Lattice Boltzmann approach for complex nonequilibrium flows

Impact of hydraulic tortuosity on microporous and nanoporous media flow

Using two-dimensional porous structures made up of homogeneously arranged solid obstacles, we examine the effects of rarefaction on the hydraulic tortuosity in the slip and early transition flow regimes via extended lattice Boltzmann method. We observed that modification in either the obstacle's arrangement or the porosity led to a power-law relation between the porosity-tortuosity. Along with this, we also found that in the slip-flow regime, the exponent of this relation contains the effect of finite Knudsen number (Kn). In addition, we observed that on properly scaling Kn with porosity and hydraulic tortuosity, a generalized correlation can be obtained for apparent permeability.

Discrete-velocity Boltzmann model: Regularization and linear stability

A discrete-velocity Boltzmann model for a nine-velocity lattice is considered. Compared to the conventional lattice Boltzmann (LB) schemes the collisions for the model are defined explicitly. Space and time discretization of the model is based on the collide and stream method; in addition, the regularization of the collision term is proposed. It is demonstrated that the regularized model can be represented as a two-relaxation-time LB model of a special type. The scheme is compared to the Onsager regularized (a specific filtered Galilean invariant model) and recursively regularized LB equations in terms of stability and dissipation properties, and linear stability analysis is performed. Several numerical experiments are carried out: double shear layer, lid-driven cavity flow, and propagation of acoustic and shear waves are considered for different grid resolutions, Mach and Reynolds numbers. It is shown that free parameters in the model corresponding to collision cross sections can be adjusted in such a way that the dissipation and stability properties can be controlled.

Multiple-relaxation-time finite-difference lattice Boltzmann model for the nonlinear convection-diffusion equation

In this paper, a multiple-relaxation-time finite-difference lattice Boltzmann method (MRT-FDLBM) is developed for the nonlinear convection-diffusion equation (NCDE). Through designing the equilibrium distribution function and the source term properly, the NCDE can be recovered exactly from MRT-FDLBM. We also conduct the von Neumann stability analysis on the present MRT-FDLBM and its special case, i.e., single-relaxation-time finite-difference lattice Boltzmann method (SRT-FDLBM). Then, a simplified version of MRT-FDLBM (SMRT-FDLBM) is also proposed, which can save about 15% computational cost. In addition, a series of real and complex-value NCDEs, including the isotropic convection-diffusion equation, Burgers-Fisher equation, sine-Gordon equation, heat-conduction equation, and Schrödinger equation, are used to test the performance of MRT-FDLBM. The results show that both MRT-FDLBM and SMRT-FDLBM have second-order convergence rates in space and time. Finally, the stability and accuracy of five different models are compared, including the MRT-FDLBM, SMRT-FDLBM, SRT-FDLBM, the previous finite-difference lattice Boltzmann method [H. Wang, B. Shi et al., Appl. Math. Comput. 309, 334 (2017)10.1016/j.amc.2017.04.015], and the lattice Boltzmann method (LBM). The stability tests show that the sequence of stability from high to low is as follows: MRT-FDLBM, SMRT-FDLBM, SRT-FDLBM, the previous finite-difference lattice Boltzmann method, and LBM. In most of the precision test results, it is found that the order from high to low of precision is MRT-FDLBM, SMRT-FDLBM, SRT-FDLBM, and the previous finite-difference lattice Boltzmann method.

Onsager-regularized lattice Boltzmann method: A nonequilibrium thermodynamics-based regularized lattice Boltzmann method

The regularized class of lattice Boltzmann methods (LBMs) leverage the potency of the standard lattice-Bhatnagar-Gross-Krook method by filtering out spurious nonhydrodynamic moments from the moment space; this is achieved through evaluating regularized populations via a multiscale or a Hermite polynomial expansion approach. In this paper, we propose an alternative approach for evaluating the lattice populations. This approach is based on a kinetic theory that is consistent with nonequilibrium thermodynamics and obeys the Onsager-reciprocity principle. The proposed method is verified and validated for a number of canonical problems such as the athermal shock tube, the double periodic shear layer, the lid driven cavity, flow past square cylinder, and Poiseuille flow at nonvanishing Knudsen numbers. Additionally, the proposed method is compared to existing regularized LBM schemes and is shown to yield significant improvement in the stability and accuracy of the simulations.

