Modified Equilibrium Distribution Function Research Articles

A thermal lattice Boltzmann model for evaporating multiphase flows

Modeling thermal multiphase flows has become a widely sought methodology due to its scientific relevance and broad industrial applications. Much progress has been achieved using different approaches, and the lattice Boltzmann method is one of the most popular methods for modeling liquid–vapor phase change. In this paper, we present a novel thermal lattice Boltzmann model for accurately simulating liquid–vapor phase change. The proposed model is built based on the equivalent variant of the temperature governing equation derived from the entropy balance law, in which the heat capacitance is absorbed into transient and convective terms. Then a modified equilibrium distribution function and a proper source term are elaborately designed in order to recover the targeting equation in the incompressible limit. The most striking feature of the present model is that the calculations of the Laplacian term of temperature, the gradient term of temperature, and the gradient term of density can be simultaneously avoided, which makes the formulation of the present model is more concise in contrast to all existing lattice Boltzmann models. Several benchmark examples, including droplet evaporation in open space, droplet evaporation on a heated wall, and nucleate boiling phenomenon, are carried out to assess numerical performance of the present model. It is found that the present model effectively improves the numerical accuracy in solving the interfacial behavior of liquid–vapor phase change within the lattice Boltzmann method framework.

Modified equilibrium distributions for Cooper-Frye particlization

We introduce a positive definite single-particle distribution that is suitable for describing the transition from a macroscopic hydrodynamic to a microscopic kinetic description during the late stages of heavy-ion collisions in the presence of moderately large viscous corrections. The modified equilibrium distribution function can be constructed with hydrodynamic input from either relativistic viscous fluid dynamics or anisotropic fluid dynamics. We test the modified equilibrium distribution's hydrodynamic output for a stationary hadron resonance gas subject to either shear stress, bulk pressure, or baryon diffusion current at a given freeze-out temperature and baryon chemical potential. While it does not reproduce all components of the net baryon current and energy-momentum tensor exactly, it significantly improves upon the customary linearized approximations for the non-equilibrium correction $\delta f_n$ which typically lead to unphysical negative distribution functions at large particle momenta. A comparison of particle spectra and $p_T$-differential elliptic flow coefficients from the Cooper-Frye formula computed with the modified equilibrium distribution and with linearized $\delta f_n$ corrections is presented, for two different (2+1)-dimensional hypersurfaces corresponding to central and non-central Pb+Pb collisions at the Large Hadron Collider (LHC).

A lattice Boltzmann model for the conjugate heat transfer

Conjugate heat transfer problems are of great importance in many applications, while the available lattice Boltzmann (LB) models have some limitations in the study of such problems with the heterogeneous media. In this paper, we develop a new LB model for the conjugate heat transfer problems where the evolution equation with a modified equilibrium distribution function and a source term is considered. Through the Chapman–Enksog analysis, the macroscopic temperature equation can be recovered correctly. In order to test the accuracy, efficiency and stability of this LB model, some simple but nontrivial benchmark problems, including the planar thermal Poiseuille flow with two immiscible fluids, the conjugate heat transfer between a stratified heterogeneous medium with variable thermophysical properties, the steady and unsteady-state planer thermal flows with the two immiscible fluids, the steady-state conjugate heat transfer problem with a vertical interface, and the natural convection in a square cavity filled with circular and square rods, are studied. It is found that the results obtained from the present LB model are in good agreement with the analytic solutions or some previous works. In addition, it is also demonstrated that the developed LB model has a second-order convergence rate in space, and is more stable than some previous LB models. Finally, it should be noted that in our work, there is no special treatment to the interface, and the continuity conditions at the interface can be satisfied automatically.

