A lattice based solution of the collisional Boltzmann equation with applications to microchannel flows This article has been downloaded from IOPscience. Please scroll down to see the full text article. J. Stat. Mech. (2013) P07016 (http://iopscience.iop.org/1742-5468/2013/07/P07016) Download details: IP Address: 146.229.56.102 The article was downloaded on 05/08/2013 at 14:11 Please note that terms and conditions apply. View the table of contents for this issue, or go to the journal homepage for more Home Search Collections Journals About Contact us My IOPscience
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A lattice based solution of the collisional Boltzmann equation with applications to
microchannel flows
This article has been downloaded from IOPscience. Please scroll down to see the full text article.
ournal of Statistical Mechanics:J Theory and Experiment
A lattice based solution of thecollisional Boltzmann equation withapplications to microchannel flows
B I Green and Prakash Vedula
School of Aerospace and Mechanical Engineering, University of Oklahoma,Norman, OK 73019, USAE-mail: [email protected] and [email protected]
Received 10 April 2013Accepted 10 July 2013Published 30 July 2013
Online at stacks.iop.org/JSTAT/2013/P07016doi:10.1088/1742-5468/2013/07/P07016
Abstract. An alternative approach for solution of the collisional Boltzmannequation for a lattice architecture is presented. In the proposed method, termedthe collisional lattice Boltzmann method (cLBM), the effects of spatial transportare accounted for via a streaming operator, using a lattice framework, and theeffects of detailed collisional interactions are accounted for using the full collisionoperator of the Boltzmann equation. The latter feature is in contrast to theconventional lattice Boltzmann methods (LBMs) where collisional interactionsare modeled via simple equilibrium based relaxation models (e.g. BGK). Theunderlying distribution function is represented using weights and fixed velocityabscissas according to the lattice structure. These weights are evolved basedon constraints on the evolution of generalized moments of velocity accordingto the collisional Boltzmann equation. It can be shown that the collisionintegral can be reduced to a summation of elementary integrals, which canbe analytically evaluated. The proposed method is validated using studies ofcanonical microchannel Couette and Poiseuille flows (both body force andpressure driven) and the results are found to be in good agreement with thoseobtained from conventional LBMs and experiments where available. Unlikeconventional LBMs, the proposed method does not involve any equilibrium basedapproximations and hence can be useful for simulation of highly nonequilibriumflows (for a range of Knudsen numbers) using a lattice framework.
The traditional lattice Boltzmann method (LBM), which is based on kinetic theory,has been used to efficiently solve problems in fluid dynamics relevant to a variety ofsub-disciplines including single- and multiphase flows, interfacial flows, porous mediaflows, microflows and biological flows [1, 2]. However, generalization of this approachto the description of highly nonequilibrium flows (where the Knudsen number is notalways a small parameter) is challenging, due to the equilibrium based assumptionsinvolved in the traditional LBM. The latter challenge may be addressed via formulationsderived from fundamental kinetic theory, based on the full collisional Boltzmannequation. Various methods have been used to obtain the solutions to the Boltzmannequation. Chapman–Enskog and Burnett analysis uses a multi timescale expansion ofthe full Boltzmann equation [3]–[5]. The linearized Boltzmann method replaces thenonlinear collision operator with a linear approximation [6]. Particle methods such asdirect simulation Monte Carlo [7] and various hybrid (particle–continuum or moleculardynamics–continuum) models [8, 9] have also been presented as methods for solving theBoltzmann equation. While all these methods have merit and have found application ina number of simulations, they can also be computationally expensive or may suffer fromlimited accuracy, statistical noise or instability issues [5, 10, 11].
In order to address some of the underlying challenges, we present in this paper anew and unified method termed the collisional lattice Boltzmann method (cLBM) foranalyzing the Boltzmann equation within a lattice framework, applicable over a broadrange of Reynolds numbers and Knudsen numbers. This proposed method shares somecommon features with the traditional lattice Boltzmann method (LBM). LBM is oftenderived from an equilibrium based approximation of the full Boltzmann equation througha discretization of space and time. This discretization results in a simplified latticegeometry and consequently a simplification of the velocity distribution functions. LBM
A lattice based solution of the collisional Boltzmann equation
has the advantage of being relatively simple to implement and may be easily extendedinto parallel computing routines [2]. There are several features of the LBM that areimportant to note. First, LBM assumes that the collisional timescale is generally muchsmaller than the transport timescale. These effects may therefore be treated separatelyand thus the influence of spatial transport becomes decoupled from that of intermolecularcollisions. Transport or streaming on a discrete lattice can be represented via a simpleoperation involving data swapping. Spatial discretization also requires the use of a discretepopulation distribution function. Traditionally, this discrete distribution function has beenobtained by means of Gauss–Hermite quadrature, but in the proposed method, quadratureis accomplished by means of a combination of analytical and numerical integration. Theadvantage here is that while Gauss–Hermite quadrature requires the use of a specific setof velocity abscissas, cLBM may be applied to any arbitrary (but presumably symmetric)set of abscissas. This allows for the lattice to be scaled as needed to simulate viscouseffects.
