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Spectral effects of entropic multi-relaxation for lattice
Boltzmann methods
Stephan Simonisa,c,∗, Benedikt Dorschnere, Ilya V. Karlind, Mathias J. Krausea,b,c
aInstitute for Applied and Numerical Mathematics,
Karlsruhe Institute of Technology (KIT), 76131 Karlsruhe, Germany
bInstitute of Mechanical Process Engineering and Mechanics,
Karlsruhe Institute of Technology (KIT), 76131 Karlsruhe, Germany
cLattice Boltzmann Research Group,
Karlsruhe Institute of Technology (KIT), 76131 Karlsruhe, Germany
dComputational Kinetics Group, Institute of Energy and Process Engineering,
ETH Zürich, 8092 Zurich, Switzerland
eNVIDIA Switzerland AG, 8004 Zürich, Switzerland
Abstract
We analyze the spectral properties of entropic stabilizers for lattice Boltzmann methods
(LBM) with multi-relaxation collision akin to Karlin–Bösch–Chikatamarla (KBC) mod-
els. In combination with a standard second-order truncated Maxwellian equilibrium
and the reduced 𝐷3𝑄19 velocity set, the method is used to numerically approximate
artificial decaying homogeneous isotropic turbulence generated by Taylor–Green vortex
flow initialization. For the first time, numerical effects of the entropy maximization
through controlled higher-order moment relaxation frequencies are analyzed in wave
space via Fourier-transforming not only the flow quantities but also the entropy con-
troller itself. First, the energy spectra of the time-evolved vortex initializations are used
to observe the effectiveness of the KBC stabilization within the turbulent regime. Thus,
we give primary numerical evidence that the entropic estimate successfully detects and
counteracts spectral energy overloads at high wavenumbers. Second, the recovery of
dissipation rates and integrated vorticity peak regions is used to determine the accu-
racy via extracting the time-dependent experimental order of convergence in space by
a brute-forced parameter study. Conclusively, based on the computationally explored
parameter spaces, we approve second-order convergence of space-time-averaged turbu-
lence quantities produced with KBC LBM in diffusive scaling up to a Reynolds number
of 𝑅𝑒 =6000 in three-dimensional incompressible fluid flow. Additionally, in this case,
we unveil for the first time a data-based convergence rate between orders one and two
of the KBC collision toward the classical Bhatnagar–Gross–Krook (BGK) model.
Keywords: lattice Boltzmann methods, Taylor–Green vortex, entropy stability,
spectral analysis, brute force method, Navier–Stokes equations
2010 MSC: 65Y05, 37M25, 35Q30, 76P05, 76D05
∗Corresponding author
Email address: stephan.simonis@kit.edu (Stephan Simonis)
Preprint submitted to Journal of Computational Physics September 18, 2024
This preprint research paper has not been peer reviewed. Electronic copy available at: https://ssrn.com/abstract=4972419
Preprint not peer reviewed
Contents
1 Introduction 2
2 Methodology 4
2.1 Targeted partial differential equations . . . . . . . . . . . . . . . . . 4
2.2 Lattice Boltzmann method . . . . . . . . . . . . . . . . . . . . . . . 4
2.3 Entropic stabilizer for natural moment collision . . . . . . . . . . . . 7
2.4 Implementation ............................. 9
3 Numerical experiments 9
3.1 Taylor–Greenvortex .......................... 9
3.2 Turbulence quantities . . . . . . . . . . . . . . . . . . . . . . . . . . 10
3.3 Fourier transformed relaxation functions . . . . . . . . . . . . . . . . 11
3.4 Discretization parameters and reference solutions . . . . . . . . . . . 12
3.5 Integral turbulence quantities . . . . . . . . . . . . . . . . . . . . . . 13
3.6 Entropy controller statistics . . . . . . . . . . . . . . . . . . . . . . . 14
3.7 Computational spectral analysis . . . . . . . . . . . . . . . . . . . . 18
3.8 Experimental order of convergence . . . . . . . . . . . . . . . . . . . 21
4 Conclusion 23
1. Introduction
Meanwhile, the lattice Boltzmann method (LBM) has become an established nu-
merical technique in computational fluid dynamics (CFD) and beyond [1]. Based on
the intrinsic combination of discretization and relaxation, the LBM offers advantageous
parallelizability. Over the years, LBMs have been found well-suited for applications
(see [2] and references therein) where good scalability on high performance comput-
ing (HPC) architectures are essential. For example, LBM enables feasible computer
simulations of turbulent fluid flow in three dimensions [3, 4, 5] and even overnight
runtime of realistic problem configurations [6]. This achievement is typically based
on combining the LBM with implicit numerical diffusion (e.g. [7]) or large eddy sim-
ulation (LES) in space and in time, respectively [8] and [9]. Especially LBMs paired
with LES show significant speedups over traditional methods such as finite volume
methods (FVM) [10, 11, 12, 6] on industry relevant scales. Besides, the mesoscopic
derivation of LBMs offers thermodynamically consistent and stable method extensions
for approximating multiphysics models. Highly efficient simulations of reactive [13],
particulate [14], turbulent [12], and thermal fluid flow models [8], coupled radiative
transport [4, 15] or melting and conjugate heat transfer [16] have been reported. In
addition, compressible fluid flows with strong discontinuities [17, 18, 19, 20] and crack
propagation in linear elastic solids [21] are meanwhile realizable. Notably, with proper
code optimization, modern LBMs are capable of saturating [22, 23] modern-day HPC
machinery. Conclusively, even standard LBM formulations can be used as easy to
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implement and mostly second order accurate, matrix-free algorithm in space-time for
approximating coupled systems of partial differential equations.
However, the standard version of LBM, based on a second order discretization of
the Boltzmann equation with Bhatnagar–Gross–Krook (BGK) [24], is only stable for
grid Reynolds numbers in and the order of magnitude 𝑅𝑒 △𝑥=𝑈c△𝑥/𝜈c⪅O(10),
where △𝑥is the cell size of the mesh, 𝑈cdenotes the characteristic velocity and 𝜈c
is the characteristic viscosity of the simulated system. Karlin et al. [25] proposed a
remedy towards unconditional stability by using entropy-controlled relaxation functions
for higher order moments. This approach has been further studied for turbulent flows in
[26] and is referred to as Karlin–Chikatamarla–Bösch (KBC) LBM. Thorough reviews
of entropic and other LBMs are given in [27, 28].
Since the relaxation principle of the LBM does come at the price of inducing a
bottom-up method, which complicates rigorous numerical analysis of LBMs in realistic
dimensions, a brute-force approach by extensive numerical testing has been proposed
by Simonis et al. [7]. Here, we follow this direction to computationally analyze the
numerical effects of KBC LBM in turbulent flow simulations in three dimensions
by using efficient implementations provided in the open source solver OpenLB [2].
In combination with a standard second-order truncated Maxwellian equilibrium and
the reduced 𝐷3𝑄19 velocity set, KBC LBM is used here to numerically approximate
artificial decaying homogeneous isotropic turbulence generated by Taylor–Green vortex
flow initialization.
In this work, we make the following contributions. For the first time, numerical
effects of the entropy maximization through controlled higher-order moment relax-
ation frequencies are analyzed in wave space via Fourier-transforming not only the
flow quantities but also the entropy controller itself. First, the energy spectra of the
time-evolved vortex initializations are used to observe the effectiveness of the KBC
stabilization within the turbulent regime. Thus, we give primary numerical evidence
that the entropic estimate successfully detects and counteracts spectral energy overloads
at high wavenumbers. Second, the recovery of dissipation rates and integrated vorticity
peak regions is used to determine the accuracy via extracting the time-dependent exper-
imental order of convergence in space by a brute-forced parameter study. Conclusively,
based on the computationally explored parameter spaces, we approve second-order
convergence of space-time-averaged turbulence quantities produced with KBC LBM in
diffusive scaling up to a Reynolds number of 𝑅𝑒 =6000 in three-dimensional incom-
pressible fluid flow. Additionally, in this case, we unveil for the first time a data-based
convergence rate between orders one and two of the KBC collision toward the classical
Bhatnagar–Gross–Krook (BGK) model. In summary, enabled by efficient implementa-
tions in OpenLB, we numerically approve the following claims:
1. KBC LBM approximates the incompressible Navier–Stokes equations with sec-
ond order in space for diffusive scaling,
2. KBC LBM is inviscidly stable due to the space-time adaptive relaxation of higher
order moments,
3. KBC LBM is consistent of order larger than one to an LBM based on BGK
collision.
