Krook Collision Operator Research Articles

Lattice Boltzmann simulations for the drying of porous media with gas–side convection–diffusion boundary

One of the most significant challenges in describing the drying of porous materials is the realistic integration of internal transport phenomena into mathematical models, coupled with the external free flow. However, the intricacies of pore-scale geometry make experimentation and observation very difficult in practice, thereby making modeling techniques a useful tool for the analysis of the drying process. Among the many modeling techniques developed for drying, the lattice Boltzmann method (LBM)-based modeling approach has gained favor in recent years due to its ability to incorporate realistic geometry and transport at the pore scale. Our previous works on Shan–Chen LBM for drying of capillary porous media were based on the Bhatnagar–Gross–Krook collision operator and diffusion interface boundary conditions. This study elucidates the drying of a capillary porous medium under the influence of convection–diffusion boundary conditions at the gas side, using Shan–Chen LBM. The pore-scale effects of convection–diffusion conditions during the drying process are presented in relation to the macroscale drying kinetics. Moreover, the differences between the convection–diffusion kinetics and purely diffusion–dominated kinetics of the drying process are also presented here. This work also aims to incorporate the convection–diffusion transport phenomena into the drying process of a porous medium under the influence of an imposed thermal gradient, establishing and studying the phenomena of stabilization and destabilization of the drying front under the influence of a temperature gradient, thereby extending the lattice Boltzmann method of modeling for the simulation of convection–diffusion drying, both for the isothermal case and the imposition of a thermal gradient.

Combination of Physics-Informed Neural Networks and Single-Relaxation-Time Lattice Boltzmann Method for Solving Inverse Problems in Fluid Mechanics

Physics-Informed Neural Networks (PINNs) improve the efficiency of data utilization by combining physical principles with neural network algorithms and thus ensure that their predictions are consistent and stable with the physical laws. PINNs open up a new approach to address inverse problems in fluid mechanics. Based on the single-relaxation-time lattice Boltzmann method (SRT-LBM) with the Bhatnagar–Gross–Krook (BGK) collision operator, the PINN-SRT-LBM model is proposed in this paper for solving the inverse problem in fluid mechanics. The PINN-SRT-LBM model consists of three components. The first component involves a deep neural network that predicts equilibrium control equations in different discrete velocity directions within the SRT-LBM. The second component employs another deep neural network to predict non-equilibrium control equations, enabling the inference of the fluid’s non-equilibrium characteristics. The third component, a physics-informed function, translates the outputs of the first two networks into physical information. By minimizing the residuals of the physical partial differential equations (PDEs), the physics-informed function infers relevant macroscopic quantities of the flow. The model evolves two sub-models that are applicable to different dimensions, named the PINN-SRT-LBM-I and PINN-SRT-LBM-II models according to the construction of the physics-informed function. The innovation of this work is the introduction of SRT-LBM and discrete velocity models as physical drivers into a neural network through the interpretation function. Therefore, the PINN-SRT-LBM allows a given neural network to handle inverse problems of various dimensions and focus on problem-specific solving. Our experimental results confirm the accurate prediction by this model of flow information at different Reynolds numbers within the computational domain. Relying on the PINN-SRT-LBM models, inverse problems in fluid mechanics can be solved efficiently.

Velocity discretization for lattice Boltzmann method for noncontinuum bounded gas flows at the micro- and nanoscale

The lattice Boltzmann (LB) method intrinsically links to the Boltzmann equation with the Bhatnagar–Gross–Krook collision operator; however, it has been questioned to be able to simulate noncontinuum bounded gas flows at the micro- and nanoscale, where gas moves at a low speed but has a large Knudsen number. In this article, this point has been verified by simulating Couette flows at large Knudsen numbers (e.g., Kn=10 and Kn=100) through use of the linearized LB models based on the popular half-range Gauss–Hermite quadrature. The underlying cause for the poor accuracy of these conventional models is analyzed in the light of the numerical evaluation of the involved Abramowitz functions. A different thought on velocity discretization is then proposed using the Gauss–Legendre (GL) quadrature. Strikingly, the resulting GL-based LB models have achieved high accuracy in simulating Couette flows, Poiseuille flows, and lid-driven cavity flows in the strong transition and even free molecular flow regimes. The numerical study in this article reveals an essentially distinct but workable way in constructing the LB models for simulating micro- and nanoscale low-speed gas flows with strong noncontinuum effects.