Study on Deposition Characteristics of Microparticles in Terminal Pulmonary Acini by IB-LBM

As an indicator of health risk, the deposition of microparticles in terminal pulmonary acini is of great significance in the medical field. To control particulate pollution and optimize aerosol delivery, it is necessary to perform an in-depth study of the microparticle deposition in terminal pulmonary acini; however, little research has been done on this topic. This paper proposes a respiratory movement model of terminal pulmonary acini using an immersed boundary–lattice Boltzmann method. In addition, we explored the effect of gravity direction, respiratory rate, microparticle diameter, and other parameters on the microparticles deposition process and distribution, under the airflow in the acinar wall. It was found that the deposition of microparticles is sensitive to gravity direction, and the growth of the respiratory rate increases the rate of microparticle migration and deposition. It was observed that the gravity effect is enhanced by increasing the diameter of microparticles, causing a high deposition and dispersion rate. The study reveals the dynamic correlation between the respiration process and the movement of microparticles, which is of reference value to figure out the pathogenicity mechanism of inhalable particles and to optimize the aerosol delivery.

Tracking droplets in soft granular flows with deep learning techniques

The state-of-the-art deep learning-based object recognition YOLO algorithm and object tracking DeepSORT algorithm are combined to analyze digital images from fluid dynamic simulations of multi-core emulsions and soft flowing crystals and to track moving droplets within these complex flows. The YOLO network was trained to recognize the droplets with synthetically prepared data, thereby bypassing the labor-intensive data acquisition process. In both applications, the trained YOLO + DeepSORT procedure performs with high accuracy on the real data from the fluid simulations, with low error levels in the inferred trajectories of the droplets and independently computed ground truth. Moreover, using commonly used desktop GPUs, the developed application is capable of analyzing data at speeds that exceed the typical image acquisition rates of digital cameras (30 fps), opening the interesting prospect of realizing a low-cost and practical tool to study systems with many moving objects, mostly but not exclusively, biological ones. Besides its practical applications, the procedure presented here marks the first step towards the automatic extraction of effective equations of motion of many-body soft flowing systems.

Simulation of collapsing cavitation bubbles in various liquids by lattice Boltzmann model coupled with the Redlich-Kwong-Soave equation of state

A computational technique based on the pseudo-potential multiphase lattice Boltzmann method (LBM) is employed to investigate the collapse dynamics of cavitation bubbles of various liquids in the vicinity of the solid surface with different wettability conditions. The Redlich-Kwong-Soave equation of state (EoS) that includes an acentric factor is incorporated to consider the physical properties of water (H_{2}O), liquid nitrogen (LN_{2}), and liquid hydrogen (LH_{2}) in the present simulations. Accuracy and performance of the present multiphase LBM are examined by simulation of the homogenous and heterogeneous cavitation phenomena. The good agreement of the results obtained based on the present solution algorithm in comparison with the available data confirms the validity and capability of the multiphase LBM employed. Then, the cavitation bubble collapse near the solid wall is studied by considering the H_{2}O, LN_{2}, and LH_{2} fluids, and the wettability effect of the surface on the collapse dynamics is investigated. The obtained results demonstrate that the collapse phenomenon for the H_{2}O is more aggressive than that of the LH_{2} and LN_{2}. The cavitation bubble of the water has a shorter collapse time with an intense liquid jet, while the collapse process in the LN_{2} takes a longer time due to the larger radius of its bubble at the rebound. Also, this study demonstrates that the increment of the hydrophobicity of the wall causes less energy absorption by the solid surface from the liquid phase around the bubble that leads to form a liquid jet with higher kinetic energy. Therefore, the bubble collapse process occurs more quickly for hydrophobic surfaces, regardless of the fluids considered. The present study shows that the pseudopotential LBM with incorporating an appropriate EoS and a robust forcing scheme is an efficient numerical technique for simulation of the dynamics of the cavitation bubble collapse in different fluids.