Besides generating insights to contemporary LBMs, this work is part of a joint commu-
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nity effort to provide efficient, open source, and sustainable software for research and
applications.
The rest of this paper is structured as follows. Section 2 introduces and mathe-
matical models and numerical methods. Section 3 proposes novel concepts for Fourier
transformed relaxation functions and summarizes and the numerical experiments. At
last, we draw a conclusion and give an outlook to future research in Section 4.
2. Methodology
2.1. Targeted partial differential equations
Let the incompressible 𝑑-dimensional force-free NSE define an initial value problem
∇ · 𝒖=0in Ω×𝐼,
𝜕𝑡𝒖+1
𝜌∇𝑝+ ∇ · (𝒖⊗𝒖)−𝜈Δ𝒖=0in Ω×𝐼 ,
𝒖(·,0)≡𝒖0in Ω,
(1)
where 𝒖:Ω×𝐼→𝑈BR𝑑,(𝒙, 𝑡)↦→ 𝒖(𝒙, 𝑡 )is the fluid velocity, 𝑝:Ω×𝐼→
R,(𝒙, 𝑡)↦→ 𝑝(𝒙, 𝑡 )denotes the pressure, 𝜌is a constant density, and 𝜈 > 0describes
a given kinematic viscosity, respectively. We assume that the fluid domain is Ω = R𝑑.
Moreover, let the initial data 𝒖0:Ω→R𝑑be weakly divergence-free and square-
integrable, i.e. 𝒖0∈𝐿2
div (Ω;𝑈). Unless stated otherwise, let 𝑑=3below.
2.2. Lattice Boltzmann method
Approximating the NSE with a discrete velocity Boltzmann equation can be achieved
from top-down design via moment matching or using a constructive ansatz [31, 32]
starting at the weakly compressible NSE [33]. Based on the commuting advective
structure [34] of relaxation systems [32], the total number of scalar moment components
equals the size of the discrete velocity stencil. Thus for 𝑞velocities, we have
𝜕𝑡𝑓𝑖+𝒄𝑖· ∇𝑓𝑖=𝐽𝑖(𝒇),(2)
where 𝒄𝑖denote discrete velocities contained in the herewith declared set 𝐷𝑑𝑄 𝑞,𝑓𝑖
are discrete velocity particle distribution functions and 𝐽𝑖is the collision operator. The
𝑞variables 𝑓𝑖contained in 𝒇∈R𝑞are named populations. Here and in the following
𝑖, 𝑗 ∈{0,1,2, . . . , 𝑞 −1}are reserved as discrete velocity counters. For the sake of cost
efficiency, we employ the symmetrically reduced set 𝐷3𝑄19 with three energy shells.
Our decision is motivated by reducing the computational costs.
Natural moment space. The MRT collision model expressed in matrix form reads
𝐽𝑖(𝒇)=−𝑲𝑖[𝒇−𝒇eq],(3)
where 𝒇eq =(𝑓eq
𝑖)𝑖∈R𝑞denotes the equilibrium population, and 𝑲𝑖=(𝐾𝑖 , 𝑗 )𝑗∈R𝑞is
the 𝑖-th row vector of the matrix
K=M−1SM ∈R𝑞×𝑞.(4)
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The relaxation matrix S=diag (𝒔)∈R𝑞×𝑞gathers the relaxation frequencies 𝒔=(𝑠𝑖)𝑖.
The moment matrix M∈GL𝑞(R)is constructed with the mapping
𝑚𝑖=⟨𝝓𝑖,𝒇⟩(5)
based on the standard inner product ⟨·,·⟩ on R𝑞, where 𝝓𝑖=(𝜙𝑖
𝑗)𝑗∈R𝑞are 𝑞vectorial
representations of linearly independent polynomials in discrete velocity vector entries.
The latter are constructed via generic moments
𝜌Γ𝑝1, 𝑝2,... , 𝑝𝑑=⟨(𝑐𝑖)𝑝1
1(𝑐𝑖)𝑝2
2· ·· (𝑐𝑖)𝑝𝑑
𝑑,𝒇⟩(6)
The compression of 𝝓𝑖as rows of a matrix defines
M=𝑀𝑖, 𝑗 𝑖 , 𝑗 =𝜙𝑖
𝑗𝑖, 𝑗 ∈R𝑞×𝑞,(7)
inducing an isomorphic map 𝒎=M𝒇from population to moment space. Further,
let 𝑽𝑗=(𝑉𝑗
𝑖)𝑖∈R𝑞denote the columns of M−1which are linearly independent by
construction. Table 1 summarizes the natural moments constructed for 𝐷3𝑄19 and lists
the notation for the corresponding relaxation frequencies. Table 2 groups the moments
into kinematic, shear, and kinetic types and assigns relaxation frequencies to the each
group.
Remark 2.1. The present moment basis (Table 1) is natural in the sense that it com-
prises linearly independent monomials except for 𝑁𝑥𝑧 ,𝑁𝑦 𝑧, and 𝑇. The additive
redefinition of these moments is necessary to decouple shear and bulk viscosity [35,
Equation (40)].
𝑥
𝑦
𝑧
Figure 1: Discrete velocity set 𝐷3𝑄19 used in the present work. Orange, blue and green denote zeroth, first
and second energy shells, respectively.
Truncated Maxwellian. Here and in the following we choose the standard (second order
truncated) Bhatnagar–Gross–Krook (BGK) [24] equilibrium
𝑓eq
𝑖(𝜌, 𝒖)=𝜌𝑤𝑖1+1
𝑐2
𝑠
𝑐𝑖 𝛼𝑢𝛼+1
2𝑐4
𝑠(𝑐𝑖 𝛼𝑐𝑖 𝛽 −𝑐2
𝑠𝛿𝛼𝛽 )𝑢𝛼𝑢𝛽(8)
to obtain (1) in the diffusion limit. Here, the lattice speed of sound is 𝑐𝑠=1/√3and the
weights 𝑤𝑖are standard, e.g. see [35]. Note that the present work exclusively focuses
on the effect of modulating relaxation frequencies of higher order (kinetic) moments in
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tensor moment
1Γ0,0,0
𝑢𝑥, 𝑢𝑦, 𝑢 𝑧Γ1,0,0,Γ0,1,0,Γ0,0,1
𝑁𝑥𝑧 ,𝑁𝑦 𝑧 Γ2,0,0−Γ0,2,0,Γ0,2,0−Γ0,0,2
Π𝑥𝑦,Π𝑦 𝑧 ,Π𝑥 𝑧 Γ1,1,0,Γ0,1,1,Γ1,0,1
𝑇Γ2,0,0+Γ0,2,0+Γ0,0,2
𝑄𝑥𝑧 𝑧 ,𝑄𝑦𝑧𝑧,𝑄𝑥 𝑥 𝑧,𝑄𝑦𝑦 𝑧 ,𝑄𝑥 𝑥𝑦,𝑄𝑥 𝑦 𝑦 Γ1,0,2,Γ0,1,2,Γ2,0,1,Γ0,2,1,Γ2,1,0,Γ1,2,0
𝐴𝑥𝑦,𝐴𝑥 𝑧 ,𝐴𝑦 𝑧 Γ2,2,0,Γ2,0,2,Γ0,2,2
Table 1: Physical tensor notation of natural moments.
type tensor order frequency
kinematic 10𝑠𝑘=0
𝒖1
shear N2𝑠𝜈=2𝛽=2𝑐2
𝑠
2𝜈+𝑐2
𝑠
𝚷2
kinetic
(hom)
T2𝑠𝜂
𝑠𝑄
𝑠𝐴)≡𝑠=𝛽𝛾 :entropy
controlled
Q3
A4
Table 2: Relaxation frequencies of physical tensors.
space-time. In this realm, we attribute no nonlinear stability gain but solely the enforcing
of thermodynamic consistency in higher expansion orders to the usage of a Lyapunov
functional for 𝒇eq [36]. That the relaxation time correction is primarily responsible for
stabilization has been discussed in previously [26]. In contrast to that, the choice of the
equilibrium population 𝒇eq can increase linear stability (see [28] and references therein).
In addition, a thermodynamically consistent equilibrium obtained from minimizing an
𝐻-function can be used to guarantee isotropy in compressible, trans- and supersonic
flows. Since the isotropy of the equilibrium is up to the order of the discretization [36],
the second order truncated equilibrium is suitable for incompressible flows. Albeit this
equilibrium does not obey a classical global 𝐻-theorem [37], we can still use the local
𝐻-value as a basis of approximation. To improve the relaxation stability in nonlinear
regimes, we modify the relaxation frequencies in Sto be space-time dependent functions
as outlined below.
