A new hybrid recursive regularised Bhatnagar–Gross–Krook collision model for Lattice Boltzmann method-based large eddy simulation

Abstract

A new Lattice Boltzmann collision model for large eddy simulation (LES) of weakly compressible flows is proposed. This model, referred to as the Hybrid Recursive Regularised Bhatnagar–Gross–Krook (HRR-BGK) model, is based on a modification of previously existing regularised collision models defined with the BGK Lattice Boltzmann method (LBM) framework. By hybridising the computation of the velocity gradient with an adequate Finite Difference scheme when reconstructing the non-equilibrium parts of the distribution function, a hyperviscosity term is introduced in the momentum equation, whose amplitude can be explicitly tuned via a weighting parameter. A dynamic version of the HRR-BGK is also proposed, in which the control parameter is tuned at each grid point and each time step in order to recover an arbitrarily fixed total dissipation. This new collision model is assessed for both explicit and implicit LES considering the flow around a circular cylinder at . The dynamic HRR-BGK is observed to yield very accurate results when equipped with Vreman's subgrid model to compute the target dissipation.

Full text

HAL Id: hal-02114308
https://hal.archives-ouvertes.fr/hal-02114308
Submitted on 29 Apr 2019
HAL is a multi-disciplinary open access
archive for the deposit and dissemination of sci-
entic research documents, whether they are pub-
lished or not. The documents may come from
teaching and research institutions in France or
abroad, or from public or private research centers.
L’archive ouverte pluridisciplinaire HAL, est
destinée au dépôt et à la diusion de documents
scientiques de niveau recherche, publiés ou non,
émanant des établissements d’enseignement et de
recherche français ou étrangers, des laboratoires
publics ou privés.
A new hybrid recursive regularised
Bhatnagar–Gross–Krook collision model for Lattice
Boltzmann method-based large eddy simulation
Jérôme Jacob, Orestis Malaspinas, Pierre Sagaut
To cite this version:
Jérôme Jacob, Orestis Malaspinas, Pierre Sagaut. A new hybrid recursive regularised Bhatnagar–
Gross–Krook collision model for Lattice Boltzmann method-based large eddy simulation. Journal of
Turbulence, Taylor & Francis, 2018, pp.1 - 26. �10.1080/14685248.2018.1540879�. �hal-02114308�

A new hybrid recursive regularised Bhatnagar–Gross–Krook
collision model for Lattice Boltzmann method-based large
eddy simulation
Jérôme Jacob a, Orestis Malaspinas band Pierre Sagaut a
aCentrale Marseille, Aix Marseille University, CNRS, Marseille, M2P2, France; bDepartment of Computer
Science, University of Geneva, Carouge, Switzerland
ABSTRACT
A new Lattice Boltzmann collision model for large eddy simulation
(LES) of weakly compressible flows is proposed. This model, referred
to as the Hybrid Recursive Regularised Bhatnagar–Gross–Krook
(HRR-BGK) model, is based on a modification of previously existing
regularised collision models defined with the BGK Lattice Boltzmann
method (LBM) framework. By hybridising the computation of the
velocity gradient with an adequate Finite Difference scheme when
reconstructing the non-equilibrium parts of the distribution func-
tion, a hyperviscosity term is introduced in the momentum equation,
whose amplitude can be explicitly tuned via a weighting parame-
ter. A dynamic version of the HRR-BGK is also proposed, in which the
control parameter is tuned at each grid point and each time step in
order to recover an arbitrarily fixed total dissipation. This new colli-
sion model is assessed for both explicit and implicit LES considering
the flow around a circular cylinder at Re =3900. The dynamic HRR-
BGK is observed to yield very accurate results when equipped with
Vreman’s subgrid model to compute the target dissipation.
KEYWORDS
Large eddy simulation;
Lattice Boltzmann method;
circular cylinder Reynolds
3900
1. Introduction
Large eddy simulation (LES) is now a mature simulation technique for high-resolution
unsteady simulation of turbulent ows [1–3]. This technique, which is under develop-
ment since the early 1960s, is now implemented in almost all computational uid dynamics
(CFD) tools, including commercial software. Among the key issues faced when develop-
ing LES-based simulation tools, one must cite the development of accurate models for
subgrid scales and adequate numerical schemes that guarantee numerical stability whose
induced dissipation does not overwhelm the eect of physical subgrid models. To this end,
a huge amount of subgrid models and numerical methods have been proposed since the
1980s. This issue is far from being a trivial one, since it is now accepted that the discreti-
sation errors and the subgrid models are implicitly coupled in a genuinely nonlinear way
CONTACT Jérôme Jacob jerome.jacob@univ-amu.fr M2P2 UMR7340, Centrale Marseille Plot 6, 38 re
Joliot-Curie 13451 Marseille, France

which escapes a full mathematical analysis. Therefore, nding an optimised pair (numeri-
cal scheme and subgrid model) is a very dicult task, since it has been observed that some
partial error cancellation may occur, leading to: (i) a kind of super-convergence of LES
results (an unexpected high accuracy is obtained in such a case since modelling errors and
numerical errors partially balance each other) and (ii) the counter-intuitive result that in
some cases an increase of the order of the numerical scheme or grid renement may lead to
adecreaseoftheglobalaccuracyoftheresults[4–6]. A promising way to solve this prob-
lem is to develop stabilised numerical schemes whose leading error terms mimic explicit
subgrid models, leading to the implicit large eddy simulation (ILES) approach [7]. Using
such a scheme, one may expect to guarantee numerical robustness and physical accuracy
atthesametimeinabettercontrolledway.
LES has been implemented within the Lattice Boltzmann method (LBM) framework
[8–12] since the 1990s [13–26],andbothclassicalLESandILESapproacheshavebeen
proposed. The most common way to incorporate an explicit subgrid viscosity model is to
modify the relaxation time in the collision model in order to recover the targeted eective
total viscosity, dened as the sum of the molecular viscosity of the uid and the subgrid
viscosity. But the need to develop stabilised LBM schemes to handle high Reynolds number
ows has also led to the denition of collision models with some ILES capabilities, such as
Entropic methods [27–30], Cascaded models [31–33] and regularised methods [34–39].
While it has been observed in numerical experiments that these methods may lead to sat-
isfactory results in turbulent ow simulations without adding an explicit subgrid model,
the direct link between their build-in dissipation and the physical subgrid one is not fully
understood.
This paper proposes a new improved recursive regularised LBM method based on
the hybridisation of the computation of the velocity gradients when evaluating the
coecients of the reconstruction of the regularised non-equilibrium component of dis-
tribution functions, leading to the denition of an Hybrid Recursive Regularised Bhatna-
gar–Gross–Krook (HRR-BGK) model. In this new model, the numerical stabilisation can
be explicitly tuned in order to recover a perfectly controlled amount of local dissipation. To
get an accurate ILES method, the free parameter appearing in this new collision kernel can
besetdynamicallyateachgridpointandtimesteptoavaluewhichallowtorecoverexactly
the same dissipation as the one provided by any classical LES subgrid model, according to
an explicit formula given below. The scope of the paper is restricted to low-Mach number
athermal ows.
This paper is organised as follows. The basic regularised BGK collision model used to
develop the new collision scheme is reminded in Section 2.ThenewdynamicRR-BGK
collisionkernelisthendiscussedinSection3. The analysis of its build-in dissipation and
thebridgewithILESarepresentedinSection4. Key features of the Vreman’s subgrid model,
whichisusedinthispaperforexplicitLESandtuningoftheILESsimulationsarereminded
in Section 5. The approach is then assessed considering the turbulent ow at Re =3900,
which is a well-documented validation case. This test case is selected because it allows for a
wall-resolving simulation without wall model in a ow conguration in which the subgrid
model plays a very important role. The decoupling from the eects of a wall model is an
important point here, since wall models may have a deeper impact on the solution than
the subgrid model, masking the eect of the later. Here, the subgrid closure is known to
have a leading eect on the development of the separated shear layers and the formation

