the constant velocity and diffusion coefficient, where the D1Q3 (three discrete velocities in one-dimensional space) lattice structure is used. We also perform the Chapman-Enskog analysis to recover the CDE from the MRT-LB model. Then an explicit four-level finite-difference (FLFD) scheme is derived from the developed MRT-LB model for the CDE. Through the Taylor expansion, the truncation error of the FLFD scheme is obtained, and at the diffusive scaling, the FLFD scheme can achieve the fourth-order accuracy in space. After that, we present a stability analysis and derive the same stability condition for the MRT-LB model and FLFD scheme. Finally, we perform some numerical experiments to test the MRT-LB model and FLFD scheme, and the numerical results show that they have a fourth-order convergence rate in space, which is consistent with our theoretical analysis.

Accuracy improvement of immersed boundary-lattice Boltzmann and finite element method by iterative velocity correction

In an analysis of the fluid–structure interaction (FSI) problem, the non-slip boundary condition at solid walls cannot be accurately satisfied by the conventional immersed boundary-lattice Boltzmann coupling schemes due to insufficient interpolation accuracy. To solve this problem, an improved iterative velocity correction procedure for the immersed boundary-lattice Boltzmann coupling scheme is proposed by introducing a modified velocity operator. The particle distribution function was modified at each time step, and the evolution governing equation of the multiple relaxation time-lattice Boltzmann method was performed. A numerical framework for coupling lattice Boltzmann and finite element methods for transient problems involving FSI was established, and the iterative velocity correction immersed boundary method was used for the partitioned approach. The solid structure was discretized with the finite element method, while the single-component fluid flows were simulated with the lattice Boltzmann method. An FSI benchmark model was employed to verify the efficiency of the proposed coupling method. The results show that the developed method guarantees the non-slip boundary condition and maintains the convergence rate of the conventional immersed boundary method. In viscous flow and strong shearing flow, the accuracy of both stationary and moving solid boundaries is obviously improved.

Numerical Investigation of Convective Heat Transfer and Fluid Flow Past a Three Square Cylinders Controlled by a Partition in Channel

This document presents a research article on the control of fluid flow around three heated square cylinders placed side by side in a 2D horizontal channel using a flat plate. The objective of this research is to examine the effect of the position, length and height of a flat plate on fluid flow and heat transfer. For this purpose, numerical simulations are performed by using the Boltzmann double relaxation time multiple network method (DMRT-LBM). The MRT-D2Q9 and MRT-D2Q5 models are used to treat the flow and temperature fields respectively. In contrast to several existing investigations in the literature in this domain which study the passive control of the flow using a horizontal or vertical plate around a single cylinder, this work presents a numerical study on the effect of the position, length and height of a flat plate (horizontal and vertical) on three heated square cylinders on the flow and temperature fields. First, the effect of the position and length of the horizontal flat plate is examined. This study shows that the implementation of a flat plate of length Lp = 4D at a position g=3 behind the central cylinder reduces the amplitude of the Von Karman Street and allows large and regular heat exchange. Thus, in the second part, the effect of the position and height of the vertical flat plate is studied. The results obtained show that the implementation of a flat plate of height h=2D at a position g=3 behind the central cylinder improves the thermal exchange between the incoming fluid and the heated cylinders. This numerical work could lead to the prediction of the cooling of the electronic components: The cooling of the obstacles is all the better when the control plate is arranged at g = 3 and its height h = 2D in the case of the vertical plate or its length Lp equal to 4D in the case where the plate is implemented horizontally

Natural convection heat transfer with anisotropic thermal diffusion for tilted two-dimensional cavities