Evolution equation. A complete discretization of second order [38, 39] with an implicit
population shift retains the collision model and yields the classical MRT LBM evolution
equation
𝑓𝑖(𝒙+ △𝑡𝒄𝑖, 𝑡 + △𝑡)stream
=𝑓★
𝑖(𝒙, 𝑡)collide
=𝑓𝑖(𝒙, 𝑡)+ △𝑡 𝐽𝑖(𝒙, 𝑡 ),(9)
where 𝑓𝑖(𝒙, 𝑡)is discrete in (𝒙, 𝑡)∈Ω△𝑥×𝐼△𝑡. Without entropic correction, the zeroth
and first order moments of 𝒇are proven to satisfy (1) up to first and second order in
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space, respectively [40, 41].
2.3. Entropic stabilizer for natural moment collision
The grouping of relaxation frequencies given in Table 2 is equivalent to the KBC-N1
scheme proposed in [36] albeit for being defined on 𝐷3𝑄19 which is computationally
less demanding. For the sake of clarity, we recall the classical derivation of KBC
collision for entropic MRT LBMs on natural moments, i.e. all moments except the ones
which are responsible for shear and bulk viscosity decoupling are raw and thus based on
monomials. Let M={𝜈, 𝑇 , 𝑄, 𝐴}. We split the population in additive modes separated
in moment space [25, 26, 42]
𝒇=
♭∈M
𝒇♭.(10)
This directly implies
𝒇eq =
♭∈M
𝒇eq
♭(11)
and
𝒇neq =
♭∈M
𝒇neq
♭,(12)
with the inversion
𝒇neq
♭=
𝑙∈𝐿♭⟨𝝓𝑙,𝒇neq⟩𝑽𝑙,(13)
where 𝐿♭⊆ {0,1, . . . , 𝑞 −1}denotes the corresponding velocity indices of the vectorial
entries of ♭∈ M. Conforming to [25, 36], we gather the post-collide nonequilibrium
contribution of higher order moments
𝑓neq
hom ≡
♭∈{𝜂, 𝑄, 𝐴}
𝑓neq
♭,(14)
such that, due to the diagonality of 𝑆, (9) becomes
𝒇★(𝒙, 𝑡)=𝒇(𝒙, 𝑡 ) − △𝑡M−1SsM𝒇neq (𝒙, 𝑡)
=𝒇(𝒙, 𝑡) − △𝑡
♭∈M
𝑠♭𝒇neq
♭(𝒙, 𝑡)
=𝒇(𝒙, 𝑡) − △𝑡𝑠𝑣𝒇neq
𝜈(𝒙, 𝑡) − △𝑡𝑠 𝒇neq
hom (𝒙, 𝑡),(15)
where Ssis the shifted relaxation matrix and all elements thereof are, here and below,
considered as shifted [43].
Definition 2.1. The higher order relaxation frequency 𝑠=𝛽𝛾 is entropy controlled
in the sense that the controller 𝛾is computed to maximize the discrete entropy of the
post-collision state 𝒇★.
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Recall that 𝒇neq
♭=𝒇♭−𝒇neq
♭, hence [25] the critical point which minimizes the Lyapunov
functional (understood as an 𝐻-function [44, 45])
𝐻𝒇★=
𝑞−1
𝑖=0
𝑓★
𝑖ln 𝑓★
𝑖
𝑤𝑖,(16)
is reached under the condition of
𝑞−1
𝑖=0
𝑓neq
hom,𝑖 ln 1+(1−𝑠)𝑓neq
hom,𝑖 + (1−𝑠𝜈)𝑓neq
𝜈,𝑖
𝑓eq
𝑖!!
=0.(17)
Definition 2.2. The entropic scalar product for 𝑿,𝒀∈R𝑞with respect to 𝒇eq ∈R𝑞is
defined as
⟨𝑿|𝒀⟩=
𝑞
𝑖=1
𝑋𝑖𝑌𝑖
𝑓eq
𝑖
.(18)
A truncation to first order in both terms of the enumerator of (17), results in an expression
for the entropy controller [36]
𝛾★=1
𝛽−2−1
𝛽⟨𝒇neq
𝜈|𝒇neq
hom⟩
⟨𝒇neq
hom |𝒇neq
hom⟩,(19)
such that (17) is approximately fulfilled, in turn approximately minimizing the Lyapunov
functional of the discrete dynamical system defined by (9). Notably, a search for
different Lyapunov functionals which ensure nonlinear stability might lead to novel
entropic stabilizer approximations (see [30, 27] and references therein).
Remark 2.2. Strictly speaking, 𝛾★=𝛾★(𝒙, 𝑡)is a space-time variable and thus the
associated relaxation times are referred to as relaxation functions in the sense of [43]
below.
From Remark 2.2 we anticipate the need for measur ing the following statistical quantities
based on 𝛾★.
Definition 2.3. We compute the mean controller via averaging over all nodes in space
𝛾★(𝑡)=1
|Ω△𝑥|
𝒙∈Ω△𝑥
𝛾★(𝒙, 𝑡).(20)
The spatial average 𝛾★is then further processed to compute the root-mean-square error
in the form of the variance
⟦𝛾★⟧(𝑡)=1
|Ω△𝑥|
𝒙∈Ω△𝑥h𝛾★(𝑡) − 𝛾★(𝒙, 𝑡)i2.(21)
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2.4. Implementation
Wang [42] restated (19) in terms of a least squares problem possibly for entropy-
controlling individual moment collisions and suggested a solution via QR-decomposition.
Here, we follow [36] and sketch the collision kernel in Algorithm 1. Note that various
computational details of the entropic SRT and MRT models were recently discussed in
[46] and [47], and in parts also extend to the present configuration.
Algorithm 1 Realization of entropic MRT collision (KBC-N1 [36]) in OpenLB [48]
procedure collideKBC-N1(𝒇)⊲Input: pre-collision 𝒇at local node (𝒙, 𝑡)
compute hydrodynamic moments: (𝜌, 𝒖)←Í𝑞
𝑖=1𝑓𝑖,Í𝑞
𝑖=1𝒄𝑖𝑓𝑖/𝜌
compute equilibrium 𝒇eq ←𝒇eq (𝜌, 𝒖)⊲(8)
compute nonequilibrium 𝒇neq ←𝒇−𝒇eq
compute all moments 𝒎←𝒎(𝒇)⊲(5)
compute shear nonequilibrium populations 𝒇neq
𝜈←𝒇neq
𝜈(𝑁, Π)⊲(13)
compute higher order nonequilibrium populations 𝒇neq
hom =𝒇neq −𝒇neq
𝜈⊲(14)
assign viscosity relaxation frequency 𝛽←𝜔/2⊲ 𝜔 =1/𝜏
compute entropy controller 𝛾★←𝛾★𝛽, 𝒇eq,𝒇neq
𝜈,𝒇neq
hom⊲(19)
local collision of 𝒇★←𝒇★𝛽, 𝛾★,𝒇,𝒇neq
𝜈,𝒇neq
hom⊲(15)
return 𝒇★⊲Output: post-collision 𝒇★at local node (𝒙, 𝑡)
end procedure
3. Numerical experiments
All following results are produced with the parallel data structure OpenLB [2, 49].
Additional packages are included into the computational framework, for example FFTW
[50] to Fourier-transform the space-time dependent variables. Computations were
executed on a maximum of four nodes with two Intel Xeon Platinum 8368 processors
each, which is regarded as fairly low for the amount of information generated in this
study. For the sake of compactness, we assume a physical SI unit system and neglect
its consistent notation unless stated otherwise.
3.1. Taylor–Green vortex
Let Ω = [0,2𝜋𝑙𝑐]3. The relevant characteristic scales are 𝑙𝑐,𝑈𝑐and 𝜌𝑐. Normal-
izing 𝑙𝑐=1m,𝑈𝑐=1m/s, the Reynolds number is defined as 𝑅𝑒 =(1m2/s)/𝜈. The
initialization for the TGV flow [51] dictates
𝒖(𝒙,0)=©«
𝑈𝑐sin 𝑥
𝑙𝑐cos 𝑦
𝑙𝑐cos 𝑧
𝑙𝑐
−𝑈𝑐cos 𝑥
𝑙𝑐sin 𝑦
𝑙𝑐cos 𝑧
𝑙𝑐
0ª®®®¬
.(22)
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Additionally, conforming to the solenoidal velocity field, the initial pressure is computed
as
𝑝(𝒙,0)=𝑝∞+𝜌𝑐𝑈2
𝑐
16 cos 2𝑥
𝑙𝑐+cos 2𝑦
𝑙𝑐cos 2𝑧
𝑙𝑐+2,(23)
where 𝑝∞is a reference.