ofthecylinderwake.Itisworthnotingthatthisisnotthecaseconsideringplanechannel
ows on LBM grids. As a matter of fact, wall-resolving LES of plane channel ow would be
based on a uniform grid with x+=y+=z+=1−2nearthewall,whichisner
than classical DNS grid resolution requirement, yielding a poor interest for the assessment
of the present subgrid closure approach.
2. Recursive regularised BGK LBM
The Boltzmann equation discretised in the velocity space reads
∂tfi(x,t)+(ξi·∇)fi(x,t)=i,(1)
where {ξi}q−1
i=0is the discrete velocity set, iis the collision operator. The most widely used
collision operator in the lattice Boltzmann community is the BGK model, where is a
relaxation towards a polynomial approximation of the Maxwell–Boltzmann distribution
equilibrium, noted f(0)
i
=−
1
τfi−f(0)
i,(2)
where f(0)
iis the equilibrium distribution function
f(0)
i=wiρ1+ξi·u
c2
s
+1
2c4
s
H(2)
i:uu,(3)
with csthe speed of sound (which is a constant) and H(2)
i=ξiξi−c2
sI(Iis the iden-
tity matrix). This model is accurate for low-Mach number, weakly compressible athermal
ows. It suers from a lack of numerical stability in high Reynolds number ows. Therefore
several successful attempts have been made to make it more stable. Among these the regu-
larisedapproach(see[35,36]) is particularly appealing due to its simplicity. The collision
operator of this model reads
i=−
1
τf(1)
i,(4)
where
f(1)
i=− 1
2c4
s
H(2)
i:P(1),(5)
with P(1)=q−1
i=0H(2)
i(fi−f(0)
i)is the deviatoric stress tensor.
In Ref. [37], a recursive procedure is introduced to add higher order terms to make
the regularisation model even more stable, which was later used and generalised in Refs.
[34,38]. It is based on the relation between the equilibrium and non-equilibrium Hermite
coecients of the distribution function fi.Byexpandingfiin Hermite polynomials up to
an order N(see [11]) one gets
fi=wi
N

n=0
1
c2n
sn!H(n)
i:a.(6)

Then by using the standard Chapman–Enskog expansion [40], one can decompose fiinto
two parts
fi=f(0)
i+f(1)
i,(7)
where f(0)
iis the equilibrium distribution function and f(1)
iis the o-equilibrium distribu-
tion function with f(0)
if(1)
i.Thesetwodistributionfunctionscanalsobeexpandedin
Hermite polynomials
f(0)
i=wi
N

n=0
1
c2n
sn!H(n)
i:a(n)
0,(8)
f(1)
i=wi
N

n=0
1
c2n
sn!H(n)
i:a(n)
1,(9)
where a(n)
0and a(n)
1are the Hermite coecients of the equilibrium and o-equilibrium
distributions respectively,
a(n)
0=
q−1

i=0
H(n)
if(0)
i, (10)
a(n)
1=
q−1

i=0
H(n)
if(1)
i. (11)
In the athermal case, the equilibrium Hermite coecients of order nare given by
a(n)
0=a(n−1)
0u, with (12)
a(0)
0=ρ. (13)
Using the Chapman–Enskog expansion as in [37], one asymptotically rst recovers the
weakly compressible Navier–Stokes equations
∂tρ+∇·(ρu)=0, (14)
ρ(∂tu+u·∇u)=−∇·P, (15)
where
ρ=
i
fi, (16)
u=
i
ξifi, (17)
P=
i
(ξi−u)(ξi−u)fi=P(0)+P(1)=Ip−2ρνS, (18)
where S=(∇u+(∇u)T)/2 is the strain-rate tensor, p=c2
sρis the pressure (this is the
perfect gas law) and ν=c2
sτis the kinematic viscosity. Another very interesting property

of the Chapman–Enskog expansion of Equation (2) is that the o-equilibrium coecients
are also related between themselves by a recursive relation (as are the equilibrium ones)
and are given by
a(n)
1,α1···αn=a(n−1)
1,α1···αn−1uαn+uα1···uαn−2a(2)
1,αn−1αn+perm(αn), (19)
where ‘perm(αn)’ stands for all the cyclic index permutations of indexes from α1to αn−1
(αnis never permuted), and where
a(2)
1,αβ =−2ρτc2
sSαβ , (20)
with Sαβ =(∂αuβ+∂βuα)/2isthestrainratetensor.Untilthispointwemadethehid-
den assumption that the discrete Hermite polynomials H(n)
ihave the same orthogonality
relations between themselves that the continuous Hermite polynomials up to an arbitrary
order n.Inrealityonly,alimitednumberofHermitepolynomialspossessthisproperty:
for the standard lattices D3Q15, D3Q19 and D3Q27 this is true up to order n=2(see[41]
for example). In [37], one uses the D3Q27 lattice because of its better isotropy and there-
fore higher resemblance with the continuous case at the expense of a higher computational
cost. To lower the amount of computations needed per grid point, we use here the D3Q19
quadrature.
The Hermite polynomials used here are a bit dierent than for the D3Q27 case. Here
we will use only combinations of third-order Hermite polynomials which have the correct
orthogonality properties. These are
H(3)
α,xxy +H(3)
α,yzz, (21)
H(3)
α,xzz +H(3)
α,xyy, (22)
H(3)
α,yyz +H(3)
α,xxz, (23)
H(3)
α,xxy −H(3)
α,yzz, (24)
H(3)
α,xzz −H(3)
α,xyy, (25)
H(3)
α,yyz −H(3)
α,xxz. (26)
These Hermite polynomials being added to the set of H(0),H(1)and H(2)the equilibrium
distribution function is given by
f(0)
i=wiρ+ξi·(ρu)
c2
s
+1
2c4
s
H(2)
i:a(2)
0+1
2c6
s
(H(3)
i,xxy +H(3)
i,yzz)(a(3)
0,xxy +a(3)
0,yzz)
+1
2c6
s
(H(3)
i,xzz +H(3)
i,xyy)(a(3)
0,xzz +a(3)
0,xyy)
+1
2c6
s
(H(3)
i,yyz +H(3)
i,xxz)(a(3)
0,yyz +a(3)
0,xxz)

