Extensive analysis of the lattice Boltzmann method on shifted stencils

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Reference

Extensive analysis of the lattice Boltzmann method on shifted stencils

HOSSEINI, S, et al.

Abstract

Standard lattice Boltzmann methods (LBMs) are based on a symmetric discretization of the phase space, which amounts to study the evolution of particle distribution functions (PDFs) in a reference frame at rest. This choice induces a number of limitations when the simulated flow speed gets closer to the sound speed, such as velocity-dependent transport coefficients. The latter issue is usually referred to as a Galilean invariance defect. To restore the Galilean invariance of LBMs, it was proposed to study the evolution of PDFs in a comoving reference frame by relying on asymmetric shifted lattices [N. Frapolli, S. S. Chikatamarla, and I. V.

Karlin, Phys. Rev. Lett. 117, 010604 (2016)]. From the numerical viewpoint, this corresponds to overcoming the rather restrictive Courant-Friedrichs-Lewy conditions on standard LBMs and modeling compressible flows while keeping memory consumption and processing costs to a minimum (therefore using the standard first-neighbor stencils). In the present work systematic physical error evaluations and stability analyses are conducted for different discrete equilibrium distribution functions [...]

HOSSEINI, S, et al . Extensive analysis of the lattice Boltzmann method on shifted stencils.

Physical Review. E , 2019, vol. 100, no. 6, p. 063301

DOI : 10.1103/PhysRevE.100.063301

Available at:

http://archive-ouverte.unige.ch/unige:128678

Disclaimer: layout of this document may differ from the published version.

Extensive analysis of the lattice Boltzmann method on shifted stencils

S. A. Hosseini ,^1,2,3C. Coreixas ,⁴N. Darabiha ,²and D. Thévenin¹

1Laboratory of Fluid Dynamics and Technical Flows, University of Magdeburg “Otto von Guericke,” D-39106 Magdeburg, Germany

2Laboratoire EM2C, CNRS, CentraleSupélec, Université Paris-Saclay, 91192 Gif-sur-Yvette Cedex, France

3International Max Planck Research School (IMPRS) for Advanced Methods in Process and Systems Engineering, Magdeburg, Germany

4Department of Computer Science, University of Geneva, 1204 Geneva, Switzerland

(Received 21 September 2019; published 4 December 2019)

Standard lattice Boltzmann methods (LBMs) are based on a symmetric discretization of the phase space, which amounts to study the evolution of particle distribution functions (PDFs) in a reference frame at rest. This choice induces a number of limitations when the simulated flow speed gets closer to the sound speed, such as velocity-dependent transport coefficients. The latter issue is usually referred to as a Galilean invariance defect.

To restore the Galilean invariance of LBMs, it was proposed to study the evolution of PDFs in a comoving reference frame by relying on asymmetric shifted lattices [N. Frapolli, S. S. Chikatamarla, and I. V. Karlin, Phys. Rev. Lett. 117, 010604 (2016)]. From the numerical viewpoint, this corresponds to overcoming the rather restrictive Courant-Friedrichs-Lewy conditions on standard LBMs and modeling compressible flows while keeping memory consumption and processing costs to a minimum (therefore using the standard first-neighbor stencils). In the present work systematic physical error evaluations and stability analyses are conducted for different discrete equilibrium distribution functions (EDFs) and collision models. Thanks to them, it is possible to (1) better understand the effect of this solution on both physics and stability, (2) assess its viability as a way to extend the validity range of LBMs, and (3) quantify the importance of the reference state as compared to other parameters such as the equilibrium state and equilibration path. The results clearly show that, in theory, the concept of shifted lattices allows the scheme to deal with arbitrarily high values of the nondimensional velocity.

Furthermore, just like the zero-Mach flow for the standard stencils, it is observed that setting the shift velocity to the fluid velocity results in optimal physical and numerical properties. In addition, a detailed analysis of the obtained results shows that the properties of different collision models and EDFs remain unchanged under the shift of stencil. In other words, by introducing a velocity shift in the stencil, the optimal operating point, in terms of physics and numerics, will also be shifted by the same vector regardless of the EDF or collision model considered. Eventually, while limited to the D2Q9 stencil with the nine possible first-neighbor shifts, the present study and corresponding conclusions can be extended to other stencils and velocity shifts in a straightforward manner.

DOI:10.1103/PhysRevE.100.063301 I. INTRODUCTION

Recent years have witnessed extensive work on the exten- sion of the lattice Boltzmann method (LBM) to high-speed compressible flows [1–21]. For such configurations and com- plex nonisothermal flows, the main limitations of the LBM lie in (1) physical errors that become important for large velocity and temperature variations with respect to the reference state, and (2) the very restrictive CFL condition associated with the classical SRT collision operator especially in the limit of van- ishing viscosities. To alleviate these restrictions, many differ- ent approaches have been proposed over the past two decades.

