Title: A Quadrature-Based Third-Order Moment Method for Dilute Gas-Particle Flows

Transcription

1 Editorial Manager(tm) for Journal of Computational Phsics Manuscript Draft Manuscript Number: JCOMP-D--R Title: A Quadrature-Based Third-Order Moment Method for Dilute Gas-Particle Flows Article Tpe: Regular Article Kewords: Quadrature method of moments; Velocit distribution function; Kinetic equation; Gas-particle flows; Boltzmann equation Corresponding Author: Dr. Rodne O. Fox, Corresponding Author's Institution: Iowa State Universit First Author: Rodne O. Fox Order of Authors: Rodne O. Fox

2 *. Manuscript A Quadrature-Based Third-Order Moment Method for Dilute Gas-Particle Flows R. O. Fox Department of Chemical and Biological Engineering, Sweene Hall, Iowa State Universit, Ames, IA -, USA Abstract Dilute gas-particle flows can be described b a kinetic equation containing terms for spatial transport, gravit, fluid drag, and particle-particle collisions. However, the direct numerical solution of the kinetic equation is intractable for most applications due to the large number of independent variables. A useful alternative is to reformulate the problem in terms of the moments of the velocit distribution function. Closure of the moment equations is challenging for flows awa from the equilibrium (Maxwellian) limit. In this work, a quadrature-based third-order moment closure is derived that can be applied to gas-particle flows at an Knudsen number. A ke component of quadrature-based closures is the moment-inversion algorithm used to find the weights and abscissas. A robust inversion procedure is proposed for moments up to third order, and tested for three example applications (Riemann shock problem, impinging jets, and vertical channel flow). Extension of the moment-inversion algorithm to fifth (or higher) order is possible, but left to future work. The spatial fluxes in the moment equations are treated using a kinetic description and hence a gradient-diffusion model is not used to close the fluxes. Because the quadraturebased moment method emplos the moment transport equations directl instead of a discretized form of the Boltzmann equation, the mass, momentum and energ are conserved for arbitrar Knudsen number (including the Euler limit). While developed here for dilute gas-particle flows, quadrature-based moment methods can, in principle, be applied to an application that can be modeled b a kinetic equation (e.g., thermal and non-isothermal flows currentl treated using lattice Boltzmann methods), and examples are given from the literature. Ke words: Quadrature method of moments, Velocit distribution function, Kinetic equation, Gas-particle flows, Boltzmann equation Corresponding author address: rofox@iastate.edu (R. O. Fox). Preprint submitted to Elsevier Science March

3 Introduction The numerical simulation of gas-particle flows is complicated b the wide range of phenomena that can occur in real applications [,,,,,,,,,,, ]. In the absence of the gas phase, the particles behave as a granular flow. In the dilute limit, a granular flow is dominated b binar collisions and can be described b a kinetic equation [] (i.e., it is a granular gas). On the other hand, in the dense limit sustained particle-particle contacts are dominant and the (local) kinetic description breaks down. As in rarefied gas dnamics [], the dilute limit can be parameterized b a dimensionless number (Knudsen number, Kn) that represents the importance of particle-particle collisions relative to free transport. In the limit of small Knudsen number, collisions are dominant and the particle velocit distribution function is ver near the equilibrium (Maxwellian) distribution. In the equilibrium limit, it is thus possible to describe a dilute granular flow b velocit moments up to second order (the so-called hdrodnamic limit []). For larger Knudsen number, the equilibrium distribution is no longer a good approximate and, eventuall, one must solve the Boltzmann kinetic equation to adequatel capture the flow phsics of a non-equilibrium granular gas (as is done for rarefied gases [,, ]). For wall-bounded granular gases, nonequilibrium effects can be important near the walls even at small Knudsen numbers because the velocit distribution function is composed of incoming and outgoing particles with velocities far from equilibrium (e.g. a bimodal distribution can be observed near the walls). Adding the fluid phase introduces new phsics, the most important of which is the fluid-particle drag term. For an isolated particle in a uniform fluid, the particle Renolds number (Re p ) determines the net force of the fluid on the particle. For gas-particle flows with moderatel large particles, we usuall have (Re p ) and the drag force is parameterized b a drag coefficient []. Another important parameter in fluid-particle flows is the Stokes number (St), which is the ratio of the characteristic response time of the particle to the characteristic time scale of the fluid flow. For example, in a particle-laden impinging-jet flow the value of the Stokes number will determine whether or not the particles cross the impingement plane. If the Stokes number is sufficientl small (St ), the particles will not cross the plane and will have nearl the same velocit as the fluid. On the other hand, for ver large Stokes numbers, the particles barel feel the fluid and thus will continue directl through the impingement plane. Note that if the flow is dilute, particles can readil cross the plane without risk of collision with other particles, i.e., particle trajector crossing (PTC) will occur. Locall (i.e., at the impingement plane) the velocit distribution function will be bimodal (even when particles do not collide) with values corresponding to the velocities originating from each side of the plane. More generall, particles with finite Stokes number will