The effects of a domain tilt (counterclockwise, γ=00,300and600) and an anisotropic thermal diffusion on the natural convection heat transfer inside a square or rectangular cavity are investigated. The multiple relaxation time Lattice Boltzmann method with a double distribution function is used. The simulations are performed for a range of Rayleigh number (Ra=104to108). It is observed that the number of eddies is reduced due to the counterclockwise inclination of the cavity for a given Rayleigh number. The flow reversal is observed with a change in γ and anisotropic thermal diffusion ratio (β). Further, the stretching and merging of the inner core eddy with changes in γ and β are discussed. The spatially averaged Nusselt number along the hot wall (Nu¯h) is calculated to study the heat transfer rate and characteristics. It shows that the rate of heat transfer increases with an increase in the inclination and the thermal diffusivity ratio. For Ra=107and108 the observed flow structures are unsteady. The heat transfer in this type of unsteady flow and corresponding thermal plume dynamics are studied by computing Nu¯h at each non-dimensional timestep.

Improved curved-boundary scheme for lattice Boltzmann simulation of microscale gas flow with second-order slip condition

An improved curved-boundary scheme with second-order velocity slip condition for multiple-relaxation-time-lattice Boltzmann (MRT-LB) simulation of microgas flow is proposed. The proposed interpolation bounce-back (IBB)-explicit counter-extrapolation (ECE) scheme adopts the IBB method to describe the curved boundary, while the ECE method is employed to predict the slip velocity on gas-solid interface. To incorporate the effect of second-order velocity slip term and the influence of boundary curvature, a slip velocity model is also derived, from which the gas slip velocity is captured by the ECE discretization method. The influence of fictitious slip velocity can be eliminated by adopting the present ECE method, and the influence of actual offset between the lattice node and the physical boundary can be well considered by the IBB method. The proposed IBB-ECE boundary scheme is then implemented with the MRT-LB model and tested by simulations of force-driven gas flow in horizontal (inclined) microchannel, gas flow around a micro-cylinder, and Couette flow between two micro-cylinders. Numerical results show that the proposed IBB-ECE scheme improves the computational accuracy of gas slip flow (0.001<Kn≤0.1) when compared with other boundary schemes reported in the literature, and provides a precise and easy implementing scheme for curved boundary with second-order slip condition.

Phase-field lattice Boltzmann model for two-phase flows with large density ratio.

In this work, a lattice Boltzmann (LB) model based on the phase-field method is proposed for simulating large density ratio two-phase flows. An improved multiple-relaxation-time (MRT) LB equation is first developed to solve the conserved Allen-Cahn (AC) equation. By utilizing a nondiagonal relaxation matrix and modifying the equilibrium distribution function and discrete source term, the conserved AC equation can be correctly recovered by the proposed MRT LB equation with no deviation term. Therefore, the calculations of the temporal derivative term in the previous LB models are successfully avoided. Numerical tests demonstrate that satisfactory accuracy can be achieved by the present model to solve the conserved AC equation. What is more, the discrete force term of the MRT LB equation for the incompressible Navier-Stokes equations is also simplified and modified in the present work. An alternative scheme to calculate the gradient terms of the order parameter involved in the discrete force term through the nonequilibrium part of the distribution function is also developed. To validate the ability of the present LB model for simulating large density ratio two-phase flows, series of benchmarks, including two-phase Poiseuille flow, droplet impacting on thin liquid film, and planar Taylor bubble are simulated. It is found that the results predicted by the present LB model agree well with the analytical, numerical, and experimental results.

Multiple Relaxation Time Lattice Boltzmann Schemes for Advection-Diffusion Equations with Application to Radar Image Processing

Multiple Relaxation Time Lattice Boltzmann Schemes for Advection-Diffusion Equations with Application to Radar Image Processing

Siamese-SR: A Siamese Super-Resolution Model for Boosting Resolution of Digital Rock Images for Improved Petrophysical Property Estimation

