Flowing Matter pp 271299  Cite as
QuadratureBased Lattice Boltzmann Models for Rarefied Gas Flow
Abstract
At the microscale, experiments show that in the vicinity of a moving boundary, rarefied fluid flows exhibit a velocity slip (i.e., the fluid and wall velocities are not equal), as well as a temperature jump (i.e., the fluid temperature is not equal to the wall temperature). Such effects can be captured within the framework of the Boltzmann equation by employing kinetic boundary conditions (i.e., diffuse reflection). In our contribution, we present the systematic construction of lattice Boltzmann (LB) models based on Gauss quadratures, which allows one to simulate gas flows between diffuse reflecting boundaries from the hydrodynamic limit to the ballistic regime. Key to the success of this approach is the use of halfrange Gauss–Hermite quadratures, which are essential in order to capture the discontinuity of the distribution function induced by the boundaries. Since the resulting models are offlattice, an overview of the appropriate numerical schemes for these models is also provided.
Keywords
Rarefied gases Diffuse reflection Halfrange Hermite polynomials Couette flow Poiseuille flow9.1 Introduction
At nonnegligible values of the Knudsen number Kn (defined as the ratio between the mean free path of the fluid particles in a gas and the characteristic length of the domain), the Navier–Stokes equations lose applicability [1, 2]. Such rarefied gas flows can be approached within the framework of the Boltzmann equation [3, 4, 5]. This equation describes the sixdimensional phasespace evolution of the distribution function f, where f(t, x, p)d^{3}xd^{3}p gives the number of particles at time t which are contained in an infinitesimal volume d^{3}x centred on x, having momenta in an infinitesimal range d^{3}p about p. Because of its complexity, the Boltzmann equation can be solved analytically only in a very limited number of cases. Alternatively, numerous wellestablished approaches to the numerical solutions of the Boltzmann equation are now currently used for academic or engineering purposes, of which we only mention the direct simulation Monte Carlo (DSMC) technique [6], the discrete velocity models (DVMs) [7, 8, 9], the discrete unified gaskinetic scheme (DUGKS) [10, 11, 12] and the lattice Boltzmann (LB) models [13, 14, 15, 16, 17, 18, 19, 20].
The LB models are a particular type of DVMs and are derived from the Boltzmann equation using a simplified version of the collision operator, as well as an appropriate discretisation of the momentum space, which ensure the recovery of the moments of the distribution function f up to a certain order N. Originally derived nearly 30 years ago from the lattice gas automata [17, 19, 20], the LB models were primarily designed to recover the hydrodynamics of fluid systems at the Navier–Stokes level. The LB models inherited the collisionstreaming concept from their ancestors, according to which the velocities of the fluid particles are aligned along the lattice links such that after one time step δt, each particle arrives at a neighbouring node [13, 14, 17, 19, 21, 22].
One disadvantage of the collisionstreaming paradigm is the increasing difficulty to approach fluid systems far from equilibrium (e.g., rarefied gases or micro/nanoscale flowing fluids) using suitable LB models. In this case, the accurate recovery of specific effects in channel flow at large values of Kn, such as the velocity slip and the temperature jump at the channel walls [1, 2], requires that higher order moments of the single particle distribution function f are ensured. Since the moments of f are derived by integration in the momentum space, their numerical computation involves the use of convenient quadrature methods. When using a quadrature method, the moments of the single particle distribution f (up to a certain order N) are exactly recovered by sums over a finite set of momentum vectors p_{k}, 1 ≤ k ≤ K [13, 14, 17, 18, 19, 21, 22, 23, 24, 25, 26]. As the fluid system is farther from the equilibrium and the characteristic value of the Knudsen number increases, the number K of the momentum vectors (i.e., the quadrature points) should also be increased, as it will be shown later. This task becomes more and more elaborated if one wants to keep the particles hopping from a lattice node to another one during a single time step [27, 28, 29, 30].