+1
6c6
s
(H(3)
i,xxy −H(3)
i,yzz)(a(3)
0,xxy −a(3)
0,yzz)
+1
6c6
s
(H(3)
i,xzz −H(3)
i,xyy)(a(3)
0,xzz −a(3)
0,xyy)
+1
6c6
s
(H(3)
i,yyz −H(3)
i,xxz)(a(3)
0,yyz −a(3)
0,xxz)(27)
and the o-equilibrium is
f(1)
i=wi1
2c4
s
H(2)
i:a(2)
1+1
2c6
s
(H(3)
i,xxy +H(3)
i,yzz)(a(3)
1,xxy +a(3)
1,yzz)
+1
2c6
s
(H(3)
i,xzz +H(3)
i,xyy)(a(3)
1,xzz +a(3)
1,xyy)
+1
2c6
s
(H(3)
i,yyz +H(3)
i,xxz)(a(3)
1,yyz +a(3)
1,xxz)
+1
6c6
s
(H(3)
i,xxy −H(3)
i,yzz)(a(3)
1,xxy −a(3)
1,yzz)
+1
6c6
s
(H(3)
i,xzz −H(3)
i,xyy)(a(3)
1,xzz −a(3)
1,xyy)
+1
6c6
s
(H(3)
i,yyz −H(3)
i,xxz)(a(3)
1,yyz −a(3)
1,xxz),(28)
where
a(2)
1,αβ =
i
H(2)
i,αβ (fi−f(0)
i)=−2ρτ c2
sSαβ , (29)
a(3)
1,αβγ =uαa(2)
1,βγ +uβa(2)
1,γα +uγa(2)
1,αβ .(30)
With these two modications to the regularised model (see Equation 4), we have now the
recursive regularised BGK Lattice-Boltzmann method (RR-BGK).
3. A new improved RR-BGK collision operator
In this paper, we propose a modied version of the regularisation procedure, which relies
on the hybridisation of the computation of the velocity gradients with nite dierence
schemes in the computation of the recursive reconstruction parameters discussed in the
preceding section. The deviatoric stress tensor in the above equation is replaced by a
stabilisinghybridstresstensor ˜
P(1)which reads
˜
P(1)=P(1)σ−(1−σ)2ρτc2
sSFD,0≤σ≤1, (31)
where σ=1isequivalenttotheRR-BGKmodel.

Using the simple second-order centred nite dierence approximation
g(x+x)−g(x−x)
2x=∂xg(x)+1
6x2∂3
xg(x), (32)
the nite-dierence strain rate tensor SFD can be evaluated as
SFD
αβ ∼
=1
2∂αuβ+1
6x2∂3
αuβ+∂βuα+1
6x2∂3
βuα,
∼
=1
2∂αuβ+∂βuα+1
6x2(∂3
αuβ+∂3
βuα),
∼
=Sαβ +1
21
6x2(∂3
αuβ+∂3
βuα), (33)
where Sαβ denotes the exact velocity gradient tensor.
The collision operator then reads
=−
1
τ
˜
f(1)
i, (34)
where
˜
f(1)
i=f(1)
iσ−(1−σ)ρτ
c2
s
H(2)
i:SFD, (35)
where f(1)
iis given by Equation (28). A more ecient way to compute this last term is to
evaluate it through the recursive relation using a following modied expression for the
coecients:
˜
f(1)
i=f(1)
i˜
a(2)
1,αβ , (36)
with f(1)
istill given in Equation (28) and where ˜
a(2)
1,αβ is
˜
a(2)
1,αβ =σ
i
H(2)
i,αβ (fi−f(0)
i)+(1−σ)(−2ρτc2
sSαβ )=σa(2)
1,αβ +(1−σ)(−2ρτc2
sSαβ ).
(37)
The actual Boltzmann equation we will solve hereafter is given by
∂tfi+ξi·∇fi=−
1
τf(1)
iσ−(1−σ)ρτ
c2
s
H(2)
i:SFD. (38)
Performing the Chapman–Enskog expansion on this equation, one gets at the lowest order
∂tf(0)
i+ξi·∇f(0)
i=−
1
τf(1)
iσ−(1−σ)ρτ
c2
s
H(2)
i:SFD. (39)
Taking the zeroth-, rst- and second-order moments of this last equation, one gets
∂tρ+∇·(ρu)=0, (40)
∂t(ρu)+∇·(ρ uu)=−∇p, (41)

∂t(ρuu)+∇·(ρ uuu)+∇(ρu)+(∇(ρu))T=−
1
τ
˜
P(1). (42)
Using the rst two equations, the third can be rewritten as
−2ρc2
sτS+O(Ma3)=˜
P(1), (43)
where Ma =u/cs1 is the Mach number. The Mach number is shown to scale like Ma ∼
t/x(see [35]amongothers).Thereforeusingthediusivelimitforthescalingoft,
t∼x2, which is the limit consistent with incompressible uids, one can safely neglect
the O(Ma3)term which scales like x3.
Therefore the previous equation can be rewritten as
σP(1)
αβ =−2ρτ c2
sSαβ +(1−σ)2ρτc2
sSFD
αβ ,
P(1)
αβ =−2ρτ c2
sSαβ +1−σ
σρτc2
s
x2
6(∂3
αuβ+∂3
βuα). (44)
With this relation, one recovers the equivalent Navier–Stokes equation
ρ∂tuα+uβ∂βuα=−∂αp+2∂β(μSαβ )
−(1−σ)x2
6σ∂β(μ(∂3
αuβ+∂3
βuα)), (45)
where μ=c2
sρτ.
After performing the standard time–space discretisation using the trapezoidal approx-
imation of Equation (38), one gets the following numerical scheme
¯
fi(x+ξi,t+1)=feq
i(ρ,u)+1−1
¯τ˜
f(1)
i, (46)
where ¯
fi=fi+1
2τ˜
f(1)
i,˜
f(1)
iis given by Equation (36) and ¯τ=τ+1/2.
The present model is discretised in space using embedded subdomains with uniform
mesh with a grid spacing ratio of two between two successive subdomains (see Figure 1).
For all the grid points far from the transition region, the collision–propagation algorithm
is applied and SFD is computed using centred second-order nite dierences schemes. As
shown in Figure 1,thecoarsegrid() overlaps the ne grid (•) making it possible to apply
the collision–propagation algorithm on the transition grid points belonging to the coarse
grid (). Because of a lack of neighbour points, the propagation step cannot be applied at
the ne grid points (•) located in the transition layer. On these grid points, the macroscopic
values (ρand u) and the deviatoric stress tensor (P(1)) are interpolated from the coarse
grid neighbours located in the transition layer and SFD is computed using non-centred
rst-order nite dierence schemes.
4. Associated kinetic energy dissipation, equivalent artificial viscosity and
dynamic collision kernel
The extra dissipation of kinetic energy associated to the use of the new hybrid collision
kernelintroducedabove,denotedεσ, can be easily obtained considering the evolution