In general, these approaches can be characterized as pertain- ing to two main categories: (a) derivation of enhanced equi- librium states and (b) enhanced equilibration paths. A third possibility would consist of using Eulerian space- and time- discretization strategies instead of the Lagrangian approach specific to the LBM. Nonetheless, this leads to a drastic decrease of the CPU efficiency and/or accuracy of the method.

The concept of optimal equilibration paths in the context of the LBM has been an area of active research and resulted in a variety of modified collision models [22]. The first

class of modified collision models is based on d’Humière’s generalized lattice Boltzmann equations [23]. This concept—

now known and referred to as the multiple relaxation time (MRT) model—relies on applying the relaxation operator in moment space, as originally proposed by Higueraet al.[24]

and further discussed by Benzi and coworkers [25]. Doing so, one gets (i) additional degrees of freedom in the form of the choice of the moments basis and (ii) an indepen- dent control over the equilibration rates of the different mo- ments. The zeroth- and first-order moments being invariants of the discrete collision operator (for the classical mass- and momentum-conserving LBM), the choice of their relaxation rates does not affect the dynamics of the solver. The relaxation rates of moments tied to the off-diagonal components of the second-order moment are fixed by the fluid kinematic viscosity. For moments related to the diagonal components, however, the relaxation rate can be used to impose the fluid bulk viscosity (contrary to the SRT where the bulk viscosity is fixed toζ =2ν/3 in three dimensions (3D) [26]). Relaxation rates of higher-order moments can be fine-tuned or defined by considering stability and/or optimal spectral dispersion-

dissipation properties [27]. Looking at the overall structure of the MRT, it is observed that better numerical properties can be achieved by adjusting error terms tied to higher-order moments [28]. As such, closure for the free parameters can only be provided a posteriori through asymptotic and/or spectral analyses of the discrete scheme [29–31]. Attempts at further reducing velocity-dependent higher-order error terms later resulted in approaches such as the cascaded method based on relaxation of central moments (evaluated in the local fluid velocity frame) instead of raw moments, and the cumulants LBM [32,33]. It is also worth noting that similar asymptotic behaviors could also be obtained by adopting extended equilibria, even with the very simple Bhatnagar- Gross-Krook (BGK) collision operator [22].

Theregularizedcollision operators, starting with the model proposed in Refs. [34,35] for standard lattices and further ex- tended to high-order ones in Refs. [36–38], go about introduc- ing an optimal equilibration path by filtering out higher-order contributions in the distribution function [f_α⁽ⁿ⁾ =0,∀n>1].

In practice, this is achieved by reconstructing the precollision populations taking into account only equilibrium and first- order nonequilibrium contributions

f_α^reg= f_α^(eq)+ f_α⁽¹⁾. (1) The first-order nonequilibrium contributions f_α⁽¹⁾ can be computed via projection onto the Hermite polynomial ba- sis and/or the CE expansion [8,39]. The latter approach, more commonly referred to as recursive regularized (RR) LBM, leads to pronounced stability improvements for high- Reynolds-number flows in both weakly and fully compress- ible regimes [9,14,39–43].

A third class of LB formulation, usually referred to as the entropic LBM (ELBM) relies on both optimization of the equilibration path and equilibrium state to achieve better sta- bility properties [7,44–48]. In the original formulation, both the attractor (equilibrium state) and equilibration rate are de- termined through a discrete entropy functional. The entropic equilibrium state construction guarantees linear stability of the scheme (as shown later in this article). The relaxation rate appearing in the fully discretized time-evolution equation is defined as:

αβ= αδt

2ν/c²s +δt, (2)

wherecsis the lattice speed of sound,δtis the time step size, and the free parameterαis determined as the positive-valued nontrivial root of the entropy estimate. This step of the ELBM can be seen as a nonlinear flux limiter as it imposes an upper limit on population change during the collision step [49].

This can also be related to the behavior of a subgrid scale model [50] whose purpose is to mimic energy transfers, from large scales to small ones, through dissipation that occur at length scales that cannot possibly be resolved by the grid cell size [51].

The discrete equilibrium state is another component of the LBM that has been extensively studied as a possible way to enhance its numerical properties. While most EDFs recover the same moments appearing at the Navier-Stokes (NS) level, they differ in their higher-order moments. Although a wide

variety of EDFs have been proposed over the years, two systematic approaches to constructing discrete EDFs have be- come widely popular [22]: (a) discretization of the continuous EDF via projection onto Hermite polynomials space [52,53]

and (b) construction of a discrete equilibrium state based on extremization of a discrete entropy functional [46,48,54].

Projecting the continuous EDF onto the Hermite polyno- mial space and keeping only terms up to order two results in the well-known second-order polynomial discrete EDF.

This form of the EDF can also be recovered using a Taylor- McLaurin expansion of the continuous Gaussian distribu- tion [55,56]. This form of the EDF is known to admit errors in the stress tensor at the NS level. These errors come from inconsistencies in the third-order moments tensor of the EDF.

