12th International Symposium on Turbulence and Shear Flow Phenomena (TSFP12)
Osaka, Japan (Online), July 19-22, 2022

ON IMPLICIT LARGE EDDY SIMULATION IN COMPRESSIBLE
TURBULENCE

Guiyu CaoSouthern University of Science andTechnology
caogy@sustech.edu.cn

Wenjin Zhao
Hong Kong University of Science andTechnology
wzhaoag@connect.ust.hk

Shiyi Chen
Southern University of Science and Technology
chensy@sustech.edu.cn

ABSTRACTet al., 2016; Moser et al., 2021).

Current research conducts the quantitative comparisons
between implicit large eddy simulation (iLES) and static explicit large eddy simulation (eLES). We start with the implementations of iLES and eLES in compressible Taylor-Greenvortex problem. Compared with the key statistical quantities
of direct numerical simulation (DNS), iLES outweighs eLES
on the exactly same unresolved grids. With DNS solution, priori analysis of compressible filtered subgrid-scale (SGS) turfbulent kinetic energy ρKsgs provides the following observations: forward and backward filtered SGS turbulent kinetic energy transfer; ensemble turbulent kinetic energy Ek is in theorder of O(104 ) to O(102 ) of ensemble filtered SGS turbuf
lent kinetic energy Ksgs ; ensemble dominant physical dissipation rate ε1 is approximately 20 times larger than ensemble filf ftered SGS dissipation rate −τi j Sei j . Then, for iLES and eLES,
the total dissipation rate is decomposed into resolved physimod ,cal dissipation rate ε phy , modeling SGS dissipation rate εsgs
num
and numerical SGS dissipation rate εsgs . Quantitative comparisons on the modeling SGS dissipation rate and numerical
SGS dissipation rate in iLES and eLES are implemented. We
confirms that the numerical dissipation in iLES can be treatedas the built-in SGS dissipation, which accounts for the reasonable performance of iLES. While the explicit modeling SGS
dissipation in eLES pollutes the resolved turbulent structuresin such low-Reynolds number turbulence. We believe that thenext generation of large eddy simulation on unresolved grids
must take both the built-in numerical SGS dissipation and its
competition explicit modeling SGS dissipation into account.

In the past decades, the finite-volume gas-kinetic scheme(GKS) based on the Bhatnagar-Gross-Krook (BGK) model(Bhatnagar et al., 1954) have been developed systematicallyfor computations from low speed flows to supersonic ones (Xu,
2001, 2015). The GKS presents a gas evolution process fromkinetic scale to hydrodynamic scale, where both inviscid andviscous fluxes are recovered from a time-dependent and multidimensional gas distribution function at a cell interface. Basedon the time-dependent flux, a reliable two-stage framework (Li& Du, 2016) was provided for developing the high-order GKS(HGKS) (Pan et al., 2016). In terms of low-Reynolds number
turbulent flows, the HGKS has been used as a direct numerical simulation (DNS) tool (Kumar et al., 2013; Cao et al.,2021a, 2022). The numerical performance (i.e., numerical accuracy, robustness) and computational cost is comparable withthe widely-used high-order finite difference method (DeBonis,2013). HGKS also shows advantage in supersonic turbulence
studies due to its reliable robustness. HGKS indeed provides avalid tool for the numerical simulation of turbulence, which ismuch less reported in finite volume scheme.
HGKS has been implemented for the iLES in compressible turbulent flows (Cao et al., 2021b), i.e., compressible turbulent flow over periodic hills with volumetric Mach number
Mav = 0.2 and cross-sectional Reynolds number Reb = 2800.
Compared with the key statistical quantities of DNS, we foundthat iLES outweighs eLES on the exactly same unresolved
grids. eLES over-predicts the normalized Reynolds stresses
and provides much stronger turbulent fluctuation than that of
DNS solution. While, the solutions from iLES agree wellwith the DNS results, and the over-predicted performance seldom appears. Thus, we speculated that the static explicit LES
model may pollute the resolved turbulent structures of lowReynolds number separated turbulent flows. To shed light on
the seemingly abnormal performance, current research conducts the quantitative comparisons between iLES and eLES,
especially focusing on the numerical SGS dissipation andmodeling SGS dissipation.