Figure 1. Visualisation of the grid arrangement around transition in resolution.
equation for the kinetic energy 1
2u·uderived from Equation (45). After some algebra,
one obtains
εσ=νσ|∇2u|2,νσ=1−σ
6σx2c2
sτ,(47)
where νσis the σ-dependent articial hyperviscosity associated to the hybrid RR-BGK col-
lision operator. It is worth noting that the induced dissipation originates in the leading error
term of the nite dierence scheme used to compute the velocity derivatives. Therefore, the
use of a second-order centred scheme leads to a bi-Laplacian-based hyperviscosity, i.e. a
νσ∇4ucorrection in the momentum equation. Higher order articial dissipation opera-
tors can be easily dened considering higher order centred nite dierence schemes to
compute Sij.Asamatteroffact,usingapth-order scheme will introduce an hyperviscous
operator proportional to ∇2pu. An increase in the order of the hyperviscosity is associated
to an concentration of the induced dissipation at small resolved scales and to a possible
way for smart control of spurious wiggles.
The above expression for the induced dissipation εσoersasmartwaytoobtainan
ILES scheme by tuning σin order to recover the same dissipation as the one provided by a
subgrid scale model, εsgs. Considering a generic subgrid viscosity νt(the same development
holds for any RANS eddy viscosity model), one has εsgs =νt|∇u|2. Therefore, equalising
the articial and the subgrid dissipation leads to
νt|∇u|2=νσ|∇ 2u|2,(48)
whose solution is
νσ=L2
VK νt,(49)

where LVK =|∇u|/|∇2u|can be interpreted as a renormalised extended denition of the
von Kármán length scale [42]. The associated value of σis
σ=1
6νt
L2
VK
x2c2
sτ+1
.(50)
These expressions can be further simplied considering subgrid viscosity models that can
be written as νt=csgs x2/τsgs where τsgs and csgs are the subgrid time scale and the subgrid
model constant, respectively. The expression for the tuning parameter is
σ=1
6csgsL2
VK
c2
sττ
sgs
+1
.(51)
It is worth noting that this last expression does not explicitly involve the mesh size x,
leading to an easy implementation in multiresolution algorithms.
ThelastrelationcanbeinvertedtondthevalueoftheSmagorinskyconstantthatyields
the same dissipation as the regularised collision kernel:
csgs =1
σ−1c2
sττ
sgs
6L2
VK
.(52)
In the case, the regularised collision kernel is used in the ILES mode, the inuence of the
implicit subgrid dissipation can be evaluated thanks to the subgrid viscosity parameter s
dened as [43]
s=εσ
εσ+εν
,(53)
where εν=ν|∇u|2denotes the molecular viscosity-induced dissipation. After some alge-
bra, one obtains
1
s=1+εν
εσ
=1+ν
νσ
L2
VK =1+6σν
(1−σ)x2c2
sτL2
VK .(54)
5. Vreman’s explicit subgrid model
The explicit subgrid model selected in the present paper for both explicit LES and ILES
simulations is the one proposed by Vreman [44], which is observed to have very interesting
self-adaptationfeatureswithoutrelyingonatestlter-basedprocedureoraprognostic
equation for a subgrid quantity. More specically, this model is observed to behave in a
very satisfactory manner in fully developed turbulent shear ows, and also in transitional
owsandnearwallregionwithouttheuseofdynamicprocedure,testlterandsemi-
empirical stabilisation step such as averaging or clipping. It is also computationally very
ecient, since it does not involve the computation of eigenvalues. A complete analysis of
its properties, including symmetry preservation, is available in [45,46].

The subgrid tensor is dened as
Rij =uiuj−uiuj,(55)
wherethebarsymboldenotestheLESlter.Itismodelledas
Rij −2νtSij +Rkk δij,(56)
where viscosity is dened as
νt=cBβ
αijαij
, (57)
with
αij =∂¯
ui
∂xj
,βij =2αijαij (58)
and
Bβ=β11β22 −β2
12 +β11β33 −β2
13 +β22β33 −β2
23.(59)
The cuto length is taken equal to the mesh size xin the present LBM-based simula-
tions, which are carried out on grids with cubic cells. The model constant is taken equal to
c=2.5C2
S,whereCS=0.18 is the Smagorinsky constant.
In the classical LES mode, this model is implemented by modifying the relaxation time
τin the BGK collision kernel according to
τ=ν+νt
c2
s
+t
2,(60)
where νdenotes the uid molecular viscosity.
6. Application to the flow around a cylinder at Re =3900
6.1. Case description
ThenewHybridRR-BGKcollisionoperatorisassessedconsideringtheowarounda
cylinder at Re =U0D/ν =3900 and Ma =U0/C0=0.0585 with U0the inow velocity,
C0the speed of sound and Dthe cylinder diameter. Four dierent computational cases
are dened, in order to assess both LES and ILES capabilities of the method (see Table 1).
Case 1 and Case 2 correspond to ILES simulations with constant values of the weighting
parameter σ. Case 3 is an explicit LES simulation, in which σ=0.99, i.e. the total amount
of numerical dissipation is expected to remain very small. The last test case (Case 4) cor-
responds to an ILES simulation with a dynamic evaluation of the parameter σ,Vreman’s
Tab le 1 . Computation parameters.
Case Subgrid model σ
Case 1 – 0.97
Case 2 – 0.985
Case 3 Vreman 0.99
Case 4 Vreman Dynamic

Figure 2. View of the computational domain and the grid points.
subgridmodelbeingusedtoevaluatethetargeteddissipation.Thecomputationofσin
that case follows the procedure described in Section 3for ρand ufor the transition in
resolution ne grid points.
The computational domain (see Figure 2) is extended 9.5 diameter upstream the cylin-
der and 49.5 diameter downstream the cylinder. A non-reective subsonic outow [47]
is used for the outlet and free slip conditions are imposed in the top and bottom bound-
aries 14.5 diameter far from the cylinder leading to a blockage ratio of 3.3%. The spanwise
extent of the computational domain is taken equal to 4Dandisusedwithperiodicbound-
aries. This spanwise size is known to prevent spurious eects due to unphysical correlations
that may be induced by periodic boundary conditions. It is shown in Breuer [48]thatno
dierences are obtained on average quantities using πDor 2πDfor the spanwise size.
The computational grid (see Figure 2)iscomposedofseveralembeddedvolumesof
uniform cartesian mesh. The grid size is reduced from D/2.5 far from the cylinder to
D/80 (ner than in Parnaudeau et al. [49]) around the cylinder and in its wake. A layer
of 15 grid points with x=D/80 is applied around the cylinder and extended 1.5 diam-
eter downstream it (x=2D) to ensure a good representation of the recirculation bubble.
The renement areas with x=D/40 and x=D/20 are applied on 10 grid point layers
around the cylinder and extended 3.5 and 5.5 diameter downstream the cylinder (x=4D
and x=6D). The distance of the rst node to the cylinder is between 0 and 0.0177Dwith an
averaged value of 0.00865Dwhich is lower than in Alkishriwi et al. [50]andOuvrardetal.
[51]. The use of cubic cells leads to 320 grid points in the spanwise direction for the mini-
mal grid size area and a total of 10 million grid points were used in this computation. The
computational time was around 47 CPU hours per vortex shedding period on 48 proces-
sors. Statistically steady state was reached after 93T∗with T∗=D/U0and all the statistics
presented in Section 6.2 were computed on 84T∗(≈17 vortex shedding periods).
6.2. Results
Key parameters related to the mean ow are reported in Table 2with reference
experimental and numerical results. It is observed that the mean drag coecient Cdand