4 tend to accumulate in regions of the fluid flow with small vorticit []. In turbulent flows, such regions correspond to local (time-dependent) impingement zones []. Because of the complex particle phsics in such regions, which lead to particle velocit distributions far from Maxwellian, equilibrium closures for the moment equations will fail. For example, an equilibrium closure for the velocit moments will predict delta-shocks [] in the local number densit when, in fact, no such behavior is present in the flow []. In order to avoid such unphsical behavior, it is necessar to extend the moment closure to handle PTC, which is ubiquitous in turbulent gas-particle flows. The range of Knudsen, Renolds, and Stokes numbers observed in real gasparticle flows is ver wide, and hence it is difficult to find a computational model that will work for all applications. Moreover, even in a given flow, the ranges of these numbers ma be large due, for example, to local dense and dilute regions or to a wide range of gas-phase velocities. In general, a model for dilute gas-particle flow must include both the kinetic description for the particle phase and a coupled momentum balance equation for the gas phase (i.e., two-wa coupling). Nevertheless, the principal modeling challenge is the treatment of the kinetic equation and, therefore, we will consider onl onewa coupling (i.e., the fluid velocit is given) in this work. Furthermore, we will consider examples where the Stokes number is large enough that velocit fluctuations (i.e., due to gas-phase turbulence) can be neglected []. These assumptions will allow us to isolate the critical role of non-equilibrium effects in moment closures, and to investigate their treatment using quadrature-based moment closures. We should note, however, that because the coupling with the fluid phase occurs at the level of the moments, the conclusions drawn in this work have direct implications on the closures used in full two-wa coupled moment methods (e.g., found using the kinetic theor of granular flow []). The quadrature method of moments (QMOM) was introduce b [] as a closure for population balance equations (PBE) involving a number densit function (NDF) f(ξ, t) with independent variable ξ representing the particle volume or mass. For example, in an aerosol ξ might correspond to the droplet volume, which evolves in time due to phsical processes such as condensation, evaporation, aggregation, and breakup. The PBE that governs the NDF is then an integro-differential equation that is difficult to solve numericall. In practice, the principal quantities of interest for comparison with data are the moments of the NDF defined b m k (t) = ξ k f(ξ, t)dξ. Thus, a closure of the PBE that can accuratel predict the lower-order moments is often sufficient. In most cases, however, the moment equations derived from the PBE are not closed (see [] for a discussion of moment closures) and a closure step is required. Using QMOM, the moments are expressed in terms

5 of a finite set of N weights n α (t) and N abscissas ξ α (t), defined such that [] N m k = n α ξα k for k =,,..., N. α= For higher-order moments (i.e. k N) and non-integer moments, the quadrature weights and abscissas provide a consistent closure. For example, let g(ξ) be a smooth function of ξ. The QMOM approximation of the moment involving g is then N g(ξ)f(ξ)dξ n α g(ξ α ). α= In a similar manner, all of the terms in the moment equations for m k with k =,,..., N can be closed, resulting in a closed sstem of N equations. A ke component of QMOM is the inversion algorithm used to find the weights and abscissas from the moments. A direct non-linear solve is poorl conditioned (and gets worse for increasing N); however, the Product-Difference (PD) algorithm introduced b [] overcomes this difficult b replacing the non-linear solve with a computationall efficient eigenvalue-eigenvector problem that is well conditioned even for large N. Using the PD algorithm, it is possible to solve for a spatiall dependent NDF using standard flow codes to transport the moments [,,,, ]. Moreover, even with highl coupled and non-linear aggregation kernels, good accurac for the lower-order moments can usuall be attained with N = nodes []. Unfortunatel, the PD algorithm onl works for uni-variate distribution functions, but not for bi-variate (or higher) NDF where f(ξ, η) depends on two (or more) internal coordinates. In order to overcome this limitation, one can work directl with transport equations the weights and abscissas (direct QMOM [] or DQMOM), which works well when ξ and η represent passive scalars (e.g. droplet volume and surface area []). The DQMOM approach has also been applied to poldisperse gas-solid flows [] to describe the fluid dnamics in the presence of multiple particle sizes. However, in that application the particle velocit is assumed to be in the equilibrium (Maxwellian) limit so that onl the mean particle velocit (albeit conditioned on particle size) suffices to describe the flow dnamics. For non-equilibrium gas-solid flows (such as those considered in this work), the velocit distribution function can deviate strongl from the Maxwellian form, and higher-order velocit moments are required to close the transport equations. In principle, DQMOM can be used with the velocit moments to rewrite the moment transport equations in terms of transport equations for the weights and abscissas []. In the absence of collisions, the resulting transport equations for the velocit abscissas [] have the form of the pressure-less gas dnamics equation [], and each abscissa evolves independentl of the others. Although the DQMOM formulation for

6 velocit works well in most of the flow domain, it fails at singular points where the velocit abscissas change discontinuousl [] (e.g., the PTC points). Remarkabl, a quadrature-based moment closure for the velocit distribution function is robust at such points [, ]. However, for multi-dimensional velocit distributions (e.g. three-dimensional phase space), the ke challenge of inverting the moments to find the weights and velocit abscissa remains. The kinetic equation for the velocit distribution function is used in man applications besides gas-particle flows [,,,,,,,,,,,,,,, ], and thus there have been man computational methods developed to find approximate solutions. In the context of quadrature methods, the most closel related techniques are those based on multiphase or multibranch solutions to the kinetic equation (i.e. without the collision term in the Boltzmann equation) [,,,,,,,,, ]. (See [] for a recent review.) In fact, the quadrature formula used in moment-based methods [] for these applications to relate the weights and abscissas to the moments is the same as in QMOM; however, in the literature on multiphase solutions a direct non-linear solver is used (and is therefore limited to small N) instead of the PD algorithm. A more significant difference between QMOM and multiphase solutions is that the latter are designed to ield exact solutions to the kinetic equation without collisions, while the former provides approximate solutions to the kinetic equation with collisions. Despite these differences, the difficult of representing multiple velocities generated b the spatial transport term is shared b man applications of the kinetic equation. At present, there are two classes of methods that can be used to find accurate solutions to the kinetic equation: (i) direct solvers that discretize velocit phase space [,,,,, ] and (ii) Lagrangian (or ra-tracing) methods [,,,,, ]. However, the computational cost of using either of these methods in man applications (such as fluid-particle flows) is prohibitive. Moreover, in most applications we are not interested in knowing the exact form of the velocit distribution function, rather knowledge of its lower-order moments is sufficient. For these reasons, there is considerable motivation to develop predictive moment closures whose accurac can be improved in a rational manner. Quadrature-based moment closures fall into this categor because, in principle, the accurac of these closures can be improved b increasing the number of quadrature nodes. Nevertheless, as mentioned previousl, a ke More generall, multiphase solutions do not consider interactions between different branches or cases where the velocit distribution function is continuous in velocit (e.g. Maxwellian). Mathematicall, the form of the kinetic equation in these applications is at most first-order derivatives in the independent variables []. Collisions and other non-local interaction terms (which are not first order) couple the velocit abscissas in a non-trivial manner [, ]. While this has been clearl demonstrated for passive moments (e.g. volume) [,, ], it remains to be shown for the velocit moments.