The challenge with LBM remains in determining the influence of intermolecularcollisions. Traditional approaches bridge this problem by simply controlling the relaxationrate to an equilibrium condition. Such methods include single relaxation methods(e.g. BGK) [1] and the multi-relaxation technique (MRT) [12]. In both these collisionalmodels, the equilibrium condition is predetermined for each lattice node at each time step.The population evolution due to collisions is then determined by a relaxation model thatutilizes the known pre-collision distribution and the calculated equilibrium distribution. Incontrast, we present here a collision based method which is built on the lattice architecture,without using any a priori knowledge of the equilibrium distribution function. Thismethod, termed the collisional lattice Boltzmann method (cLBM) is derived from astatistical analysis of the full Boltzmann collision integral and as such it is inherentlynonlinear. This method utilizes the underlying symmetries of the collision operator(e.g. with a hard sphere model) to evolve local distributions, subject to constraints onthe evolution of generalized moments. As a result the full Boltzmann collision integralis expressed as a summation of elementary integrals with analytical solutions. As cLBMis based on a lattice structure, it can be easily implemented into existing LBM codes(e.g. the MUPHY code [13]) and shares many of the same advantages as LBM, includingthe ease of code implementation for parallel computer architectures and treatment ofcomplex geometries.
It is important to note that direct lattice collisions were implemented in the seminalwork (in the context of LBM) of McNamara and Zanetti [14]. In their work, they pointedout that the hydrodynamic properties of lattice-gas automata (LGA) could be determinedvery efficiently using an associated Boltzmann equation and hence eliminated the problemof noise in LGA. The review by Benzi et al [15] also discusses the salient features of manyother collision models. While McNamara and Zanetti [14] derived the collision operatorbased on Boolean models of collision (that were specific to the lattice), Higuera et al [16]proposed an enhanced collision model where the collision operator was chosen such thatthe desired macroscopic equations were reproduced without worrying about the underlyingBoolean microdynamics. In the latter case, a collision matrix whose coefficients could betuned to recover the macroscopic equations was used and an eigenvector analysis of thissystem was later proposed [15].
A lattice based solution of the collisional Boltzmann equation
The rest of this paper is organized as follows. Section 2 first outlines the developmentof cLBM from the Boltzmann equation and then compares cLBM to LBM approacheswhile highlighting much of the commonality between the two methods. Issues surroundingboundary conditions and flow forcing methods and how they relate to the proposedcollision operator are examined in section 3. Section 4 examines the results obtainedusing cLBM for various canonical flows with comparison to experimental values and toresults obtained from BGK simulations. Conclusions are presented in section 5.
2. Mathematical formulation
2.1. Collisional lattice Boltzmann method (cLBM)
In developing the collisional lattice Boltzmann method (cLBM), we start with the fullBoltzmann equation, which is a nonlinear, integro-differential equation that describes theevolution of a particle distribution function f ≡ f(x,u, t) in the position x and velocityu phase space [17], as
∂f
∂t+ u ·∇f = J(f, f). (1)
Here, u is the microscopic velocity and J(f, f) is the collision operator. The particledistribution function f , which is a function of time t and three-dimensional vectors ofposition x and velocity u, is defined such that the integral over the entire velocity spacegives the (mass) density of particles (e.g. atoms or molecules) at a given point in spaceand time. The right-hand side of equation (1) represents the collision term. In conceptualterms, this equation accounts for the time evolution of the particle distribution functionin terms of (a) contributions due to spatial transport of particles (given by the u · ∇fterm) and (b) contributions due to inter-particle collisions (denoted by J(f, f)).
The collisional term J(f, f), which contains a five-fold integral comprised of a volumeintegral over a three-dimensional velocity space and a two-dimensional surface integralover the sphere of influence, is given by
where σ is the differential collision cross section, dΩ ≡ sinχ dχ dφ is the differential solidangle and χ is the deflection angle between the pre-collision relative velocity vectorg ≡ u − u1 and the post-collision relative velocity vector g′ ≡ u′ − u′1. The relationbetween the pre-collision velocities (u,u1) and post-collision velocities (u′,u′1) will bediscussed later in this section. The magnitude of the pre-collision relative velocity isdenoted by g ≡ |u − u1| and φ denotes the azimuthal angle. Inherent in equation (2)are the assumptions of molecular chaos and restriction to binary collisions [18, 17], bothof which are appropriate for a dilute gas. In our analysis, the particles are considered ashard spheres and hence the so-called hard sphere model for the collision cross section isused [18]. While direct numerical solutions of the Boltzmann equation are cumbersome,we seek a more efficient approach to obtain numerical solutions of the Boltzmann equationusing a lattice framework.
A lattice based solution of the collisional Boltzmann equation
In our collisional lattice Boltzmann method, we consider a discrete model for theparticle distribution function using a finite number of Dirac delta functions, as
f(x,u, t) =N∑i=1
fi(x, t)δ(u− ui(x)), (3)
where δ(u − ui(x)) ≡ δ(u − ui)δ(v − vi)δ(w − wi) and fi is a quadrature weight to bedetermined, u ≡ (u, v, w) denotes the microscopic velocity and ui ≡ (ui, vi, wi) denotesthe ith discrete velocity abscissa. Note that there are N quadrature weights to bedetermined at each point in space x and time t. In our cLBM approach, these quadratureweights will be determined using constraints on the evolution of generalized moments ofvelocity, while preserving certain underlying symmetries of the collision operator. Thesesymmetries (to be discussed later in this section) not only ensure that the collisioninvariants (such as mass, momentum and total energy) are preserved but also ensurethat important symmetries regarding property changes (for high-order moments) are alsocorrectly represented in our numerical approach.