3.2. Turbulence quantities
We compute the following classical quantities to assess the simulation quality. The
kinetic energy
𝑘(𝑡)=1
|Ω|∫
Ω
1
2𝑢𝛼𝑢𝛼d𝒙(24)
and the enstrophy
𝜁(𝑡)=1
|Ω|∫
Ω
𝑟𝛼𝑟𝛼d𝒙,(25)
where 𝒓=∇ × 𝒖is the vorticity field, as well as the total dissipation rate
𝜖tot (𝑡)=−d𝑘
d𝑡(26)
are calculated, respectively. The maximum vorticity magnitude
𝜔(𝑡)=max
𝒙∈Ω|𝒓(𝒙, 𝑡)|2(27)
is used to measure the recovery of initial peak regions (IPR) for the present type of
vortex initializations [52]. Further, we compute the instantaneous nondirectional energy
spectrum
𝐸(𝜅, 𝑡 )=∯
𝑆(𝜅)
1
2Φ(𝒌, 𝑡)d𝑆(𝜅) ≈
𝒌∈N𝑑
𝜅−1<|𝒌|≤𝜅
1
2Φ(𝒌, 𝑡)(28)
on spherical wave shells 𝑆(𝜅)={𝒌∈ K :|𝒌|=𝜅}, where 𝜅is the scalar wavenumber
and
Φ(𝒌, 𝑡)=∥˜
𝒖(𝒌, 𝑡)∥2
2(29)
squares the spatially Fourier-transformed velocity
˜
𝒖(𝒌, 𝑡)=1
√2𝜋𝑑∫R𝑑
𝒖(𝒙, 𝑡)exp (−i𝒌·𝒙)d𝒙.(30)
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The latter is approximated by the spatial discrete Fourier transform according to
˜
Υ𝛼(𝒌, 𝑡)=(𝑁−1)11×𝑑
𝒙=01×𝑑
Υ𝛼(𝒙, 𝑡)exp −2𝜋i
𝑁𝒌·𝒙(31)
for all 𝛼=1,2, . . . , 𝑑, defined on the wave nodes 𝒌=(𝑘1, 𝑘2, . . . , 𝑘 𝑑)with 𝑘𝑖=
0,1, . . . , 𝑁 for all 𝑖, where 𝚼:Ω△𝑥×𝐼△𝑡→R𝑑denotes a computed vectorial quantity
on the grid nodes 𝒙=(𝑛1, 𝑛2, . . . , 𝑛𝑑)and 𝑛𝑖=0,1, . . . , 𝑁 for all 𝑖. Due to the
real input symmetry, we mostly postprocess 𝑘𝑖=0,1, . . . , ⌊𝑁/2⌋below. Notably,
for postprocessing the computed velocity field in a turbulent flow simulation, central
difference stencils of order eight have been found suitable, e.g. by Simonis et al. [7, 9]
and Geier et al. [53]. Hence, given an approximate solution 𝒖on 𝑍ℎ= Ω△𝑥×𝐼△𝑥to
(1), we approximate its gradient with
𝜕𝛽𝑢𝛼(𝒙)=1
△𝑥4
5𝑢𝛼𝒙+ △𝑥𝒆𝛽−𝑢𝛼𝒙− △𝑥𝒆𝛽
+1
5𝑢𝛼𝒙−2△𝑥𝒆𝛽−𝑢𝛼𝒙+2△𝑥𝒆𝛽
+4
105 𝑢𝛼𝒙+3△𝑥𝒆𝛽−𝑢𝛼𝒙−3△𝑥𝒆𝛽
+1
280 𝑢𝛼𝒙−4△𝑥𝒆𝛽−𝑢𝛼𝒙+4△𝑥𝒆𝛽+ O △𝑥8,(32)
where the temporal argument is neglected for the purpose of illustration.
3.3. Fourier transformed relaxation functions
The above objects are classical and have been studied extensively in the literature.
In addition, motivated by preliminary results in [54], we propose the following novel
quantity to specifically measure the relaxation in wave-time and with it, the entropic
stabilization.
Definition 3.1. The instantaneous entropy control spectrum
𝐶(𝜅, 𝑡 )=∯
𝑆(𝜅)
1
2Ψ(𝒌, 𝑡)d𝑆(𝜅)(33)
where
Ψ(𝒌, 𝑡)=˜𝛾★(𝒌, 𝑡)(34)
correlates the discrete Fourier transform of the space-time dependent entropy control
𝛾★(𝒙, 𝑡)reading
˜𝛾★(𝒌, 𝑡)=1
√2𝜋𝑑∫R𝑑
˜𝛾★(𝒙, 𝑡)exp (−i𝒌·𝒙)d𝒙.(35)
Upon discretization (31), we thus obtain a discrete variant of the control spectrum.
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Definition 3.2. The instantaneous discrete control spectrum approximates
𝐶(𝜅, 𝑡 )≈
𝒌∈N𝑑
𝜅−1<|𝒌|≤𝜅
1
2Ψ(𝒌, 𝑡),(36)
where
Ψ(𝒌, 𝑡)=ˆ𝛾★(𝒌, 𝑡)(37)
denotes the absolute value of the discrete Fourier transformed wave-time dependent
entropy control
ˆ𝛾★(𝒌, 𝑡)=(𝑁−1)11×𝑑
𝒙=01×𝑑
𝛾★(𝒙, 𝑡)exp −2𝜋i
𝑁𝒌·𝒙.(38)
As 𝛾★defines a relaxation function of the grid nodes 𝒙and the time steps 𝑡, the control
spectrum (36) is occasionally called relaxation spectrum below.
3.4. Discretization parameters and reference solutions
The parameter spaces used in the following computations are summarized in Table 3.
To cover acoustic (AS) and diffusive scaling (DS) for three Reynolds numbers 𝑅𝑒 =
1600,3000,6000, the tested Mach numbers 𝑀𝑎 =0.2,0.1,0.05 are paired with all
resolutions 𝑁=32,64,128. The tested resolutions resemble a strongly underresolved
setting with a grid Reynolds number 𝑅𝑒 △=𝑅𝑒/𝑁∈ [50,200]on purpose which
typically leads to divergence of the SRT BGK scheme in the standard formulation.
The LBM results are assessed in terms of an error computation with respect to a
Table 3: Parameter grid for the numerical experiments with KBC-N1 on 𝐷3𝑄19 with natural moments
(Table 1), entropy controlled relaxation (19) and truncated equilibrium (8).
𝑁△𝑡 𝑀𝑎 𝜏
𝑅𝑒 =1600 𝑅𝑒 =3000 𝑅𝑒 =6000
32
2.34 ×10−20.2 0.501068 0.500570 0.500285
1.17 ×10−20.1 0.500534 0.500285 0.500142
5.85 ×10−30.05 0.500267 0.500142 0.500071
64
1.15 ×10−20.2 0.502171 0.501158 0.500579
5.75 ×10−30.1 0.501085 0.500579 0.500289
2.87 ×10−40.05 0.500543 0.500289 0.500146
128
5.71 ×10−30.2 0.504376 0.502334 0.501167
2.85 ×10−30.1 0.502188 0.501167 0.500582
1.42 ×10−40.05 0.501094 0.500583 0.500292
psDNS reference solution. The presently used grid-converged reference solution of
Duponcheel (see for example [55, 56]) has been produced with a dealiased psDNS on
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5123spatial grid points with a three-step Runge–Kutta scheme for time integration at
△𝑡=1.0×10−3. The data is freely available at [57]. For simplicity, we will refer to [56]
for this reference solution below. The error computation below is done for 𝑅𝑒 =1600
only. For 𝑅𝑒 =3000, the DNS data from [9] computed with the SEM (Nek5000) is
used for qualitative comparison.
3.5. Integral turbulence quantities
Simulations of the TGV flow are computed with the KBC-N1 LBM for the parameter
grid in Table 3. The obtained flow fields should approximate weak solutions to (1). To
assess the quality of approximation, each flow field is post-processed via computing the
kinetic energy (24), the enstrophy (25), the total dissipation rate (26) and the maximum
vorticity (27). Concerning the latter for the TGV flow at high Reynolds numbers,
Thantanapally et al. [52] have observed that the maximum vorticity shows a single
separated peak at 𝑡≈6.5after an initial virtually inviscid phase. This IPR is followed
by a series of sub-peaks with similar magnitude. For Reynolds numbers smaller than
𝑅𝑒 =5000, the IPR is shifted forward in time. For example at 𝑅𝑒 =800 the separated
peak is observed at 𝑡≈7[43]. Hence, here the computed vorticity is expected to
recover an isolated peak in the IPR ranging between [6,7]. For 𝑅𝑒 =1600, the psDNS
reference [56] provides kinetic energy, enstrophy and total dissipation rate results for
comparison.