While diagonal components of this tensor cannot be directly corrected due to insufficient symmetry of the first-neighbor stencils, the off-diagonal components can be correctly re- covered using higher-order Hermite expansions of the EDF that are naturally accounted for in the most sophisticated collision models [22]. For the D2Q9 stencil, for example, keeping up to third-order terms of the expansion is sufficient to correctly recover the off-diagonal components. Keeping higher-order terms, however, while not affecting the NS level terms directly, has been observed to improve the stability of the scheme in the small viscosity region [9,29,43].

Entropic equilibria are another systematic approach to derive the discrete EDF based on the maximum entropy principle [57,58]. As previously mentioned, starting from the abscissae of the Gauss-Hermite quadrature, contrary to the previous approach where the discrete EDF is a truncated projection of the continuous EDF onto a set of orthonormal functions, here the discrete EDF is directly reconstructed by solving a conditional extremization problem. In the isothermal case, the discrete EDF can be found as the (exact) solution extremizing the discrete entropy under constraints of mass and momentum conservation [46]. For thermal LBMs, it is usually not possible to find exact solutions, and instead, approximated ones are obtained using root-finding algorithms. In addition, this further frees the resulting LBM from the restrictive choice of discrete velocities imposed by the Gauss-Hermite quadra- ture [7,16].

Another way to extend the stability domain of the LBM is based on the concept of adaptive stencils [1–3,10,11,59]. It is well known that the optimal operation point of the LBM is at velocity zero (as error terms are minimized at this point [7]).

As such, discretizing phase space around and as a function of local fluid velocity and temperature places the solver in its optimal operation point and minimizes errors. To better understand this approach and quantify its effect on stability, the present work focuses on a theoretical and numerical analysis of shifted stencils applied to standard first neighbor stencils only. In other words, only constant velocity shiftU with integer-valued components (Ux,Uy∈ {0,±δx/δt}) will be considered in the present paper.

To put forward the applicability of the concept to pre- existing equilibria and collision operators, three different ex- act EDFs and three different collision operators will be stud- ied. More specifically, Sec.IIpresents the detailed derivation of the LBM with shifted stencils along with the different

EDFs, i.e., second- and fourth-order polynomial Hermite ex- pansions and entropic and collision operators (SRT), lattice kinetic scheme (LKS), and fourth-order recursive regularized (RR). The effect of the stencil shift on both physical and nu- merical properties is studied by looking at the deviation from higher-order equilibrium moments and using the von Neu- mann analysis method, in Secs.IIIandIV, respectively. These results are then corroborated with numerical validations, using the isothermal vortex convection test case (Sec. V). Final conclusions are drawn in Sec.VI. For the sake of complete- ness, detailed derivations of the Jacobians used for the von Neumann analysis are also provided in Appendix.

II. SHIFTED STENCILS FORMALISM A. Phase-space discretization and equilibrium states

1. Hermite expansion with a shift velocity

Starting with the usual expression for the Maxwell- Boltzmann EDF inD-dimensional space:

f^(eq)(ξ,u,T)=ρ

2πkBT m

₋D/2

e^{−m(ξ−u)22kBT} , (3) where ξ is the particle speed, mthe molecular mass, T the local temperature, u the fluid velocity, and kB Boltzmann’s constant, and introducing a “reference” sound speed c²_s = kBT0/m0and a shift velocityU, we write the EDF in nondi- mensional form [60] (nondimensionalized usingcs):

f^(eq)(ξ^∗,u^∗, θ^∗)=ρ(2πθ^∗)^−D/2exp−(ξ^∗−u^∗)² 2θ^∗ , (4) whereθ^∗= ^kBT_c2^/m

s ,ξ^∗= ^ξ−_cs^U, andu^∗= ^u−_cs^U. The Maxwell- Boltzmann EDF can be expanded in terms of Hermite poly- nomials as [52,53,61]:

f^(eq)(ξ^∗,u^∗, θ^∗)=w(ξ^∗) ∞ n=0

1

n!a^(eq)_n (u^∗, θ^∗) :Hn(ξ^∗), (5) where the weight functionw(ξ^∗) is defined as:

w(ξ^∗)=(2π)^−D/2exp

−ξ^∗2 2

, (6)

and the rank n tensor corresponding to the nth Hermite polynomial is computed as:

H⁽ⁿ⁾(ξ^∗)= (−1)ⁿ

w(ξ^∗)∇_ξ^∗w(ξ^∗). (7) The first few Hermite polynomials can be readily computed using Eqs. (6) and (7) as:

H0=1, (8a)

