Elsevier

Journal of Computational Physics

Volume 234, 1 February 2013, Pages 263-279
Journal of Computational Physics

Unified framework for a side-by-side comparison of different multicomponent algorithms: Lattice Boltzmann vs. phase field model

https://doi.org/10.1016/j.jcp.2012.09.029Get rights and content

Abstract

Lattice Boltzmann models (LBM) and phase field models (PFM) are two of the most widespread approaches for the numerical study of multicomponent fluid systems. Both methods have been successfully employed by several authors but, despite their popularity, still remains unclear how to properly compare them and how they perform on the same problem. Here we present a unified framework for the direct (one-to-one) comparison of the multicomponent LBM against the PFM. We provide analytical guidelines on how to compare the Shan–Chen (SC) lattice Boltzmann model for non-ideal multicomponent fluids with a corresponding free energy (FE) lattice Boltzmann model. Then, in order to properly compare the LBM vs. the PFM, we propose a new formulation for the free energy of the Cahn–Hilliard/Navier–Stokes equations. Finally, the LBM model is numerically compared with the corresponding phase field model solved by means of a pseudo-spectral algorithm. This work constitute a first attempt to set the basis for a quantitative comparison between different algorithms for multicomponent fluids. We limit our scope to the few of the most common variants of the two most widespread methodologies, namely the lattice Boltzmann model (SC and FE variants) and the phase field model.

Introduction

Since the beginning of the last century several approaches have been developed for theoretical analysis and numerical simulation of multicomponent and multiphase flows. The most common theoretical approach to study such complex systems is based on the sharp-interface assumption, in which the interface between the different fluids is considered of zero-thickness. Each component is characterized by its own physical properties, such as density, concentration, and viscosity which are uniform over the portion of the domain occupied by the single fluid component. These physical properties may change in a discontinuous way across the interface and their jumps are determined by the equilibrium conditions at the interface. The evolution of the system is described by a set of conservation laws (mass, momentum, energy, etc.) separately written for each component. Their solution requires also a set of boundary conditions at the interface, which ultimately has to be tracked. The specific numerical difficulties involved with the interface tracking can be circumvented by the adoption of diffuse interface methods, like the ones focus of the present study: the multicomponent Lattice Boltzmann method (LBM) and the phase field model (PFM). The drawback of these models is the high numerical resolution necessary to model real interface thickness that require a fictitious enlargement of the interface thickness. Despite this limitation, in recent years several researchers worked on the development and refinement of PFM as well as different variants of the multicomponent LBM.

The kinetic theory for multicomponent fluids and gas mixtures has received a lot of attention in literature [1], [2], [3], [4], [5], [6], [7]. Many of the kinetic models developed for the study of mixtures are based on the linearized Boltzmann equations, especially the single-relaxation-time model due to Bhatnagar et al. [8], also named BGK-model:f(x,v,t)t+v·f(x,v,t)+a·vf(x,v,t)=Ω(x,t)==-1τf(x,v,t)-f(eq)(x,v,t),where f(x,v,t) is the probability density function to find at the space–time location (x,t) a particle with velocity v. The collisional kernel, on the right hand side of Eq. (1), stands for the relaxation (with a characteristic relaxation time τ) towards the local equilibrium f(eq)(x,v,t) which, in turn, depends on the local coarse grained variables, as density and momentum:ρ(x,t)=f(x,v,t)dvρu(x,t)=f(x,v,t)vdv.a·vf(x,v,t) represents the effect of a volume/body force density, a, on the kinetic dynamics. Modern discrete-velocity counterparts of (1), the so-called Lattice Boltzmann methods (LBM), are able to simulate multiphase and multicomponent fluids and have attracted considerable attention from the scientific community [9], [10], [11], [12], [13], [14], [15], [16], [17], [18]. The LBM is an discrete form of Boltzmann kinetic equation describing the dynamics of a fictitious ensemble of particles [19], [20], [21], [22], whose motion and interactions are confined to a regular space–time lattice. This approach consists in the following evolution:fi(x+ciΔt,t+Δt)-fi(x,t)=-Δtτ[fi(x,t)-fi(eq)(x,t)],where fi(x,t) is the probability density function of finding a particle at site x and time t, moving in the direction of the i-th lattice speed ci with i=0,,b. Systematic ways to derive the discrete set of velocities in these models are either the discretization of the Boltzmann equation on the roots of Hermite polynomials [23], [68], [24], [25], [26], [27], [28], [29] or the construction of high-order lattices for more stable LBM based on entropic approaches [30], [31]. At the same time, the translation of the body/volume force a·vf(x,v,t) onto the discrete-lattice framework represented one of the most challenging issues in the last years of Lattice Boltzmann research [11], [12], [13], [14], [15], [16], [17], [32], [33], [34], [35], [36]. Through one of the first approaches proposed in the literature, the so called Shan–Chen (SC) approach [11], [12], the non-ideal interactions have been introduced directly at the discrete lattice level among the constituent (kinetic) particles [11], [12], [37]. These lattice forces embed the essential features and are able to produce phase separation (i.e. a non-ideal equation of state and a non-zero surface tension) as well as a detailed diffuse interface structure. The application of the SC models has been particularly fruitful for many applications [38], [36], [39], [40], [41], [42]. Nevertheless, its theoretical foundations have been object of debate in the recent years [32], [37], [40], [41], mainly because of the thermodynamic consistency of the mesoscopic interactions involved. On the other hand, in the so called free-energy (FE) models [15], [16], [17], the collisional properties of the model have been chosen in such a way that the large scale equilibrium is consistent with an underlying free energy functional, embedding both hard core effects and weak interacting tails. In this case, more traceable theoretical foundations have been provided, at least from the point of view of a continuum theory [43]. Among others, some studies have also performed where more elaborated lattice models, including the effect of an exclusion volume based on Enskog theory [32], [34], [33], effective equilibria [39], or even effective SC forces, were designed to match the desired bulk pressure of a given fluid [35], [36].