INTRODUCTION
Large eddy simulation (LES) is proposed to solve the filtered Navier-Stokes equations (NSE) with resolvable turbulent
structures above the inertial scale (Manabe et al., 1965). Static
explicit large eddy simulation (eLES) has been widely used inunsteady separated turbulent flows (Nicoud & Ducros, 1999;
Vreman, 2004; Sagaut, 2006; Garnier et al., 2009). Different with the eLES, implicit large eddy simulation (iLES) takes
the built-in numerical dissipation as the subgrid-scale (SGS)
dissipation (Boris et al., 1992; Grinstein et al., 2007). Dueto the lower computational costs and reasonable performance,
iLES has been gradually utilized in LES community (Lombard

This paper is organized as follows. After the introduction, HGKS is introduced briefly. Followed by the numericalresults and quantitative analysis, the last section presents theconclusion and discussion.

1

12th International Symposium on Turbulence and Shear Flow Phenomena (TSFP12)
Osaka, Japan (Online), July 19-22, 2022
NUMERICAL SCHEME

viscosity model according to the following relation between
turbulent eddy viscosity µt and turbulent relaxation time τt asµ + µtτ + τt =,(8)p

The three-dimensional BGK equation (Bhatnagar et al.,1954) reads
g− f
,
(1)
ft + u f x + v f y + w f z =τwhere static Smagorinsky model (S-model) and Vreman-typemodel (V-model) (Vreman, 2004) are used to obtain the turbulent eddy viscosity µt in Eq.(8). The detailed implementations and comparisons among different eddy viscosity models
within the HGKS framework has been conducted in Zhao et al.
(2021). In following study, the S-model is equipped with thecoefficient Cs = 0.1, and V-model adopts corresponding coefficient as Cv = 2.5Cs2 = 0.025.

where u = (u, v, w)T is the particle velocity, f is the gas distribution function, g is the three-dimensional Maxwellian distribution and τ is the particle collision time. Chapman-Enskog
expansion (Chapman & Cowling, 1990) provides τ = µ/p,
where µ is the molecular viscosity and p is the pressure. The
collision term satisfies the compatibility conditionZg− fψ dΞ = 0,(2)τwith ψ = (1, u, v, w, (u2 + v2 + w2 + ξ 2 )/2)T , ξ 2 = ξ12 + ... +ξN2 , dΞ = dudvdwdξ1 . . . dξN . N = (5 − 3γ)/(γ − 1) is the internal degree of freedom, and γ is the specific heat ratio.
Taking moments of Eq.(1) and integrating with respect tospace, the finite volume scheme can be expressed as
Qi jk )d(QQi jk ),
= L (Q(3)dt
where Q i jk is the vector of conservative variables, namely,
density ρ, momentum ρUi , and total energy ρE. The operator L is defined as
Qi jk ) = −
L (Q

6
1
F p (t),∑|Ωi jk | p=1

RESULTS AND ANALYSIS
In this section, iLES and eLES are implemented in compressible Taylor-Green vortex problem (TGV). Priori coarsegrained analysis of DNS solution sheds light on expected modeling SGS dissipation, and the quantitative SGS dissipation
analysis will be presented subsequently.

Numerical Simulation in TGV
Taylor-Green vortex is a classical problem in fluid dynamics developed to study vortex dynamics, turbulent transition,
turbulent decay and energy dissipation process (Brachet et al.,
1983; Gallis et al., 2017). The flow is computed within a periodic square box defined as −πL ≤ x, y, z ≤ πL. Fluid is aperfect gas with γ = 1.4 and Prandtl number Pr = 1. Mach
number takes Ma = 0.1 and Reynolds number is Re = 1600.
iLES and eLES are simulated in 2563 grids. The detailed setup can be found in Cao et al. (2022).
To evaluate the performance of iLES and eLES, several
diagnostic dynamic quantities are computed. The ensemble(volume-averaged) turbulent kinetic energy Ek is
Z
1
1
Ek =
ρUi ·Ui dΩ,(9)ρ0 Ω Ω 2(4)

where control volume Ωi jk = xi × y j × zk with xi = [xi −∆x/2, xi + ∆x/2], y j = [y j − ∆y/2, y j + ∆y/2], zk = [zk −∆z/2, zk + ∆z/2], F p (t) is the numerical flux across the cellinterface Σ p .
The numerical flux in x-direction is given as an example2