Digital Rock Physics leverages advances in digital image acquisition and analysis techniques to create 3D digital images of rock samples, which are used for computational modeling and simulations to predict petrophysical properties of interest. However, the accuracy of the predictions is crucially dependent on the quality of the digital images, which is currently limited by the resolution of the micro-CT scanning technology. We have proposed a novel Deep Learning based Super-Resolution model called Siamese-SR to digitally boost the resolution of Digital Rock images whilst retaining the texture and providing optimal de-noising. The Siamese-SR model consists of a generator which is adversarially trained with a relativistic and a siamese discriminator utilizing Materials In Context (MINC) loss estimator. This model has been demonstrated to improve the resolution of sandstone rock images acquired using micro-CT scanning by a factor of 2. Another key highlight of our work is that for the evaluation of the super-resolution performance, we propose to move away from image-based metrics such as Structural Similarity (SSIM) and Peak Signal to Noise Ratio (PSNR) because they do not correlate well with expert geological and petrophysical evaluations. Instead, we propose to subject the super-resolved images to the next step in the Digital Rock workflow to calculate a crucial petrophysical property of interest, viz. porosity and use it as a metric for evaluation of our proposed Siamese-SR model against several other existing super-resolution methods like SRGAN, ESRGAN, EDSR and SPSR. Furthermore, we also use Local Attribution Maps to show how our proposed Siamese-SR model focuses optimally on edge-semantics, which is what leads to improvement in the image-based porosity prediction, the permeability prediction from Multiple Relaxation Time Lattice Boltzmann Method (MRTLBM) flow simulations as well as the prediction of other petrophysical properties of interest derived from Mercury Injection Capillary Pressure (MICP) simulations.

On the performance of a MRT lattice Boltzmann algorithm for transient radiative transfer problems

An exact multiple-relaxation-time lattice Boltzmann (MRT-LB) algorithm without any assumptions or approximations is raised for solving time-dependent radiative transfer problems (RTPs). The macroscopic radiative transfer equation (RTE) is exactly recovered from the mesoscopic lattice Boltzmann equation (LBE). The proposed MRT-LB algorithm is utilized to obtain the time-dependent radiation distribution with strong variations. Analysis of its numerical performance, including the accuracy and convergence order, is carried out. Results demonstrate that the present MRT-LB is accurate for solving the RTE, with a convergence order of two. The MRT-LB has a good ability to capture the time-dependent properties, including the radiation intensity, radiative heat flux, as well as wavefront, for the radiation propagation problems.

Multiple-relaxation-time lattice Boltzmann model-based four-level finite-difference scheme for one-dimensional diffusion equations.

In this paper, we first present a multiple-relaxation-time lattice Boltzmann (MRT-LB) model for one-dimensional diffusion equation where the D1Q3 (three discrete velocities in one-dimensional space) lattice structure is considered. Then through the theoretical analysis, we derive an explicit four-level finite-difference scheme from this MRT-LB model. The results show that the four-level finite-difference scheme is unconditionally stable, and through adjusting the weight coefficient ω_{0} and the relaxation parameters s_{1} and s_{2} corresponding to the first and second moments, it can also have a sixth-order accuracy in space. Finally, we also test the four-level finite-difference scheme through some numerical simulations and find that the numerical results are consistent with our theoretical analysis.

Pore-scale convection-conduction heat transfer and fluid flow in open-cell metal foams: A three-dimensional multiple-relaxation time lattice Boltzmann (MRT-LBM) solution

Flow and heat transfer in porous foams are usually simulated based on averaged properties that are far from the actual values and can lead to substantial inaccuracies in the results. The pore-scale simulation could provide accurate and realistic models for such properties. Accordingly, a multiple-relaxation time lattice Boltzmann (MRT-LBM) code is extended for pore-scale direct numerical simulation (DNS) of fluid flow and conductive-convective heat transfer in open-cell metal foams (OCMFs) using the Palabos code, an open-source parallel LB solver. Three foam models with different numbers of pores per inch (PPI) are generated, and through simulations, the permeability, the non-Darcy coefficient β, the onset of non-Darcy flow, and Nusselt number (Nud) are determined. The results show that by increasing the PPI of the samples from 20 to 40, the flow becomes more complex, and velocity and permeability are reduced by up to 70%. On the other hand, the heat transfer coefficient increases significantly so that in the same Reynolds number (Re), the overall heat transfer coefficient of sample 3 (40 PPI) is on average 110% higher than that of sample 1 (20 PPI). Finally, the global Nusselt number (Nud) is determined as a function of Re, and the related correlations are presented.