Lattice Boltzmann method for fractional Cahn-Hilliard equation

Fractional phase field models have been reported to suitably describe the anomalous two-phase transport in heterogeneous porous media, evolution of structural damage, and image inpainting process. It is commonly different to derive their analytical solutions, and the numerical solution to these fractional models is an attractive work. As one of the popular fractional phase-field models, in this paper we propose a fresh lattice Boltzmann (LB) method for the fractional Cahn-Hilliard equation. To this end, we first transform the fractional Cahn-Hilliard equation into the standard one based on the Caputo derivative. Then the modified equilibrium distribution function and proper source term are incorporated into the LB method in order to recover the targeting equation. Several numerical experiments, including the circular disk, quadrate interface, droplet coalescence and spinodal decomposition, are carried out to validate the present LB method. It is shown that the numerical results at different fractional orders agree well with the analytical solution or some available results. Besides, it is found that increasing the fractional order promotes a faster evolution of phase interface in accordance with its physical definition, and also the system energy predicted by the present LB method conforms to the energy dissipation law.

Lattice Boltzmann advection-diffusion model for conjugate heat transfer in heterogeneous media

Abstract Many practical flow configurations involve energy transfer in fluids, or in solids and fluids with different thermo-physical properties. The classical advection-diffusion lattice Boltzmann (LB) solver admits some errors when dealing with such configurations. Given that the macroscopic equation recovered by this model is only valid in the limit of incompressible flows with constant heat capacities, one would, for example, observe inconsistent fluxes at the interface of a fluid and solid with different densities or specific heat capacities. This inconsistency being second-order in space, it will have non-negligible effects on the final results. In this work, a modified equilibrium distribution function (EDF) is proposed to overcome these issues. The proposed scheme recovers the correct partial differential equation (PDE) describing energy transfer, as shown by a multi-scale Chapman–Enskog analysis. The performance of the model is checked through a variety of test-cases, involving conjugate heat transfer and variable specific heat capacities in both steady and unsteady configurations. In all cases the obtained results are in excellent agreement with reference data.

A novel two-dimensional coupled lattice Boltzmann model for thermal incompressible flows

A novel two-dimensional coupled lattice Boltzmann model is developed for thermal incompressible fluid flows. A modified equilibrium distribution function is proposed in the present model. A mesoscopic discrete force is coupled into the modified equilibrium distribution function based on the Boussinesq approximation. The outstanding advantages of the standard lattice Boltzmann method are retained in present model besides better numerical stability. The present model is validated by the numerical simulation of the natural and Rayleigh–Benard convection at a wide range of Rayleigh numbers. Excellent agreement between the present results and previous lattice Boltzmann method or theoretical prediction demonstrates that present model is an efficient numerical method for natural and Rayleigh–Bénard convection. Further, present model is also successfully assessed considering Rayleigh–Taylor instability. It is also easier and convenient to be implemented as compared with the previous thermal models.

A unified thermal lattice Boltzmann equation for conjugate heat transfer problem

In the present paper, an optimal two-relaxation-time (OTRT) thermal lattice Boltzmann equation (TLBE) is proposed to simulate conjugate heat transfer. The derivation process is based on a developed idea of virtual heat capacity correction method. Finally, it is found that a unified TLBE can be achieved. The derivation process shows how the modified equilibrium distribution function works in simulating conjugate heat transfer. It also reveals the relation between the virtual heat capacity correction method and the method of reconstructing the equilibrium distribution function. By using Chapman-Enskog expansion, the proposed OTRT TLBE can recover the standard energy equation up to second-order accuracy. For validation, six test cases including both steady and transient conjugate heat transfer with flat or curved interfaces are calculated. The results show that the OTRT TLBE performs well in simulating conjugate heat transfer problems with both flat and curved interfaces, at both transient and steady states, and the OTRT TLBE could have better numerical accuracy than the single-relaxation-time (SRT) TLBE.

Cosmological bounds on neutrino statistics

We consider the phenomenological implications of the violation of the Pauli exclusion principle for neutrinos, focusing on cosmological observables such as the spectrum of Cosmic Microwave Background anisotropies, Baryon Acoustic Oscillations and the primordial abundances of light elements. Neutrinos that behave (at least partly) as bosonic particles have a modified equilibrium distribution function that implies a different influence on the evolution of the Universe that, in the case of massive neutrinos, can not be simply parametrized by a change in the effective number of neutrinos. Our results show that, despite the precision of the available cosmological data, only very weak bounds can be obtained on neutrino statistics, disfavouring a more bosonic behaviour at less than 2σ.