In cLBM, similarly to LBM, we use an operator splitting approach, where operatorscorresponding to streaming of populations (i.e. the left-hand side of the Boltzmannequation) and the collision step are treated in separate stages. The discretized populationdistribution is evaluated at the center (node) of each lattice cell and individual populationsare assumed to stream only from node to node in a given time step. This requires a setof discrete velocities ui which arise naturally from the discretization of space and time.To illustrate our approach, we use the standard D3Q27 lattice (i.e. a three-dimensionallattice with 27 discrete velocities), whose velocity components are given as [19]
The underlying transport process is represented via a streaming of populations basedon the corresponding discrete velocities as
fi(x, t+ δt) = fi(x− uiδt, t), (5)
where i denotes the index for a discrete velocity abscissa. In cLBM, the evolution ofpopulations during the collision step (owing to operator splitting) is given by ∂tf = J(f, f)and is computed via a time integration procedure, as
fi(x, t+ δt) = fi(x, t) +
∫ t+δt
t
dfidt
dt, (6)
where the source term dfi/dt ≡ J(fi, fi) (for each velocity abscissa ui appearing on theright-hand side of the above equation) accounts for the time rate of change of populations(corresponding to ui) due to molecular collisions. As described earlier in this section,these source terms (and hence the integral in equation (6)) will be evaluated based onconstraints on the evolution of generalized moments of velocity (or other generalizedfunctions of velocity) and some symmetries of the collision operator.
A lattice based solution of the collisional Boltzmann equation
2.2. Source terms due to collisions
In cLBM, as in conventional LBM, macroscopic quantities are obtained as weightedaverages of the corresponding microscopic properties. For any microscopic property φ(u)expressed as a function of the microscopic velocity vector u, the corresponding macroscopicproperty is defined by the weighted average,
〈ψ(u)〉 (x, t) =1
ρ
∫ψ(u)f(x,u, t) du, (7)
where ρ ≡ ρ(x, t) =∫f(x,u, t) du is the mass density and the integration is over the
entire velocity space. Based on this definition, any generalized moment of velocity (withCartesian components (u, v, w)) can be obtained by selecting φ(u) = upvqwr for any set ofnon-negative real numbers (p, q, r). It may be noted that based on the particular discretelattice based representation of the distribution function (according to equation (3)), ageneralized moment Mp,q,r corresponding to ψ(u) = upvqwr can be expressed in termsof the discrete weights of the distribution function fi corresponding to lattice velocities(ui, vi, wi) as
Mp,q,r =N∑i=1
upi vqiw
ri fi. (8)
The evolution of the macroscopic quantity 〈ψ〉 can be derived from the Boltzmannequation, by multiplying both sides of the equation by ψ(u) and integrating over thevelocity space, as
∂
∂t(ρ 〈ψ〉) +
∂
∂x· (ρ 〈uψ〉) = I(ψ). (9)
The second term on the left-hand side of equation (9) represents the contribution tothe population evolution due to spatial transport, while the right-hand side term I(ψ) ≡∫
duψ(u) J(f, f) represents the contribution due to particle collisions. This evolutioncan be approximated in terms of streaming and collision operators. While the streamingoperator in cLBM is similar to that in LBM (as described in equation (5)), the evolutiondue to the collision step (in operator split form) can be obtained by ignoring the spatialgradients in equation (9) and can be expressed as
N∑i=1
upi vqiw
ri
dfidt
= I(ψ)∣∣∣ψ=upvqwr
. (10)
Note that there are N unknown source terms dfi/dt, which can be determined via solutionof a system of equations, given by equation (10), and denoted as
[A]
[df
dt
]= [b] . (11)
Here A is an N ×N matrix whose rows are generated from equation (10) for a particularchoice of (p, q, r) characterizing the distinct generalized moments selected. If the right-hand side of equation (10) can be evaluated, then the source terms needed for evolvingthe populations in the collision step (in equation (6)) can be obtained via solution of thesystem of equations. The N moment constraints (i.e. N selections for (p, q, r)) should be
carefully considered, as an arbitrary set of N moment constraints could lead to singularor ill-conditioned numerical behavior. This is particularly important in cLBM as thelattice velocity vector components are generally zero, unity or their integer multiples.The selection of an acceptable set of moments, or (p, q, r) values, should be made suchthat the matrix A is non-singular and not ill-conditioned. A sample selection of momentconstraints for the D3Q27 lattice is given in table 1. This selection of moments not onlyensures a correct representation of the behavior of the evolution of low-order moments(including mass, momentum and total energy conservation during collisions) but alsoensures that the matrix A is non-singular and well conditioned. The condition number ofmatrix A is used to check whether or not the matrix is ill-conditioned.
In order to evaluate the right-hand side of equation (10) (also referred to in thispaper using the symbol Ip,q,r) representing the contribution of the collision operator tothe evolution of the generalized moments, we first consider certain symmetries of thecollision operator, using which the integral I(ψ) can be expressed as
I(ψ) =D2
2m
∫du
∫du1 f(x,u, t) f(x,u1, t)×
∫dn Θ(g · n) (g · n) ∆ [ψ(u) + ψ(u1)] ,
(12)
where D denotes the particle diameter, m denotes the mass of the particle, n denotesthe unit vector along the line of centers of particles and g ≡ u − u1 denotes the relativevelocity vector before a binary collision. Note that Θ(·) represents the Heaviside functionand the operator ∆ is defined such that
denotes the change in the property ψ(u) + ψ(u1) due to collisions. The post-collisionvelocities can be expressed in terms of pre-collision velocities, using the law of conservationof momentum, as
u′ = u − (g · n)n
u′1 = u1 + (g · n)n.(14)
We can see from the integrals in equation (12) that for any conserved or collision-invariantphysical quantity, the integral I(ψ) is zero, as ∆[ψ(u) + ψ(u1)] = 0. For instance, mass,momentum and energy are conserved properties in a system with purely elastic collisions
A lattice based solution of the collisional Boltzmann equation
and hence the integral in equation (12) is zero. For more general tensor functions ofvelocity u, the appropriate symmetry regarding the property change is ensured and thecorrect structural form of the rate of change (in the form of production or dissipationrate) of any property ψ due to collisions is considered via such integrals.
