Double Distribution Functions Research Articles

Numerical study of fluid flow and convective heat transfer over multi-cylindrical obstacles in a channel using the lattice Boltzmann method

The aim of this work is devoted to study the hydrodynamic and thermal behavior of fluid flowing around multi-cylindrical obstacles confined inside a 2D plane horizontal channel with constant-cool parallel plates. For this purpose, numerical simulations using the lattice Boltzmann method (LBM) based on double-distribution functions (DDFs) with a D2Q9 scheme is used to compute both velocity and temperature fields. The developed model is validated against established analytical and numerical results available in literature and the deviation does not exceed 2%. For a comprehensive parametric study, numerous calculations have been done for several parameters such as Reynolds number ( 10 ≤ Re ≤ 200 ), Prandtl number ( 0.1 ≤ Pr ≤ 2 ), blockage ratio ( 1 / 8 ≤ β ≤ 1 / 2 ) and number of obstacles. Effects on heat transfer evolution and flow characteristics are analyzed. Results are presented in terms of streamlines, isotherms and Nusselt numbers. The findings reveal a significant influence of the number of cylinders on both temperature and velocity distributions inside the channel. The average Nusselt number exhibits a linear correlation with Reynolds number, reaching a peak value at Re = 200 . These observations highlight the potential of LBM as an accurate and efficient tool to simultaneously predict the dynamic and thermal behavior of fluid flows in heat exchangers.

Mesoscopic Lattice Boltzmann Modeling of the Liquid-Vapor Phase Transition.

We develop a mesoscopic lattice Boltzmann model for liquid-vapor phase transition by handling the microscopic molecular interaction. The short-range molecular interaction is incorporated by recovering an equation of state for dense gases, and the long-range molecular interaction is mimicked by introducing a pairwise interaction force. Double distribution functions are employed, with the density distribution function for the mass and momentum conservation laws and an innovative total kinetic energy distribution function for the energy conservation law. The recovered mesomacroscopic governing equations are fully consistent with kinetic theory, and thermodynamic consistency is naturally satisfied.

Numerical analysis of mixed convection heat transfer and laminar flow in a rectangular inclined micro-channel totally filled with Water/Al2O3 Nano fluid

A numerical analysis was carried out of mixed convection heat transfer for a laminar flow in a rectangular inclined microchannel totally filled with a water/Al2O3 nanofluid. The governing conservation equations are translated into a dimensionless form using the thermal single relaxation time and they modify the lattice Boltzmann method with double distribution functions. The viscous dissipation effects are adapted to the energy equation. The effects of nanoparticle volume fractions ϕ (0 ≤ ϕ ≤ 0.04) and inclination angles γ (0° ≤ γ ≤ 60°) on the flow of the nanofluid and the heat transfer are investigated. The obtained results are presented in terms of streamlines, isotherms, slip velocity, wall temperature and Nusselt number. The results show that the higher values of the volume fraction of Al2O3 and the large values of inclination angles improve the heat transfer rate.

Numerical analysis of forced convection heat transfer in a rectangular micro-channel totally filled with Ag/ water nano fluid in slip flow regime using the lattice Boltzmann method

Numerical simulation reported on heat transfer and fluid flow in a two-dimensional rectangular micro channel totally filled with Ag/water. The first –order slip/jump boundary conditions were uniformly imposed to the up and bottom walls. The governing conservation equations are translated in dimensionless form using the thermal Single Relaxation Time (T-SRT) modified Lattice Boltzmann Method (LBM) with double distribution functions (DDFs). The viscous dissipations effects are adopted into the energy equation. Effects of nanoparticle volume fraction φ, slip coefficient, B, on the flow of Nano fluid and heat transfer were studied. The results were interpreter in terms of slip velocity; temperature jump and Nusselt number. Based on the results found, it can be concluded that decreasing the values of slip coefficient enhances the convective heat transfer coefficient and consequently the Nusselt number (Nu) but increases the slip velocity at the wall and temperature jump values.