An alternative to the collisionstreaming paradigm is provided by the offlattice LB models, where the distribution functions are evolved in the lattice nodes using finitedifference, finitevolume or interpolation schemes [31, 32, 33, 34]. A fourthorder, offlattice LB model for the simulation of thermal flows in the continuum regime (small values of the Knudsen number), where the fluid density, velocity and temperature fields are derived from a single set of distribution functions, was proposed by Watari and Tsutahara [35] for 2D flows and subsequently extended to the 3D case [36, 37, 38]. Offlattice LB models of any order N can be easily constructed using the Gauss quadrature method in the velocity space [18, 23, 25, 26, 39, 40].
Another challenge for microfluidics simulations is due to the implementation of boundary conditions. In general, the particle–wall interaction governs the distribution of particles emerging from the wall back into the fluid. Since the distribution of particles travelling from the fluid towards the wall is essentially arbitrary, the distribution function becomes discontinuous near the wall [41, 42]. Examples of boundary conditions include the diffusespectral [9] and Cercignani–Lampis [43] particle–wall interaction models; however, for simplicity, we restrict the analysis to the simpler diffuse reflection model, which is a limiting case of both models mentioned above. According to the diffuse reflection paradigm, the reflected particles follow a Maxwell–Boltzmann distribution corresponding to the wall temperature and velocity. In order to accurately compute the incident and emergent fluxes required to impose diffuse reflection (kinetic) boundary conditions, it is convenient to discretise the velocity set based on halfrange Gauss quadrature methods. Such techniques were also used in the frame of DVMs [44, 45] and more recently, they were adapted for the LB method [24, 26, 46, 47].
9.2 Generalities
 1.
discretising the momentum space;
 2.
replacing the equilibrium distribution function f^{eq} in the collision term of Eq. (9.1) by a truncated polynomial with respect to the particle velocity ;
 3.
replacing the momentum derivative of the distribution function f in Eq. (9.1) using a suitable expression [see Eq. (9.23)];
 4.
choosing a numerical method for the time evolution and spatial advection;
 5.
implementation of the boundary conditions.
9.3 OneDimensional QuadratureBased LB Models
9.3.1 FullRange Gauss–Hermite Quadrature
9.3.2 HalfRange Gauss–Hermite Quadrature
The momentum derivative of f can be projected on the space of the halfrange Hermite polynomials as discussed in Sect. 9.3.1. Since this projection is not relevant for the further development of this chapter, we refer the reader to Refs. [49, 50] for further details.
9.4 LB Models in the ThreeDimensional Momentum Space
In the threedimensional (3D) momentum space, the discretisation procedure can be conducted using a direct product rule. On each Cartesian axis α ∈{x, y, z}, one can choose a specific Gauss–Hermite (fullrange or halfrange) quadrature of order Q_{α}, depending on the characteristics of the flow (e.g., the existence of a noticeable wallinduced discontinuity of the distribution function along the α axis). Let \(p_{\alpha ,k_{\alpha }}\), \(1\leq k_{\alpha } \leq \mathcal {Q}_{\alpha }\), be the quadrature points on the Cartesian axis α (note that \(\mathcal {Q}_{\alpha } \in \{Q_{\alpha }, 2Q_{\alpha }\}\) as mentioned in Sect. 9.3). These quadrature points are the components of the 3D vectors p_{k}, \(k = (k_{z}1){\mathcal {Q}_{x}}{\mathcal {Q}_{y}} + (k_{y}1){\mathcal {Q}_{x}} + k_{x}\), \(1\leq k \leq K = {\mathcal {Q}_{x}}{\mathcal {Q}_{y}}{\mathcal {Q}_{z}} \). Following Refs. [25, 26], we generally refer to the resulting models as mixed quadrature LB models. The numerical solution of the discretised form Eq. (9.10) of the Boltzmann equation can be obtained following the steps described in Sect. 9.3, which will be detailed further.