While evaluation of integrals I(ψ) corresponding to arbitrary order moments isin general extremely difficult due to the integro-differential nature of the Boltzmannequation, we can use analytic integration to partially integrate over the directionaldependence (n) of the line(s) joining the centers of particles in a binary collision. Followingthis, we can use quadrature to provide closed form expressions for the rest of the integralsin terms of the quadrature weights fi and discrete velocity abscissas ui.
In order to evaluate integrals of type I(ψ), where ψ = upvqwr, we consider the followingexpression for the property change in ψ:
where (ex, ey, ez) denote the unit vectors along the axes of a fixed coordinate system. Toevaluate this property change, we need to compute the dot products of each of the unitvectors ex, ey and ez with (a) the unit vector n, (b) the pre-collision velocities u and (c) thepost-collision velocities, according to equations (14), (13) and (15). Another dot productthat also needs to be evaluated for the computation of property changes is (g · n). Thesevarious dot products may be obtained by establishing a local coordinate system (x∗, y∗, z∗)for each collision. This local coordinate system is defined such that the z∗ unit vector liesalong the pre-collision relative velocity vector g. To facilitate conversions between globaland local coordinate systems, an orthonormal transformation matrix L is constructed. Thismatrix can be assembled using a Gram–Schmidt orthonormalization procedure (using therelative velocity vector g, a random velocity vector and their cross product as basis vectorsthat span the vector space). It should be noted that owing to the restrictions of the lattice,this random vector (with almost arbitrary orientation) cannot be aligned along any of thelattice directions. This choice for the construction of the transformation matrix L alsoensures rotational invariance about the z∗ axis, so long as the z∗ axis is aligned along thedirection of the relative velocity vector, g. Using this transformation matrix any vectorcan be expressed in either of the reference frames (collision frame or fixed frame) using thetransformations u∗ = Lu or u = LTu∗, where u and u∗ denote vectors expressed in thefixed coordinate reference frame and the collision frame of reference, respectively. Basedon these transformations, the dot products appearing in equation (15), like n · ex, can beexpressed as n · ex = n∗TLex, where n∗ corresponds to the unit vector n expressed in thecollision frame of reference. From this point, the preceding dot products can be obtainedin the form n · ex = n∗TLex, where n∗ is the unit vector n expressed in the collisioncoordinate system. The final integral requires integration over a sphere (since the collisionis assumed to occur between idealized elastic spherical particles) and we thus introduce aspherical coordinate system. Within this spherical coordinate system, the dot products inthe integral are reduced to simple identities (i.e. n∗ = (sin(θ) cos(φ), sin(θ) sin(φ), cos(θ)),n = LT n∗, (g ·n) = g cos(θ), and dn∗ = sin(θ) dθ dφ) which are used to express the variousdot products as functions of the polar and azimuthal angles (θ, φ). It can therefore beshown that the property change due to collision can be expressed in terms of a sum ofproducts of multinomial expansions. The innermost integral over n is expressed as a sum
The terms a, b, c and d are the sums of the loop variables (as shown below). Multinomialexpansions are used to obtain the post-collision velocity components (and their powers)in the discretized space, which in turn constitute the generalized moments. For instance,the monomial u′p can be expressed using a multinomial expansion as
u′p
=
p∑i1=0
i1∑j1=0
j1∑k1=0
(−1)i1(p
i1
)(i1j1
)(j1k1
)gi1Li1−j111 Lj1−k121 Lk131
× up−i1 (cos (θ))i1 (n∗1)i1−j1 (n∗2)
j1−k1 (n∗3)k1 . (18)
Similar expansions can be generated for the property ψ ≡ upvqwr, for non-negative integers(p, q, r). The integral Ip,q,r can hence be evaluated by enumerating the terms of themultinomial expansions, as
Note that the above integral is obtained via evaluation of elementary integrals of typeKa,b,c,d in terms of Gamma functions. The Kronecker delta functions in equation (19) areused to define the appropriate indices (a, b, c, d).
The complete form of the Ip,q,r integral is straightforward to compute, requiring onlysummation over multiple loops. As it turns out, the nine inner summation loops ofequation (19) are purely dependent on the lattice geometry. Hence, for a fixed latticegeometry, the inner summation loops need to be evaluated only once (i.e. at the start ofthe simulation).
It is also interesting to note the similarities between the cLBM and the DQMOM [20]approaches. Unlike in cLBM, in DQMOM the velocity abscissas can evolve in time (inaddition to the quadrature weights for populations). While there are some advantages tothis adaptive nature of DQMOM, both the streaming and the collision operators are morecomputationally intensive compared to cLBM. The simplicity of the streaming operatorpresent in cLBM and LBM is absent in DQMOM as the discrete velocities are not confined
A lattice based solution of the collisional Boltzmann equation
to any lattice. In the evaluation of the collisional contributions, the analog of matrix Afor DQMOM has to be computed for each time step and the velocity abscissas evolve intime. This requirement appears to makes DQMOM less efficient than cLBM.