Suitability of 2D modelling to evaluate flow properties in 3D porous media

The employment of 2D models to investigate the properties of 3D flows in porous media is ubiquitous in the literature. The limitations of such approaches are often overlooked. Here, we assess to which extent 2D flows in porous media are suitable representations of 3D flows. To this purpose, we compare representative elementary volume (REV) scales obtained by 2D and 3D numerical simulations of flow in porous media. The stationarity of several quantities, namely porosity, permeability, mean and variance of velocity, is evaluated in terms of both classical and innovative statistics. The variance of velocity, strictly connected to the hydrodynamic dispersion, is included in the analysis in order to extend conclusions to transport phenomena. Pore scale flow is simulated by means of a Lattice Boltzmann model. The results from pore scale simulations point out that the 2D approach often leads to inconsistent results, due to the profound difference between 2D and 3D flows through porous media. We employ the error in the evaluation of REV as a quantitative measure for the reliability of a 2D approach. Moreover, we show that the acceptance threshold for a 2D representation to be valid strongly depends on which flow/transport quantity is sought.

Extensive analysis of the lattice Boltzmann method on shifted stencils

Standard lattice Boltzmann methods (LBMs) are based on a symmetric discretization of the phase space, which amounts to study the evolution of particle distribution functions (PDFs) in a reference frame at rest. This choice induces a number of limitations when the simulated flow speed gets closer to the sound speed, such as velocity-dependent transport coefficients. The latter issue is usually referred to as a Galilean invariance defect. To restore the Galilean invariance of LBMs, it was proposed to study the evolution of PDFs in a comoving reference frame by relying on asymmetric shifted lattices [N. Frapolli, S. S. Chikatamarla, and I. V. Karlin, Phys. Rev. Lett. 117, 010604 (2016)].PRLTAO0031-900710.1103/PhysRevLett.117.010604 From the numerical viewpoint, this corresponds to overcoming the rather restrictive Courant-Friedrichs-Lewy conditions on standard LBMs and modeling compressible flows while keeping memory consumption and processing costs to a minimum (therefore using the standard first-neighbor stencils). In the present work systematic physical error evaluations and stability analyses are conducted for different discrete equilibrium distribution functions (EDFs) and collision models. Thanks to them, it is possible to (1) better understand the effect of this solution on both physics and stability, (2) assess its viability as a way to extend the validity range of LBMs, and (3) quantify the importance of the reference state as compared to other parameters such as the equilibrium state and equilibration path. The results clearly show that, in theory, the concept of shifted lattices allows the scheme to deal with arbitrarily high values of the nondimensional velocity. Furthermore, just like the zero-Mach flow for the standard stencils, it is observed that setting the shift velocity to the fluid velocity results in optimal physical and numerical properties. In addition, a detailed analysis of the obtained results shows that the properties of different collision models and EDFs remain unchanged under the shift of stencil. In other words, by introducing a velocity shift in the stencil, the optimal operating point, in terms of physics and numerics, will also be shifted by the same vector regardless of the EDF or collision model considered. Eventually, while limited to the D2Q9 stencil with the nine possible first-neighbor shifts, the present study and corresponding conclusions can be extended to other stencils and velocity shifts in a straightforward manner.

Microvorticity fluctuations affect the structure of thin fluid films

The dynamic interaction of complex fluid interfaces is highly sensitive to near-contact interactions occurring at the scale of ten of nanometers. Such interactions are difficult to analyze because they couple self-consistently to the dynamic morphology of the evolving interface, as well as to the hydrodynamics of the interstitial fluid film. In this work, we show that, above a given magnitude threshold, near-contact interactions trigger nontrivial microvorticity patterns, which in turn affect the effective near-contact interactions, giving rise to persistent fluctuating ripples at the fluid interface. In such a regime, near-contact interactions may significantly affect the macroscopic arrangement of emulsion configurations, such as those arising in soft-flowing microfluidic crystals.