F p (t) =

∑ ωmn

Z

ψ u f (xxi+1/2, jm ,kn ,t, u , ξ )dΞ∆y∆z. (5)m,n=1The Gaussian quadrature is used over the cell interface, whereωmn is the quadrature weight, x i+1/2,m,n = (xi+1/2 , y jm , zkn )and (y jm , zkn ) is the Gauss quadrature point of cell interface
y j × zk . The gas distribution function f (xxi+1/2, jm ,kn ,t, u, ξ ) in
the local coordinate can be given by the integral solution ofEq.(1) asZ0
1 tg(xx0 ,t 0 , u , ς )e−(t−t )/τ dt 0
f (xxi+1/2, jm ,kn ,t, u , ξ ) =τ 0(6)where ρ0 is the initial density, Ω the volume of whole computational domain, (·) the inner product. The ensemble enstrophy
dissipation rate ε(ζ ) is related to the enstrophy ζ as
2µε(ζ ) =
ζ,
ρ0
(10)
Z
1
1
ζ=
ρωi · ωi dΩ,ρ0 Ω Ω 2+ e−t/τ f0 (−uut, ξ ),where vorticity is ωi = εi jkUk, j with εi jk the alternating tensor
and Uk, j = ∂Ui /∂ x j . ε(ζ ) is utilized to evaluate the resolutionof resolved turbulent structures. For the compressible flow, theensemble physical dissipation rate ε phy obtained from the NSE
is the sum of three contributions
ε phy = ε1 + ε2 + ε3 ,
Z
2µ
ε1 =
S∗ : S∗ dΩ,
ρ0 Ω Ω i j i jZ(11)µε2 = bθ 2 dΩ,
ρ0 Ω Ω
Z1
ε3 = −pθ dΩ,
ρ0 Ω Ω

where x 0 = x i+1/2, jm ,kn − u (t − t 0 ) is the trajectory of particles,
f0 is the initial gas distribution function, and g is the corresponding equilibrium state. In GKS framework, the secondorder gas distribution function at cell interface has been constructed (Xu, 2001). After the gas distribution function being
determined, the numerical flux can be obtained by taking moments of it as Eq.(5). To achieve high-order accuracy in space
and time, the high-order spatial reconstruction and the multistage time discretization has been systematically developed in
Li & Du (2016); Pan et al. (2016). In current study, the welldeveloped in-house three-dimensional (3D) parallel code will
be used to provide the high-accuracy flow-fields for iLES andeLES. More details in HGKS can be found in Cao et al. (2022).
When implementing the eLES models, an extended BGKequation has been proposed asg− fft + u f x + v f y + w f z =,(7)
τ + τtwhere Si∗j is the deviatoric part of the strain rate tensor Si j ,with Si∗j = Si∗j − δi j Skk /3, Si j = (Ui, j + U j,i )/2. (:) denotes
the product for second-order tensor. µb is the bulk viscosity,and the inherent bulk viscosity is µb = 4µ/15 for BGK model(Xu, 2015). θ = Ui,i denotes the divergence of turbulent velocity. To eliminate the post-processing error from numerical
discretization, all spatial derivatives in Eq.(10) and Eq.(11) are

where τt is an enlarged turbulent relaxation time. Based on
the Chapman-Enskog expansion, Eq.(7) can recover the eddy2

12th International Symposium on Turbulence and Shear Flow Phenomena (TSFP12)
Osaka, Japan (Online), July 19-22, 2022
computed by sixth-order central difference.
Figure 1 presents the time history of ensemble turbulent
kinetic energy Ek . Compared with the DNS solution (Caoet al., 2022), we observe that the first-order statistical variable Ek is not sensitive to the grid resolution, as well as iLES
or eLES models. Figure 2 clearly presents that iLES performs
better than eLES, entailing iLES resolve much accurate turbulent structures. In current compressible simulation with small
Mach number Ma = 0.1, Figure 3 shows that primary dissipation rate ε1 is the dominant contribution to ensemble physicaldissipation rate ε phy . Meanwhile, dominant ensemble dissipation rate ε1 in Figure 3 again to show that iLES outweighs
eLES on the exactly same unresolved grids. The similar conclusion has been drawn in previous simulation of compressible separated turbulent flow (Cao et al., 2021b), where iLES
outweighs eLES in compressible turbulent flow over periodic
hills. In terms of ignorable ensemble dissipate rate ε2 and ε3 ,iLES and eLES are in the quite similar performance.

0.015
DNS
iLES
S-ModelV-Model

ε1

0.01

0.005

0
0

5

10

t

15

20

1.2E-07
1E-07

DNS

0.12

DNS
iLES
S-ModelV-ModeliLES
S-ModelV-Model

8E-08
6E-08

Ek

ε2

0.08

4E-08
2E-08
0.04
0
-2E-08
0

0
0

5

10

t

15

5

10

15

20

t

20

Figure 1. Time history of ensemble turbulent kinetic energy
Ek . The DNS solution with 10243 grids is provided in Cao
et al. (2022).