Figures 2, 3, 4 visualize the integral turbulence quantities computed for 𝑅𝑒 =
1600,3000,6000, respectively, with the Mach numbers and resolutions summarized in
Table 3. The only stable SRT BGK results are obtained with the configurations 𝑁=128
with 𝑀𝑎 =0.2,0.1,0.05 for 𝑅𝑒 =1600. A higher Reynolds number or lower resolution
led to divergence of the SRT BGK scheme used for the TGV flow. In contrast to that,
all computations with KBC-N1 are stable and convergent.
The present results suggest that the reasonable stability bound 𝑅𝑒△𝑥=O(10)of
the SRT BGK LBM for approximating the incompressible NSE (1) [58] is removed or
at least drastically increased by the entropy controlled relaxation of kinetic moments.
This observation has been predicted by Lyapunov stability for a suitable equilibrium
formulation. The novelty in the present results is the numerically indicated validity
for the computationally more efficient configuration with a second order truncated
Maxwellian and 𝐷3𝑄19. As 𝑅𝑒△𝑥increases further, accuracy is however still not
preserved. For very coarse resolutions (𝑁=32), the kinetic energy is overpredicted
(see Figures 2a, 2b, 2c, 3a, 3b, 3c, 4a, 4b, 4c), which shifts the total dissipation
rate forward in time (see Figures 2g, 2h, 2i, 3g, 3h, 3i, 4g, 4h, 4i). It is remarkable
however, that the shapes and magnitudes of both curves are well recovered. This is not
the case for the enstrophy and the maximum vorticity, which are heavily reduced, see
Figures 2d, 2e, 2f, 3d, 3e, 3f, 4d, 4e, 4f and Figures 2j, 2k, 2l, 3j, 3k, 3l, 4j,
4k, 4l, respectively. In addition, for decreasing the Mach number at fixed resolution,
the reduction of enstrophy and vorticity increases (see Figures 2d, 2e, 2f, 3d, 3e, 3f,
4d, 4e, 4f and Figures 2j, 2k, 2l, 3j, 3k, 3l, 4j, 4k, 4l). The enstrophy reduction
has also been reported by Geier et al. [53, Figures 2 and 3] for the K15 regularized
cumulant LBM with unit relaxation times. In a previous study [7], a Mach decrease
under constant resolution even led to instabilities. Another noticeable feature of the
results for 𝑁=32 are strong oscillations of the total dissipation rate in the initial time
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steps 𝑡 < 5(see Figures 2g, 2h, 2i, 3g, 3h, 3i, 4g, 4h, 4i). The oscillations vanish
for larger resolutions, which has been just recently observed also by [59]. Here, the
effects are present for both SRT BGK and KBC-N1 collision. Notably, the derivative
of the kinetic energy is approximated with second order central differences, where the
oscillations are independent on the edge stencil. The root of these numerical artifacts
is investigated currently, and meanwhile attributable to the initialization procedure.
All of the observed effects vanish with increasing resolutions according to DS and
the numerical KBC-N1 solution visibly approximates the DNS reference. Particularly
the IPR of the maximum vorticity is rendered increasingly pronounced already at
𝑁=64,128. In summary, all tested integral turbulence quantities are approximated
very well measured against the very coarse resolutions. In regard to the limit consistent
approximation, the reduction of the entropy controller in the relaxation limit as well as
the EOC of the total dissipation rate are further discussed below.
3.6. Entropy controller statistics
The relaxation function defined by the entropy controller (19) is measured in each
time step of the simulation in terms of a statistical distance from the SRT BGK value
𝛾★=2. The mean (20) and the variance (root-mean-square error) of 𝛾★(21) computed
in all simulations with the parameters summarized in Table 3 are plotted in Figure 5.
The asymptotic decrease of |2−𝛾★|for increasing 𝑁is clearly visible for all Reynolds
numbers (cf. Figures 5a, 5b, 5c, Figures 5d, 5e, 5f, and Figures 5g, 5h, 5i,
respectively). In addition, the variance ⟦𝛾★⟧decreases with increasing resolution.
Notably, the configuration where the SRT BGK relaxation is stable (𝑁=128) still
shows 𝛾★≠2. However, for increasing 𝑅𝑒 at fixed 𝑁and 𝑀𝑎 an increase in |2−𝛾★|is
visible, where the effect is more pronounced for large 𝑁. Similarly to the correlation of
𝑅𝑒 and |2−𝛾★|, the latter decreases with decreasing 𝑀𝑎. In summary, the stabilization
via 𝛾★according to (19) is more active for large 𝑅𝑒 and large 𝑀𝑎 and vanishes in the
smallness limit of both. The point in time, where the entropy controller is most active
in terms of the steepness of |2−𝛾★|is visibly matching with the divergence point of
the SRT BGK collision (see for example Figure 4a) on the one hand, and with the first
peak region of the maximum vorticity (see for example Figure 4j) on the other. The
connection of 𝛾★to the latter seems even more natural when comparing the shape of the
slopes. The entropy controller seems to track the vorticity and in case of low resolutions
compensate its effect within the flow field, similarly to a turbulence model. In addition,
an exemplary EOC computation of the entropy controller statistics at 𝑡≈5.1in DS
yields 2−𝛾★(𝑡)=O𝑁1.34 ,(39)
⟦𝛾★⟧(𝑡)=O𝑁1.61 ,(40)
which suggests convergence with order larger than one of the KBC-N1 relaxation under-
stood as an multi-relaxation-function model [43] toward the SRT BGK configuration.
The crucial difference to classical explicit turbulence models with a grid coupled filter
width is with respect to the order of viscosity. The relaxation functions only affect
kinetic moments uncoupled from the viscosity. Since the latter in turn appear as higher
14
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0510 15 20
0
10
20
30
40
𝑡
𝑘|Ω|
(a) Kinetic energy, 𝑀𝑎 =0.2
0510 15 20
0
10
20
30
40
𝑡
𝑘|Ω|
(b) Kinetic energy, 𝑀𝑎 =0.1
0510 15 20
0
10
20
30
40
𝑡
𝑘|Ω|
(c) Kinetic energy, 𝑀𝑎 =0.05
0510 15 20
0
5
10
𝑡
𝜁
(d) Enstrophy, 𝑀𝑎 =0.2
0510 15 20
0
5
10
𝑡
𝜁
(e) Enstrophy, 𝑀𝑎 =0.1
0510 15 20
0
5
10
𝑡
𝜁
(f) Enstrophy, 𝑀𝑎 =0.05
0510 15 20
0
1
2
3
4
𝑡
𝜖tot |Ω|
(g) Total diss. rate, 𝑀𝑎 =0.2
0510 15 20
0
1
2
3
4
𝑡
𝜖tot |Ω|
(h) Total diss. rate, 𝑀𝑎 =0.1
0510 15 20
0
1
2
3
4
𝑡
𝜖tot |Ω|
(i) Total diss. rate, 𝑀𝑎 =0.05
0510 15 20
100
101
𝑡
𝜔
(j) Max. vorticity, 𝑀𝑎 =0.2
0510 15 20
100
101
𝑡
𝜔
(k) Max. vorticity, 𝑀𝑎 =0.1
0510 15 20
100
101
𝑡
𝜔
(l) Max. vorticity, 𝑀𝑎 =0.05
𝑁=32 KBC-N1 𝑁=64 KBC-N1 𝑁=128 KBC-N1
𝑁=32 SRT BGK 𝑁=64 SRT BGK 𝑁=128 SRT BGK
SEM DNS Simonis et al. [9] IPR [52]
Figure 2: Integral turbulence quantities of KBC-N1 and SRT BGK collision for 𝑅𝑒 =1600 at several
resolutions 𝑁=32,128,256 and Mach numbers 𝑀𝑎 =0.2,0.1,0.05.