Local vorticity computation approach in double distribution functions based lattice Boltzmann methods for flow and scalar transport

Computation of vorticity, or the skew-symmetric velocity gradient tensor, in conjunction with the strain rate tensor, plays an important role in the flow classification, in vortical structure identification and in the modeling of various complex fluids and flows. For the simulation of flows accompanied by the advection-diffusion transport of a scalar field (e.g., temperature), double distribution functions (DDF) based lattice Boltzmann (LB) methods, involving a pair of LB schemes are commonly used. We present a new local vorticity computation approach by introducing an intensional anisotropy of the scalar flux in the third order, off-diagonal moment equilibria of the LB scheme for the scalar field, and then combining the second order non-equilibrium components of both the LB methods. As such, any pair of lattice sets in the DDF formulation that can independently support the third order off-diagonal moments would enable local determination of the complete flow kinematics, with the LB methods for the fluid motion and the transport of the passive scalar respectively providing the necessary moment relationships to determine the symmetric and skew-symmetric components of the velocity gradient tensor. Since the resulting formulation is completely local and do not rely on any finite difference approximations for velocity derivatives, it is by design naturally suitable for parallel computation. As an illustration of our approach, we formulate a DDF-LB scheme for local vorticity computation using a pair of multiple relaxation times (MRT) based collision approaches on two-dimensional, nine velocity (D2Q9) lattices, where the necessary moment relationships to determine the velocity gradient tensor and the vorticity are established via a Chapman-Enskog analysis. Simulations of various benchmark flows demonstrate good accuracy of the predicted vorticity fields using our approach against available solutions, including numerical results, with a second order convergence. Furthermore, extensions of our formulation for a variety of collision models, including those based on cascaded and non-cascaded central moments, to enable local vorticity computation are presented.

A NOTE ON PROPERTIES OF EXPONENTIAL POWER DISTRIBUTION

The most common generalization of the normal, Kotz-symmetric and double exponential distribution functions was the exponential power distribution. This distribution had been found useful in modeling real life data as studied in the literature. The present study found it necessary to fill the void in the literature by presenting some properties which characterized exponential power distribution and further made it useful in applications.

WITHDRAWN: Modelling of free convection heat transfer in a triangular cavity equipped using double distribution functions (DDF) lattice Boltzmann method (LBM)

WITHDRAWN: Modelling of free convection heat transfer in a triangular cavity equipped using double distribution functions (DDF) lattice Boltzmann method (LBM)

Electron acoustic envelope solitons in non-Maxwellian plasmas

The NASA Van Allen Probes spacecraft has recently observed broadband electrostatic turbulence in the inner magnetosphere which has been conjectured to be produced by large amplitude electrostatic solitary waves of generally two types [D.M. Malaspina, J.R. Wygant, R.E. Ergun, G.D. Reeves, R.M. Skoug, B.A. Larsen, J. Geophys. Res.: Space Phys. 120, 4246 (2015); C.S. Dillard, I.Y. Vasko, F.S. Mozer, O.V. Agapitov, J.W. Bonnell, Phys. Plasmas 25, 022905, (2018)]. The solitary waves with highly asymmetric bipolar parallel electric field have been recently shown to correspond to the electron-acoustic plasma mode (existing due to two-temperature electron population) [C.S. Dillard, I.Y. Vasko, F.S. Mozer, O.V. Agapitov, J.W. Bonnell, Phys. Plasmas 25, 022905, (2018)]. These findings along with the observations of non-Maxwellian electrons in terrestrial magnetosheath and planetary magnetospheres [W. Masood, S.J. Schwartz, M. Maksimovic, A.N. Fazakerley, Ann. Geophys. 24, 1725 (2006); W. Masood, S.J. Schwartz, J. Geophys. Res. 113, A01216 (2008); M.N.S. Qureshi, W. Nasir, W. Masood, P.H. Yoon, H.A. Shah, S.J. Schwartz, J. Geophys. Res.: Space Phys. 119, 10059 (2014); M.N.S. Qureshi, W. Nasir, R. Bruno, W. Masood, MNRAS 488, 954 (2019)] have prompted us to theoretically investigate the problem of amplitude modulation of the electron acoustic waves (EAWs) in plasmas whose ingredients are stationary ions, inertial cold electrons and warm (r,q) distributed inertialess electrons. The nonlinear Schrodinger equation (NLSE) that governs the modulational instability (MI) of the EAWs has been derived using the standard reductive perturbation technique (RPT). The presence of the warm (r,q) distributed electrons has been shown to influence the existence conditions of MI of the EAWs and both the indices have been found to cause a decrease in the growth rate of MI. A detailed comparison has also been drawn between double spectral index (r,q), kappa and Maxwellian distribution functions and the differences are highlighted. Additionally, the nondimensional parameter α=nc0/ne0 which is the equilibrium density ratio of the cold to hot electron component, has been shown to play an important role in the formation of envelope solitary excitations.