Mass-conserving advection–diffusion Lattice Boltzmann model for multi-species reacting flows

Given the complex geometries usually found in practical applications, the Lattice Boltzmann (LB) method is becoming increasingly attractive. In addition to the simple treatment of intricate geometrical configurations, LB solvers can be implemented on very large parallel clusters with excellent scalability. However, reacting flows and especially combustion lead to additional challenges and have seldom been studied by LB methods. Indeed, overall mass conservation is a pressing issue in modeling multi-component flows. The classical advection–diffusion LB model recovers the species transport equations with the generalized Fick approximation under the assumption of an incompressible flow. However, for flows involving multiple species with different diffusion coefficients and density fluctuations – as is the case with weakly compressible solvers like Lattice Boltzmann –, this approximation is known not to conserve overall mass. In classical CFD, as the Fick approximation does not satisfy the overall mass conservation constraint a diffusion correction velocity is usually introduced. In the present work, a local expression is first derived for this correction velocity in a LB framework. In a second step, the error due to the incompressibility assumption is also accounted for through a modified equilibrium distribution function. Theoretical analyses and simulations show that the proposed scheme performs much better than the conventional advection–diffusion Lattice Boltzmann model in terms of overall mass conservation.

A lattice Boltzmann model for interphase conjugate heat transfer

ABSTRACTA lattice Boltzmann model is proposed with a newly modified equilibrium distribution function for solving the conservation form of the energy equation to treat the interphase conjugate heat transfer problems under both steady state and unsteady state. The temperature and heat flux continuity conditions at the interface can be inherently satisfied without needing any additional treatments, such as iterative computation, correcting procedure for the incoming distribution function, and the complicated calculation procedure for the source term, to account for the interphase conjugate heat transfer. The implementation of the present LB model, therefore, is more straightforward and more efficient than those in most previous models, especially for problems with complex interfaces. The applicability and accuracy of the proposed LB model were evaluated by some benchmark problems including both simple flat interface and complex interface geometry. The results show excellent agreements with analytical solutions or finite volume results, demonstrating that the present model can serve as a promising numerical technique for dealing with fluid flow and heat transfer in complex heterogeneous systems.

A novel two-dimensional coupled lattice Boltzmann model for incompressible flow in application of turbulence Rayleigh–Taylor instability

A novel coupled lattice Boltzmann model is developed for two-dimensional incompressible Rayleigh–Taylor instability. a modified equilibrium distribution function (D2Q13)is proposed in this paper. The present model is stable and reliable up to temperature jumps between top and bottom walls of the order of 50 ϰ the averaged bulk temperature. The regimes of mixing Rayleigh–Taylor instability are discussed using a simple scaling and the scaling relations obtained are validated by the present model. It is demonstrated that excellent agreement between the present results and the other numerical method or analytical solution shows that the present model is an efficient numerical method for Rayleigh–Taylor instability.

A novel lattice Boltzmann model for the solid–liquid phase change with the convection heat transfer in the porous media