Tab le 2 . Overview of numerical and experiment results.
Case Model CdC
lSt Lr/D−Cpb
Parnaudeau et al.
[49]
PIV 0.208 1.51
Dong et al. [52] DNS 0.206–0.210 1–1.18 0.93–1.04
Ma et al. [53] DNS 0.84–1.04 0.203–0.219 1–1.59
Alemi et al. [54] LES-Smag 0.92–1.01 0.07–0.14 0.205–0.225
Alkishriwi et al. [50] LES 1.05 0.217 1.31
Breuer [48] LES-Smag 0.969–1.486 0.397–1.686 0.687–1.665
LES-DynSmag 1.016–1.071 1.197–1.372 0.941–1.011
Mani et al. [55] LES 0.99 0.206 0.86
Franke and Frank
[56]
LES 0.978–1.005 0.209 1.34–1.64 0.85–0.94
Kravchenko and
Moin [57]
LES 1.04 0.210 1.35 0.94
Lysenko et al. [58] LES-TKE 0.97 0.09 0.209 1.67 0.91
LES-Smag 1.18 0.44 0.19 0.9 0.8
Meyer et al. [59] LES 1.05–1.07 0.21–0.215 1.18–1.38 0.92–1.05
Ouvrard et al. [51] LES-Smag 0.99 0.125 0.218 1.54 0.85
LES-Vreman 0.92 0.054 0.227 1.83 0.78
LES-WALE 1.02 0.219 0.221 1.22 0.94
ILES 0.92 0.052 0.225 1.85 0.77
Parnaudeau et al.
[49]
LES 0.208 1.56
Abrahamsen Prsic
et al. [60]
LES 1.0784–1.2365 0.1954–0.4490 0.1956–0.2152 1.27
Wormon et al. [61] LES-WALE 0.99 0.108 0.21 1.45 0.88
Zhang et al. [62] LES 1.001–1.098 0.125–0.345 0.21–0.22
D’Alessandro et al.
[63]
SA DES 1.205–1.2776 0.428–0.6140 0.204–0.215 0.7172–0.85 1.077–1.289
SA IDDES 1.0235–1.4106 0.1458–0.8283 0.205–0.222 0.5137–1.4270 0.8780–1.4688
¯
v2−fDES 0.9857–1.2553 0.1088–0.5719 0.205–0.214 0.7270–1.6780 0.8290–1.2570
Case 1 ILES 0.936 0.044 0.212 2.05 0.779
Case 2 ILES 0.973 0.077 0.210 1.835 0.828
Case 3 LES-Vreman 0.954 0.048 0.209 2.04 0.779
Case 4 ILES 1.047 0.165 0.212 1.425 0.925
Note: Smag stands for Smagorinsky model, DynSmag for Dynamic Smagorinsky model, TKE for Turbulent Kinetic Energy
model, SA for Spalart Allmaras and IDDES for Improved Delayed DES.
the Strouhal number St associated to the main frequency of aerodynamic forces exerted
on the cylinder are very accurately recovered in all cases. The predicted values of the rms
value of the uctuations of the lift coecient C
l, the base pressure coecient Cpband the
normalised recirculation bubble length Lr/Dexhibitmoredispersion,butalwaysremain
within the range of variation of previous DNS (Direct Numerical Simulation), LES and
DES(DetachedEddySimulation)results.Butitisworthnotingthatatoolargevalueof
Lrisfoundincases1,2and3showingthatthetransitiontoturbulenceintheseparated
shear layers present in the formation region is delayed when compared to Parnaudeau’s
experiments. On the overall, case 4 (ILES with dynamic tuning of the weighting parameter
σ) yields a very good prediction of all parameters, followed by case 2 (ILES with a small
constant value of σ).
A deeper insight into the results is obtained considering Figure 3, which displays the
streamwise evolution of the mean longitudinal velocity along the symmetry axis in the
cylinder wake. It is observed that in all cases the maximum amplitude of the reverse ow
in the recirculation bubble is accurately predicted when compared to experimental data. It
must also be noticed that the velocity in the far wake is also recovered in all cases, showing

Figure 3. Mean streamwise velocity in the wake centreline: Parnaudeau et al. [49] PIV 1 • and PIV 2 *,
present case 1 , present case 2 , present case 3 , present case 4 .
that both domain size and boundary conditions are adequately chosen since there is no
spurious mass/momentum leaks on upper and lower boundaries and that outow bound-
ary conditions do not induce unphysical mass/momentum ux. As mentioned above, case
4 leads to an almost perfect agreement with experimental results. Moreover it is obser ved in
Figure 3that the transition in resolution located at x/D=2andx/D=4doesnotinclude
spurious eects on the velocity elds which demonstrates the good implementation of the
present method.
The topology of the wake is now investigated looking at vertical proles of the mean
velocityatdierentlocationinthecylinderwake(seeFigures4and 5). Once again, the
simulation based on the dynamic version of the new HRR-BGK collision model (case 4)
is in almost perfect agreement with experimental data. An important point is that a
V-shaped prole is recovered on the mean longitudinal velocity prole in agreement with
experimental data, while many less accurate LES predict a U-shaped prole [49]. Discrep-
ancies observed in the three other cases are mainly due to the error on the prediction of
the length of Lr, but the global topology of both the mean recirculation bubble and near
wake is recovered in all cases.
Since the size of the recirculation bubble is governed by the transition process in the
separated shear layers, it is interesting to analyse the resolved Reynolds stresses. Vertical
proles of the longitudinal and vertical resolved Reynolds stresses at several locations in
the wake are displayed in Figures 6and 7. Once again, the very good accuracy of the case
4 is observed. In that case, the shear layer dynamics is very well recovered, since both the
maximum value and the shear layer thickness and spreading rate are accurately predicted.
In other cases, the separated shear layer spreading rate is underpredicted, leading to the
prediction of a too long recirculation bubble.
This is further conrmed looking at the streamwise evolution of the resolved longi-
tudinal Reynolds stress uualong the wake symmetry line that is displayed in Figure 8.
Case 4 is in very good agreement with experimental data, since both the location and the
amplitude of the peak located near the end of the recirculation bubble are satisfactorily
predicted.Inthethreeothercases,peaksaredampedandshifteddownstream,whichis

Figure 4. Mean streamwise velocity at x=1.06D(top), x=1.54D(middle) and x=2.02D(bottom) in
the wake of the cylinder: Parnaudeau et al. [49] PIV 1 • and PIV 2 *, present case 1 , present case
2, present case 3 , present case 4 .