2.3. Traditional LBM approach
Now that cLBM has been fully derived, we can compare the derivation to that of otherLBM approaches. The collisional timescale is generally treated independently from thetransport timescale and thus the spatial transport process is decoupled from the collisionoperator. This fact permits cLBM to replace only the collisional calculation in anysimulation while the boundary conditions and streaming algorithm are effectively thesame as in traditional LBM approaches. Traditional LBM therefore consists of iteratingbetween streaming and collision processes with boundary conditions and flow forcingapplied according to the problem specifications. In this generic sense, cLBM is not verydifferent from the traditional approach, as space is discretized onto a regular lattice andthe collisional and transport processes are uncoupled and treated separately. Unlike incLBM, where the form of the equilibrium distribution is not assumed in the a priorisense, most LBM approaches start with a known form of the equilibrium distribution,which is also used in the BGK model for collisional contributions. LBM utilizes a discreteform of the equilibrium Maxwell–Boltzmann distribution
f eq(x,u, t) =ρ
(2πRT )3/2exp
(−(u− uµ)2
2RT
), (20)
where ρ, uµ and T are the macroscopic mass density, mean velocity and temperaturerespectively. The symbol R is the gas constant. This equilibrium distribution functionf eq(x,u, t) is generally expanded as a series (in the limit of constant temperature T andsmall macroscopic mean velocity uµ) containing terms up to order O(u2
µ) as
f eq =ρ
(2πRT )3/2exp
(− u2
2RT
)×[1 +
(u · uµ)
RT+
(u · uµ)2
2RT 2− uµ
2
2RT
]. (21)
Note that by dropping the higher order terms, the above distribution is only valid forlower mean velocities (the so-called low Mach number approximation). However, by thisapproximation, we recognize that the moments of this distribution can be obtained usingGauss quadrature (using Hermite polynomials).
After applying Gauss quadrature to this specific lattice architecture, the discreteequilibrium distribution is shown to be
f eqi,BGK = fiρ
[1 +
3(ei · uµ)
c2+
9(ei · uµ)2
2c4− 3uµ
2
2c2
]. (22)
Equation (21) utilizes the identity RT = c2s = c2/3, where cs is the lattice speed ofsound, and c = δx/δt (the discrete velocity set). It is generally convenient to take c = 1.The fi terms are the equilibrium weights and are found as a result of the quadrature. Fora D3Q27 lattice, these equilibrium weights are 8
27for the node, 2
27for the faces, 1
54for the
edges and 1216
for the corners. For a more detailed derivation of this equilibrium expressionsee He and Luo [19].
Now that the discrete equilibrium distribution has been defined, the BGK collisionoperator assumes that the populations relax to their equilibrium values through the
A lattice based solution of the collisional Boltzmann equation
following model:
JBGK = −1
τ[fi − f eq
i,BGK]. (23)
In practical terms, τ controls the rate of relaxation to equilibrium, and 1/τ is related to thecollision frequency. In many respects, the MRT model uses many of the same mechanicsbut provides more control by allowing the individual populations to be evolved on differenttimescales. The only requirement is that the timescales be coupled in such a way as notto violate the conservation laws.
These conventional models work well within certain constraints; issues may arise whenthe flow being modeled is far from equilibrium. It is hoped that cLBM will have someadvantages for these types of flows. Given that all these methods use the lattice structure,and that the collision step is decoupled from the rest of the algorithm, it is certainlypossible to develop algorithms that switch between cLBM and these traditional approachesdepending on flow conditions.
3. Boundary conditions and forcing
As already stated, cLBM is fully capable of utilizing many of the existing methods (inLBM) for implementation of appropriate boundary conditions. Many boundary conditionimplementations such as full bounce back, mid-plane bounce back, periodic and diffuseboundary schemes have been employed appropriately in our simulations, and adaptedto the cLBM framework. In order to investigate the behavior of slip flow near the wall,we use fully diffuse boundary conditions [18, 11, 21, 22] at the solid wall boundaries.In this boundary scheme, the incoming (post-collision) populations are scattered backinto the flow in proportion to the equilibrium distribution for a given wall velocity. Thusthe distribution is divided into three categories: populations leaving the flow domain(f oi ), populations in the plane of the wall (fpi ), and populations streaming back into theflow (f ei ). These populations are computed from the relationships shown in the followingequations:
f oi (x, y, z, t+ δt) = 12(f oi (x, y, z, t) + fi(x− uiδt, y − viδt, z − wiδt, t)), (24)
fpi (x, y, z, t+ δt) = fi(x− uiδt, y − viδt, z − wiδt, t) (25)
and
f ei (x, y, z, t+ δt) =f e,eqi (ρ, uwall)∑j f
e,eqj (ρ, uwall)
∑j
f oj (x, y, z, t). (26)
Note that this boundary condition must use the equilibrium weights set correspondingto the full collision operator. These equilibrium weights also ensure that the selectedgeneralized moments remain unchanged at equilibrium.
In the presence of body force, the cLBM collision operator can be augmented toaccount for the influence of forcing. One approach is to account for the effects of forcingon moments via an additive term on the right-hand side of equation (11), where the
A lattice based solution of the collisional Boltzmann equation
additive term Fp,q,r takes the form
Fp,q,r =N∑k=1
fk[pup−1
k vqkwrkax + qupkv
q−1k wrkay + rupkv
qkw
r−1k az
], (27)
where ax, ay and az represent the fluid acceleration. This term is obtained from theevolution equation for a macroscopic quantity using the Boltzmann equation in thepresence of a body force term and using the discrete velocity approximation on the lattice.To the best of the authors’ knowledge, such forcing techniques have not been used inconventional LBM (based on BGK) and hence a direct comparison of the results fromcLBM and LBM based on this type of forcing scheme is difficult.