H1,i1 =ξi^∗1, (8b) H2,i₁i₂ =ξi^∗₁ξi^∗₂−δi₁i₂, (8c) H3,i1i2i3 =ξi^∗1ξi^∗2ξi^∗3−

ξi^∗3δi1i2+ξi^∗2δi1i3+ξi^∗1δi2i3

, (8d) H4,i₁i₂i₃i₄ =ξi^∗1ξi^∗2ξi^∗3ξi^∗4+

δi₁i₂δi₃i₄+δi₁i₃δi₂i₄+δi₂i₃δi₁i₄

−

ξi^∗₃ξi^∗₄δi₁i₂+ξi^∗₂ξi^∗₄δi₁i₃+ξi^∗₂ξi^∗₃δi₁i₄

+ξi^∗₁ξi^∗₄δi2i3+ξi^∗₁ξi^∗₃δi2i4+ξi^∗₁ξi^∗₂δi3i4

, (8e)

whereδi jis the Diracδfunction. Using these polynomials and the EDF, the corresponding coefficients defined as:

a^(eq)_n (u^∗, θ^∗)= H⁽ⁿ⁾(ξ^∗)f^(eq)(ξ^∗,u^∗, θ^∗)dξ^∗, (9) can be obtained:

a₀^(eq)=ρ, (10a)

a₁^(eq)_,i1 =ρu^∗i₁, (10b)

a_2,^(eq)_i1i2 =ρu^∗i₁u^∗_i2+ρ(θ^∗−1)δi1i2, (10c) a^(eq)_3,i1i2i3 =ρu^∗i1u^∗_i2u^∗_i3

+ρ(θ^∗−1)

u^∗_i3δi1i2+u^∗_i2δi1i3+u^∗_i1δi2i3

, (10d) a^(eq)_4,i

1i₂i₃i₄ =ρu^∗i₁u^∗_i2u^∗_i3u^∗_i4 +ρ(θ^∗−1)²

δi1i2δi3i4+δi1i3δi2i4+δi2i3δi1i4

+ρ(θ^∗−1)

u^∗_i3u^∗_i4δi1i2+u^∗_i2u^∗_i4δi1i3+u^∗_i2u^∗_i3δi1i4

+u^∗_i1u^∗_i4δi2i3+u_i^∗₁u^∗_i3δi2i4+u^∗_i1u^∗_i2δi3i4

. (10e) Then the phase space can be discretized using the Gauss- Hermite quadrature [52,53], resulting in the discrete nondi- mensional abscissaec^∗_αwith (in the case of the D2Q9 stencil) c^∗_α,i∈ {−√

3,0,√

3}. In addition, the discrete weightsw_α= i=x,yw_α,i are now defined asw_α,i∈ {1/6,2/3,1/6}, which leads to the discrete EDF:

f_α^(eq),N(u^∗, θ^∗)=w_α N

n=0

1

n!a^(eq)_n (u^∗, θ^∗) :Hn,α(c^∗_α). (11) Eventually, the dimensional form of discrete shifted velocities reads asc_α,i∈ {−√

3kBT₀/m₀+Ui,Ui,√

3kBT₀/m₀+Ui}. 2. Entropic equilibrium

In the context of the entropic lattice Boltzmann method as described in Ref. [46], the discrete equilibrium state is found as the minimizer of a convex discrete entropy functional under mass and momentum conservation constraints only. The derivation starts with the roots of the third-order Hermite polynomials as the discrete abscissae and considering the following conservation constraints:

α

f_α^(eq) =ρ, (12)

α

c_α^∗f_α^(eq)=ρu^∗. (13) Under previously set constraints the EDF is derived as the function extremizing the discrete entropy function:

Hw_α,c^∗_α =

α

f_αln f_α

w_α

. (14)

Given the Galilean invariance of the weights, the expression for the entropy function is also Galilean invariant [59]. The EDF can be expressed as:

f_α^(eq)=w_αexp(χ0) D i=1

exp(c^∗_α,iχi), (15)

whereχ0andχiare the Lagrange multipliers associated with constraints on the zeroth and first-order moments. Introducing the following changes of variables,X =exp (−χ0) andZi= exp (χi), the EDF is rewritten as:

f_α^(eq)=w_αX⁻¹ D i=1

Z^cα,i∗ . (16) Writing down the conservation equations using the new vari- ables for the D2Q9 stencil, the following algebraic system of equations is obtained:

ρX =

α

w_α

i=x,y

Z^c∗α,i, (17a) ρu^∗xX =

α

w_αc^∗_α,x

i=x,y

Z^cα,i∗ , (17b) ρu^∗_yX =

α

w_αc^∗_α,y

i=x,y

Z^c∗α,i. (17c) Solving this system of equation forZx,Zy, andX and keeping positive roots one gets:

Zi= 2u^∗_i +

3u^∗_i²+1

1−u^∗_i , (18)

X⁻¹=ρ

i=x,y

2−

3u^∗_i²+1

. (19)

Therefore the entropic discrete equilibrium can be expressed as:

f_α^(eq)=w_αρ

i=x,y

2−

3u^∗_i²+1⎛

⎝2u^∗_i +

3u^∗_i²+1 1−u^∗_i

⎞

⎠

c^∗_α,i

.