Figure 5. Mean normal velocity at x=1.06D(top), x=1.54D(middle) and x=2.02D(bottom) in the
wake of the cylinder: Parnaudeau et al. [49] PIV 1 • and PIV 2 *, present case 1 , present case 2
, present case 3 , present case 4 .

Figure 6. Vertical profiles of the streamwise resolved Reynolds stress uuat x=1.06D(top), x=1.54D
(middle) and x=2.02D(bottom) in the wake of the cylinder: Parnaudeau et al. [49] PIV 1 • and PIV 2 *,
present case 1 , present case 2 , present case 3 , present case 4 .

Figure 7. Vertical profiles of the normal resolved Reynolds stress wwat x=1.06D(top), x=1.54D
(middle) and x=2.02D(bottom) in the wake of the cylinder: Parnaudeau et al. [49] PIV 1 • and PIV 2
*, present case 1 , present case 2 , present case 3 , present case 4 .

Figure 8. Variance of the streamwise velocity in the wake centreline: Parnaudeau et al. [49]PIV1•
and PIV 2 *, present case 1 , present case 2 , present case 3 , present case 4
.
Figure 9. Iso contours of normalised Q criterion (Q∗=Q(D2/U2
0)=1) coloured by velocity magnitude:
Case 1 (top left), Case 2 (top right), Case 3 (bottom left) and Case 4 (bottom right).
coherent with previous comments dealing with the damping of the shear layer dynamics
in these simulations.
TheeectsofthedissipationontheowstructuresareillustratedinFigures9and 10,
which display iso-surfaces of instantaneous Q criterion. While it is seen that the ow
physicsisqualitativelywellpredictedinallcases(laminarboundarylayersalongthecylin-
der with laminar separation and transition in the separated shear layers, large-scale roll-up
oftheshearlayersandappearanceofsmallscaleworm-likevorticesinthewake),somesub-
tle dierences can be detected. As a matter of fact, it can be seen that small-scale structures

Figure 10. Iso contours of normalised Q criterion (Q∗=Q(D2/U2
0)=100) coloured by velocity magni-
tude: Case 1 (top left), Case 2 (top right), Case 3 (bottom left) and Case 4 (bottom right).
Figure 11. Iso contours of the σvalue coloured by the velocity magnitude for σ=0.97 (left) and σ=
0.985 (right) for the Case 4.
are mode developed in case 4 and that the separated shear layer transition occurs earlier
in that case. The same worm-like vortices are observed in Figure 11 which display the
iso-value of σ=0.97 (Case 1 value) and σ=0.985 (Case 2 value) for the case 4. The σ
parameter tends to decrease down to 0.5 or lower values for a few grid points in areas where
dissipation is needed to ensure a good representation of physical phenomena or stays at
values close to 1 when no dissipation is needed.
A last quality criterion may be obtained analysing the frequency content of the wake.
This is done looking at the frequency spectrum of both streamwise and normal velocity
uctuations in the wake at x/D=3, see Figures 12 and 13, respectively. Power spectra are
computed using 3 sequences of 10 vortex shedding cycles with 50% of overlapping using
the periodogram technique of Welch [64] and Hanning window. It is seen that the main
frequency peak is recovered in all cases, showing that the vortex shedding frequency fvsis
accurately predicted. More interestingly, the existence of a secondary peak at a frequency

Figure 12. Power spectra density of the streamwise velocity at x=3D: present case 1 ,present
case 2 , present case 3 , present case 4 .
Figure 13. Power spectra density of the normal velocity at x=3D: present case 1 , present case
2, present case 3 , present case 4 .
three times larger than the primary peak, i.e. f=3fvsin the normal velocity spectrum is
accurately captured in case 4, showing the very good quality of this simulation [49]. An
inertial range with a slope is observed in all cases, whose width is about 1 decade, as in
high-resolution LES and experiments presented in Ref. [49].
7. Concluding remarks
A new regularised collision model for Lattice Boltzmann-based LES of weakly compress-
ible ows, referred to as the HRR-BGK collision model, has been presented. It relies on
the modication of previously existing recursive regularised BGK models, which consists
of hybridising the computation of the velocity gradient with a centred nite dierence
evaluation when reconstructing the regularised non-equilibrium part of the distribution

functions. The resulting eect is the introduction of a stabilising hyperviscosity term,
whose amplitude can be explicitly tuned via a control parameter σ.Theresultingmodel
can be used in both LES and ILES modes.
Thenewcollisionmodelhasbeenassessedconsideringtheowaroundacylinderat
Re =3900. The results obtained with a xed uniform value of σexhibit a delay in the tran-
sition of the separated shear layers, a phenomenon which is also observed when an explicit
subgrid viscosity term is added. On the opposite, very satisfactory results are obtained
when using the dynamic version of the HRR-BGK collision kernel equipped with the Vre-
man’s subgrid model. It is important noting that ILES based on the dynamic HRR-BGK is
notstrictlyequivalenttoaclassicalBGKcoupledtotheVreman’smodel,sincethescale-by-
scale distributions of the total dissipation are not equivalent. While the original Vreman
model is associated to a classical Laplacian-based dissipation, the present ILES method
introduces an bi-Laplacian-based dissipation. The use of higher order dissipation is known
to preserve large-scale and inertial-range dynamics [65,66], and has been successfully used
in several ILES methods for Navier–Stokes equations, e.g. Refs. [67–73], with a few existing
extensions to the LBM framework [74–77].
The signicant dierences between the dynamic version of the present ILES method
and non-dynamic ones and also the classical explicit LES are due to the fact that most
key features of the ow are governed by the transition in separated shear layers. This phe-
nomenon is very sensitive to viscous and hyperviscous damping. A fully turbulent ow is
less sensitive to ne details of the dissipative mechanisms, and much smaller dierences
would certainly occur in such a ow, as observed in ILES results based on Navier–Stokes
equations.
Acknowledgments
This work was carried out using the ProLB solver. Professor Eric Lamballais is warmly acknowledged
for providing experimental data.
Disclosure statement
No potential conict of interest was reported by the authors.
Funding
This work was supported by the French project CLIMB, with the nancial support of BPIFrance
(Project No. P3543-24000), in the framework of the program ‘Investissement d’Avenir: Calcul
Intensif et Simulation Numérique’. This work was performed using HPC resources from GENCI-
TGCC/CINES (Grant 2017-A0012A07679).
ORCID
Jérôme Jacob http://orcid.org/0000-0001-9287-4167
Orestis Malaspinas http://orcid.org/0000-0001-9427-6849
Pierre Sagaut http://orcid.org/0000-0002-3785-120X
References
[1] Garnier E, Adams N, Sagaut P. Large-eddy simulation for compressible ows. Berlin: Springer;
2009.