Multilevel asymptotic-preserving Monte Carlo for kinetic-diffusive particle simulations of the Boltzmann-BGK equation

We develop a novel multilevel asymptotic-preserving Monte Carlo method, called Multilevel Kinetic-Diffusion Monte Carlo (ML-KDMC), for simulating the kinetic Boltzmann transport equation with a Bhatnagar–Gross–Krook (BGK) collision operator. This equation occurs, for instance, in mathematical models of the neutral particles in the plasma edge of nuclear fusion reactors. In this context, the Kinetic-Diffusion Monte Carlo method is known to maintain accuracy both in the low-collisional and the high-collisional limit, without an exploding simulation cost in the latter. We show that, by situating this method within a Multilevel Monte Carlo (MLMC) framework, using a hierarchy of larger time step sizes, the simulation cost is reduced even further. The different levels in our ML-KDMC method are connected via a new and improved recipe for correlating particle trajectories with different time step sizes. Furthermore, a new and more general level selection strategy is presented. We illustrate the efficiency of our ML-KDMC method by applying it to a one-dimensional test case with non-homogeneous and anisotropic plasma background. Our method yields significant speedups compared to the single-level KDMC scheme, both in the low- and high-collisional regime. In the high-collisional case, our ML-KDMC outperforms the single-level KDMC method by several orders of magnitude.

Driven electrostatic phase space vortices in a 1D weakly dissipative Vlasov–Poisson system

The effect of collisions on driven electrostatic phase space vortices is analyzed by means of Eulerian simulation for two different collision models. It was demonstrated recently [P. Trivedi and R. Ganesh, Phys. Plasmas 23, 062112 (2016)] that in the absence of collisions, at late times, steady state phase space vortices manifest to form a plateau in the resonant region of the particle velocity distribution function, due to trapping of particles supporting multiextrema giant phase space vortices (PSVs). In the presence of collisions, over long time, this multiextrema plateau are found to smooth out, since collisions drive the velocity distribution toward Maxwellian, irrespective of how weak the collisions are as long as they are non-zero. In these conditions, kinetic processes and collisionality are found to be in competition, and the evolution of the plasma is found, therefore, to be a result of nontrivial combination of these two effects. An attempt has been made by means of numerical simulations to study the effect of weak collisionality on the electrostatic driven phase space vortices with two types of collision operators: (1) Bhatnagar–Gross–Krook (Krook) collision operator, where the colliding particles can be treated as isolated pairs and, (2) Fokker–Planck (FP) type collision operator (Zakharov–Karpman) in one dimension, where many weak collisions lead to particle diffusion in velocity space. It is shown that depending on the collision model used, the nature of smoothing in velocity space of giant PSVs results in qualitatively very different phase space structures. However, irrespective of the collision model used, excess density fractions over 10% are retained.

Large-eddy simulation coupled with wall models for turbulent channel flows at high Reynolds numbers with a lattice Boltzmann method — Application to Coriolis mass flowmeter

The numerical prediction of pressure drop within wall-bounded flow domains at high Reynolds numbers (Re) using a large-eddy simulation (LES) approach is a challenging task for industrial applications because the fluid domain is usually underresolved. A lattice Boltzmann method (LBM) with Bhatnagar, Gross and Krook (BGK) collision operator coupled with the Smagorinsky–Lilly turbulence model is used to model these wall-bounded turbulent flows. The near wall region is modelled using wall functions to decrease the required mesh resolution for high Re. The influence of different velocity boundary approaches and wall functions is investigated for the benchmark bi-periodic fully developed turbulent channel flow for friction Reynolds numbers (Reτ) of 1000, 2000 and 5200. This benchmark case is validated against direct numerical simulation (DNS) results for turbulent statistics of 1st and 2nd order. Based on this validation, the pressure drop of an industrial Coriolis mass flowmeter is compared to experimental data for Re up to 127 800. The error of the pressure drop calculation in underresolved grids is reduced by two orders of magnitude in comparison to a no-slip approach for curved boundaries.