The phase field model is based on the idea that the interface between two fluids is a layer of finite thickness rather than a sharp discontinuity. Across the interfacial layer the physical properties of the mixture components vary in a smooth and continuous way, from one fluid to the other. This approach is based on the pioneering work of van der Waals [44], who first determined the interface thickness of a critical liquid–vapor mixture. In the PFM the state of the system is described, at any time, by an order parameter ϕ=ϕ(x), which is a function of the position vector x. The order parameter directly represents a physical properties of the fluid, such as its density, molar concentration, etc.; all the remaining properties are in turn modeled as proportional to ϕ(x) [45], [46]. According to the diffuse-interface approach, the order parameter is continuous over the entire domain and it shows smooth variations across the interface between single fluid regions, where ϕ(x) assumes approximately uniform values. Coupling the continuous and diffuse representation of the system with a transport equation of the order parameter, the system evolution can be resolved in time. One of the best-known PFM is the Cahn–Hilliard equation [47], [48], an extension to binary mixtures of the work of van der Waals [44]. This equation is a transport equation for the order parameter, where the evolution of ϕ(x) is proportional to the gradient of the chemical potential, μ. The chemical potential is defined in terms of the free energy functional, F[ϕ]:μ=δF[ϕ]δϕ,where the free energy, F[ϕ], assumes suitable definitions according to the problem under analysis (and also depending on which physical quantity has to be described) and is a conservative, thermodynamically consistent functional. The most common free energy formulation is given by the sum of an ideal part, Fid[ϕ], and a non-local part, Fnl[ϕ]. The ideal part accounts for the tendency of the system to separate in two pure components and is derived from the thermodynamics of mixtures. The non-local part accounts for the diffusive interfacial region. Through the Cahn–Hilliard equation, the evolution of the order parameter is thermodynamically consistent and subject to a phase field conservation. As a result, the prediction of the interfacial layer does not deteriorate. In the case of density-matched fluid systems, the convective Cahn–Hilliard equation is coupled with a modified Navier–Stokes equation, where a surface tension (or capillary) forcing term, which is derived from the Korteweg stress [49], is introduced. This contribution yields to the Cahn–Hilliard/Navier–Stokes coupled equations system [50], [51], [52], which is also known as Model-H, according to the classification of Hohenberg and Halperin [53], who studied the convective phase separation of a partially miscible fluid mixture. This model, originally developed to study critical phenomena [54], [55], [56], [57], [58], was subsequently used by many authors to study two-phase flows of Newtonian fluids [59], [50], [60], [46], [61]. In this case, even if the fluids are in fact immiscible, molecular diffusion between the two species is allowed in the interfacial region. Thus, the thinner is the interface, the more realistic is the numerical solution. Realistic interface thickness requires high numerical resolution, which is usually beyond current computational limits. For this reason the interface is often kept larger than corresponding physical value (this approximation holds also for lattice Boltzmann approaches). However, in spite of this approximation, the method has shown capabilities to capture complex interfacial dynamics in a wide range of real physical problems [45], [50], [62], [63].