7 technical challenge with quadrature-based moment closures is the development of efficient moment-inversion algorithms for multidimensional velocit distribution functions. We should note that because the weights are non-negative and the abscissas are located in velocit phase space, a quadrature-based moment method provides an adaptive discretization of velocit phase space that is consistent with the underling moments. Compared to direct solvers, this discretization is ver sparse (equal to the number of nodes). An important open question is thus to determine the range of accurac that can be achieved using quadrature in comparison to direct solvers. In this work, we address the problem of finding a moment-inversion method for velocit moments up to third order, and leave the question of increased accurac using more nodes to future work. The remainder of the paper is organized as follows. In Sec. we introduce the kinetic equation for dilute gas-particle flows. In Sec. we present the moment transport equations up to third order derived from the kinetic equation. Section describes in detail the moment-inversion method for the velocit moments for one-, two-, and three-dimensional velocit phase space (using,, and nodes, respectivel) and compares the quadrature formulas to other methods available in the literature. In Sec. we present the kinetic-based numerical algorithm used to solve the moment equations. Section is devoted to example applications to test the numerical implementation. Finall, in Sec. conclusions are drawn and the ke characteristics of the proposed moment method and numerical algorithm are discussed. Kinetic Description of Dilute Gas-Particles Flows Consider the following kinetic equation for the velocit distribution function f(v;x, t) of dilute monodisperse solid particles in a gaseous flow []: t f + v x f + v (fg) + v (f Fmp ) = C, () where v is the particle velocit vector, g is the gravit force, F is the drag force from the gas phase acting on a particle, m p is the particle mass, and C is the particle-particle collision term. In this work, we will assume that the collision term is closed. For example, using the Bhatnagar-Gross-Krook (BGK) approximation [], the collision term becomes C = τ (f eq f) () The discretization is adaptive because it changes with the moments.

8 where τ is a collision time constant and f eq is the equilibrium (Maxwellian) distribution. In d dimensions, f eq is given b M f eq (v) = (πσ eq ) d/ exp ( v U p σ eq where M = f dv is the particle number densit (zero-order moment), U p and σ eq are the mean particle velocit and equilibrium variance, respectivel. For real particles, collisions are usuall inelastic (i.e. σ eq is not conserved). However, in this work, we will treat σ eq as a conserved quantit. Note that the Knudsen number is proportional to the collision time τ, so that the velocit distribution function is equal to f eq when Kn =. In the opposite limit, the particles are collision-less and the velocit distribution function will be determined b the remaining terms in Eq. () (and hence can be ver far from equilibrium). For dilute gas-particle flows, the drag force can be approximated b F(U f,v) = m pρ f d p ρ p C d U f v (U f v), () where U f (x, t) is the fluid velocit (assumed to be known in this work), d p is the particle diameter, ρ f the fluid densit, ρ p the particle densit, and C d the particle drag coefficient given b the following correlation due to []: C d = ( ) +.Re. p, () Re p with Re p = U f v d p /ν f being the particle Renolds number where ν f is the fluid kinematic viscosit. In some cases, we will also use the simpler Stokes drag coefficient: C d = /Re p. The drag term can then be expressed as F(U f,v) = m p τ p (U f v), () where τ p = ρ p d p/(µ f ). The particle Stokes number can then be defined as St = τ p /τ f where τ f is the characteristic time for the fluid velocit U f. For ver small St the particles will follow the fluid (v U f ) and hence the will not collide in dilute flows. Moreover, in this limit, flow-induced particle segregation is not observed in turbulent flows. On the other hand, for ver large St the particles respond ver slowl to changes in the fluid velocit and act as a granular gas. In between, for St flow-induced segregation in turbulent Inelastic collisions can be introduced using a restitution coefficient e []. Then, in the moment equations, one simpl replaces σ eq with e σ eq. For example, the deca rate for σ eq is then equal to (e )σ eq. Isothermal flows can be treated b setting σ eq to a constant value. ) ()