Another forcing technique is to add a body force term to the local population weightsduring the iterative loop. This results in the modified Boltzmann equation
fi(x + uiδt, t)− fi(x, t) =
∫ t+δt
t
J(f, f) dt+ Fi. (28)
Here, Fi is the body force term. Kim et al [23] related this force term to an accelerationapplied directly to the microscopic populations. The forcing term thus takes the formFi = piρ(aαuiα/c
2s ). This method works equally well for both the conventional LBM (based
on BGK) and the cLBM collision operator and as such is sufficient for comparing the twoapproaches directly.
We shall also examine pressure driven Poiseuille flows using the method outlinedin Kim et al [24]. In this method, periodic boundary conditions are used where thepopulations leaving the computational domain are reintroduced at the opposite boundary.The flow is driven by scaling the outgoing populations to create a density gradient (andthus a pressure gradient) across the domain. In this scheme, the populations are consideredin two parts, the equilibrium portion and the nonequilibrium portion (i.e. fi = f eq
i +fneqi ).
Two arrays of ghost nodes reside on either side of the computational domain and aredesignated as fi,0 and fi,N+1 (the domain covers spatial nodes indexed 1 to N in thestreamwise direction, where periodic boundary conditions are used). Before the streamingstep, the ghost populations are scaled such that
fi,0 = f eqi (ρ0, jN) + fneq
i,N (29)
and
fi,N+1 = f eqi (ρN+1, j1) + fneq
i,1 . (30)
As the Kim and Pitsch [24] boundary conditions require separate treatment of theequilibrium and nonequilibrium portions of the populations and since cLBM does notinternally utilize an equilibrium calculation, it is necessary to evolve the populations toequilibrium at the boundaries and then subtract the equilibrium values from the initialvalues to determine the nonequilibrium populations. This permits the equilibrium (andthus nonequilibrium) distributions to be calculated and applied to the ghost nodes; thedistribution of the boundary nodes remains unaltered through this step.
Driving a flow with a moving boundary such as in the case of a Couette flow typeproblem is accomplished by replacing the populations of the first wetted boundary nodewith an equilibrium distribution that is biased with the desired velocity component. Forthe BGK collision operator, the equilibrium condition is the Maxwellian equilibrium
A lattice based solution of the collisional Boltzmann equation
Figure 1. Evolution of density, mean velocity and temperature in a spatiallyhomogeneous flow undergoing collisional relaxation.
obtained through Gauss–Hermite quadrature. Since cLBM does not utilize the sameequilibrium distribution, it is necessary to replace the populations of the wetted nodeswith the cLBM equilibrium populations. As already outlined, these populations may beobtained by initializing the homogeneous cell with a BGK distribution that has beenbiased to a desired mean velocity value. The weights are then evolved to equilibriumusing cLBM and may then be applied to the desired nodes for a moving boundary. Forthe microflows, the boundary is only initialized in this manner since the diffuse conditionrequires the collision operator to be applied to the boundary nodes in order to capturethe slip velocity.
4. Results
For most of our analysis, we use the D3Q27 lattice (as a test lattice). Using this lattice,we first investigate the collisional relaxation of a spatially homogeneous flow. This is agood test case that enables us to (i) investigate the relaxation to equilibrium (where themoments of the distribution are constant) and (ii) obtain the equilibrium populationweights. For this test case, the distribution function is independent of the spatialcoordinate and hence the contribution due to the convective term (in the Boltzmannequation) is zero.
Figure 1 shows that some macroscopic quantities (which can be related to certainmoment of the distribution function) like density (ρ), mean velocity (uµ) and the variance(σ2) (which is related to the internal energy of the fluid) do not change with time as a resultof collisions (as expected). This demonstrates that collision invariants are also preservedin the numerics of cLBM. Note that figure 1 was generated using a relaxation diameter
A lattice based solution of the collisional Boltzmann equation
Figure 2. Weight evolution of the node vector for a homogeneous cell with zeromean velocity. Values have been normalized to the initial (BGK) weight value.
of one (in simulation units). Note that this parameter controls the rate of relaxation toequilibrium while the collision invariants (including mass, momentum and temperature)are preserved during relaxation to equilibrium. In figure 1, the mean velocity components〈u〉, 〈v〉 and 〈w〉 are all set to zero. The initial weights are set to the BGK values. Thedensity is set to one. The variance is c2/3. As pointed out in the discussion of the LBMdevelopment, c = δx/δt. Taking c = 1 implies that δt = 1 in equations (6) and (28). Thesedefinitions give rise the lattice unit dimensions for space, time and velocity.
Using the weights obtained from traditional LBM as a starting point, it is now possibleto calculate the cLBM equilibrium weights with any desired mean velocity. In other words,the weights are initialized to those of a Maxwellian distribution (from traditional LBM)based on the desired mean velocity (by equation (22)) and are allowed to evolve accordingto the dynamics of the collisional relaxation. Note that the D3Q27 lattice utilizes fourenergy shells: the node (or center), face, edge and corner. Due to the symmetry of thelattice, the distribution weights will be the same for all velocity vectors with the samemagnitude (for a distribution with zero mean). Thus it is only necessary to examine theweight evolution of one member of each energy shell; these results are presented in figures2–5. Given the range of weights involved, it was convenient to normalize the weights totheir initial BGK values. These figures demonstrate the effect of diameter (scaled with√m) on the rate of relaxation of the weights, and, as is apparent, the smaller the (scaled)
diameter is the longer the time to relax to equilibrium is. These figures also highlight thenonlinear capabilities of cLBM; the rate of relaxation changes in time and is proportionalto the deviation from the equilibrium condition.