A novel lattice Boltzmann (LB) model with double distribution functions is proposed to simulate the solid–liquid phase change problems with the convection heat transfer in the porous media at the representative elementary volume scale. A generalized LB model is adopted to simulate the velocity field of the liquid region in the presence of the porous media. Based on the framework of a LB model for nonlinear convection–diffusion equation, an LB evolution equation without a phase change source term, as well as a newly modified equilibrium distribution function including the effective total enthalpy and thermal capacity ratio, are constructed to solve the effective total enthalpy energy conservation equation for the porous media. The macroscopic effective total enthalpy energy conservation equation can be exactly recovered from the present LB model by the Chapman–Enskog procedure. As compared with the most previous models, the present model can provide a higher computational efficiency, because the iterative step or linear equations solving procedure for the latent heat source term can be avoided. Furthermore, the temperature and heat flux continuity conditions at the interface between phases of different thermophysical properties are inherently satisfied for the present model. In such a case, the differences in both thermal conductivity and specific heat between phases, as well as the variation of porosity can be tackled by independently adjusting the relaxation time and the thermal capacity ratio included in the equilibrium distribution function, respectively. Additionally, the numerical diffusion across the phase interface for temperature, induced by solid–liquid phase change, can be dramatically reduced in the present model through the MRT collision scheme with a special configuration for the relaxation matrix. Numerical tests for several benchmark problems are carried out to validate the wide practicability and the accuracy of the present model. Very good agreements with analytical solutions or previous experimental and numerical results can be achieved, demonstrating the present model can be served as a promising numerical technique for simulating the transient solid–liquid phase change problems in the porous media.

Improved lattice Boltzmann modeling of binary flow based on the conservative Allen-Cahn equation.

The primary and key task of binary fluid flow modeling is to track the interface with good accuracy, which is usually challenging due to the sharp-interface limit and numerical dispersion. This article concentrates on further development of the conservative Allen-Cahn equation (ACE) [Geier et al., Phys. Rev. E 91, 063309 (2015)10.1103/PhysRevE.91.063309] under the framework of the lattice Boltzmann method (LBM), with incorporation of the incompressible hydrodynamic equations [Liang et al., Phys. Rev.E 89, 053320 (2014)10.1103/PhysRevE.89.053320]. Utilizing a modified equilibrium distribution function and an additional source term, this model is capable of correctly recovering the conservative ACE through the Chapman-Enskog analysis. We also simulate four phase-tracking benchmark cases, including one three-dimensional case; all show good accuracy as well as low numerical dispersion. By coupling the incompressible hydrodynamic equations, we also simulate layered Poiseuille flow and the Rayleigh-Taylor instability, illustrating satisfying performance in dealing with complex flow problems, e.g., high viscosity ratio, high density ratio, and high Reynolds number situations. The present work provides a reliable and efficient solution for binary flow modeling.

A modified lattice Bhatnagar–Gross–Krook model for convection heat transfer in porous media

The lattice Bhatnagar–Gross–Krook (LBGK) model has become the most popular one in the lattice Boltzmann method for simulating the convection heat transfer in porous media. However, the LBGK model generally suffers from numerical instability at low fluid viscosity and effective thermal diffusivity. In this paper, a modified LBGK model, which incorporates the shear rate and temperature gradient in the equilibrium distribution functions, is developed for incompressible thermal flows in porous media at the representative elementary volume scale. With two additional parameters, the relaxation times in the collision process can be fixed at a proper value invariable to the viscosity and the effective thermal diffusivity. In addition, by constructing a modified equilibrium distribution function and a source term in the evolution equation of temperature field, the present model can recover the macroscopic equations correctly through the Chapman–Enskog analysis, which is another key point different from previous LBGK models. Several benchmark problems are simulated to validate the present model with the local computing scheme for the shear rate and temperature gradient, and the numerical results agree well with analytical solutions and/or those well-documented data in previous studies. It is also shown that the present model and the computational schemes for the gradient operators have a second-order accuracy in space, and better numerical stability of the present modified LBGK model than previous LBGK models is demonstrated.

Imposition of the no-slip boundary condition via modified equilibrium distribution function in lattice Boltzmann method

A novel scheme for implementation of the no-slip boundary conditions in the lattice Boltzmann method is presented. In detail, we have substituted the classical bounce-back idea by the direct velocity boundary condition specification employing geometric-based manipulation of the equilibrium distribution functions. In this way we have constructed the equilibrium density function in such a way that it imposes the desired Dirichlet boundary conditions at numerical boundary points. Therefore, in fact a kind of equilibrium boundary condition is made. This specification for general curved solid surfaces is made by means of immersed boundary concepts, but without any need to interpolating density distribution values. On the other hand, the results show that the method presents a faster solution procedure in comparison to the bounce-back scheme.