Stress-Dependent Pore Deformation Effects on Multiphase Flow Properties of Porous Media

Relative permeability and capillary pressure are the governing parameters that characterize multiphase fluid flow in porous media for diverse natural and industrial applications, including surface water infiltration into the ground, CO₂ sequestration, and hydrocarbon enhanced recovery. Although the drastic effects of deformation of porous media on single-phase fluid flow have been well established, the stress dependency of flow in multiphase systems is not yet fully explored. Here, stress-dependent relative permeability and capillary pressure are studied in a water-wet carbonate specimen both analytically using fractal and poroelasticity theory and experimentally on the micro-scale and macro-scales by means of X-ray computed micro-tomography and isothermal isotropic triaxial core flooding cell, respectively. Our core flooding program using water/N₂ phases shows a systematic decrease in the irreducible water saturation and gas relative permeability in response to an increase in effective stress. Intuitively, a leftward shift of the intersection point of water/gas relative permeability curves is interpreted as an increased affinity of the rock to the gas phase. Using a micro-scale proxy model, we identify a leftward shift in pore size distribution and closure of micro-channels to be responsible for the abovementioned observations. These findings prove the crucial impact of effective stress-induced pore deformation on multiphase flow properties of rock, which are missing from the current characterizations of multiphase flow mechanisms in porous media.

Numerical Simulation of Darcy–Forchheimer 3D Unsteady Nanofluid Flow Comprising Carbon Nanotubes with Cattaneo–Christov Heat Flux and Velocity and Thermal Slip Conditions

A mathematical model comprising Darcy Forchheimer effects on the 3D nanofluid flow with engine oil as a base fluid containing suspended carbon nanotubes (CNTs) is envisioned. The CNTs are of both types i.e., multi-wall carbon nanotubes (MWCNTs) and single-walled carbon nanotubes (SWCNTs). The flow is initiated by an exponentially stretched surface. The impacts of Cattaneo–Christov heat flux along with velocity and thermal slip conditions are key factors in the novelty of the defined model. The boundary layer notion is designed to convert the compact form of equations into the component shape. Appropriate transformations lead to differential equations with high nonlinearity. The final non-dimensional system is solved numerically by a “MATLAB” function known as bvp4c. For both CNTs, different graphical sketches are drawn to present the influence of arising parameters versus related profiles. The outcomes show that higher slip parameter boosts the axial velocity, whereas fluid temperature lowers for a sturdier relaxation parameter.

Mesoscale modelling of soft flowing crystals

We outline the main ideas behind the numerical modelling of soft flowing crystals, paying special attention to their application to microfluidic devices for the design of novel mesoscale porous materials. This article is part of the theme issue 'Multiscale modelling, simulation and computing: from the desktop to the exascale'.

Consistent lattice Boltzmann modeling of low-speed isothermal flows at finite Knudsen numbers in slip-flow regime. II. Application to curved boundaries

Gaseous flows inside microfluidic devices often fall in the slip-flow regime. According to this theoretical description, the Navier-Stokes model remains applicable in bulk, while at solid walls a slip velocity boundary model shall be considered. Physically, it is well established that, to properly account for the wall curvature, the wall slip velocity must be determined by the shear stress, rather than the normal component of the velocity derivative alone, as commonly applied to planar surfaces. It follows that the numerical transcription of this type of boundary condition is generally a challenging task for standard computational fluid dynamics (CFD) techniques. This paper aims to show that the realization of the slip velocity condition on arbitrarily shaped boundaries can be accomplished in a natural way with the lattice Boltzmann method (LBM). To substantiate this conclusion, this work undertakes the following three studies. First, we examine the conditions under which the generic reflection-type boundary rules used by LBM become consistent models for the slip velocity boundary condition. This effort makes use of the second-order Chapman-Enskog expansion method, where we address both planar and curved boundaries. The analysis also clarifies the capabilities and limitations behind the considered reflection-type slip schemes. Second, we revisit the family of parabolic accurate LBM slip boundary schemes, originally formulated in [Phys. Rev. E 96, 013311 (2017)2470-004510.1103/PhysRevE.96.013311] on the basis of the multireflection framework, and discuss their characteristics when operating on curved boundaries as well as the limitations of other less accurate LBM slip boundary formulations, such as the linearly accurate slip schemes and the widely popular "kinetic-based" boundary schemes. In addition, we also discuss the numerical stability of the parabolic slip schemes previously developed, providing an heuristic strategy to improve their stable range of operation. Third, we evaluate the performance of the several slip boundary schemes debated in this paper. The numerical tests correspond to two classical 2D benchmark flow problems of slip over non-planar solid surfaces, namely: (i) the velocity profile of the cylindrical Couette flow, and (ii) the permeability of a slow rarefied gas over a periodic array of circular cylindrical obstacles. The obtained numerical results confirm the competitiveness of the LBM when equipped with slip boundary schemes of parabolic accuracy as CFD tool to simulate slippage phenomena over arbitrarily non-planar surfaces. Indeed, although operating on a simple uniform mesh discretization, the LBM yields a similar, or even superior, level of accuracy compared to state-of-the-art FEM simulations conducted on hardworking body-fitted meshes. This conclusion establishes the LBM as a very appealing CFD technique for simulating microfluidic flows in the slip-flow regime, a result that deserves further exploration in future studies.