Note that the cell was initialized to the BGK equilibrium weight set, but the systemevolved to a different set of equilibrium weights. The difference in equilibrium is relatedto the different quadrature methods used between the two systems: traditional BGKutilizes Gauss–Hermite quadrature while cLBM uses a combination of analytical and
A lattice based solution of the collisional Boltzmann equation
Figure 3. Weight evolution of the face vector for a homogeneous cell with zeromean velocity. Values have been normalized to the initial (BGK) weight value.
Figure 4. Weight evolution of the edge vector for a homogeneous cell with zeromean velocity. Values have been normalized to the initial (BGK) weight value.
numerical quadrature. The resulting equilibrium weights for cLBM are also different fromthe traditional LBM weights, due to the nature of the moment constraints used (whichare different from those used in traditional LBM).
Although not analyzed extensively here, it should be pointed out that there isnothing to preclude lattices with higher microscopic velocities (so long as the velocitiesare multiples of unity and thereby permit appropriate streaming). As a point of
A lattice based solution of the collisional Boltzmann equation
Figure 5. Weight evolution of the corner vector for a homogeneous cell with zeromean velocity. Values have been normalized to the initial (BGK) weight value.
demonstration, the authors examined the D3Q125-ZOT (zero–one–three) lattice [25].Again, using the equilibrium weights presented by Chikatamarla et al as the initializationpoint, it is possible to evolve the populations to the cLBM equilibrium weights. Theconstraint set for this case was the full permutation of 0:4 for p, q, r.
From kinetic theory, we know that the Knudsen number is [26] Kn = 1/√
2Lπd2n0,where L is the characteristic length and n0 is the number density. Given that ρ = m · n0,then the relation between the Knudsen number and the relaxation diameter D is Kn =1/√
2LπD2ρ, where D ≡ δ/√m and δ is the particle diameter. It is also known [23] that the
Knudsen number can be related to the BGK relaxation parameter as Kn = (τcs/L)√π/2.
While traditional LBM controls relaxation to equilibrium through the parameter τ , cLBMcontrols relaxation based on properties of the particles (mass, particle diameter), thecharacteristic length scale and the collision cross-sections.
The results for a microchannel Couette flow with slip (and fully diffuse boundaryconditions) are given in figure 6, where the effects of the Knudsen number are alsoillustrated. As explained earlier, the flow was initialized using the cLBM equilibriumdistribution. Most of the flow was set to a zero mean but the moving boundary node wasinitialized to a distribution with a desired mean. In the analysis here, we used 33 spatialnodes in the wall-normal direction. The density was set to unity (in lattice units) and theflow was then fully evolved in time. The output velocity profile was then normalized tothe moving wall velocity and plotted using the midstream as the datum. The results forfive Knudsen numbers are presented in figure 6.
In figure 6, the lines were generated with cLBM and compared to our own BGKalgorithm implementation on the D3Q27 lattice. As is apparent, cLBM closely followsthe BGK results. Kim et al [23] showed that the BGK algorithm produces higher slipvelocities than predicted by DSMC. The discrepancy is small for a Knudsen numberof 0.1, but increases with increasing Knudsen number. As others have pointed out,
A lattice based solution of the collisional Boltzmann equation
Figure 6. cLBM and BGK results for a Couette driven flow over a range ofKnudsen numbers. The flow has been normalized to the wall velocity and thedatum is taken at midstream. ρ = 1.
the error between LBM and DSMC is due to the accuracy of the diffused boundarycondition [27, 28].
Examination of the body force driven Poiseuille flow produces a similar observation.Here, as before, the flow was analyzed with an initial (reference) density of one. As before,we used 33 spatial nodes in the wall-normal direction. Note that these profiles have beennormalized with the mean bulk velocity given by
ub =1
L
∫ L/2
−L/2u(y) dy. (31)
Here, as before, the results from cLBM and traditional LBM (based on BGK) agree wellwith each other for the range of Knudsen numbers considered (as shown in figure 7).
Now we turn to the subject of pressure driven Poiseuille flow. The proposed collisionoperator was tested with both Zhang and Kwok [29] and Kim and Pitsch [24] type pressuredriven periodic boundaries. The Zhang and Kwok method is more straightforward toimplement but the Kim and Pitsch boundary conditions provided a more linear densityvariation over the computational domain (as they themselves noted), and thus we haverestricted the results presented here to the Kim and Pitsch forcing technique.
The domain dimensions were taken to be 33 nodes (transverse) by 99 nodes(streamwise) by seven nodes (cross stream). The pressure driven boundary conditionswere applied in the streamwise direction while periodic conditions were used cross stream,and fully diffused boundary conditions used in the transverse dimension. The velocityprofile was evaluated in the transverse direction along a locus of points passing through themidstream of the channel length. The input and output density was taken to be ρin = 1.01and ρout = 0.99 respectively. As discussed in the literature [30]–[32], we expect a velocityprofile of the form u(Y ) = u0(Y −Y 2−Vs), where u0 and Vs are obtained by regression tothe simulation results (as discussed in [30, 31]). The velocity profiles were then normalized
A lattice based solution of the collisional Boltzmann equation
Figure 7. cLBM and BGK results for a Poiseuille flow driven by a body force forvarious Knudsen numbers. The profiles have been normalized to the mean bulkvelocity. ρ = 1.
Figure 8. Comparison of BGK and cLBM results driven with Kim and Pitschpressure boundary conditions to experimental values [28].
to u0 and plotted against experimental results (referenced in Zhou et al [28]). As figure 8demonstrates, our cLBM based results closely agree with those obtained from traditionalLBM (based on the BGK model) and the experimental results. This further demonstratesthe validity of the proposed lattice based numerical approach, which accounts for therepresentation for the full collision integral, for prediction of microchannel flows.