Multicomponent LBM and PFM have demonstrated excellent performances to predict the dynamics of multiphase and multicomponent flows. Yet, both methods show their own peculiar characteristics and drawbacks which can limit their use, performances and range of validity. A particular unexpected, and unwanted, feature of multiphase and multicomponent solvers is the manifestation of non physical velocities near equilibrium interface, present even for systems at rest. From a physical viewpoint the velocity should clearly vanish at equilibrium but, as it has been observed by many authors, small spurious currents most often exist in the proximity of the interfaces. In an attempt to remove these unwanted features several improvement to the LBM have been proposed in recent years [64], [65], [66], [67], [23], [68], [69]. It worths noticing that some of these improvement are capable to remove these spurious current to machine precision [67]. Spurious currents have also been observed in other numerical methods including the PFM [70], [71], [72].

Because of the magnitude of these spurious current drastically depend on the actual variant of the LBM or PFM, the comparison between the two methods may be somewhat ill defined. Here we aim at comparing the LBM vs. the PFM for their most widespread and used variants. Our answer will thus not provide a general statement valid for the two methods as such, but will still provide some extremely useful insight in what can be expected from the most employed variations of the approaches. As a side results, we will also quantitatively compare two of the most widely used lattice Boltzmann variants, the SC-LBM and the FE-LBM, under the same conditions (i.e. same diffuse interface model, same surface tension, same chemical potential, etc.). In order to achieve our goal, the problem of a one-to-one matching of the PFM with SC/FE multicomponent LBM needs to be addressed first. The one-to-one matching of the two methods gives also the opportunity to clarify how they compare with respect to the computational costs. In order to address these issues we start by analyzing the SC model for two population with inter-particle repulsion; the large scale continuum limit is reviewed and formulated in terms of a diffuse interface model with an underlying thermodynamic FE functional. In this way one of the crucial issues in the matching of SC model vs. the corresponding FE model is being solved. Then, starting from the matched SC/FE multicomponent LBM, a new formulation for the free-energy of the PFM is derived in order to directly compare them. Finally, a comparison of the numerical results obtained, on the same problem, from both LBM and PFM is presented, focusing in particular on unwanted spurious currents or mass leakage in sheared suspensions. This work focuses only the case of binary mixtures, even if modeling more than two components is nowadays far from trial extension [73], [74], [75], [76], [77].

Section snippets

The multicomponent Shan–Chen model

In this section the multicomponent model introduced by Shan & Chen [11], [12] is reviewed. First the main properties of the model are recalled, then the equilibrium features (diffusive current and pressure tensor) relevant on the hydrodynamic scales are analyzed. Starting from a kinetic lattice Boltzmann equation [19], [20], [22] for a multicomponent fluid with Ns species [13], [14], the evolution equations over a characteristic time lapse Δt read as follows:fis(x+ciΔt,t+Δt)-fis(x,t)=-Δtτs[fis(x

The phase field model

In this section the phase field model for a multicomponent fluid system is analyzed. Starting from the features of the PFM proposed by Badalassi et al. [46] and Yue et al. [61], a new formulation of the free energy functional is derived to match the SC/FE multicomponent LBM reported in Section 2. For an isothermal binary mixture, where ϕ is the relative concentration of the mixture components, the free energy functional reads:F[ϕ]=F[ϕ]id+F[ϕ]nl=V(ϕ)+κ2|ϕ|2dx,where V(ϕ) is the ideal part of

Numerical tests

In this section the numerical results obtained from both the PFM and the LBM approaches are discussed. Two different tests have been performed with the three models, SC-LBM, FE-LBM and PFM, as described in Sections 2.1, 2.2, and 3 respectively. First, the equilibration of a two dimensional static droplet has been simulated until the steady state has been reached. Then, starting from the settled droplet, its deformation under a Kolmogorov flow has investigate. For the numerical analysis a SC-LBM

Conclusions

In this work two of the most widely adopted approaches for the numerical study of multicomponent fluid systems, the Lattice Boltzmann models (LBM) and phase field models (PFM) have been compared on equal footing. First the Shan–Chen (SC) multicomponent LBM and the free energy (FE) LBM have been reviewed and analyzed. Focusing on the specific case of phase separating fluids with two species, the long-wavelength limit (i.e. the hydrodynamical limit) of both lattice Boltzmann models has been

Acknowledgements

We thank L. Busolini for the help in the development of the phase field model algorithm. This work was carried out under the HPC-EUROPA2 project (Project No.: 228398), with the support of the European Community – Research Infrastructure Action of the FP7. This work is part of the research programme of the Foundation for Fundamental Research on Matter (FOM), which is part of the Netherlands Organisation for Scientific Research (NWO). We acknowledge the COST Action MP0806 for support. The

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