9 flow is significant and has a strong effect on collision statistics [, ]. In an case, the Stokes number has a strong effect on the velocit distribution function, which can be described b Eq. (), and makes the behavior of gasparticle flows significantl different than molecular gases (i.e. St = ). Note that drag can be a source or a sink for σ eq, depending on how the particles respond to fluid velocit fluctuations. However, at large St the drag will lead to dissipation of σ eq because fluid velocit fluctuations will have no effect on the particle velocit fluctuations []. In practice, due to the complexit of the kinetic description, the velocit distribution function must be found using a Lagrangian particle tracking code (or DSMC) coupled to a turbulent flow solver for U f. Although accurate, this solution method is rather expensive and subject to statistical errors due to finite sample sizes. Our interest here is to approximate solutions to Eq. () for dilute gas-particle flows [,,,,, ], such as the vertical channel flows described in [,, ], using Eulerian moment methods. However, the numerical method developed in this work can, in principle, be used for other phsical situations that can be described b Eq. () (e.g., dilute spras []). For example, neglecting the terms involving the gravit and drag force, Eq. () is the simplest version of the Boltzmann equation, for which there exists an extensive literature on numerical methods for its solution in various limiting cases [, ], including small-scale gaseous hdrodnamics [] and the semiconductor Boltzmann-Poisson sstem []. We will thus use the Riemann shock problem from gas dnamics [,,, ] as one of the test cases in Sec.. Another important test case for dilute gas-particle flows is impinging particle jets [, ] with or without collisions, which is a model for particle trajector crossing (PTC) at finite Stokes numbers. For this example, the velocit distribution function is locall bimodal and hence far removed from f eq. Moment methods developed for flows near equilibrium (i.e. collision dominated or small Kn) cannot be used to describe PTC because the cannot handle locall multi-valued velocities. However, quadrature-based moment closures perform well for this case in the collision-less limit [, ]. As noted in the Introduction, the kinetic equation (i.e. without collisions) also appears in other applications such as geometric optics []. Moment Transport Equations In this work we develop a third-order moment closure using quadrature. In three dimensions, the twent moments up to third order will be denoted b W = (M, M, M, M, M, M, M, M, M, M, M, M, M, M, M, M, M, M, M, M ).

10 The moments are found from the velocit distribution function b integration: M = f dv, Mi = v i f dv, Mij = v i v j f dv, Mijk = v i v j v k f dv. Thus the transport equations for the moments can be easil found starting from Eq. () (repeated indices impl summation): M t M i t M ij t M ijk t + M i x i =, + M ij x j = g i M + D i, + M ijk x k + M ijkl x l = g i M j + g jm i + M τ (σ eqδ ij σ ij ) + D ij, = g i M jk + g j M ik + g k M ij + τ ( ijk M ijk where the final terms on the right-hand sides are due to drag. The moments of the equilibrium distribution are defined b ) + D ijk, ijk = M (δ ij U pk + δ ik U pj + δ jk U pi )σ eq + M U pi U pj U pk, () where the mean particle velocit is U pi = Mi /M. Note that the collision term onl affects directl the second- and third-order moments, moving them towards the equilibrium values. Nevertheless, the collisions affect all moments indirectl through the transport terms. The unclosed terms in the moment transport equations (Mijkl, D i, D ij and D ijk ) are closed using quadrature as described in Sec.. Of particular importance is the treatment of the spatial fluxes, which are represented b a kinetic description [,,, ]. For example, the flux term for the zero-order moment is separated into two contributions: Mi = Q i + Q + i ( ) = v i f(v)dv j dv k dv i + ( v i f(v)dv j dv k )dv i. () () () The fluxes for the higher-order moments are constructed in an analogous manner. The quadrature-based closure of the right-hand side of Eq. () is discussed in Sec...

11 Quadrature-Based Third-Order Moment Closure As noted in the Introduction, quadrature-based moment methods distinguish themselves from other moment methods b the use of quadrature weights and abscissas to model unclosed terms in the moment transport equations. Thus, when developing a quadrature method, an important task is to define the algorithm for computing the weights and abscissas from the moments. In onedimensional sstems, this is easil accomplished using the PD algorithm []. The extension to higher dimensions is less straightforward and is the main objective of this section. Here we limit ourselves to quadrature formulas for moments up to third order, and use one-dimensional product formulas []. Thus, the number of quadrature nodes in each direction of velocit phase space will be two.. Relationship between moments and quadrature nodes Let d (,, ) denote the number of phase-space dimensions, and define β = d. Let V β = [(n α,u α )] with α (,..., β) denote the set of weights and abscissas for the β-node quadrature approximation of f. Note that the set of quadrature nodes V β contains ( + d)β unknowns (i.e. β weights, and β d-component velocit vectors). To find the components of V β, we work with the velocit moments up to third order (for i j k,..., d), which are related to the quadrature weights and abscissas b β β M = n α, Mi = n α U αi, α= α= β β Mij = n α U αi U αj, Mijk = n α U αi U αj U αk. α= α= () In the following, we will let W d denote the set of third-order moments in d dimensions. The principal objective of this section is to describe an algorithm for finding V β from W d. The inverse operation (finding W d from V β ) is Eq. (), which we will refer to as projection for reasons that will become apparent later. In general, it will not be possible to represent all possible moment sets in W d using weights and abscissas in V β. We will therefore define the set of representable moments as W d W d.