In this contribution we restrict ourselves to the LB simulation of Couette and forcedriven Poiseuille flows of rarefied gases between parallel plates. In these cases, the flow is homogeneous along the z axis and the computational effort can be significantly decreased by taking advantage of the reduced distribution functions introduced in Sect. 9.4.1. Section 9.4.2 discusses the construction of the mixed quadrature LB models for the investigation of rarefied Couette and forcedriven Poiseuille flows using the reduced distribution functions and the resulting evolution equations are presented in Sect. 9.4.3. The section ends with a discussion of our nondimensionalisation convention, presented in Sect. 9.4.4.
9.4.1 Reduced Distributions
9.4.2 Mixed Quadrature LB Models with Reduced Distribution Functions
In the mixed quadrature LB models, the momentum space is constructed using a direct product rule. This allows the quadrature on each axis to be constructed independently by taking into account the characteristics of the flow. When the gas flow is homogeneous along the z axis, the reduced distribution functions evolve in a twodimensional space and thus, the elements of the discrete set of momentum vectors can be written as p_{ij} = (p_{x,i}, p_{y,j}). The indices i and j run from 1 to \(\mathcal {Q}_{\alpha }\) (α ∈{x, y}), where \(\mathcal {Q}_\alpha = Q_\alpha \) or \(\mathcal {Q}_{\alpha } = 2Q_\alpha \) when a fullrange or a halfrange quadrature of order Q_{α} is employed on the α axis. As shown in Refs. [25, 26], a fullrange Gauss–Hermite quadrature of order Q_{y} = 4 is sufficient on the y axis in order to capture exactly the evolution of the velocity, temperature and of heat flux fields. For low Mach flows, the quadrature order Q_{x} can be taken to be Q_{x} = 4 in the Navier–Stokes regime, where the fullrange Gauss–Hermite quadrature is efficient. As Kn is increased, Q_{x} must also be increased in order to retain the accuracy of the simulation results. In the case of the channel flows considered in this chapter, the discontinuity in the distribution functions ϕ and χ induced by the diffusereflective walls becomes significant for sufficiently large Kn. Hence, the fullrange Gauss–Hermite quadrature on the x axis becomes inefficient compared to the halfrange Gauss–Hermite quadrature, as demonstrated in Refs. [25, 26]. In this chapter, we only consider the halfrange Gauss–Hermite quadrature of order Q_{x} on the x axis. The resulting models are denoted HHLB(Q_{x}) ×HLB(4) following the convention in Ref. [26], employing 8Q_{x} velocities and 16Q_{x} distinct populations (ϕ_{ij} and χ_{ij}), as discussed below.
9.4.3 The Lattice Boltzmann Equation
9.4.4 NonDimensionalisation Procedure

The wall temperature, such that T_{w} = 1.

The particle mass, such that m = 1.

The reference speed \(c_{\mathrm {ref}} = \sqrt {k_B T_w / m}\).

The channel width, such that L = 1.
 The average particle number density, such that the total number of particles obeys:$$\displaystyle \begin{aligned} N_{\mathrm{tot}} = \int_{1/2}^{1/2} dx\, n(x) = 1. \end{aligned} $$(9.56)
9.5 Simulation Results
The advantage of the quadraturebased approach to LB modelling quickly becomes apparent when considering rarefied flows. An excellent arena for this type of tests is represented by channel flows. In particular, we will restrict the discussion to the Couette and the forcedriven Poiseuille flows between parallel plates, which have become canonical benchmark problems in the microfluidics community. In the context of these flows, the distribution function becomes discontinuous due to the diffuse reflection interaction with the boundary. Thus, at large values of Kn, halfrange quadratures are much more efficient than the more traditional fullrange ones [25, 26, 52, 53]. More complex flows, where the application of halfrange Gauss–Hermite quadrature is essential are investigated in Refs. [50, 54, 55].