(20) B. Collision models

1. Single relaxation time

With the above-mentioned definitions of the equilibrium state, the space- and time-continuous partial differential equa- tions of the discrete force-free Boltzmann equation, with the SRT approximation, are as follows:

∂tf_α+c_α·∇f_α=_α^(SRT), (21) where the collision operator_α^(SRT)is defined as [62]:

^(SRT)_α = −1 τ

f_α−f_α^(eq),N

, (22)

and the relaxation coefficientτis tied to the fluid viscosity as:

τ = ν

kBT/m. (23)

For isothermal flows, studied in this article:

τ = ν

c²_s. (24)

TABLE I. Values ofλused for the LKS collision model.

νδt/δ²x λ

0 0.503

5×10⁻⁴ 0.503

1×10⁻³ 0.53

5×10⁻³ 0.53

1×10⁻² 0.53

5×10⁻² 0.53

0.1 0.8

0.5 2

2. Lattice kinetic scheme

For the LKS, the collision operator is written as:

^(LKS)_α = −1 λ

f_α−f_α^(eq,LKS,N)

. (25)

The second relaxation coefficient λ is related to the SRT relaxation coefficient as:

λ−A=τ, (26)

whereAis a constant fixed by the choice of the free parameter.

The EDF is then defined as:

f_α^(eq,LKS,N)= f_α^(eq),N−A τ

w_α

2 a^(neq)₂ :H_2,α. (27) As demonstrated in Ref. [29], the LKS is a minimalist MRT collision model (with two relaxation coefficients) based on Hermite moments. It further reduces to the original regular- ized collision operator [34,35] when λ=1. In the context of the present study, parameters maximizing the stability domain, as derived in Ref. [29], are used. For the sake of readability these parameters are listed in TableI.

3. Recursive regularized collision operator

In the context of the SRT-RR model, the collision term can be written as:

^(RR)_α = f_α^(eq),N+

1−1 τ

f_α^(neq,N), (28) where the nonequilibrium part of the distribution function

f_α^(neq,N)is replaced by f_α^(1,N)and reconstructed as:

f_α^(1,N) u^∗, θ^∗

=w_α N n=2

1

n!a⁽¹⁾_n (c^∗_α) :Hn(u^∗, θ^∗). (29) For standard and isothermal LBMs, Malaspinas [39] proposed to approximatea⁽¹⁾₂ (c^∗_α) using the nonequilibrium part of the EDF as:

a⁽¹⁾_2,i1i2≈

α

Hi₁i₂

f_α−f_α^(eq),N

, (30)

while higher-order components are computed using the fol- lowing recursive formula:

a⁽¹⁾_n,i

1...i_n=a⁽¹⁾_n−1,i

1...i_n−1u^∗_in+

u^∗_i1. . .u^∗_in−2a⁽¹⁾_2,i

n−1i_n+perm(in) , (31)

where “perm” stands for the cyclic permutation of indices. For the D2Q9 stencil with a fourth-order EDF, the higher-order terms can therefore be computed as:

a₃⁽¹⁾_,xyy =u_x^∗a₂⁽¹⁾_,yy+2u^∗_ya₂⁽¹⁾_,xy, (32a) a_3,⁽¹⁾_xxy=u_y^∗a_2,⁽¹⁾_xx+2u^∗_xa⁽¹⁾_2,xy, (32b) a₄⁽¹⁾_,xxyy =u_y^∗a₃⁽¹⁾_,xxy+u^∗_x²a⁽¹⁾_2,yy+2u^∗_xu^∗_ya⁽¹⁾_2,xy. (32c) By reconstructing the nonequilibrium part, one ensures the proper macroscopic behavior of LBMs (up to the NS level) and filters out higher-order contributions (Burnett, Super- Burnett, etc.) which are usually not properly recovered due to the low symmetry of the lattice considered. It is worth noting that high-order contributions are still hidden ina⁽¹⁾_2,i

1i₂

due to the approximation f_α⁽¹⁾≈ f_α^(neq) for its computation.

One way to entirely filter out these high-order contributions is to, once again, rely on the Chapman-Enskog expansion (with the isothermal approximation [8])

a⁽¹⁾_2,i

1i₂= −τc²_s[∂iu^∗_j+∂ju^∗_i], (33) as demonstrated in Ref. [42]. One could also consider a hybrid computation of these terms to further improve the stability of the SRT-RR collision model [41]. It is also worth nothing that depending on the discretization scheme used to evaluate the right-hand side of Eq. (33) additional errors (hyperviscosity) will be introduced. Hereafter, we will restrict our study to the SRT-RR with the standard formulation (30), since the impact ofa⁽¹⁾_2,i

1i₂on the stability is out of the scope of the present paper.

C. Space and time discretization

Integrating the discrete (in phase space) force-free Boltzmann equation:

∂tf_α+c_α·∇f_α=_α (34) along the characteristic lines, and using the trapezoidal rule (to evaluate the collision term) and a change of variable (to make the scheme explicit in time), one gets the lattice Boltz- mann equation with the shifted lattices based on the D2Q9 formulation [26,63]:

f_α(x+c_αδt,t+δt)−f_α(x,t)=δtα(x,t), (35) whereδx/δt =√

3kBT0/m0. It is worth noting that even though the same notation is used for populations f_α in Eqs. (34) and (35), they share different properties when it comes to their nonequilibrium part [26,64]. To recover the correct macro- scopic behavior, it is then mandatory to redefine the relaxation coefficient as:

τ¯ = 3ν δx²/δt²

+δt

2. (36)

For the remainder of this article, the overbar will be dropped for the sake of readability. It is obvious that setting Ui= Niδx/δt withNi∈Zresults in shifted stencils with on-lattice streaming. Some of the resulting on-lattice shifted stencils are shown in Fig.1.