12 . Quadrature-based closure of the moment transport equations The moment transport equations derived in Sec. contain unclosed terms. Using quadrature, these terms can be expressed in terms of the weights and abscissas: where D i = β α= D ij = β D ijk = M ijkl = α= β α= β n α m p F iα, () α= n α m p (U iα F jα + F iα U jα ), () n α m p (U iα U jα F kα + F iα U jα U kα + U iα F jα U kα ), () n α U iα U jα U kα U lα () F α = F(U f,u α ) = m p τ α (U f U α ) () and τ α is the characteristic drag time for node α. Quadrature is also used to write the spatial fluxes in terms of the weights and abscissas [, ]. The fluxes are based on the kinetic description given in Eq. () using a delta-function representation of the distribution function: β f(v) = n α δ (v U α ). () α= For example, the negative and positive contributions to the flux terms for the zero-order moment are expressed as β β Q i = n α min(, U iα ) and Q + i = n α max (, U iα ). () α= α= Likewise, the fluxes for higher-order moments have analogous forms [, ]. This flux definition has important ramifications on the numerical algorithm used to solve the moment transport equations. For one thing, it is obvious that the fluxes are tightl coupled to the quadrature algorithm because the latter determines the abscissas. In particular, the fluxes will var smoothl in space onl if the abscissas are smoothl varing. It can also be noted that because the fluxes are written in terms of a finite set of velocities, the closed model equations do not contain gradient-diffusion terms as is usuall the case in other moment methods. Moreover, because of the form of the fluxes, there

13 is no advantage gained b rewriting the transport equations in terms of the central velocit moments (i.e. moments of the fluctuating velocit). Finall, we should note that fluxes as defined above are not guaranteed to produce moments that can be represented b the proposed quadrature algorithm (i.e. the moments lie in W d but not necessaril in W d, and onl the latter can be represented b the weights and abscissas). For this reason, after advancing the moments due to the fluxes (or an other process that does not remain in W d ), it is necessar to project the moments back into W d. This is accomplished simpl b using the moments to compute the weights and abscissas, and then using Eq. () to recompute the moments. We will refer to this operation as a projection step. Our principal objective is to show that if (i) M > and (ii) for all i (,..., d) M ii > (M i ) /M, then V β can be found from W d in a well-defined manner. For clarit, we will first describe the one-dimensional case with d = and two nodes: β =. The higher-dimensional cases (β = and ) are constructed in a similar manner, and are discussed later.. Two-node quadrature: V from W For the one-dimensional case, we begin b defining the mean particle velocit: U p = M /M, () and the velocit variance: σ U = σ = M /M U p, () so that σ eq = σ. With a two-node quadrature approximation, we introduce a new variable X defined b a linear transformation of v : X = (v U p )/σ so that v = σx + U p. () Note that X is a scalar representing the normalized distance (in velocit phase space) from the mean velocit in the direction of the velocit fluctuations. Denoting the four normalized moments with respect to X up to third order b W = (m, m, m, m ), Introducing X when d = is superfluous. However, for consistenc with higher dimensions, we do so anwa.

14 it is straightforward to show that these moments are related to the velocit moments b m =, m =, m =, m = h ( U p, σ, M /M ), () where h denotes a function of its arguments. This function is rather long (see Appendix A), but can easil be written out starting from the definition of X in terms of v given in Eq. (). Likewise, the weights and abscissas in X phase space, denoted b V = [(n α, X α ), α (, )], are related to V b n α = M n α and U α = σx α + U p. () Thus, we can compute V in terms W, and then use Eq. () to find V. The moment set W is related to V b m = n α, m = n αx α, m = n αxα, m = n αxα. () α= α= α= α= Note that W contains four moments. Thus, if W is known, we can invert Eq. () to find the four unknowns in V, from which we can compute V using Eq. (). Using two-node quadrature [], the moments of X can be easil inverted to find V : ( ) / γ n =. + γ, X =, + γ ( ) / + γ n =. γ, X =, γ where ( / < γ < /) γ = m / [(m ) + ] () /. () It is then possible to compute W, defined b W = (M, M, M, M ), using V in Eq. (). These operations can be expressed as W W V V W. Note that in the case of d = all four moments in W can be exactl reproduced b V. This will not be the case for higher dimensions.

15 . Four-node quadrature: V from W In two dimensions, the set of ten moments up to third order is defined b W = (M, M, M, M, M, M, M, M, M, M ), and we seek to define a four-node quadrature based on these moments. Again we begin b defining the mean particle velocit vector: U p = M /M, () M /M and the velocit covariance matrix: σ U = [σ ij ] = M /M Up M/M U p U p, () M/M U p U p M/M Up so that σ eq = (σ + σ )/. The next step is to introduce a linear transformation A to diagonalize σ U. The choice of the linear transformation is not unique, but for reasons that will be described later, we choose to use the Cholesk decomposition defined such that L T L = σ U and L is upper triangular: L = σ /σ /. () (σ σ /σ ) / σ/ With this choice we set A = L T and introduce a two-component vector X = [X X ] T defined b X = A (v U p ) so that v = AX + U p. () If we denote the first four moments of X i b m k i, k (,,, ); then the are related to the velocit moments b m i =, m i =, m i =, m i = h i ( A,Up, M /M,..., M /M ), () Strictl speaking, it is not necessar to diagonalize the covariance matrix. However, if this is not done, it can be shown that certain realizable moment sets lead to negative weights. The transformation is not unique because we can alwas introduce an orthogonal matrix C with the propert C T = C such that L T C T CL = σ U. The linear transformation can then be defined as A = (CL) T. There are an infinite number of choices for C, e.g., the rotation matrices.