Application of a lattice Boltzmann method combined with a Smagorinsky turbulence model to spatially resolved heat flux inside a refrigerated vehicle

In this work the simulation of velocity and temperature distributions inside a refrigerated vehicle is evaluated. For this purpose a 3D double distribution lattice Boltzmann method (LBM) with the Bhatnagar–Gross–Krook (BGK) collision operator is coupled by the buoyancy force calculated with the Boussinesq approximation. This LBM is extended by a Smagorinsky subgrid method, which numerically stabilizes the BGK scheme for low resolutions and high Reynolds and Rayleigh numbers. Besides validation against the two benchmark cases porous plate and natural convection in a square cavity evaluated at resolutions of y+≈2 for Ra numbers between 103 and 1010, the method and its implementation are tested via comparison with experimental data for a refrigerated vehicle at Re≈53000.The aim of the investigation is to provide a deeper understanding of the refrigerated vehicle’s insulation processes including its thermal performance under turbulent flow conditions. Therefore, we extend this method by the half lattice division scheme for conjugate heat transfer to simulate in the geometry of a refrigerated vehicle including its insulation walls. This newly developed method combination enables us to accurately predict velocity and temperature distributions inside the cooled loading area, while spatially resolving the heat flux through the insulation walls. We simulate the time dependent heating process of the open door test and validate against measurements at four characteristic velocity and 13 temperature positions in the truck.

Advanced lattice Boltzmann scheme for high-Reynolds-number magneto-hydrodynamic flows

ABSTRACTIs the lattice Boltzmann method suitable to investigate numerically high-Reynolds-number magneto-hydrodynamic (MHD) flows? It is shown that a standard approach based on the Bhatnagar–Gross–Krook (BGK) collision operator rapidly yields unstable simulations as the Reynolds number increases. In order to circumvent this limitation, it is here suggested to address the collision procedure in the space of central moments for the fluid dynamics. Therefore, an hybrid lattice Boltzmann scheme is introduced, which couples a central-moment scheme for the velocity with a BGK scheme for the space-and-time evolution of the magnetic field. This method outperforms the standard approach in terms of stability, allowing us to simulate high-Reynolds-number MHD flows with non-unitary Prandtl number while maintaining accuracy and physical consistency.

Improved coupling of time integration and hydrodynamic interaction in particle suspensions using the lattice Boltzmann and discrete element methods

This paper introduces improvements to the simulation of particle suspensions using the lattice Boltzmann method (LBM) and the discrete element method (DEM). First, the benefit of using a two-relaxation-time (TRT) collision operator, instead of the popular Bhatnagar–Gross–Krook (BGK) collision operator, is demonstrated. Second, a modified solid weighting function for the partially saturated method (PSM) for fluid–solid interaction is defined and tested. Results are presented for a range of flow configurations, including sphere packs, duct flows, and settling spheres, with good accuracy and convergence observed. Past research has shown that the drag, and consequently permeability, predictions of the LBM exhibit viscosity-dependence when used with certain boundary conditions such as bounce-back or interpolated bounce-back, and this is most pronounced when the BGK collision operator is employed. The improvements presented here result in a range of computational viscosities, and therefore relaxation parameters, within which drag and permeability predictions remain invariant. This allows for greater flexibility in using the relaxation parameter to adjust the LBM timestep, which can subsequently improve synchronisation with the time integration of the DEM. This has significant implications for the simulation of large-scale suspension phenomena, where the limits of computational hardware persistently constrain the resolution of the LBM lattice.