Inverse Radon transform and the transverse-momentum dependent functions

We revisit the standard representation of the (inverse) Radon transform which is well-known in the mathematical literature. We extend this representation to the case involving the parton distributions. We have found the new additional contribution which is essentially related to the generalized transverse-momentum dependent parton distribution and double-distribution functions. We discuss the possible relationship of this term with the Sivers function.

Deicing performances of a road unit driven by a hydronic heating system in severely cold regions of China

The deicing performances of a road unit driven by a hydronic heating system in severely cold regions of China were investigated by using the Open Source Lattice Boltzmann Code. The model used the enthalpy-based method and double distribution functions for the velocity and temperature fields to solve the ice melting problem. The average road surface temperatures and melting conditions (melting ratio and mass) as affected by ice thickness (3 ∼20 mm), pipe spacing (80 ∼240 mm), and air temperature (252 ∼268 K) were detailed, and the parameters were analyzed by the orthogonal test method. The results show that for the ice thickness increasing from 3 mm to 20 mm, the heating rate was slightly increased from 6.3 K/h to 6.45 K/h in the preheating and initial melting stages, and from 1.32 K/h to 2.06 K/h in the rapid melting stage, the melting mass was increased from 360 g/m to 762.06 g/m whereas the melting ratio was decreased from 1.00 to 0.32. The pipe spacing of 120 mm was suitable for the road heating whereas ≥ 160 mm was not feasible in the severely cold regions. For the air temperature was increased from 252 K to 268 K, the preheating time was decreased from 3.02 h to 0.82 h, the heating rate in the melting stages was increased from 0.68 K/h to 2.94 K/h and the melting ratio was linearly increased. The parameter analysis reveals that the air temperature had the most important influence on the melting ratio whereas the pipe spacing had the most significant impact on the average road surface temperature, and the ice thickness had a slight influence on the melting ratio and average road surface temperature.

A lattice Boltzmann model for the fractional advection–diffusion equation coupled with incompressible Navier–Stokes equation

The fractional advection–diffusion problem coupled with incompressible Navier–Stokes equations is important in science and engineering. In this paper, a fresh lattice Boltzmann model with double-distribution functions for this problem is presented. One distribution function is used for pressure and velocity. The other is applied for concentration. Through Chapman–Enskog expansion, the macroscopic continuum equations can be recovered from the LB model. Numerical simulations are carried out to verify the numerical accuracy and efficiency.

Rayleigh-Bénard type natural convection heat transfer in two-dimensional geometries

The natural convection of Rayleigh-Bénard type for various Rayleigh numbers, in two-dimensional L shape and diagonally flipped-L shape enclosures is studied using multiple relaxation time (MRT) based lattice Boltzmann method (LBM). Two separate particle distribution functions (i.e. the use of double distribution functions (DDF) approach) are employed to obtain the flow- and temperature-fields. The variables are intrinsically coupled by considering the effect of density difference in the buoyancy term and neglecting the viscous heat dissipation and compression work due to pressure. The benchmark simulations are first performed for differentially heated square enclosure to ascertain the correctness of in-house build code. Then, simulations are carried out for L shape and diagonally flipped-L shape 2D enclosures by varying Rayleigh number from 103 to 108. The steady Rayleigh-Bénard convection (RBC) is observed for both the enclosures at low Rayleigh numbers (i.e. Ra = 103,104 and 105). However, for higher Rayleigh numbers, unsteadiness in the flow-structure is observed. This unsteady behavior of the flow is then characterized by using time-dependency signal of spatially averaged Nusselt number on the hot wall and its corresponding frequency analysis.