FIG. 1. Shifted D2Q9 stencils: (a)Ux=δx/δt, (b)Ux= −δx/δt, (c)Uy=δx/δt, and (d)Uy= −δx/δt (bottom right).

III. EFFECT OF SHIFT ON PHYSICS

In this section, the physical behavior of the solver with shifted stencils will be studied at the asymptotic level by evaluating the deviation in equilibrium moments.

A. Asymptotic behavior

Through the Chapman-Enskog expansion [65], it is possi- ble to show that LBMs based on both Hermite polynomial and entropic EDFs are not able to correctly recover the Navier- Stokes level stress tensor because of errors in the diagonal components of the third-order equilibrium moments tensor

^(eq)xxx =ρ

u³_x+3c²_sux

, yyy^(eq)=ρ

u³_x+3c²_suy

. (37) In addition, the second-order Hermite expansion and the entropic formulation also fail in recovering the correct off- diagonal components

^(eq)xxy =ρ

u²_x+c²_s

uy, ^(eq)xyy =ρux

u²_y+c²_s

, (38) while both third- and fourth-order Hermite expansion do not suffer from such a defect. The entropic EDF, as presented in the previous section, further admits error in the diagonal components of the second-order equilibrium moments tensor,

^(eq)xx =ρ u²_x+c²_s

, ^(eq)yy =ρ u²_y+c²_s

, (39) since its derivation does not account for the conservation of the isothermal energy as compared to its polynomial counter- part [22].

B. Error in equilibrium moments

In all cases, it is known that the above deviations become non-negligible at high Mach numbers. As such, it is interest- ing to look at the effect of the shift on the behavior of these errors. In the following, deviations are computed in terms of

FIG. 2. Deviation of the^(eq)xxx moment of the discrete (a) Hermite and (b) entropic EDFs from their continuous counterpart as a function of the Mach number (Ma) for different shift velocities in thexdirection: ( )Ux=0, ( )Ux=δx/δt, and ( )Ux=2δx/δt.

relative error =

1−

αc_α,^pxc^q_α,yf_α^{(eq),N (eq)}

, (40) where ^(eq) is the macroscopic equilibrium moment of interest—Eqs. (37), (38), or (39)—and (p,q) are integers cor- responding to the equilibrium moment one wants to compute thanks to one of the available EDF. For example, if^(eq) = ^(eq)xyy, then (p,q)=(1,2). Results regarding deviations from ^(eq)xxx, ^(eq)xyy, and ^(eq)xx are compiled in Figs. 2, 3, and 4, respectively. While errors on diagonal components [^(eq)xxx and ^(eq)xx ] will be used to highlight the impact of the stencil shift (through the longitudinal Mach number), the influence of the spanwise Mach number will further be investigated by looking at deviations from off-diagonal componentsxyy^(eq).

First, and as expected from the Chapman-Enskog expan- sion, for nonshifted lattices (1) all EDFs have Galilean in- variance issues that prevent them from recovering the proper

behavior of^(eq)xxx, (2) the second-order Hermite expansion and the entropic formulations also suffer from such defects with ^(eq)xyy, and eventually (3) the entropic EDF cannot reproduce the correct evolution ofxx^(eq). It is worth noting that similar conclusions can also be drawn from ^(eq)xxy and ^(eq)yy . Inter- estingly, it was also confirmed that third- and fourth-order Hermite polynomial equilibria lead to a perfect match ofxyy^(eq)

andxxy^(eq)with≈10⁻¹⁶, which corresponds to the machine precision.

Second, the position of the optimal operation point (where errors are minimized) corresponds to the stencil shift velocity.

Such a behavior is observed for all EDFs and equilibrium moments. In other words, the use of asymmetric lattices does enforce the Galilean invariance principle with respect to the shifted reference frame, and this is confirmed for all EDFs considered in the present work.

Consequently, it is possible toartificiallyincrease the va- lidity range of a given velocity discretization by choosing the

FIG. 3. Deviation of the^(eq)xyy moment of (a) the second-order and (b) entropic discrete EDFs from their continuous counterpart as a function of the Mach number (Ma) for a streamwise shift ofUx=δx/δt and different spanwise Mach numbers: Bottom to top ( ) 0.173, ( ) 0.346, and ( ) 0.519.