Modeled Boltzmann Equation and Its Application to Shock-Capturing Simulation

A modified equilibrium distribution function for the Bhatnagar-Gross-Krook-type modeled Boltzmann equation has recently been proposed. The function was deduced using acoustics scaling to normalize the equation and allowed a correct recovery of similarly normalized Euler equations. It is a combination of a Maxwellian distribution plus three other terms that are moments of particle velocity. The lattice counterpart of the modified equilibrium distribution function also led to an exact recovery of the Euler equations; therefore, there is no Mach number limitation in the entire approach. This lattice counterpart was able to replicate aeroacoustics problems involving vorticity-acoustic and entropy-acoustic interactions correctly, and the simulations were carried out using a finite difference lattice Boltzmann method employing only a two-dimensional, nine-velocity lattice. Thus formulated, the numerical scheme has no arbitrary constants and all calculations were carried out using one single relaxation time and a set of constants derived from the analysis. This paper investigates the validity and extent of the formulation to capture shocks and resolve contact discontinuity and expansion waves in one- and two-dimensional Riemann problems. The simulations are carried out using the same two-dimensional, nine-velocity lattice, and identical set of constants and relaxation time; they are compared with theoretical results and those obtained by solving the Euler equations directly using Harten's first-order numerical scheme. Good agreement is obtained for all test cases. However, the modified equilibrium distribution function is not suitable for shock structure simulation; for that, an exact recovery of the Navier-Stokes equations is required.

ASSESSMENT OF MAGNETOHYDRODYNAMIC LATTICE BOLTZMANN SCHEMES IN TURBULENCE AND RECTANGULAR JETS

Two lattice Boltzmann method (LBM) formulations are possible to account for the effect of the magnetic field on the velocity field in magnetohydrodynamic (MHD) flows. In the body-force formulation (BFF), the magnetic field effects manifest as an external acceleration. In the extended equilibrium formulation (EEF), the effect appears through a modified equilibrium distribution function. Further, for the velocity field itself, the available choices are the single-relaxation time (SRT) and multi-relaxation time (MRT) models. Thus, for MHD-LBM, there are four possible permutations: SRT-BFF, SRT-EEF, MRT-BFF and MRT-EEF. Numerical implementation of the first three have already been presented in the literature. In this work, we, (i) develop the numerical implementation of MRT-EEF and (ii) perform an assessment of the four possible approaches. Our results indicate that the MRT-EEF is the most robust and accurate of the MHD-LBM computational schemes examined.

Modeled Boltzmann Equation and Its Application to Direct Aeroacoustic Simulation

The Bhatnagar, Gross, and Krook modeled Boltzmann equation has been applied to simulate different fluid dynamics problems with varying degrees of success. However, its application to direct aeroacoustic computation is less successful. One possible reason could be its inability to recover the state equation correctly for a diatomic gas and hence an inaccurate determination of the speed of sound. The present study reports on the development of an improved modeled Boltzmann equation for aeroacoustics simulation. The approach is to modify the Maxwellian distribution normally assumed for the equilibrium particle distribution function. Constraints imposed are the exact recovery of the state equation for a diatomic gas and the Euler equations without invoking the small Mach number assumption. Thus formulated, a distribution function consisting of the Maxwellian distribution plus three other terms that attempt to account for particle-particle collisions is obtained. A velocity lattice method is used to solve the improved modeled Boltzmann equation using an equivalent lattice equilibrium distribution function. The simulations are validated against benchmark aeroacoustic problems whose solutions are deduced from a direct numerical simulation of the Euler equations. The results of the improved modeled Boltzmann equation obtained using a smaller computational domain are in excellent agreement with those deduced from direct numerical simulation using a larger computational domain, thus verifying the viability and correctness of the modified equilibrium distribution function.