6E-05
DNS
iLES
S-ModelV-Model

4E-05
0.015
2E-05

ε3

DNS
iLES
S-ModelV-Model

0

0.01

ε(ζ)

-2E-05-4E-05

0.005

0

5

10

15

t

20

Figure 3. Time history of three contributions of ensemble
physical dissipation rate ε1 , ε2 and ε3 .
0
0

5

10

t

15

20

Figure 2. Time history of ensemble enstrophy dissipation
rate ε(ζ ).

f

the filtered SGS production of Ksgs quantitatively. The following DNS flow-fields have been obtained as case T G3 in
Cao et al. (2022), where 23 grids are coarsen to 1 grid with
Box filter. Positive definite kernel of Box filter can guarantee
positive filtered SGS turbulent kinetic energy (Vreman et al.,1994). After filtering process, the flow variable can be decom0
posed into resolved and SGS terms as φ (xx) = φ (xx) + φ (xx). To
avoid SGS term appearing in the filtered continuity equation,f
Priori Analysis of K sgs

To study the seemingly abnormal performance of iLES
and eLES, we firstly focus on the priori coarse-grained analf
ysis of filtered SGS turbulent kinetic energy Ksgs to evaluate312th International Symposium on Turbulence and Shear Flow Phenomena (TSFP12)
Osaka, Japan (Online), July 19-22, 2022
Favre (1965) proposed the Favre (density-weighted) filteringfas φ̃ = ρφ /ρ. Favre filtered SGS stress τi j and compressf

ible filtered SGS turbulent kinetic energy ρKsgs are defined
f
f
f
f
e fU
e f ), ρKsgs
as τi j = ρ(Ug= τkk /2.
iU j − UijCompressible filtered SGS turbulent kinetic energy transport equation (Cao et al., 2021a) has been derived as
f
f ef
f
f
f
f
(ρKsgs ),t + (ρKsgsUj ), j = Psgs − Dsgs + Πsgs + Tsgs ,
f

(12)

f

where Psgs is the filtered SGS production term, Dsgs the filteredf
SGS dissipation term, Πsgs the filtered SGS pressure dilationfterm, and the last term Tsgs the sum of filtered SGS diffusion
terms. The detailed right-hand-side terms in Eq.(12) can befound in Cao et al. (2021a). Be of scientific interest, we pay
special attention to the filtered SGS production term asf
f f
Psgs = − τi j Sei j ,(13)f
e f +Ue f ). We know that filtered SGS
with filtered Sei j = 1/2(U
i, j
j,i
f f
production term −τi j Sei j represents the inter-scale transfer as-

Figure 5. 3D and two-dimensional (2D) forward filtered SGS
f fturbulent kinetic energy transfer with positive −τi j Sei j at t =2.5 and 15. Corresponding 2D slice is located at z = 0 plane.

sociated with the interaction of the resolved and unresolved
scales (Piomelli et al., 1991).

Figure 6. 3D and 2D backward filtered SGS turbulent kinetic
f fenergy transfer with negative −τi j Sei j at t = 2.5 and 15. Corresponding 2D slice is located at z = 0 plane.

fFigure 4. Filtered SGS turbulent kinetic energy Ksgs at t =2.5, 5, 10 and 15.
Figure 4 qualitatively presents the contours of filteredf
SGS turbulent kinetic energy Ksgs . We clearly observe thatfKsgs is time-dependent during the unsteady evolution process.
Contours of forward and backward filtered SGS turbulent energy transfer are presented in Figure 5 and Figure 6. Forward and backward filtered SGS turbulent kinetic energy transfer is clearly observed. Filtered SGS turbulent kinetic energy backscatter illustrates the SGS turbulent kinetic energy
transfer from subgrid scales to resolved scales (Piomelli et al.,1991). Unfortunately, we know that backward energy transfer
cannot be modeled by both iLES and eLES with static coefficients.
Table 1 quantitatively D
presents
E that Ek is in the order offO(104 ) to O(102 ) of the Ksgs during the evolution process. Dominant physical dissipate rate ε1 is approximately 20
times
largerD
E than the ensemble filtered SGS production ratef ef
−τi j Si j . When constructing eLES models, the modeling
SGS dissipation of eLES models is expected to be pointwise
f fequivalent to the filtered SGS production term −τi j Sei j . As
presented in Table 1, the ensemble filtered SGS production