Collisional damping of the geodesic acoustic mode with poloidal rotation

The collisional damping of the geodesic acoustic mode is investigated by using the gyrokinetic equation with a poloidal rotation flow and a number-conserving Krook collision operator. A dispersion relation of the geodesic acoustic mode with poloidal rotation and collision terms is derived and solved both analytically and numerically. It is shown that poloidal rotation has a significant influence on the collisional damping rate of the geodesic acoustic mode within a certain collision rate and safety factor domain. The variation of the damping rate induced by poloidal rotation depends on the safety factor and collision rate. In addition, the effect of poloidal rotation on the geodesic acoustic mode would be weakened by the safety factor, especially when the safety factor is relatively large. The ratio of the collisional damping rate to the collisionless damping rate of the geodesic acoustic mode in the large safety factor domain is performed, and the critical collision rate is estimated. Besides, the physical mechanism of the results is briefly discussed.

High‐order discontinuous Galerkin nonlocal transport and energy equations scheme for radiation hydrodynamics

SummaryThe nonlocal theory of the radiative energy transport in laser‐heated plasmas of arbitrary ratio of the characteristic inhomogeneity scale length to the photon mean free paths is applied to define the closure relations of a hydrodynamic system. The corresponding transport phenomena cannot be described accurately using the Chapman–Enskog approach, that is, with the usual fluid approach dealing only with local values and derivatives. Thus, we directly solve the photon transport equation allowing one to take into account the effect of long‐range photon transport. The proposed approach is based on the Bhatnagar–Gross–Krook collision operator using the photon mean free path as a unique parameter. Such an approach delivers a calculation efficiency and an inherent coupling of radiation to the fluid plasma parameters in an implicit way and directly incorporates nonequilibrium physics present under the condition of intense laser energy deposition due to inverse bremsstrahlung. In combination with a higher order discontinuous Galerkin scheme of the transport equation, the solution obeys both limiting cases, that is, the local diffusion asymptotic usually present in radiation hydrodynamics models and the collisionless transport asymptotic of free‐streaming photons. In other words, we can analyze the radiation transport closure for radiation hydrodynamics and how it behaves when deviating from the conditions of validity of Chapman–Enskog method, which is demonstrated in the case of exact steady transport and approximate multigroup diffusion numerical tests. As an application, we present simulation results of intense laser‐target interaction, where the radiative energy transport is controlled by the mean free path of photons. Copyright © 2016 John Wiley & Sons, Ltd.

An Expanded Lattice Boltzmann Method for Dual Phase Lag model

Fourier law can be solved by means of the classical Lattice Boltzmann Method (LBM) with Bhatnagar–Gross–Krook (BGK) collision operator. However, classical LBM is challenged in the micro-scale conduction. In present work, we proposed a new collision operator based on Dual Phase Lag (DPL) model, a generalized conduction law, and developed an Expanded Lattice Boltzmann Method (ELBM). To validate our derivation, a numerical example is presented. It is found that the results predicted by the ELBM coincide with the analytical solutions of DPL model quite well. ELBM has a simpler differential form than other classical simulation methods.

Collisional damping of the geodesic acoustic mode

The frequency and damping rate of the geodesic acoustic mode (GAM) is revisited by using a gyrokinetic model with a number-conserving Krook collision operator. It is found that the damping rate of the GAM is non-monotonic as the collision rate increases. At low ion collision rate, the damping rate increases linearly with the collision rate; while as the ion collision rate is higher than vti/R, where vti and R are the ion thermal velocity and major radius, the damping rate decays with an increasing collision rate. At the same time, as the collision rate increases, the GAM frequency decreases from the (7/4+τ)vti/R to (1+τ)vti/R, where τ is the ratio of electron temperature to ion temperature.

Lattice ellipsoidal statistical BGK model for thermal non-equilibrium flows

AbstractA thermal lattice Boltzmann model is constructed on the basis of the ellipsoidal statistical Bhatnagar–Gross–Krook (ES-BGK) collision operator via the Hermite moment representation. The resulting lattice ES-BGK model uses a single distribution function and features an adjustable Prandtl number. Numerical simulations show that using a moderate discrete velocity set, this model can accurately recover steady and transient solutions of the ES-BGK equation in the slip-flow and early transition regimes in the small-Mach-number limit that is typical of microscale problems of practical interest. In the transition regime in particular, comparisons with numerical solutions of the ES-BGK model, direct and low-variance deviational Monte Carlo simulations show good accuracy for values of the Knudsen number up to approximately $0. 5$. On the other hand, highly non-equilibrium phenomena characterized by high Mach numbers, such as viscous heating and force-driven Poiseuille flow for large values of the driving force, are more difficult to capture quantitatively in the transition regime using discretizations chosen with computational efficiency in mind such as the one used here, although improved accuracy is observed as the number of discrete velocities is increased.