Improved thermal multiple-relaxation-time lattice Boltzmann model for liquid-vapor phase change

In this paper, an improved thermal multiple-relaxation-time (MRT) lattice Boltzmann (LB) model is proposed for simulating liquid-vapor phase change. A temperature equation is first derived for liquid-vapor phase change, where the latent heat of vaporization is decoupled with the equation of state. Therefore, the latent heat of vaporization can be arbitrarily specified in practice, which significantly improves the flexibility of the present LB model for liquid-vapor phase change. The Laplacian term of temperature is avoided in the proposed temperature equation and the gradient term of temperature is calculated through a local scheme. To solve the temperature equation accurately and efficiently, an improved MRT LB equation with nondiagonal relaxation matrix is developed. The implicit calculation of the temperature, caused by the source term and encountered in previous works, is avoided by approximating the source term with its value at the previous time step. As demonstrated by numerical tests, the results by the present LB model agree well with analytical results, experimental results, or the results by the finite difference method where the fourth-order Runge-Kutta method is employed to implement the discretization of time.

Study of pore-scale coke combustion in porous media using lattice Boltzmann method

In-situ combustion (ISC) for heavy oil recovery is a high-risk process and thus calls for a deep understanding of the coke combustion front. However, simulations for describing pore-scale coke combustion are challenging because of complicated combustion dynamics. This study proposes a multiple-relaxation-time (MRT) lattice Boltzmann (LB) method to simulate solid coke combustion in porous media at the pore scale. The MRT LB model can enhance numerical stability compared with the single-relaxation-time counterpart. This new LB model offers two major advances over existing LB models. First, thermal expansion effects are considered via temperature-dependent density, satisfying the low Mach number flow condition. Second, conjugate heat transfer and species conservation at the coke-fluid interface are modelled by a source term and a bounce-back scheme, respectively, without resorting to iterative methods. Simulations of pore-scale coke combustion are conducted to validate the superior performance of the developed LB model. The results show that this model can reproduce coke combustion dynamics and capture variations in isotherms from the fluid phase to the solid phase at the pore scale. In contrast, without considering effects of thermal expansion or conjugate heat transfer as in previous models, simulations would underestimate the burning temperature and even fail to predict instability caused by a high driving force. Moreover, using the proposed LB model, a parametric study is performed to assess the impact of the inlet air and porous structure on coke combustion. The findings suggest that inlet air temperature and driving force should be controlled within certain ranges, otherwise their influences may become negligible or even negative. Large porosity accelerates coke combustion and leads to a desired combustion front. Heterogeneous coke distribution affects pore-scale combustion details but not the overall combustion intensity, while a random matrix structure may influence combustion dynamics both locally and globally. As a whole, this study develops a reliable LB model to investigate pore-scale coke combustion, which contributes to advancing the knowledge base for heavy oil recovery using the ISC process.

Assessment of multiple relaxation time-lattice Boltzmann method framework for non-Newtonian fluid flow simulations

A kinetic-theory based approach known as the lattice Boltzmann method (LBM) with multiple-relaxation-time (MRT) collision is employed to assess the non-Newtonian fluid models for standard two-dimensional benchmarks flow configurations. A two-dimensional nine-velocity square lattice (D2Q9) is used as a basic space-microscopic-velocity coupled LBM structure. The MRT-LBM formulation for the non-Newtonian fluids is validated for flow inside a lid-driven cavity. The non-Newtonian fluid viscosity is estimated using standard models as, Power-law, Carreau, Carreau–Yasuda, and Cross. These models are incorporated in the MRT-LBM framework to solve four standard benchmark flow configurations, namely, (a) lid-driven square cavity flow; (b) flow over a circular cylinder placed inside the lid-driven square cavity; (c) flow through two parallel plates, i.e. through a straight channel; and (d) flow over an asymmetrically placed circular cylinder in a straight channel. The simulation results for various benchmark cases found to be comparable with the earlier published data.