A Lattice Boltzmann Model for Two-Phase Flow in Porous Media

In this paper, a lattice Boltzmann (LB) model with double distribution functions is proposed for two-phase flow in porous media where one distribution function is used for pressure governed by the ...

Assessment of thermal heat loss from solar cavity receiver with Lattice Boltzmann method

Efficiency of concentrating solar power system (CSP) can be significantly reduced by heat losses from the receiver. So, the assessment of heat losses is the key to improve its thermal performance. This paper attempts to present a numerical simulation to assess the heat losses from a cavity receiver by using Lattice Boltzmann Method (LBM) coupled with net radiation method. Combined natural convection and surface radiation heat losses in open rectangular solar cavity receiver is presented. The wall facing the opening is subjected to parabolic temperature profile. The other walls are insulated. Cavity walls are assumed to be diffuse, gray and opaque while the open boundary is assumed to be a fictitious black surface at ambient temperature. Double distribution functions (D2Q9-D2Q4) with different relaxation times are used to predict dynamic and thermal fields. Influence of surface radiation, operating temperature, inclination angle and aspect ratio on heat losses from the hot wall inside the cavity is analyzed and discussed.It was found that convective heat loss is largely increasing by increasing the inclination angle. Concerning surface emissivity effect, an amplification by coefficient of 4 between ε = 0.2 and ε = 1.0 is accomplished for thermal radiation heat loss whereas convective heat loss is marginally reduced. It was also found that total heat loss from the cavity absorber is influenced by operating temperature and aspect ratio. In fact, increasing of operating temperature enhances heat loss while the increasing of aspect ratio reduces it.

Develop the lattice Boltzmann method to simulate the slip velocity and temperature domain of buoyancy forces of FMWCNT nanoparticles in water through a micro flow imposed to the specified heat flux

Lattice Boltzmann method ability is improved to simulate the mixed convection of Water / FMWCNT nanofluid inside a two dimensional microchannel. The influences of gravity on hydrodynamic and thermal domains are studied while the microchannel walls are imposed by a constant thermal heat flux at three different case studies as no-gravity, Ri=1 and Ri=10. The flow Reynolds number is chosen as one and the liquid micro flow conditions are involved by B = 0.005, B = 0.01 and B = 0.02. The mass fraction of carbon nanotubes in water are selected as ϕ=0, ϕ=0.1% and ϕ=0.2%. Double population distribution functions of “f” and “g” are used in lattice Boltzmann method. To the best of author’s knowledge, there is no article concerned the way of heat flux boundary condition simulation by LBM considering the buoyancy forces effects on nanofluid slip velocity. Generate a rotational cell due to gravity in entrance region which leads to observe the negative slip velocity phenomenon can be presented as the several interesting achievements of this work.

Pore-scale investigation of gravity effects on phase change heat transfer characteristics using lattice Boltzmann method