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0510 15 20
0
10
20
30
40
𝑡
𝑘|Ω|
(a) Kinetic energy, 𝑀𝑎 =0.2
0510 15 20
0
10
20
30
40
𝑡
𝑘|Ω|
(b) Kinetic energy, 𝑀𝑎 =0.1
0510 15 20
0
10
20
30
40
𝑡
𝑘|Ω|
(c) Kinetic energy, 𝑀𝑎 =0.05
0510 15 20
0
5
10
𝑡
𝜁
(d) Enstrophy, 𝑀𝑎 =0.2
0510 15 20
0
5
10
𝑡
𝜁
(e) Enstrophy, 𝑀𝑎 =0.1
0510 15 20
0
5
10
𝑡
𝜁
(f) Enstrophy, 𝑀𝑎 =0.05
0510 15 20
0
1
2
3
4
𝑡
𝜖tot |Ω|
(g) Total diss. rate, 𝑀𝑎 =0.2
0510 15 20
0
1
2
3
4
𝑡
𝜖tot |Ω|
(h) Total diss. rate, 𝑀𝑎 =0.1
0510 15 20
0
1
2
3
4
𝑡
𝜖tot |Ω|
(i) Total diss. rate, 𝑀𝑎 =0.05
0510 15 20
100
101
𝑡
𝜔
(j) Max. vorticity, 𝑀𝑎 =0.2
0510 15 20
100
101
𝑡
𝜔
(k) Max. vorticity, 𝑀𝑎 =0.1
0510 15 20
100
101
𝑡
𝜔
(l) Max. vorticity, 𝑀𝑎 =0.05
𝑁=32 KBC-N1 𝑁=64 KBC-N1 𝑁=128 KBC-N1
𝑁=32 SRT BGK 𝑁=64 SRT BGK 𝑁=128 SRT BGK
SEM DNS Simonis et al. [9] IPR [52]
Figure 3: Integral turbulence quantities of KBC-N1 and SRT BGK collision for 𝑅𝑒 =3000 at several
resolutions 𝑁=32,128,256 and Mach numbers 𝑀𝑎 =0.2,0.1,0.05.
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0510 15 20
0
10
20
30
40
𝑡
𝑘|Ω|
(a) Kinetic energy, 𝑀𝑎 =0.2
0510 15 20
0
10
20
30
40
𝑡
𝑘|Ω|
(b) Kinetic energy, 𝑀𝑎 =0.1
0510 15 20
0
10
20
30
40
𝑡
𝑘|Ω|
(c) Kinetic energy, 𝑀𝑎 =0.05
0510 15 20
0
5
10
𝑡
𝜁
(d) Enstrophy, 𝑀𝑎 =0.2
0510 15 20
0
5
10
𝑡
𝜁
(e) Enstrophy, 𝑀𝑎 =0.1
0510 15 20
0
5
10
𝑡
𝜁
(f) Enstrophy, 𝑀𝑎 =0.05
0510 15 20
0
1
2
3
4
𝑡
𝜖tot |Ω|
(g) Total diss. rate, 𝑀𝑎 =0.2
0510 15 20
0
1
2
3
4
𝑡
𝜖tot |Ω|
(h) Total diss. rate, 𝑀𝑎 =0.1
0510 15 20
0
1
2
3
4
𝑡
𝜖tot |Ω|
(i) Total diss. rate, 𝑀𝑎 =0.05
0510 15 20
100
101
𝑡
𝜔
(j) Max. vorticity, 𝑀𝑎 =0.2
0510 15 20
100
101
𝑡
𝜔
(k) Max. vorticity, 𝑀𝑎 =0.1
0510 15 20
100
101
𝑡
𝜔
(l) Max. vorticity, 𝑀𝑎 =0.05
𝑁=32 KBC-N1 𝑁=64 KBC-N1 𝑁=128 KBC-N1
𝑁=32 SRT BGK 𝑁=64 SRT BGK 𝑁=128 SRT BGK
Figure 4: Integral turbulence quantities of KBC-N1 and SRT BGK collision for 𝑅𝑒 =6000 at several
resolutions 𝑁=32,128,256 and Mach numbers 𝑀𝑎 =0.2,0.1,0.05.
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order gradients in the closed form equation for the conserved moments, the KBC-N1
scheme is rather interpretable as a consistent, space-time adaptive hyperviscosity model.
3.7. Computational spectral analysis
Due to the apparent energy cascade along the vortical scales in decaying homo-
geneous isotropic turbulence for large 𝑅𝑒, the above observed similarity between the
entropy controller mean value and the maximum vorticity motivates the novel Defi-
nition 3.2 of the control spectrum (36) and its computational analysis. For the latter,
all time dependent flow fields obtained with the computational parameter grid (Ta-
ble 3) are Fourier transformed. The control spectrum 𝐶(𝜅, 𝑡)is computed over the
discrete wavenumber range at several points in time, which results in an array of three-
dimensional data sets also for the energy spectra 𝐸(𝜅, 𝑡)of the KBC-N1 solution and the
SRT BGK solution. For DS, this data is visualized as waterfall plots in the Figures 6, 7, 8
for 𝑅𝑒 =1600,3000,6000, respectively.
The activity of the entropy controller on particular wave lengths as well as its effect
on the energy spectrum are observed for all tested Reynolds numbers and discretization
parameters along DS. For small 𝜅and initial times, 𝐶(𝜅, 𝑡 )is minimal on average. With
increasing wavenumbers, the control spectrum increases as well, which approves the
above observed correlation of the entropy controller 𝛾★and the maximum vorticity 𝜔.
Similarly to the power law 𝐸(𝜅, ·) =O(𝜅−5/3)predicted by Kolmogorov [60, 61], the
relaxation spectrum 𝐶(𝜅, ·) reaches an asymptotic shape toward the end of the simulated
time horizon. The shape of course differs from the one of the energy spectra and scales
inversely. For the most resolved configuration (𝑅𝑒 =1600,𝑁=128), where the SRT
BGK relaxation is stable (Figures 6i), the control spectrum landscape in 𝜅and 𝑡is
mostly flattened (see Figures 6g, 6h). In addition, the wave-time averaged magnitude
of the control spectra decreases heavily with increasing the resolution. Notably, the
correlation between the maximum vorticity and the entropy controller mean over time
reappears for the largest wavenumbers in each grid configuration (the visible edge of
the waterfall marked in red). At the cutoff wavenumber 𝜅c, the slope of the energy
spectrum 𝐸(𝜅c, 𝑡)over time has substantial similarity to the maximum vorticity curve
in Figures 2, 3, 4. Likewise, the control spectrum 𝐶(𝜅c, 𝑡)tracks the initial vorticity
minimum and upholds an asymptotically constant level as soon as the IPR is reached.
The above observed reduction to SRT BGK collision in the sense of a multi-relaxation-
function scheme [43] is thus also observed for Fourier transformed quantities. Further,
the influence of the entropy controlled stabilization is found to be connected to the
underresolved vorticity via being maximal at large wavenumbers near the cutoff.
For the purpose of cross-comparing the energy spectra produced by the SRT BGK
and the KBC-N1 relaxation, Figure 9 shows waterfall sections at the latest stable time
steps of the SRT BGK simulations before divergence occurs. Only DS parameter
configurations are shown for each Reynolds number. The reduction of the magnitude
of the control spectrum is clearly visible (row-wise, left to right). For highly resolved
settings, the control spectrum is thus expected to be constantly zero in machine precision
as already indicated by the convergence of |2−𝛾★| ↘ 0. Additionally, a common
intersection point of the control spectrum and the energy spectrum of the KBC-N1
scheme at the respective cutoff wavenumber for almost all shown parameters is observed.
The only exception for this observation is the configuration where the SRT BGK
18
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Preprint not peer reviewed
0510 15 20
1.6
1.8
2
𝑡
𝛾★,⟦𝛾★⟧
(a) 𝑅𝑒 =1600,𝑀𝑎 =0.2
0510 15 20
1.6
1.8
2
𝑡
𝛾★,⟦𝛾★⟧
(b) 𝑅𝑒 =1600,𝑀𝑎 =0.1
0510 15 20
1.6
1.8
2
𝑡
𝛾★,⟦𝛾★⟧
(c) 𝑅𝑒 =1600,𝑀𝑎 =0.05
0510 15 20
1.6
1.8
2
𝑡
𝛾★,⟦𝛾★⟧
(d) 𝑅𝑒 =3000,𝑀𝑎 =0.2
0510 15 20
1.6
1.8
2
𝑡
𝛾★,⟦𝛾★⟧
(e) 𝑅𝑒 =3000,𝑀𝑎 =0.1
0510 15 20
1.6
1.8
2
𝑡
𝛾★,⟦𝛾★⟧
(f) 𝑅𝑒 =3000,𝑀𝑎 =0.05
0510 15 20
1.6
1.8
2
𝑡
𝛾★,⟦𝛾★⟧
(g) 𝑅𝑒 =6000,𝑀𝑎 =0.2
0510 15 20
1.6
1.8
2
𝑡
𝛾★,⟦𝛾★⟧
(h) 𝑅𝑒 =6000,𝑀𝑎 =0.1
0510 15 20
1.6
1.8
2
𝑡
𝛾★,⟦𝛾★⟧
(i) 𝑅𝑒 =6000,𝑀𝑎 =0.05
SRT BGK KBC-N1 𝑁=32 KBC-N1 𝑁=64 KBC-N1 𝑁=128
Figure 5: Spatial mean (20) and standard deviation (21) of KBC-N1 entropy controller 𝛾★(19) for TGV flow
simulations at several Reynolds number 𝑅𝑒 =1600,3000,6000 and Mach numbers 𝑀𝑎 =0.2,0.1,0.05.