[2] Sagaut P. Large eddy simulation for incompressible ows: an introduction. Berlin: Springer;
2005.
[3] Sagaut P, Deck S, Terracol M. Multiscale and multiresolution approaches in turbulence.
London: Imperial College Press; 2006.
[4] Meyers J, Sagaut P. Is plane channel ow a friendly case for testing of large-eddy simulation
subgrid-scale models?. Phys Fluids. 2007;19:048105.
[5] Meyers J, Sagaut P, Geurts BJ. Optimal model parameters for multi-obective large-eddy
simulations. Phys Fluids. 2006;18:095103.
[6] Meyers J, Sagaut P, Geurts BJ. A computational error assessment of central nite-volume
discretizations in large-eddy simulation using a Smagorinsky model. J Comput Phys.
2007;227(1):156–173.
[7] Grinstein FF, Margolin LG, Rider WJ. Implicit large eddy simulation: computing turbulent uid
dynamics. Cambridge: Cambridge University Press; 2007.
[8] Chen S, Doolen GD. Lattice Boltzmann method for uid ows. Annu Rev Fluid Mech.
1998;30(1):329–364.
[9] Guo Z, Shu C. The lattice Boltzmann method and its applications in engineering. Singapore:
Worl d S c i e n t ic; 2013.
[10] Kruüger T, Kusumaatmaja H, Kuzmin A. The lattice Boltzmann method. Principles and
practice. Berlin: Springer; 2017.
[11] ShanX, Yuan X-F, Chen H. Kinetic theory representation of hydrodynamics: a way beyond the
Navier–Stokes equation. J Fluid Mech. 2006;550:413–441.
[12] Succi S. The lattice Boltzmann equation for uid dynamics and beyond. Oxford: Oxford
University Press; 2001.
[13] Ansumali S, Karlin IV, Succi S. Kinetic theory of turbulence modeling: smallness param-
eter, scaling and microscopic derivation of Smagorinsky model. Physica A. 2004;338(3–4):
379–394.
[14] Chen H, Kandasamy S, Orszag S, et al. Extended Boltzmann kinetic equation for turbulent
ows. Science. 2003;301(5633):633–636.
[15] Chen H, Orszag SA, Staroselsky I, et al. Expanded analogy between Boltzmann kinetic theory
of uids and turbulence. J Fluid Mech. 2004;519:301–314.
[16] Chen H, Succi S, Orszag S. Analysis of subgrid scale turbulence using the Boltzmann Bhatna-
gar–Gross–Krook kinetic equation. Phys Rev E. Mar 1999;59(3):R2527–R2530.
[17] Chen S. A large-eddy-based lattice Boltzmann model for turbulent ow simulation. Appl Math
Comput. 2009;215(2):591–598.
[18] Eggels JGM. Direct and large-eddy simulation of turbulent uid ow using the lattice-
Boltzmann scheme. Int J Heat Fluid Flow. 1996;17(3):307–323.
[19] Filippova O, Succi S, Mazzocco F, et al. Multiscale lattice Boltzmann schemes with turbulence
modeling. J Comput Phys. 2001;170(2):812–829.
[20] Girimaji SS. Boltzmann kinetic equation for ltered uid turbulence. Phys Rev Lett.
2007;99(3):034501.
[21] HouS,SterlingJ,ChenS,etal.AlatticeBoltzmannsubgridmodelforhighReynoldsnumber
ows. Fields Inst Comm. 1996;6:151–66.
[22] Krafczyk M, Tölke J, Luo L-S. Large-eddy simulations with a multiple-relaxation-time LBE
model. Int J Mod Phys B. 2003;17:33–39.
[23] Malaspinas O, Sagaut P. Consistent subgrid scale modelling for lattice Boltzmann methods. J
Fluid Mech. JUN 2012;700:514–542.
[24] Premnath KN, Pattison MJ, Banerjee S. Dynamic subgrid scale modeling of turbulent ows
using lattice-Boltzmann method. Physica A. 2009;388(13):2640–2658.
[25] Teixeira CM. Incorporating turbulence models into the lattice-Boltzmann method. Int J
Modern Phys C. 1999;9:1159–1175.
[26] Weickert M, Teike G, Schmidt O, et al. Investigation of the LES WALE turbulence model within
the lattice Boltzmann framework. Comput Math Appl. 2010;59(7):2200–2214. Mesoscopic
MethodsinEngineeringandScience,InternationalConferencesonMesoscopicMethodsin
Engineering and Science.

[27] Bösch F, Chikatamarla SS, Karlin IV. Entropic multi-relaxation models for simulation of uid
turbulence. ESAIM: Proc Surveys. 2015;52:1–24.
[28] Bösch F, Chikatamarla SS, Karlin IV. Entropic multi-relaxation models for turbulent ows. Phys
Rev E. 2015;92:43309.
[29] Di Ilio G, Dorschner B, Bella G, et al. Simulation of turbulent ows with the entropic mul-
tirelaxation time lattice Boltzmann method on body-tted meshes. J Fluid Mech. 2018;849:
35–56.
[30] Dorschner B, Chikatamarla SS, Karlin IV. Transitional ows with the entropic lattice Boltz-
mann method. J Fluid Mech. 2017;824:388–412.
[31] Geier M. De-aliasing and stabilization formalism of the cascaded lattice Boltzmann
automaton for under-resolved high Reynolds number ow. Int J Numer Methods Fluids.
2008;56:1249–1254.
[32] Geier M, Greiner A, Korvink JG. Cascaded digital lattice Boltzmann automata for high
Reynolds number ow. Phys Rev E. 2006;73:066705.
[33] Geller S, Upho S, Krafczyk M. Turbulent jet computations based on MRT and cascaded lattice
Boltzmann models. Comput Math Appl. 2013;65:1956–1966.
[34] Coreixas C, Wissocq G, Puigt G, et al. Recursive regularization step for high-order lattice
Boltzmann methods. Phys Rev E. 2017;96:033306.
[35] Latt J. HydrodynamiclimitoflatticeBoltzmannequations.PhDdissertation,Universityof
Geneva, Geneva, Switzerland, 2007.
[36] Latt J, Chopard B. Lattice Boltzmann method with regularized pre-collision distribution
functions. Math Comp Sim. 2006;72:165–168.
[37] Malaspinas O. Increasing stability and accuracy of the lattice Boltzmann scheme: recursivity
and regularization. arXiv, pages 1–31, 2015
[38] Mattila KK, Philippi PC, HegeleJr LA. High-order regularization in lattice-Boltzmann equa-
tions. Phys Fluids. 2017;29(4):046103.
[39] Montessori A, Falcucci G, Prestininzi P, et al. Regularized lattice Bhatnagar–Gross–Krook
model for two- and three-dimensional cavity ow simulations. Phys Rev E. 2014;89:
053317.
[40] Chapman S, Cowling TG. The mathematical theory of nonuniform gases. Cambridge: Cam-
bridge University Press; 1960.
[41] Shan X. The mathematical structure of the lattices of the lattice Boltzmann method. J Comput
Sci. 2016;17:475–481.
[42] Menter FR, Egorov Y. The scale-adaptive simulation method for unsteady turbulent ow
predictions. Part 1: theory and model description. Flow Turbul Combust. 2010;85:113–138.
[43] Geurts BJ, Frölich J. A framework for predicting accuracy limitations in large-eddy simulation.
Phys Fluids. 2002;14:L41–L44.
[44] Vreman AW. An eddy-viscosity subgrid-scale model for turbulent shear ow: algebraic theory
and applications. Phys Fluids. 2004;16:3670–3681.
[45] Silvis MH, Remmerswaal RA, Verstappen R. Physical consistency of subgrid-scale model for
large-eddy simulation of incompressible turbulent ows. Phys Fluids. 2017;29:015106.
[46] Trias FX, Folch D, Gorobets A, et al. Building proper invariants for eddy-viscosity subgrid-scale
models. Phys Fluids. 2015;27:065103.
[47] Poinsot TJ, Lele SK. Boundary conditions for direct simulations of compressible viscous ows.
JComputPhys.1992;101(1):104–129.
[48] Breuer M. Numerical and modeling inuences on large eddy simulations for the ow past a
circular cylinder. Int J Heat Fluid Flow. 1998;19(5):512–521.
[49] Parnaudeau P, Carlier J, Heitz D, et al. Experimental and numerical studies of the ow over a
circular cylinder at Reynolds number 3900. Phys Fuids. 2008;20(8):085101.
[50] Alkishriwi N, Meinke M, Schröder W. A large-eddy simulation method for low Mach number
ows using preconditioning and multigrid. Comput Fluids. 2006;35(10):1126–1136.
[51] Ouvrard H, Koobus B, Dervieux A, et al. Classical and variational multiscale LES of the ow
around a circular cylinder on unstructured grids. Comput Fluids. 2010;39(7):1083–1094.