16 where h i depends, in general, on all ten third-order velocit moments (see Appendix A). Using the two-node quadrature formulas (Eq. ), the moments of X i can be inverted for i (, ) to find (n (), n (), X (), X () ) i : ( ) / γi n (i) =. + γ i, X (i) =, + γ i ( ) / + γi n (i) =. γ i, X (i) =, γ i where ( / < γ i < /) γ i = m i/ [(m i) + ] () /. () The locations of the nodes before and after the linear transformation are illustrated in Fig. for a Maxwellian distribution with zero mean velocit. The four-node quadrature approximation is then defined using the tensor product of the one-dimensional abscissas as V = [(n, X (), X () ), (n, X (), X () ), (n, X (), X () ), (n, X (), X () )] () where the (as et) unknown weights n α must obe the linear equations n + n = n () n + n = n () n + n = n () n + n = n () = n n = n () n () n + n = n (). () n + n = n () The equations on the left-hand side are the weight constraints for each node, and the equations on the right-hand side are a linearl independent set. The right-hand sides of Eq. () are known, and have the propert that n () + n () = and n () + n () =. The linear sstem in Eq. () has rank three. We must therefore add another linear equation to define the four weights. For this purpose, we will use the cross moment m = X X =, the value of which follows from the definition of A. In terms of the weights and abscissas in Eq. (), we have X () X () n + X () X () n + X () X () n + X () X () n =. () Recall that the abscissas (X (i)α ) are known; hence, this expression is linear in the weights (n α), and we can use it with Eq. () to form a full-rank linear

17 sstem. The resulting sstem can be inverted analticall to find n = n ()n () = (. + γ )(. + γ ) n = n () n () = (. + γ )(. γ ) n = n ()n () = (. γ )(. + γ ) n = n () n () = (. γ )(. γ ). Note that these weights are alwas non-negative. () In summar, the weights and abscissas in V are found from those in V using Eq. () to invert the abscissas and n α = M n α. The eight moments controlled in this process are W = (m, m, m, m, m, m, m, m ). Note that the two third-order moments in W are a linear combination of the four third-order moments in W. Hence, W is a subset of W containing eight independent moments (instead of ten). However, given moments in W it is straightforward to project them (using the weights and abscissas) into W, i.e., the eight-dimensional moment subspace that can be represented b V is W. The overall procedure can be represented as W W V V W W, where a projection step is used to define W. We will return to this point when discussing the numerical implementation below.. Eight-node quadrature: V from W In three dimensions, the set of twent moments up to third order is W, and we seek to define a eight-node quadrature based on these moments. We begin again b defining the mean particle velocit vector: M/M U p = M /M, () M/M and the velocit covariance matrix: M /M Up M /M U p U p M /M U p U p σ U = M/M U p U p M/M Up M/M U p U p, () M/M U p U p M/M U p U p M/M Up

18 so that σ eq = (σ + σ + σ )/. The eight-node extension of the procedure described above uses the threecomponent vector X = [X X X ] T defined b X = A (v U p ) so that v = AX + U p, () where A = L T is again defined in terms of the Cholesk decomposition of σ U : σ / σ /σ / σ /σ / L = (σ σ/σ ) / σ σ σ σ σ /. () (σ σ σ ( )/ ) σ σ/σ (σ σ σ σ ) / σ (σ σ σ ) As before, the two-node quadrature formulas (Eq. () with i (,, )) are used to find the weights and abscissas in each direction. The eight-node quadrature approximation is then defined using the tensor product of the one-dimensional abscissas b V = [(n, X (), X (), X () ), (n, X (), X (), X () ), (n, X (), X (), X () ), (n, X (), X (), X () ), (n, X (), X (), X () ), (n, X (), X (), X () ), (n, X (), X (), X () ), (n, X (), X (), X () )]. The linear sstem for the weights: n + n + n + n = n () n + n + n + n = n () n + n + n + n = n () n + n + n + n = n () n + n + n + n = n () n + n + n + n = n () = n + n + n + n = n () n + n + n + n = n () n + n + n + n = n () n + n + n + n = n () () () where n (i) +n (i) =, i (,, ); has rank. Thus, in general, four additional equations are required to determine n α. Three of these equations come from the second-order moments X i X j : X () X () n + X () X () n + X () X () n + X () X () n +X () X () n + X () X () n + X () X () n + X () X () n = m =, X () X () n + X ()X () n + X ()X () n + X ()X () n +X () X () n + X () X () n + X () X () n + X () X () n = m =, X () X () n + X () X () n + X () X () n + X () X () n +X () X () n + X ()X () n + X ()X () n + X ()X () n = m = ; ()

19 and the fourth equation comes from the third-order moment X X X : X () X () X () n + X ()X () X () n + X ()X () X () n +X () X () X () n + X () X () X () n + X () X () X () n +X () X () X () n + X ()X () X () n = m. () The moment m and the abscissas are known. The linear sstem of eight equations (Eqs. () ()) can be solved analticall. In order to ensure that the weights are non-negative, we define ρ = m [a( a)b( b)c( c)] / () where a = n () =. + γ, b = n () =. + γ, c = n () =. + γ. Then, if m, we let ρ = min[ρ, abc, ( a)( b)c, ( a)b( c), a( b)( c)]. () Otherwise, if m <, we let ρ = min[ ρ, ( a)bc, a( b)c, ab( c), ( a)( b)( c)]. () The weights are then given b n = abc ρ, n = ( a)bc + ρ, n = a( b)c + ρ, n = ( a)( b)c ρ, n = ab( c) + ρ, n = ( a)b( c) ρ, n = a( b)( c) ρ, n = ( a)( b)( c) + ρ. () Note that for cases where ρ ρ, one of the weights will be zero. Hence, in order to ensure non-negative weights, we essentiall drop Eq. () and set the offending weight to zero. The remaining seven weights are determined b the other seven linear equations. In summar, the fourteen moments controlled with eight-node quadrature Obviousl this choice is not unique. An linear combination of the seven uncontrolled third-order moments could be used: m, m, m, m, m, m, m. The choice of m has the advantage that is smmetric in the indices and it is eas to control the resulting weights so that the are non-negative. Onl thirteen moments are controlled when one weight is null. However, this never occurs for an of the examples considered in this paper.