19
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Preprint not peer reviewed
100
101010 20
10−10
10−6
10−2
κt
E(κ, t)
(a) KBC-N1, 𝑀𝑎 =0.2,𝑁=32
100
101010 20
10−10
10−6
10−2
κt
C(κ, t)
(b) KBC-N1, 𝑀𝑎 =0.2,𝑁=32
100
101010 20
10−10
10−6
10−2
κt
E(κ, t)
(c) SRT BGK, 𝑀𝑎 =0.2,𝑁=32
100
101010 20
10−10
10−6
10−2
κt
E(κ, t)
(d) KBC-N1, 𝑀𝑎 =0.1,𝑁=64
100
101010 20
10−10
10−6
10−2
κt
C(κ, t)
(e) KBC-N1, 𝑀𝑎 =0.1,𝑁=64
100
101010 20
10−10
10−6
10−2
κt
E(κ, t)
(f) SRT BGK, 𝑀𝑎 =0.1,𝑁=64
100
101010 20
10−10
10−6
10−2
κt
E(κ, t)
(g) KBC-N1, 𝑀𝑎 =0.05,𝑁=128
100
101010 20
10−10
10−6
10−2
κt
C(κ, t)
(h) KBC-N1, 𝑀𝑎 =0.05,𝑁=128
100
101010 20
10−10
10−6
10−2
κt
E(κ, t)
(i) SRT BGK, 𝑀𝑎 =0.05,𝑁=128
Figure 6: Energy spectra (28) and control spectra (36) for KBC-N1 scheme (left and middle column) as well
as energy spectra (28) for SRT BGK collision (right column) of the TGV flow simulations at 𝑅𝑒 =1600 in
DS (top to bottom) for (𝑀𝑎, 𝑁 ) ∈ {(0.2,32),(0.1,64),(0.05,128)}. The red circles denote the values
of the spectra at the respective cutoff wavenumber 𝜅c=𝑁/2.
collision is stable (Figure 9c) and the intersection point is shifted to smaller 𝜅 < 𝜅c. For
this less underresolved setting the production of smoothed energy and control spectra
for both collision schemes is also noticeable. In conclusion, the asymptotic power-law
form of all three spectra is well pronounced and suggests that the control spectrum
approximates 𝐶(𝜅, ·) =O(𝜅5/3)for a further refinement of the space-time mesh and an
inverse increase of the Reynolds number at the same time.
20
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100
101010 20
10−10
10−6
10−2
κt
E(κ, t)
(a) KBC-N1, 𝑀𝑎 =0.2,𝑁=32
100
101010 20
10−10
10−6
10−2
κt
C(κ, t)
(b) KBC-N1, 𝑀𝑎 =0.2,𝑁=32
100
101010 20
10−10
10−6
10−2
κt
E(κ, t)
(c) SRT BGK, 𝑀𝑎 =0.2,𝑁=32
100
101010 20
10−10
10−6
10−2
κt
E(κ, t)
(d) KBC-N1, 𝑀𝑎 =0.1,𝑁=64
100
101010 20
10−10
10−6
10−2
κt
C(κ, t)
(e) KBC-N1, 𝑀𝑎 =0.1,𝑁=64
100
101010 20
10−10
10−6
10−2
κt
E(κ, t)
(f) SRT BGK, 𝑀𝑎 =0.1,𝑁=64
100
101010 20
10−10
10−6
10−2
κt
E(κ, t)
(g) KBC-N1, 𝑀𝑎 =0.05,𝑁=128
100
101010 20
10−10
10−6
10−2
κt
C(κ, t)
(h) KBC-N1, 𝑀𝑎 =0.05,𝑁=128
100
101010 20
10−10
10−6
10−2
κt
E(κ, t)
(i) SRT BGK, 𝑀𝑎 =0.05,𝑁=128
Figure 7: Energy spectra (28) and control spectra (36) for the KBC-N1 scheme (left and middle column) as
well as energy spectra (28) for SRT BGK collision (right column) of the TGV flow simulations at 𝑅𝑒 =3000
in DS (top to bottom) for (𝑀𝑎, 𝑁 ) ∈ {(0.2,32),(0.1,64),(0.05,128)}. The red circles denote the values
of the spectra at the respective cutoff wavenumber 𝜅c=𝑁/2.
3.8. Experimental order of convergence
The convergence of the KBC-N1 LBM is experimentally studied via computing the
time dependent relative error
err(𝜖tot (𝑡)) =𝜖tot (𝑡) − 𝜖★
tot (𝑡)
𝜖★
tot(41)
21
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100
101010 20
10−10
10−6
10−2
κt
E(κ, t)
(a) KBC-N1, 𝑀𝑎 =0.2,𝑁=32
100
101010 20
10−10
10−6
10−2
κt
C(κ, t)
(b) KBC-N1, Ma=0.2, 𝑁=32
100
101010 20
10−10
10−6
10−2
κt
E(κ, t)
(c) SRT BGK, 𝑀𝑎 =0.2,𝑁=32
100
101010 20
10−10
10−6
10−2
κt
E(κ, t)
(d) KBC-N1, 𝑀𝑎 =0.1,𝑁=64
100
101010 20
10−10
10−6
10−2
κt
C(κ, t)
(e) KBC-N1, 𝑀𝑎 =0.1,𝑁=64
100
101010 20
10−10
10−6
10−2
κt
E(κ, t)
(f) SRT BGK, 𝑀𝑎 =0.1,𝑁=64
100
101010 20
10−10
10−6
10−2
κt
E(κ, t)
(g) KBC-N1, 𝑀𝑎 =0.05,𝑁=128
100
101010 20
10−10
10−6
10−2
κt
C(κ, t)
(h) KBC-N1, 𝑀𝑎 =0.05,𝑁=128
100
101010 20
10−10
10−6
10−2
κt
E(κ, t)
(i) SRT BGK, 𝑀𝑎 =0.05,𝑁=128
Figure 8: Energy spectra (28) and control spectra (36) for KBC-N1 scheme (left and middle column) as well
as energy spectra (28) for SRT BGK collision (right column) of the TGV flow simulations at 𝑅𝑒 =6000 in
DS (top to bottom) for (𝑀𝑎, 𝑁 ) ∈ {(0.2,32),(0.1,64),(0.05,128)}. The red circles denote the values
of the spectra at the respective cutoff wavenumber 𝜅c=𝑁/2.
and the temporal 𝐿2-error
err𝐿2(𝜖·)=Í𝑚
𝑛=0|𝜖★
·(𝑡𝑛) − 𝜖·(𝑡𝑛)|2
Í𝑚
𝑛=0|𝜖★
·(𝑡𝑛)|2(42)
with respect to the psDNS solution 𝜖★
tot of [56]. Figure 10 visualizes the strong temporal
dependence of the error values for discretization parameters in DS. Figure 11 plots the
22
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section lines of for several resolutions in one plane. The error behavior is visibly more
structured in before and in the IPR (𝑡∈ (0,7)). The peak region of the total dissipation
rate (𝑡≈9) induces a very pronounced disturbance of the error landscape. A second
increased error region in this sense is located from 𝑡≈14 onward. These observations
indicate that a time averaged error of an integral quantity might not be representative
for the error behavior in subintervals in the time domain. The effect of the temporal
variation in the error magnitude on the EOC is visible in Figure 12. The time dependent
EOC oscillates between superquadratic and even divergence in the region around the
peak of the total dissipation rate. Still, on average the EOC ranges between one and
two. Table 4 gives sample data for the local in time EOC and the EOC computed with
Table 4: Sample error data of the KBC-N1 scheme with respect to the psDNS reference [56] with computed
EOC values locally 𝐸𝑂𝐶 (𝑡)(for dedicated points in time 𝑡1=1,𝑡2=6,𝑡3=11,𝑡4=16) and on average
𝐸𝑂𝐶 for 𝑡∈ [0.1,19.8].
err(𝜖tot (𝑡)) err𝐿2(𝜖tot)
(𝑁, 𝑀𝑎)𝑡=𝑡1𝑡=𝑡2𝑡=𝑡3𝑡=𝑡4𝑡∈ [0.1,19.8]
(32,0.2)3.75 ×10−11.00 ×1003.35 ×10−12.46 ×10−13.97 ×10−1
(64,0.1)1.33 ×10−12.63 ×10−11.39 ×10−12.53 ×10−21.41 ×10−1
(128,0.05)4.30 ×10−25.18 ×10−25.40 ×10−21.86 ×10−24.95 ×10−2
𝐸𝑂𝐶 (𝑡)1.56 2.13 1.31 1.86 —
𝐸𝑂𝐶 — — — — 1.50
the 𝐿2-error.