Collision effects on high-n ballooning modes with a full Lorentz collision operator

The dispersion relation for high-n ballooning modes is obtained from the customary drift gyro-kinetic equation employing the energy-dependent Lorenz collision operator. There are significant differences between our results and those with the Krook collision operator. The present results show that collision seems to have much weaker stabilizing effects, if any, on the high-n ballooning modes than those with the Krook collision operator when the collisionality, υ*e, is smaller than a critical value. However, contrary to the phenomenon, the collision shows strong stabilizing effects on the ballooning mode once the collisionality exceeds a critical value. These differences are supposed to arise from boundary layer behaviour which refers to the electron perturbation at the boundary between trapped and passing electrons in velocity space, but the physical process cannot be described by non-conserving Krook collision operators. This destabilizing mechanism of boundary layer behaviour is discussed in this paper. In addition, the critical value shows a considerably strong dependence on ηe (ηe ≡ d ln Te/d ln Ne).

A viscosity counteracting approach in the lattice Boltzmann BGK model for low viscosity flow: Preliminary verification

Due to numerical instability, the lattice Boltzmann model (LBM) with the Bhatnagar–Gross–Krook (BGK) collision operator has some limitations in the simulation of low viscosity flows. In this paper, we propose a viscosity counteracting approach for simulating a moderate viscosity flow. An extra negative viscosity term is introduced to counteract part of the moderate viscosity by using the lattice Boltzmann equation with a source term. The counteracting viscosity term is treated as a non-uniform unsteady source. The stability is enhanced; thus small viscosity flows can be simulated. Model verification consists of benchmark cases such as those of Poiseuille flow, Couette flow, waterhammer waves, Taylor–Green vortex flow, and lid-driven cavity flow. The flow patterns, error characteristics, and representative parameters are carefully analyzed. It is shown that this approach can simulate flows with lower viscosities than may be simulated using the normal LBGK model; the second-order accuracy of the LBGK model is definitely retained, although a little dissipation is added. These preliminary studies prove the effectiveness and accuracy of the model. Sophisticated analysis and further verification of the stability mechanism will be done in the near future.

Non-negativity and stability analyses of lattice Boltzmann method for advection–diffusion equation

Stability is one of the main concerns in the lattice Boltzmann method (LBM). The objectives of this study are to investigate the linear stability of the lattice Boltzmann equation with the Bhatnagar–Gross–Krook collision operator (LBGK) for the advection–diffusion equation (ADE), and to understand the relationship between the stability of the LBGK and non-negativity of the equilibrium distribution functions (EDFs). This study conducted linear stability analysis on the LBGK, whose stability depends on the lattice Peclet number, the Courant number, the single relaxation time, and the flow direction. The von Neumann analysis was applied to delineate the stability domains by systematically varying these parameters. Moreover, the dimensionless EDFs were analyzed to identify the non-negative domains of the dimensionless EDFs. As a result, this study obtained linear stability and non-negativity domains for three different lattices with linear and second-order EDFs. It was found that the second-order EDFs have larger stability and non-negativity domains than the linear EDFs and outperform linear EDFs in terms of stability and numerical dispersion. Furthermore, the non-negativity of the EDFs is a sufficient condition for linear stability and becomes a necessary condition when the relaxation time is very close to 0.5. The stability and non-negativity domains provide useful information to guide the selection of dimensionless parameters to obtain stable LBM solutions. We use mass transport problems to demonstrate the consistency between the theoretical findings and LBM solutions.