Multiple-relaxation-time lattice Boltzmann method for the Navier-Stokes and nonlinear convection-diffusion equations: Modeling, analysis, and elements.

In this paper, we first present a unified framework of multiple-relaxation-time lattice Boltzmann (MRT-LB) method for the Navier-Stokes and nonlinear convection-diffusion equations where a block-lower-triangular-relaxation matrix and an auxiliary source distribution function are introduced. We then conduct a comparison of the four popular analysis methods (Chapman-Enskog analysis, Maxwell iteration, direct Taylor expansion, and recurrence equations approaches) that have been used to obtain the macroscopic Navier-Stokes and nonlinear convection-diffusion equations from the MRT-LB method and show that from mathematical point of view, these four analysis methods can give the same equations at the second-order of expansion parameters. Finally, we give some elements that are needed in the implementation of the MRT-LB method and also find that some available LB models can be obtained from this MRT-LB method.

Lattice Boltzmann model with adjustable equation of state for coupled thermo-hydrodynamic flows

A novel lattice Boltzmann (LB) model with adjustable equation of state (EOS) is developed in this work for simulating coupled thermo-hydrodynamic flows. The velocity field is solved by the recently developed multiple-relaxation-time (MRT) LB equation for density distribution function (DF), by which an adjustable EOS can be recovered. As to the temperature field, a novel MRT LB equation for total energy DF is directly developed at the discrete level. By introducing a density-DF-related term into this LB equation and devising the equilibrium moment function for total energy DF, the viscous dissipation and compression work are consistently considered, and by modifying the collision matrix, the targeted energy conservation equation is recovered without deviation term. The full coupling of thermo-hydrodynamic effects is achieved via the adjustable EOS and the viscous dissipation and compression work. The present LB model is developed on the basis of the standard lattice, and various EOSs can be adopted in real applications. Moreover, both the Prandtl number and specific heat ratio can be arbitrarily adjusted. Furthermore, boundary condition treatment is also proposed on the basis of the judicious decomposition of DF into its equilibrium, force (source), and nonequilibrium parts. The local conservation of mass, momentum, and energy can be strictly satisfied at the boundary node. Numerical simulations of thermal Poiseuille and Couette flows are carried out with three different EOSs, and the numerical results are in good agreement with the analytical solutions. Then, natural convection in a square cavity with a large temperature difference is simulated for the Rayleigh number from 103 up to 108. Good agreement between the present and previous numerical results is observed, which further validates the present LB model for coupled thermo-hydrodynamic flows.

Application of a Steady Meandering River with Piers Using a Lattice Boltzmann Sub-Grid Model in Curvilinear Coordinate Grid

A sub-grid multiple relaxation time (MRT) lattice Boltzmann model with curvilinear coordinates is applied to simulate an artificial meandering river. The method is based on the D2Q9 model and standard Smagorinsky sub-grid scale (SGS) model is introduced to simulate meandering flows. The interpolation supplemented lattice Boltzmann method (ISLBM) and the non-equilibrium extrapolation method are used for second-order accuracy and boundary conditions. The proposed model was validated by a meandering channel with a 180° bend and applied to a steady curved river with piers. Excellent agreement between the simulated results and previous computational and experimental data was found, showing that MRT-LBM (MRT lattice Boltzmann method) coupled with a Smagorinsky sub-grid scale (SGS) model in a curvilinear coordinates grid is capable of simulating practical meandering flows.