The present observation of local in time second order spatial accuracy of the KBC-
N1 LBM toward the incompressible NSE solution approximated with a psDNS matches
the results from Bösch et al. [36]. In the latter, a local consistency error to high resolution
SRT BGK results has been computed and found to be of second order for initial time
steps 𝑡 < 1. In contrast to that, the present work shows a local and an averaged accuracy
error toward the actually targeted PDE (1). It thus becomes clear from the present
results, that averaging the error over time results in an order reduction by approximately
0.5. The variation of magnitude in the peak region of the computed total dissipation
rate is exemplarily responsible for the error reduction. We attribute this time delay
in the approximation of the time dependent total dissipation rate to the consistency
reduction of the SRT BGK collision to first order in time in DS. Nonetheless, based on
the numerically observed consistency of order two in [36], and the here measured local
and averaged accuracy, it can be concluded that the space-time dependent relaxation
functions for the kinetic moments increase the stability drastically and do not overly
affect the EOC. Supporting this observation, an EOC measurement over a shorter time
interval 𝑡∈ [0.1,10], that is typically found in the literature, results in an order of two.
4. Conclusion
In this work, we made the following contributions. For the first time, numerical
effects of the entropy maximization through controlled higher-order moment relax-
23
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ation frequencies are analyzed in wave space via Fourier-transforming not only the
flow quantities but also the entropy controller itself. First, the energy spectra of the
time-evolved vortex initializations are used to observe the effectiveness of the KBC
stabilization within the turbulent regime. Thus, we give primary numerical evidence
that the entropic estimate successfully detects and counteracts spectral energy overloads
at high wavenumbers. Second, the recovery of dissipation rates and integrated vorticity
peak regions is used to determine the accuracy via extracting the time-dependent exper-
imental order of convergence in space by a brute-forced parameter study. Conclusively,
based on the computationally explored parameter spaces, we approve second-order
convergence of space-time-averaged turbulence quantities produced with KBC LBM in
diffusive scaling up to a Reynolds number of 𝑅𝑒 =6000 in three-dimensional incom-
pressible fluid flow. Additionally, in this case, we unveil for the first time a data-based
convergence rate between orders one and two of the KBC collision toward the classical
Bhatnagar–Gross–Krook (BGK) model. In summary, enabled by efficient implementa-
tions in OpenLB, we numerically approve the following claims:
1. KBC LBM approximates the incompressible Navier–Stokes equations with sec-
ond order in space for diffusive scaling,
2. KBC LBM is inviscidly stable due to the space-time adaptive relaxation of higher
order moments,
3. KBC LBM is consistent of order larger than one to an LBM based on BGK
collision.
Based on these findings, it is especially promising to apply the KBC LBM for
numerical computations along newly proven weak-strong limits of statistical Navier–
Stokes solutions to statistical Euler solutions in three dimensions [62]. This approach
has been recently proposed in [43] and more closely validated in [63] which further
underlines the suitability of consistently stabilizing LBMs with entropic multi-relaxation
for turbulent fluid flow simulations, even with uncertainties.
Acknowledgments
S. Simonis would like to thank Louis Kronberg for prototyping the code in OpenLB
and for computing preliminary results. S. Simonis gratefully acknowledges a ConYS
grant by KHYS at KIT. S. Simonis and M. J. Krause acknowledge support by the state
of Baden-Württemberg through bwHPC.
Author contribution statement
S. Simonis: Conceptualization, Methodology, Software, Validation, Formal Analy-
sis, Investigation, Resources, Data Curation, Writing - Original Draft, Writing - Review
& Editing, Visualization, Project administration, Funding Acquisition; B. Dorschner:
Conceptualization, Methodology, Software, Writing - Review & Editing; I. V. Karlin:
Methodology, Writing - Review & Editing; M. J. Krause: Software, Writing - Review
& Editing, Funding Acquisition. All authors have read and approved the final version
of this manuscript.
24
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Data availability statement
All present LBM computations have been conducted with the parallel open source
code OpenLB [2]. Code contributions (also unreleased) are based on commit b09a3bbd.
The implementation of the KBC-N1 model in OpenLB is provided in the publicly avail-
able gitlab repository https://gitlab.com/openlb/release/ and showcased in the example
tgv3d therein.
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100101102
10−10
10−5
100
𝜅
(a) 𝑅𝑒 =1600,𝑀𝑎 =0.2,
𝑁=32,𝑡≈8.00
100101102
10−10
10−5
100
𝜅
(b) 𝑅𝑒 =1600,𝑀𝑎 =0.1,
𝑁=64,𝑡≈8.26
100101102
10−10
10−5
100
𝜅
(c) 𝑅𝑒 =1600,𝑀𝑎 =0.05,
𝑁=128,𝑡≈19.99
100101102
10−10
10−5
100
𝜅
(d) 𝑅𝑒 =3000,𝑀𝑎 =0.2,
𝑁=32,𝑡≈7.58
100101102
10−10
10−5
100
𝜅
(e) 𝑅𝑒 =3000,𝑀𝑎 =0.1,
𝑁=64,𝑡≈7.05
100101102
10−10
10−5
100
𝜅
(f) 𝑅𝑒 =3000,𝑀𝑎 =0.05,
𝑁=128,𝑡≈8.20
100101102
10−10
10−5
100
𝜅
(g) 𝑅𝑒 =6000,𝑀𝑎 =0.2,
𝑁=32,𝑡≈7.37
100101102
10−10
10−5
100
𝜅
(h) 𝑅𝑒 =6000,𝑀𝑎 =0.1,
𝑁=64,𝑡≈6.45
100101102
10−10
10−5
100
𝜅
(i) 𝑅𝑒 =6000,𝑀𝑎 =0.05,
𝑁=128,𝑡≈6.60
𝐸(𝜅, 𝑡 )KBC-N1 𝐸(𝜅, 𝑡)SRT BGK 𝐶(𝜅, 𝑡)KBC-N1
O(𝜅−5/3) O(𝜅5/3)
Figure 9: Computed energy spectra 𝐸(𝜅 , 𝑡 )and control spectra 𝐸(𝜅 , 𝑡 )of KBC-N1 and energy
spectra 𝐸(𝜅 , 𝑡 )of SRT BGK plotted at the last stable SRT BGK time step, respectively for several
𝑅𝑒 =1600,3000,6000 (top to bottom) in DS (𝑀𝑎, 𝑁 ) ∈ { (0.2,32),(0.1,64),(0.05,128)} (left to
right).
31
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Preprint not peer reviewed
101
102
0510 15 20
10−4
10−3
10−2
10−1
100
𝑁𝑡
err(𝜖tot (𝑡))
Figure 10: Time dependent error of total dissipation rate err (𝜖tot )of the KBC-N1 scheme with respect to
the psDNS reference [56]. Discretization parameters are chosen from Table 3 in DS, hence (𝑀𝑎, 𝑁 ) ∈
{(0.2,32 ),(0.1,64),(0.05,128 )}. Only a subsequence of time steps 𝑡∈ [0.1,19.7]is shown with a step
size of △𝑡=0.2.
02468 10 12 14 16 18 20
10−4
10−2
100
𝑡
err(𝜖tot)
𝑁=32 KBC-N1
𝑁=64 KBC-N1
𝑁=128 KBC-N1
Figure 11: Time dependent error of total dissipation rate err (𝜖tot (𝑡)) of the KBC-N1 scheme with respect to
the psDNS reference [56].
02468 10 12 14 16 18 20
−2
0
2
4
6
𝑡
𝐸𝑂𝐶 (𝑡)
𝐸𝑂𝐶 (𝑡)KBC-N1
𝐸𝑂𝐶 (𝑡)=2
𝐸𝑂𝐶 (𝑡)=1
Figure 12: Time dependent EOC of total dissipation rate err (𝜖tot )of KBC-N1 with respect to psDNS
reference [56].
32
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Preprint not peer reviewed