We would like to emphasize here that in contrast to some of the earlier works onLBM [15], we use a weak form of the full collision operator of the Boltzmann equation(e.g. equation (12) along with equation (9)) in cLBM. The underlying collision invariantsare accounted for in equation (12) via the ∆ operator. Unlike the collision matrix described
A lattice based solution of the collisional Boltzmann equation
in equation (71) of Benzi et al [15], we have a nonlinear operator of the form shown inequation (19). Note that the last integral in equation (12) has been analytically expressedin equation (19) via the terms that follow the × symbol in the first line of equation (19).In view of the computational complexity of the collision approximation in cLBM, we notethat these terms are pre-computed and do not have to be evaluated at each time step(assuming that the lattice structure does not change with time). Hence, the change dueto collisions can be expressed (using equation (11)) as
dfidt
= [A−1]ijCjαβfαfβ, (32)
using the Einstein summation convention. The underlying computational complexity inour collision model (using N discrete velocities) can be analyzed from the right-handside of the above equation, which involves (a) matrix–vector multiplication (i.e. [A−1][C],where Cj = Cjαβfαfβ) with complexity O(N2) and (b) evaluation of the vector C withcomplexity O(N3). Hence, the overall computational complexity involved in evaluationof the contribution due to collisions (obtained from the right-hand side of the aboveequation) is O(N5). Our cLBM simulation results indicate that this complexity can bemanaged for the D3Q27 lattice, where N = 27 (or N generalized moment constraintsare used). The computational complexity due to collisions (O(N5)) in cLBM is indeedgreater than that in standard LBM based on a single-time relaxation model (whosecorresponding complexity is O(N)) and in LBE based on an enhanced collision model(whose corresponding computational complexity is O(N2), due to the collision matrix).In spite of this increased computational cost, the proposed cLBM approach could be usefulfor analysis of nonequilibrium flows as the contributions due to collisions are derived fromfirst principles based the full collision operator of the Boltzmann equation and are devoidof any equilibrium based assumptions. We found that for the range of Knudsen numbersconsidered, the cLBM also has a significantly lower computational cost than the standardDSMC approach, as the latter contains significant statistical noise, especially in collisiondominated regions of the flow, while cLBM has no statistical noise. For instance, weobserved that our cLBM code (run on a single processor) was nearly 55 times slower thanstandard LBM, but nearly 60 times faster than DSMC (for a reasonably small noise level)for a Knudsen number of 0.1.
5. Summary
In this paper, we have presented a novel algorithm for treatment of the full collisionoperator of the Boltzmann equation using a lattice framework, similar to LBM. Ourapproach, termed the collisional lattice Boltzmann method (cLBM) is designed with thegoal of handling nonequilibrium flows on a lattice structure. Unlike most conventionalapproaches based on LBM, cLBM does not make use of any equilibrium based collisionmodels (e.g. BGK). In other words, the advantage of the proposed method is that anarbitrary distribution can be evolved based on the collisional Boltzmann equation, withouta priori knowledge of the equilibrium distribution function.
Similarly to LBM, the underlying distribution function in cLBM is represented usingweights and fixed velocity abscissas. In cLBM, the former are determined according tothe Boltzmann equation with the full collision operator. Our approach is constructed
A lattice based solution of the collisional Boltzmann equation
to preserve the structure of the rates of evolution of higher order moments (includingconservation of mass, momentum, energy, and other selected generalized moments),according to the symmetries of the collision operator. The authors’ intent is to provide animproved model using a lattice framework (via cLBM) for prediction of nonequilibriumflow behavior that often cannot be accurately simulated with conventional LBM (due to itsequilibrium based assumptions). Due to the operator splitting approach used to separatecollisional and streaming operators in cLBM (and LBM), the collisional interactions incLBM can be separately simulated using a code module which can be easily interfaced withexisting (serial and parallel) LBM based codes. These code modules can be used for furtherdetailed studies on the effects of nonequilibrium flow behavior where needed, by switchingbetween cLBM and LBM based collision operators as needed. The proposed collisionaloperator is fully compatible with a number of existing boundary methods and forcingtechniques. In principle, the philosophy used to derive cLBM may also be used to derivemoment based boundary conditions. It may also be noted that the improvements in theprediction capabilities of cLBM for nonequilibrium flows involve a greater computationalcost than conventional LBM. This is due to the fact that the treatment of the collisionaloperator in cLBM involves numerical solution of a system of coupled, nonlinear ordinarydifferential equations, while the corresponding treatment in conventional LBM involvesbasic function evaluations based on the Maxwellian distribution.
The proposed collisional lattice Boltzmann method (cLBM) was validated forcanonical flows including homogeneous relaxation flow, microchannel Couette flows and(pressure driven and body force driven) microchannel Poiseuille flows and the results werefound to be in good agreement with those obtained from conventional LBM (using theBGK collision operator) and experimental results where available.
Although the cLBM formulation addresses the need for nonequilibrium flowpredictions using a lattice framework, other limitations like the treatment of (strong)compressibility effects remain common to both cLBM and conventional LBM. The issueof compressibility can be addressed using high-order lattices, which in the context ofcLBM can also be helpful in the preservation of constraints on evolution of much higherorder generalized moments (which can further characterize highly nonequilibrium flowbehavior).
Acknowledgments
This material is based upon work supported in part by the US Army Research Laboratoryand the US Army Research Office under grant number W911NF-12-1-0192. The authorswould like to thank Mr Leonhard Striz for performing some initial computations usingthe cLBM approach.
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