Numerical study of linear dissipative drift electrostatic modes in tokamaks

The linear stability of dissipative drift electrostatic modes in tokamak plasmas is studied numerically with an extended version of the gyrokinetic code Kinezero [C. Bourdelle et al., Nucl. Fusion 42, 892 (2002)] including a modified Krook collision operator to account for collisional effects on the trapped electron response. This new version of Kinezero has been successfully tested and benchmarked against the results of a more sophisticated and complete gyrokinetic solver, GS2 [M. Romanelli, C. Bourdelle, and W. Dorland, Phys. Plasmas 11, 3845 (2004)]. The critical density and temperature gradients for ion gradient driven modes (ITG) and trapped electron modes (TEM) have been computed for different values of collisionality. The threshold for TEM disappears as the collisionality increases. A detailed study of the dependence of the growth rate of the dissipative TEM on collisionality and density gradient is presented in the paper. The range of parameters where the dissipative trapped electron modes are destabilized by increasing collisionality corresponds to very steep density gradients that can hardly be achieved in today's large tokamaks.

An investigation of water-gas transport processes in the gas-diffusion-layer of a PEM fuel cell by a multiphase multiple-relaxation-time lattice Boltzmann model

In order to study water-gas transport processes in the gas-diffusion-layer (GDL) of a proton exchange membrane (PEM) fuel cell system, a multiphase, multiple-relaxation-time lattice Boltzmann model is presented in this work. The model is based on the mean-field diffuse interface theory and can handle the multiphase flows with large density ratios and various viscosities. By using the standard bounce back boundary condition and an approximate average scheme for the non-slip and wetting boundary walls, respectively, detailed liquid-gas transportation in the GDL, in which exact boundary condition is difficult to be implemented, can be simulated. Unlike most of lattice Boltzmann methods based on the Bhatnagar–Gross–Krook collision operator, the present model shows a viscosity-independent velocity field, which is very important in simulating multiphase flows where various viscosities coexist. We validate our model by simulating a static droplet on a wetting wall and compare with theoretical predictions. Then, we simulate a water-gas flow in the GDL of a PEM fuel cell and investigate the saturation-dependent transport properties under different conditions. The results are shown to be qualitatively consistent with the previous numerical and theoretical works.

Direct numerical and large eddy simulation of longitudinal flow along triangular array of rods using the lattice Boltzmann method

Results of direct numerical (DNS) and large eddy simulation (LES) of turbulent longitudinal flow in rod bundles are presented using the lattice Boltzmann method with the Bhatnagar–Gross–Krook collision operator [P.L. Bhatnagar, E.P. Gross, M. Krook, A model for collision processes in gases. I. Small amplitude processes in charged and neutral one-component systems, Phys. Rev. 94 (1954) 511; Y.H. Qian, d’Humiéres, P. Lallemand, Lattice BGK models for Navier-Stokes equation, Europhys. Lett. 17 (1992) 479] as a computational framework. The problem requires the accurate modeling of curved walls, to which the method proposed by Yu et al. [D. Yu, M.R. Luo, W. Shyy, Viscous flow computations with the method of lattice Boltzmann equation, Prog. Aerospace Sci. 39 (2003) 329] has been applied. The computational domain is a regular hexagonal prism around the rod. Opposite sides of the prism are coupled periodically. In the longitudinal direction periodical boundary conditions are applied and the flow is driven by a body force. Simulations were carried out using two three-dimensional lattices. It has been found that the application of the model with 19 velocities (D3Q19) gives qualitatively false result. However, we have found that the application of the model with 27 links (D3Q27) can provide the proper mean axial velocity profile, and it also predicts the secondary flow patterns deduced from measurements [A.C. Trupp, R.S. Azad, The structure of turbulent flow in triangular array rod bundles, Nucl. Eng. Des. 32 (1975) 47]. Flow pulsation phenomenon is also observed in our simulations just like in some recent measurements of Krauss and Meyer [T. Krauss, L. Meyer, Experimental investigation of turbulent transport of momentum and energy in heated rod bundle, Nucl. Eng. Des. 180 (1998) 185].