Discrete Boltzmann trans-scale modeling of high-speed compressible flows

We present a general framework for constructing trans-scale discrete Boltzmann models (DBMs) for high-speed compressible flows ranging from continuum to transition regime. This is achieved by designing a higher-order discrete equilibrium distribution function that satisfies additional nonhydrodynamic kinetic moments. To characterize the thermodynamic nonequilibrium (TNE) effects and estimate the condition under which the DBMs at various levels should be used, two measures are presented: (i) the relative TNE strength, describing the relative strength of the (N+1)th order TNE effects to the Nth order one; (ii) the TNE discrepancy between DBM simulation and relevant theoretical analysis. Whether or not the higher-order TNE effects should be taken into account in the modeling and which level of DBM should be adopted is best described by the relative TNE intensity and/or the discrepancy rather than by the value of the Knudsen number. As a model example, a two-dimensional DBM with 26 discrete velocities at Burnett level is formulated, verified, and validated.

Impact of the kinetic boundary condition on porous media flow in the lattice Boltzmann formulation

To emphasize the importance of the kinetic boundary condition for micro- to nanoscale flow, we present an ad hoc kinetic boundary condition suitable for torturous geological porous media. We found that the kinetic boundary condition is one of the essential features which should be supplemented to the standard lattice Boltzmann scheme in order to obtain accurate continuum observables. The claim is validated using a channel flow setup by showing the agreement of mass flux with analytical value. Further, using a homogeneous porous structure, the importance of the kinetic boundary condition is shown by comparing the permeability correction factor with the analytical value. Finally, the proposed alternate to the kinetic boundary condition is validated by showing its capability to capture the basic feature of the kinetic boundary condition.

Recursive regularization step for high-order lattice Boltzmann methods

A lattice Boltzmann method (LBM) with enhanced stability and accuracy is presented for various Hermite tensor-based lattice structures. The collision operator relies on a regularization step, which is here improved through a recursive computation of nonequilibrium Hermite polynomial coefficients. In addition to the reduced computational cost of this procedure with respect to the standard one, the recursive step allows to considerably enhance the stability and accuracy of the numerical scheme by properly filtering out second- (and higher-) order nonhydrodynamic contributions in under-resolved conditions. This is first shown in the isothermal case where the simulation of the doubly periodic shear layer is performed with a Reynolds number ranging from 10^{4} to 10^{6}, and where a thorough analysis of the case at Re=3×10^{4} is conducted. In the latter, results obtained using both regularization steps are compared against the Bhatnagar-Gross-Krook LBM for standard (D2Q9) and high-order (D2V17 and D2V37) lattice structures, confirming the tremendous increase of stability range of the proposed approach. Further comparisons on thermal and fully compressible flows, using the general extension of this procedure, are then conducted through the numerical simulation of Sod shock tubes with the D2V37 lattice. They confirm the stability increase induced by the recursive approach as compared with the standard one.