0 1 2 3 4 5 10^-10

10^-5 10⁰

FIG. 4. Deviation of thexx^(eq)moment of the discrete entropic EDF from its continuous counterpart as a function of the Mach number (Ma) for different shift velocities in thexdirection: left to right ( )Ux=0, ( )Ux=δx/δt, and ( )Ux=2δx/δt. appropriate shift. This explains why compressible flows can be simulated using low-order velocity discretizations, which are usually restricted in terms of velocity and/or temperature ranges [1–3,11,66]. Nevertheless, in order to properly recover the macroscopic behavior of Navier-Stokes-Fourier equations, it is mandatory to have velocity- and temperature-dependent shift velocities which will eventually require (nonconserva- tive) space interpolation. Mass, momentum, and energy con- servation issues were recently reported in Refs. [11,15]. The latter is currently being investigated in more depth, and corre- sponding results will be presented in a forthcoming paper.

In order to further quantify the impact of shifted lattices on the numerical properties of LBMs, the von Neumann stability analysis of such models will be conducted below.

IV. IMPACT OF SHIFT ON NUMERICS

Hereafter, the spectral properties of shifted LBMs will be studied through a von Neumann analysis. The impact of the shifting velocity on linear stability domains will also be quantified for EDFs and collision models of interest.

A. Von Neumann stability analysis formalism

The von Neumann (VN) stability analysis [9,27,29–

31,42,43,67–78] is used to study the time evolution of a perturbation f_α injected into the shifted lattice Boltzmann equation. The perturbation is assumed to be a monochromatic plane wave. The linear stability of the scheme at a given point in the considered phase space will be assessed by monitoring the maximum attenuation rate. The VN yields meaningful results as long as its assumptions (small amplitude of the perturbation, periodic domain, uniform grid) remain valid. To linearize the time-evolution equations, a first-order Taylor-McLaurin expansion around a mean flow state ¯f_α is performed:

f_α≈ f¯_α+f_α, (41) _α(f_α)≈_α|f¯_α+J_αβf_β, (42)

where Einstein’s summation rule over β is used. J_αβ is the Jacobian of the collision operator evaluated about ¯f_β, i.e., J_αβ =∂f_βα|f¯_β. Placing these expressions into the discrete LB time-evolution equation, one gets:

f_α(x+c_αδt,t+δt)=(δαβ+J_αβ)f_β(x,t), (43) where δ_αβ is the Kronecker δ function. The detailed expressions of the Jacobian for the different EDFs and collision models are given in Appendix. Writing the perturbations f_α as monochromatic plane waves:

f_α =F_αexp [j(k·x−ωt)], (44) where F_α is the wave amplitude, j is the imaginary unit,

||k|| =k is the wave number, and ω is the complex time frequency of the wave, one then gets the following eigenvalue problem of sizeV (the number of discrete velocities):

MF =exp (−jω)F. (45)

F is the eigenvector composed of all amplitudes. M is the matrix associated to the linearized time-evolution equation and is written as:

M =E[δ+J], (46) with

E_αβ =exp[−j(c_α·k)]δαβ. (47) k is related to the wave length of f_α, whereas Im(ω) and Re(ω) are related to its attenuation and propagation speed, respectively. The latter are tightly related to the eigenvalue exp (−jω).

B. Linear stability domain

To show that the concept of shifted stencil can be used to extend the operation domain of the LBM, by using the previously introduced von Neumann method, the stability domain is evaluated for three different EDFs (using the SRT collision operator) and three different collision mod- els (using the fourth-order EDF). The linear stability do- mains are evaluated for seven different values of the nondi- mensional viscosity, i.e., νδt/δx²∈ {0.5,0.1,0.05,0.01,5× 10⁻³,1×10⁻³,5×10⁻⁴}.

The corresponding linear stability domains on standard stencils (nonshifted) are given in Figs.5 and6. From them, it is clear that both the EDF and the collision model have an impact on the linear stability domain. The latter admits an upper limit in terms of maximal Mach number which seems to be reached forνδt/δx²=0.5 for all considered configurations.

Interestingly, the stability domain reaches a square shape for the most stable approaches, which seems to be directly related to the lattice shape.

For the Hermite polynomial formulations, one obtains the upper limit Ma^max=√

3−1≈0.73, with Ma^max obtained by considering all possible orientations of the mean flow and keeping the minimal value. Interestingly, one cannot increase this critical Mach number by changing the collision model [43,79] or the numerical scheme [80]. In fact, this can only be achieved by modifying the velocity discretization and the equilibrium state [9,16,43]. The latter point is further confirmed by looking at results obtained with the entropic EDF, for which the maximal achievable velocity δx/δt is

0 /6 /3

/2 2 /3

5 /6

0 0.5 1 0

/6 /3

/2

0 1 2

FIG. 5. Stability domains of (a) Hermite polynomial EDFs of order two and four and (b) the entropic EDF on nonshifted stencils for different values ofνδt/δ²x: ( ) 0.5, ( ) 0.1, ( ) 0.05, ( ) 0.01, ( ) 5×10⁻³, ( ) 1×10⁻³, and ( ) 5×10⁻⁴.

reached, i.e., Ma^max=√

3≈1.73. Even though the entropic EDF leads to a less interesting macroscopic behavior, due to the fact that it does not recover the correct second-order equilibrium moment, it definitely leads to the widest linear stability domain. Regarding the collision models considered in the context of fourth-order Hermite polynomial EDF, which is the one leading to the best macroscopic behavior and stability domain as compared to the second-order formulation, the RR-LBM exhibits the widest stability domain.