20 are W = (m, m, m, m, m, m, m, m, m, m, m, m, m, m ). The overall procedure can be represented as W W V V W W, where a projection step is used to define W. Similar to two dimensions, onl four of the ten third-order moments appear in W. These four moments are independent linear combinations of the ten third-order velocit moments.. Comparison to other methods In the two-node quadrature method of [, ], a different approach was used to fix the weights and abscissas. The difference lies essentiall in which moments are controlled. In [, ] the eight moments (M, M, M, M, M, M, M, M + M + M ) were used. Note that this moment set does not include cross moments of second order (e.g. M ). In contrast, in the new formulation the covariance matrix is exactl reproduced. This of course comes at the expense of transporting more moments (twent instead of eight). Note that in the previous method the number of moments is equal to the number of unknown weights and abscissas. It might be argued that the new method could be improved for d > because the number of moments controlled is less than the number of unknowns. For example, if we consider the eight-node case ( unknowns), we can observe that the fourteen moments in W do not include all third-order moments. However, [] has shown that an optimal moment set exists wherein moments, including all moments of third order, can be used to find the weights and abscissas in V. Likewise, the eight moments in W can be augmented b four additional moments: m, m, m, and m, to form an optimal set for computing V. In practice, the use of the optimal moment sets introduces two new issues. First, the weights and abscissas must be found numericall b solving a nonlinear sstem of equations that can be poorl conditioned. Second, the optimal moment sets contain fourth- and fifth-order moments, all of which would have to be added to W (i.e., more moment transport equations are needed). In order to test the feasibilit of using optimal moments, we have implemented a nonlinear (NL) solver to compute V using as initial guess the results given in Sec... We found that in most cases considered (e.g., the Riemann problem), the NL solver was able to converge (albeit ver slowl), but not alwas. The

21 non-convergent cases tpicall have weights that are ver unequal. However, it was not possible to predict in advance which moment sets would not converge. In all likelihood, the convergence problems will be more acute for V given the larger number of moments in the optimal set that are not controlled b the new method. Thus, given that the computational cost is orders of magnitude larger and convergence is not guaranteed, it is unlikel that a NL solver will provide a computationall efficient method for inverting moment sets. This conclusion is consistent with the findings of []. In general, since moment inversion in one dimension is well established and reliable [], it will likel be much more fruitful to increase the number of nodes and the order of moments (e.g. computing β = d nodes requires fifth-order moments) to achieve greater accurac. We should note that moment-inversion method proposed here is ver similar in spirit to the method proposed b [] for aerosol applications (i.e., passive transport of a distribution function). The principal differences are () in [] the define the linear transformation matrix A in terms of the eigenvectors of the covariance matrix, and () the compute the weights without attempting to control m (i.e., their formula is equivalent to setting ρ = ). Of these differences, the first is the most important for the applications considered in this work. We have, in fact, attempted to define A as done b []. However, because the velocit is a dnamic variable (i.e., the spatial fluxes are computed using the velocit abscissas), the properties of the eigenvectors make them a poor choice for d =. The fundamental difficult is the fact that the eigenvectors of σ U do not var smoothl with its components. In other words, small changes in the correlation structure cause the orientation of the eigenvectors to jump in a discontinuous manner. Thus, at two neighboring spatial locations, the resulting velocit abscissas can be completel different (even though the moments are continuous). The net result is that the fluxes computed from the abscissas are then discontinuous, leading to random fluctuations in the moments. In contrast, the Cholesk matrix L varies smoothl with the components of σ U and, hence, the fluxes are well behaved. The onl known disadvantage of using the Cholesk matrix is that it depends on the ordering of the covariance matrix, and is thus different for each of the six permutations of the coordinates. In essence, one direction is chosen as the principal direction (i.e., X in Fig. ). For the examples considered in this work, this turns out not to be a problem because of smmetr. However, it would be desirable to replace L with a smoothl varing, permutation-invariant linear transform that generates abscissas close to those found using L (if it exists) and still diagonalizes σ U. Finall, we can note that quadrature-based moment closures bear some resem- The problem can be fixed for d = because the eigenvectors can be constructed to be independent of σ.

22 blance to discrete velocit models [,, ] and to the lattice Boltzmann method (LBM) [, ], including multispeed off-lattice Boltzmann methods []; however, the differences are significant. (See Appendix B for more details.) For example, in LBM the discretizations of velocit space, phsical space and time are strongl coupled and determine the numerical algorithm and properties [], while quadrature-based moment methods do not directl impl a phsical-space and time discretization scheme. This can be seen from the fact that the moment equations closed b quadrature are still written as continuous functions of phsical space and time. The principal similarit between quadrature methods and LBM is that both represent velocit phase space b a discrete set of velocities. However, a ver significant difference between the two methods is that in LBM the discrete velocities are fixed (depending on the lattice, spatial dimensions, etc. [] ), while in quadrature methods the are variables (i.e., fields) that var in space (velocit and phsical) and time according to the local flow phsics (i.e., the velocit moments). Another important difference is the number of unknowns available to control the moments. In LBM, onl the weights are used to construct a linear mapping into the moments [] and, hence, the maximum number of moments that can be controlled is equal to the number of weights (e.g., the thirteen-velocit model DQ corresponds to thirteen moments up to third order in three dimensions). In contrast, we have seen that b using quadrature eight velocit abscissas could, in principle, control moments if all the weights and abscissas were allowed to var. However, because of the difficult with inverting the nonlinear mapping between the moments and the quadrature nodes, the full quadrature procedure is intractable. Nevertheless, the partial quadrature method presented here still allows us to control fourteen moments with onl eight nodes (i.e., essentiall the same number of moments as with DQ). Another difference is that LBM uses a linear relaxation process to represent collisions [], while quadrature can be developed for an closed collision operator (e.g. the Boltzmann hard-sphere collision operator []). Finall, we can note that as shown in [] quadrature-based methods can be used for arbitrar collision times (or even collision-less cases), while LBM methods are designed for collision-dominated flows []. Based on these differences, we can expect that the range of flow phenomena that can be captured using quadrature methods will be significantl larger than with LBM (or with multispeed off-lattice Boltzmann methods []). Numerical Algorithm The sstem of moment transport equations can be solved numericall for three-dimensional problems b extending the numerical methods described in [, ]. Here, however, we are interested in solutions that depend on onl one