Latent heat thermal energy storage (LTES) has been recommended to complicated aerospace systems due to the excellent ability of storing thermal energy in recent years. Metal foam, as an effective additive to enhance the effective thermal conductive coefficient of phase change materials, has been wildly used in the LTES system. In support of the future space exploration, solid–liquid phase change within metal foam under different gravitational accelerations and corresponding heat transfer characteristics are investigated in this work to enhance understanding of heat transfer capability in reduced gravity conditions. A pore-scale lattice Boltzmann method with the double distribution functions is developed to study the solid–liquid phase change problems with the natural convection heat transfer in metal foam. Here, microstructure of metal foam is experimentally obtained by the use of X-ray computed tomography. Two numerical validation cases are performed, one for fluid–solid conjugate heat transfer problem, and the other for melting process in a square cavity without the porous media. The presented lattice Boltzmann results are in good agreement with previous analytical and numerical results. Based on the analysis of temperature and velocity fields during melting within metal foam, it is found that the clockwise circulation, caused by the temperature gradient, leads to the inhomogeneous distribution of liquid–solid interface along vertical direction. What’s more, gravity effects on the temperature and velocity fields together with the average Nusselt number and liquid fraction coefficient are investigated. Results show that with the decrease of gravitational accelerations from 1g to 0g, the maximum vertical velocity reduces and the thickness of velocity boundary layer thickens accordingly, and results in the pronounced reduction of average Nusselt number and melting rate. The natural convection appears to be even more tenuous as gravitational acceleration decreases, leading to the dominant heat transfer mechanism transited from convection to conduction, and finally alleviating the phase change process. The liquid fraction under 0g, 0.01g and 0.1g conditions are 54.6%, 58.9% and 67.9% respectively of that under 1g conditions at Fo = 0.06. In addition, porosity effects on phase change heat transfer characteristics are discussed by comparing the liquid fraction with porosity of 0.9, 0.94 and 1. It is found that metal foam with lower porosity shows higher effective thermal conductivity and outstanding heat transfer performance, while the restriction on natural convection should not be neglected, especially for melting region which is dominated by natural convection.

Lattice Boltzmann simulation of heat transfer with phase change in saturated soil during freezing process

ABSTRACTA lattice Boltzmann model is presented for simulating heat transfer with phase change in saturated soil. The model includes a quartet structure generation set for creating soil structure, double distribution functions for simulating temperature field evolution of soil particles and water, respectively, and an enthalpy-based method for tracing phase interface. The model is validated by two cases with analytical solutions. Then, we investigate the influence of porosity on freezing process in saturated sandy loam soil. The results demonstrate that porosity is the predominant factor when the location is far from the cold source; otherwise, thermal gradient is more important.

Implementation of the direct [formula omitted] method in the KENO Monte Carlo code

The Monte Carlo code KENO contains thermal scattering data for a wide variety of thermal moderators. These data are processed from Evaluated Nuclear Data Files (ENDF) by AMPX and stored as double differential probability distribution functions. The method examined in this paper uses S(α,β) probability distribution functions derived from the ENDF data files directly instead of being converted to double differential cross sections. This allows the size of the cross section data on the disk to be reduced substantially amount. KENO has also been updated to allow interpolation in temperature on these data so that problems can be run at any temperature. Results are shown for several simplified problems for a variety of moderators. In addition, benchmark models based on the KRITZ reactor in Sweden were run, and the results are compared with the previous versions of KENO without the direct S(α,β) method. Results from the direct S(α,β) method compare favorably with the original results obtained using the double differential cross sections. Sampling the data increases the run-time of the Monte Carlo calculation, but memory usage is decreased substantially.

Lattice Boltzmann simulation of transient natural convection of air in square cavity under a magnetic quadrupole field

Purpose The purpose of this paper is to apply the lattice Boltzmann method (LBM) with multiple distribution functions model, to simulate transient natural convection of air in a two-dimensional square cavity in the presence of a magnetic quadrupole field, under non-gravitational as well as gravitational conditions. Design/methodology/approach The density-temperature double distribution functions and D2Q9 model of LBM for the momentum and temperature equations are currently employed. Detailed transient structures of the flow and isotherms at unsteady state are obtained and compared for a range of magnetic force numbers from 1 to 100. Characteristics of the natural convection at initial moment, quasi-steady state and steady state are presented in present work. Findings At initial time, effects of the magnetic field and gravity are both relatively limited, but the effects become efficient as time evolves. Bi-cellular flow structures are obtained under non-gravitational condition, while the flow presents a single vortex structure at first under gravitational condition, and then emerges a bi-cellular structure with the increase of magnetic field force number. The average Nusselt number generally increases with the augment of magnetic field intensity. Practical implications This paper will be useful in the researches on crystal material and protein growth, oxygen concentration sensor, enhancement or suppression of the heat transfer in micro-electronics and micro-processing technology, etc. Originality/value The current study extended the application of LBM on the transient natural convective problem of paramagnetic fluids in the presence of an inhomogeneous magnetic field.

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.