Looking now at the stability domains of the different models on shifted stencils (Fig. 7), and comparing them to those shown in Figs. 5 and 6, it is observed that the shift in stencil results in an equivalent shift of the linear stability domain. Hence, shifted lattices not only transfer the macro- scopic properties of LBMs in the new inertial reference frame, but they also transfer their numerical ones. Clearly, based on these results, the concept of shifted stencils seems to be an effective tool in extending the operability domain of the LBM to larger Mach numbers from both the physical and numerical viewpoints.

Eventually, in agreement with the derivation presented in the first section, all collision models developed in the context

2 /3 /2

5 /6

7 /6

4 /3 3 /2 5 /3

11 /6

0 0.5 1 2

FIG. 6. Stability domains of different collision models on non- shifted stencils for different values ofνδt/δ²x: ( ) 0.5, ( ) 0.1, ( ) 0.05, ( ) 0.01, ( ) 5×10⁻³, ( ) 1×10⁻³, and ( ) 5×10⁻⁴.

of the classical stencils [22] can be transposed and used (as is) on shifted stencils.

C. Spectral properties

While shown to extend the linear stability domain of the LBM, and to recover the correct physics in the limit of van- ishing wave numbers, it is also interesting to look at the effect of the shift on spectral properties. The spectral dispersion of the different models with classical nonshifted and shifted stencils for an inviscid flow at a Mach number equivalent to an entire lattice length (Ma=1.73) are plotted in Fig.8. On the classical stencil, the second- and fourth-order EDFs clearly exhibit the effects of modal interaction (acoustic-kinetic and shear-kinetic for the second-order EDF and only acoustic- kinetic for the fourth-order EDF), while recovering the correct propagation speeds for small wave numbers. For the entropic EDF, on the other hand, the acoustic modes propagate at erroneous speeds even in the continuum limit (vanishing wave numbers). On shifted stencils, however, all models converge to a similar spectral behavior, resulting in the correct propagation speed without any sign of modal interaction. The convergence in spectral behavior, at a velocity corresponding to the stencil shift, is to be expected as on standard stencils. In addition, the different models are supposed to recover the same behavior in the limit of vanishing Mach number.

To further illustrate the effect of the stencil shift, the spec- tral dissipation of the RR collision operator, for Ma=0.519 andνδt/δx²=0.01, is shown in Fig. 9 on both standard and shifted stencils. While stable on both stencils, it is observed that the shifted stencil (chosen to be equal to the fluid velocity) results in much better spectral dissipation properties over all wave numbers. While on the standard stencil one observes overdissipation of the shear mode, and noticeable errors at wave numbers as small asπ/8 for the acoustic modes, the shifted stencil results in more consistent spectral behavior and considerably reduced numerical dissipation.

V. NUMERICAL VALIDATION

In order to further quantify the impact of shifted stencils on the numerical properties of LBMs, the transport of an isothermal vortex is studied.

- /2 - /3 - /6

0 /6 /2 /3

0 2 4

- /4 - /6

- /12 0

/12 /6 /4

0

1

2

- /4 - /6

- /12 0

/12 /6 /4

0

1

2

- /4 - /6

- /12 0

/12 /6 /4

0

1

2

- /4 - /6

- /12 0

/12 /6 /4

0

1

2

- /4 - /6

- /12 0

/12 /6 /4

0

1

2

- /4 - /6

- /12 0

/12 /6 /4

0

1

2

- /4 - /6

- /12 0

/12 /6 /4

0

1

2

FIG. 7. Linear stability domains of the shifted stencil (Ux=δx/δt) for different EDFs, i.e., (a) entropic, (b) second-order, and (c) fourth- order, and different collision models (withe the fourth-order EDF), i.e., (d) LKS and (e) RR, for different values of the nondimensional viscosity: ( ) 0.5, ( ) 0.1, ( ) 0.05, ( ) 0.01, ( ) 5×10⁻³, ( ) 1×10⁻³, and ( ) 5×10⁻⁴.

A. Set-up

This test case consists of a vortex convected by a uniform background velocity field. The vortex is initialized as:

ux=u0−β0u0

y−yc

r₀ exp− r²

2r₀², (48a)

uy=β0u₀x−xc

r0

exp− r²

2r₀², (48b)

ρ =1−β₀²u²₀

2c²_s exp−r²

r₀², (48c)

Re Re

FIG. 8. Spectral dispersion of the acoustic (Ac+and Ac-) and shear modes of different EDFs, i.e., ( ) entropic, ( ) second-order, and ( ) fourth-order, for Ma=1.73 andνδt/δx²=0 on (a) nonshifted and (b) shifted stencil withU_x=δx/δt.