Consistent lattice Boltzmann modeling of low-speed isothermal flows at finite Knudsen numbers in slip-flow regime: Application to plane boundaries

The first nonequilibrium effect experienced by gaseous flows in contact with solid surfaces is the slip-flow regime. While the classical hydrodynamic description holds valid in bulk, at boundaries the fluid-wall interactions must consider slip. In comparison to the standard no-slip Dirichlet condition, the case of slip formulates as a Robin-type condition for the fluid tangential velocity. This makes its numerical modeling a challenging task, particularly in complex geometries. In this work, this issue is handled with the lattice Boltzmann method (LBM), motivated by the similarities between the closure relations of the reflection-type boundary schemes equipping the LBM equation and the slip velocity condition established by slip-flow theory. Based on this analogy, we derive, as central result, the structure of the LBM boundary closure relation that is consistent with the second-order slip velocity condition, applicable to planar walls. Subsequently, three tasks are performed. First, we clarify the limitations of existing slip velocity LBM schemes, based on discrete analogs of kinetic theory fluid-wall interaction models. Second, we present improved slip velocity LBM boundary schemes, constructed directly at discrete level, by extending the multireflection framework to the slip-flow regime. Here, two classes of slip velocity LBM boundary schemes are considered: (i) linear slip schemes, which are local but retain some calibration requirements and/or operation limitations, (ii) parabolic slip schemes, which use a two-point implementation but guarantee the consistent prescription of the intended slip velocity condition, at arbitrary plane wall discretizations, further dispensing any numerical calibration procedure. Third and final, we verify the improvements of our proposed slip velocity LBM boundary schemes against existing ones. The numerical tests evaluate the ability of the slip schemes to exactly accommodate the steady Poiseuille channel flow solution, over distinct wall slippage conditions, namely, no-slip, first-order slip, and second-order slip. The modeling of channel walls is discussed at both lattice-aligned and non-mesh-aligned configurations: the first case illustrates the numerical slip due to the incorrect modeling of slippage coefficients, whereas the second case adds the effect of spurious boundary layers created by the deficient accommodation of bulk solution. Finally, the slip-flow solutions predicted by LBM schemes are further evaluated for the Knudsen's paradox problem. As conclusion, this work establishes the parabolic accuracy of slip velocity schemes as the necessary condition for the consistent LBM modeling of the slip-flow regime.

Maxwell iteration for the lattice Boltzmann method with diffusive scaling

In this work, we present an alternative derivation of the Navier-Stokes equations from Bhatnagar-Gross-Krook models of the lattice Boltzmann method with diffusive scaling. This derivation is based on the Maxwell iteration and can expose certain important features of the lattice Boltzmann solutions. Moreover, it will be seen to be much more straightforward and logically clearer than the existing approaches including the Chapman-Enskog expansion.

Lattice Boltzmann accelerated direct simulation Monte Carlo for dilute gas flow simulations

Hybrid particle-continuum computational frameworks permit the simulation of gas flows by locally adjusting the resolution to the degree of non-equilibrium displayed by the flow in different regions of space and time. In this work, we present a new scheme that couples the direct simulation Monte Carlo (DSMC) with the lattice Boltzmann (LB) method in the limit of isothermal flows. The former handles strong non-equilibrium effects, as they typically occur in the vicinity of solid boundaries, whereas the latter is in charge of the bulk flow, where non-equilibrium can be dealt with perturbatively, i.e. according to Navier-Stokes hydrodynamics. The proposed concurrent multiscale method is applied to the dilute gas Couette flow, showing major computational gains when compared with the full DSMC scenarios. In addition, it is shown that the coupling with LB in the bulk flow can speed up the DSMC treatment of the Knudsen layer with respect to the full DSMC case. In other words, LB acts as a DSMC accelerator.This article is part of the themed issue 'Multiscale modelling at the physics-chemistry-biology interface'.

Chimaera simulation of complex states of flowing matter

We discuss a unified mesoscale framework (chimaera) for the simulation of complex states of flowing matter across scales of motion. The chimaera framework can deal with each of the three macro-meso-micro levels through suitable 'mutations' of the basic mesoscale formulation. The idea is illustrated through selected simulations of complex micro- and nanoscale flows.This article is part of the themed issue 'Multiscale modelling at the physics-chemistry-biology interface'.