[52] Dong S, Karniadakis GE, Ekmekci A, et al. A combined direct numerical simulation-particle
image velocimetry study of the turbulent near wake. J Fluid Mech. 2006;569:185–207.
[53] Ma X, Karamanos G-S, Karniadakis GE. Dynamics and low-dimensionality of a turbulent near
wake. J Fluid Mech. 2000;410:29–65.
[54] Alemi M, Pêgo JP, Maia R. Numerical investigation of the ow behavior around a single
cylinder using large Eddy simulation model. Ocean Eng. 2017;145:464–478.
[55] Mani A, Moin P, Wang M. Computational study of optical distortions by separated shear layers
and turbulent wakes. J Fluid Mech. 2009;625:273–298.
[56] Franke J, Frank W. Large eddy simulation of the ow past a circular cylinder at ReD =3900.
JWindEngIndAerodyn.2002;90(10):1191–1206. 3rd European-African Conference on Wind
Engineering.
[57] Kravchenko AG, Moin P. Numerical studies of ow over a circular cylinder at ReD =3900.
Phys Fluids. 2000;12(2):403–417.
[58] Lysenko DA, Ertesvåg IS, Rian KE. Large-eddy simulation of the ow over a circular
cylinder at Reynolds number 3900 using the OpenFOAM toolbox. Flow Turbul Combust.
2012;89(4):491–518.
[59] MeyerM,HickelS,AdamsNA.Assessmentofimplicitlarge-eddysimulationwithacon-
servative immersed interface method for turbulent cylinder ow. Int J Heat Fluid Flow.
2010;31(3):368–377. Sixth International Symposium on Turbulence and Shear Flow Phenom-
ena.
[60] Abrahamsen Prsic M, Ong MC, Pettersen B, et al. Large eddy simulations of ow around
a smooth circular cylinder in a uniform current in the subcritical ow regime. Ocean Eng.
2014;77:61–73.
[61] Wornom S, Ouvrard H, Salvetti MV, et al. Variational multiscale large-eddy simulations of the
ow past a circular cylinder: Reynolds number eects. Comput Fluids. 2011;47(1):44–50.
[62] Zhang H, Yang J-M, Xiao L-F, Large-eddy simulation of the ow past both nite and innite
circular cylinders at Re =3900. J Hydrodyn Ser B. 2015;27(2):195–203.
[63] D’Alessandro V, Montelpare S, Ricci R. Detached-eddy simulations of the ow over a cylinder
at Re =3900 using OpenFOAM. Comput Fluids. 2016;136:152–169.
[64] Welch P. The use of fast fourier transform for the estimation of power spectra: a method based
on time averaging over short, modied periodograms. IEEE Trans Audio Electroacoust. Jun
1967;15(2):70–73.
[65] Borue V, Orszag SA. Forced three-dimensional homogeneous turbulence with hyperviscosity.
Europhys Lett. 1995;29:687–692.
[66] BorueV,OrszagSA.Self-similardecayofthree-dimensionalhomogeneousturbulencewith
hyper viscosity. Phys Rev E. 1995;51:R856–R859.
[67] Berland J, Lafon P, Daude F, et al. Filter shape dependence and eective scale separation in
large-eddy simulations based on relaxation ltering. Comput Fluids. 2011;47:65–74.
[68] Daude F, Berland J, Emmert T, et al. A high-order nite-dierence algorithm for direct
computation of aerodynamic sound. Comput Fluids. 2012;61:46–63.
[69] Hickel S, Adams NA. On implicit subgrid-scale modeling in wall-bounded ows. Phys Fluids.
2007;19:105106.
[70] Hickel S, Adams NA, Domaradzki JA. An adaptive local deconvolution method for implicit
LES. J Comput Phys. 2006;213:413–436.
[71] Stolz S, Adams NA. An approximate deconvolution procedure for large-eddy simulation. Phys
Fluids. 1999;11(7):1699–1701.
[72] Stolz S, Adams NA, Kleiser L. An approximate deconvolution model for large-eddy simulation
with application to incompressible wall-bounded ows. Phys Fluids. 2001;13(4):997–1015.
[73] StolzS,AdamsNA,KleiserL.Theapproximatedeconvolutionmodelforlarge-eddysimula-
tions of compressible ows and its application to shock-turbulent-boundary-layer interaction.
Phys Fluids. 2001;13(10):2985–3001.
[74] Malaspinas O, Chopard B, Latt J. General regularized boundary condition for multi-speed
lattice Boltzmann models. Comput Fluids. 2011;49(1):29–35.

[75] Marié S, Gloerfelt X. Adaptive ltering for the lattice Boltzmann method. J Comput Phys.
2017;333:212–226.
[76] Ricot D, Marié S, Sagaut P, et al. Lattice Boltzmann method with selective viscosity lter.
JComputPhys.2009;228(12):4478–4490.
[77] Sagaut P. Toward advanced subgrid models for Lattice-Boltzmann-based large-eddy sim-
ulation: theoretical formulations. Comput Math Appl. 2010;59(7):2194–2199. Mesoscopic
Methods in Engineering and Science, International Conference on Mesoscopic Methods in
Engineering and Science.