23 spatial dimension and time. Hence we will describe the method in the context of one-dimensional fluxes.. Transport equations The examples considered in this work have onl one inhomogeneous direction (x ), and we can take gravit (when needed) to be in the x direction. Thus we need to define the numerical scheme in one spatial dimension (albeit with a -dimensional velocit phase space) for the following moment equations: M ij t M ijk t M t M i t + M x =, + M i x = D i, + M ij x + M ijk x = D ij + C ij, = D ijk + C ijk. The source terms due to gravit and drag are defined b D i = D ij = β α= β α= D ijk = β α= ( ) Fiα n α δ i g, m p n α [( Fiα m p gδ i n α [( Fiα m p gδ i ) ( Fjα and the source terms due to collisions b )U iα ] U jα + gδ j, m p ) ( ) Fjα U jα U kα + gδ j U kα U iα + m p () ( ) ] Fkα gδ k U iα U jα, m p () Cij = M τ (σ eqδ ij σ ij ), Cijk = ( () ijk Mijk). τ We shall see below that dividing the source terms into two contributions follows naturall from the effects of gravit and drag on the weights and abscissas (i.e. the are convective processes in phase space).

24 . The solution algorithm The steps in the proposed solution algorithm are as follows: () Initialize the weights and abscissa in V β and the moments in W d. () Advance in time ( t) the moments in W d using the moment transport equations (time split): advance moments due to fluxes, advance moments due to gravit and drag, advance moments due to collisions. () Invert the moments in W d to find the weights and abscissas in V β. () Compute moments in W d using V β (projection step). () Return to step. The time step t is set to a fraction of the smallest characteristic time (convection, drag, or collisions).. A first-order, one-dimensional scheme Because of the conservative form of Eq. (), the finite-volume method [] is a natural candidate for its discretization. The underling kinetic equation (Eq. ()) can be used for the derivation of a numerical flux formula that ensures the robustness of the corresponding scheme. We begin b introducing the following notation for the vector of velocit moments up to third order, their spatial flux terms, and their source terms [, ]: M M M M D W = M, H(W) = M, S(W) = D + C. M M D + C Using classical notation, a fractional two-step, first-order, explicit, finite-volume scheme for Eq. () reads Wi = Wi n t x W n+ i = W(W [ G(W n i, Wi+) n G(Wi, n Wi n ) ], () i, t),

25 where G is the corresponding numerical flux function (described below) and W(W i, t) is an approximate solution at time t = t of the differential sstem dw dt = S(W) with W() = W i. () The advantage of a fractional-step algorithm is the possibilit of using a quasianaltic solution for the second step of the scheme to handle the stiffness of the source terms. Due to their forms, we will split the contributions to the source terms into two parts (i) drag and gravit and (ii) collisions. Drag and gravit To compute the first part, we note that the drag and gravit terms in Eq. () are equivalent to dn α =, dt du iα = F iα gδ i. dt m p () These expressions can be explicitl solved to find the changes in the weights and abscissas due to drag and gravit: n α = n α, U iα = exp( t/τ α )U iα + [ exp( t/τ α )](U fi gδ i τ α ), () where n α and U iα are the weights and abscissas found from Wi using quadrature, and τ α is the corresponding drag time (see Eq. ()) evaluated at the beginning of the step. The results in Eq. () are then emploed to compute the moments Wi using projection. Collisions The next step is to compute the contribution due to collisions: dw dt = C(W) with W() = W i. () This can be done explicitl: W i = exp( t/τ i )W i + [ exp( t/τ i )] (W i ) () where τ i is the local collision time and (W) denotes the vector of equilibrium moments. To finish up, quadrature is used to compute n n+ α and Uiα n+ from Wi, and Wi n+ is computed using projection.

26 . Kinetic-based definition of the fluxes Using the ideas of kinetic-based schemes [, ], it is natural to adopt the following expression for the numerical flux function (see Eq. ()): with and G(W l, W r ) = Q + (W l ) + Q (W r ) () β Q + U (W l ) = n αl max(, U αl ) iαl α= U iαl U jαl U iαl U jαl U kαl β Q U (W r ) = n αr min(, U αr ) iαr α= U iαr U jαr U iαr U jαr U kαr () () where the indices (i, j, k) on the velocit abscissas in the column vector (of length twent) on the right-hand side are chosen to represent the moments up to third order in W d. Note that although these expressions have been written for fluxes in the x direction, the extension to other directions is straightforward. As noted earlier, the use of kinetic-based fluxes is a ke difference between quadrature-based moment methods and standard moment methods.. Boundar conditions In order to evaluate the fluxes in Eq. (), we will need to specif boundar conditions. Without loss of generalit, we can consider the left-hand side of the domain and denote the boundar cell b i =. From the definition of the fluxes, we can observe that it suffices to specif the weights and abscissas at