9.5.1 Couette Flow Between Parallel Plates
In the stationary state of Couette flow, rarefied gases exhibit a nonlinear velocity profile in the proximity of the moving walls. This nonlinearity originates from the wallinduced discontinuity of the particle distribution function and its spatial extension (i.e., the width of the socalled Knudsen layer, where the discontinuity induced through interparticle collisions) is of the order of the mean free path of the fluid particles [1, 2]. Diffuse reflection boundary conditions are used to capture this wallinduced discontinuity [24, 25, 26], as shown in Fig. 9.1 (right).
Mathematically, diffuse reflection boundary conditions entail that the distribution functions for the particles emerging from the walls back into the fluid satisfy f(x = ±L∕2, p, t) = f^{eq}(p;±u_{w}), which is valid for ± p_{x} < 0, respectively. Noting that f^{eq}(−p;u) = f^{eq}(p;−u), it can be seen that the solution of the Boltzmann equation (9.12) possesses the symmetry f(−x, p, t) = f(x, −p, t). This symmetry allows only the right half of the channel to be considered, provided that bounce back boundary conditions are implemented at the channel centreline [i.e., f(0, −p, t) = f(0, p, t)]. This simplification effectively halves all computation times. Moreover, since the system is homogeneous along the y axis, no advection is performed in this direction and a discretisation using a single node is sufficient. In fact, this corresponds to implementing periodic boundary conditions along the y axis. The p_{z} degree of freedom is integrated out, as explained in Sect. 9.4.1, and no advection is performed along the z direction. The right panel of Fig. 9.1 presents schematically the implementation of the Couette flow geometry.
9.5.2 ForceDriven Poiseuille Flow Between Parallel Plates
The second remarkable microfluidics specific effect occurring in the forcedriven Poiseuille flow refers to the development of a dip (local minimum) in the temperature profile T(x) at the centre x = 0 of the channel. The dip occurrence was predicted by the kinetic theory at the superBurnett level and observed by DSMC simulations [1, 65, 66, 67, 68, 69, 70, 71].
9.6 Conclusions
In this chapter, we presented a systematic procedure for the construction of high order mixed quadrature LB models based on the fullrange and halfrange Gauss–Hermite quadratures. A particular attention was given to the case when the flow is homogeneous along the z axis, when reduced distribution functions can be used in order to minimise the computational effort. The capabilities of these models are demonstrated in the context of the Couette and forcedriven Poiseuille flows between parallel plates at various values of Kn. Excellent agreement is found between our results and benchmark data available in the literature from the Navier–Stokes level up to the transition regime.
The success of our models relies on the accurate recovery of halfrange integrals required for the implementation of diffuse reflection. Such integrals are exactly recovered by employing the halfrange Gauss–Hermite quadrature.
Our numerical method for solving the LB evolution equations employs finitedifference techniques. In particular, we implemented the advection using the fifthorder weighted essentially nonoscillatory (WENO5) scheme and the timestepping was performed using the thirdorder Runge–Kutta (RK3) method. This allowed us to obtain accurate results using a small number of nodes (16 for the Couette flow and 32 for the forcedriven Poiseuille flow).
Taking advantage of the homogeneity of the flows studied in this chapter with respect to the z axis, we eliminated the z axis degree of freedom by integrating the Boltzmann–BGK equation with respect to p_{z}. In order to correctly track the evolution of the temperature and heat flux fields, we employed two reduced distribution functions, ϕ and χ, obtained by integrating with respect to p_{z} the Boltzmann distribution f multiplied by 1 and \(p_z^2 / m\), respectively. The extension of the methodology presented in this chapter to more complex flow domains is straightforward since the mixed quadrature paradigm allows the type of quadrature and the quadrature orders to be adjusted for each axis separately. The treatment of complex boundaries can be performed by using the standard staircase approximation [72, 73] or the more recent vielbein approach [50]. Finally, more complex relaxation time models, such as the Shakhov model [74, 75], can be implemented as described in, e.g., Refs. [54, 55].
We conclude that the models described in this chapter can be used to obtain numerical solutions of the Boltzmann–BGK equation for channel flows at arbitrary values of the Knudsen number.
Footnotes
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