Table 1. Ensemble turbulent kineticDenergy
E Ek , ensemble filf
tered SGS turbulent kinetic energy Ksgs , ensemble dominant dissipation
D
E rate ε1 and ensemble filtered SGS production
f fterm −τi j Sei j at t = 2.5, 5, 10 and 15. Ek and ε1 are obtained
from case T G3 in Cao et al. (2022).
Time

Ek

D
E
f
Ksgs

ε1

D
E
f f
−τi j Sei j

2.5

0.123

2.6 × 10−5

8.8 × 10−4

1.3 × 10−5

5

0.118

1.2 × 10−4

4.1 × 10−3

1.3 × 10−4

10

0.074

3.3 × 10−4

1.1 × 10−2

5.1 × 10−4

15

0.036

1.3 × 10−4

4.5 × 10−3

1.3 × 10−4

term is far smaller than the dominant ensemble physical dissipation. We know eLES always provides the positive modeling SGS dissipation without considering the numerical dissipation. However, in practical simulation, the built-in numer-4

12th International Symposium on Turbulence and Shear Flow Phenomena (TSFP12)
Osaka, Japan (Online), July 19-22, 2022
ε(Ek ). Again, the discrete error in post-processing procedure
is assumed to be neglected. ε phy denotes the ensemble resolved
physical dissipation rate as Eq.(11).

ical dissipation always get involved with the process of total
SGS dissipation. On unresolved grids, the magnitude of builtin numerical dissipation rate may be larger than the filteredf f
SGS production rate −τi j Sei j , thus the explicit modeling SGS
dissipation is not required under such circumstance. Up to thispoint, we have limited our discussion in priori analysis, andthe posteriori performances will be evaluated thereafter.

0.015
DNS

Quantitative Analysis of SGS Dissipation

0.01

ε(Ek)

Before trusting eLES models without thinking, above priori analysis has reminded us to evaluate the built-in numericaldissipation quantitatively. We follow the procedures for iLES
analysis in incompressible LES (Moser et al., 2021). For compressible LES, without considering the numerical discrete error, the momentum equation reads∂ ρ Ũi ∂ ρ ŨiŨ j∂ p ∂ σ i j ∂ τi j
+
=−
+
−
,
∂t
∂xj
∂ xi
∂ xi
∂xj

0.005

(14)
0
0

eiU
e j ), and σi j = 2µS∗ with
with SGS stress τi j = ρ(UgiU j − Uij∗
Si j defined as Eq.(11). Conceptually, we can define the discrete derivative operator as δit the temporal discrete derivative operator, and δis the spatial discrete derivative operator.
Considering the numerical discrete error, the modified equation (numerical resolved equation) of Eq.(14) readsδ s τimodδ t ρ Ũi δ s ρ ŨiŨ j
δ s p δ sσ i j
j
+
=
−
+
−
(
δtt
δ sx jδ s xiδ sx jδ sx j+δ s τinumjδ sx j5

Figure 7.
ε(Ek ).

20

0.002
DNS
iLES

(15)

0.0015S-ModelV-Model0.001

ε0.000500

Figure 8.
εsgs .

5

t

10

15

Time history of ensemble total SGS dissipation rate

Table 2. Ensemble total SGS dissipation rate εsgs with iLES
and eLES at t = 2.5, 5, 10 and 15.

p

mod
num
εsgs = εsgs
+ εsgs,(16)

Eq.(16) shows that the total SGS dissipative behavior is demod
termined by both explicit modeling SGS dissipation rate εsgs
numand built-in numerical SGS dissipation rate εsgs . We noticethat there is no explicit modeling SGS dissipation in iLES.
While, the built-in numerical dissipation is usually ignored in
eLES. In terms of the free decaying TGV, ensemble total dissipation rate ε(Ek ) of turbulent kinetic energy and ensemble
total SGS dissipation rate εsgs can be computed by
dEk
,
dt
p
εsgs ≡ εsgs = ε(Ek ) − ε phy ,15Time history of ensemble total dissipation rate

),

mod
εsgs
= −τimodj S̃i j ,num
εsgs = −τinumj S̃i j .

10twhere τimodis the explicit modeling SGS stress in wellj
developed eLES (Nicoud & Ducros, 1999; Vreman, 2004;
Sagaut, 2006; Garnier et al., 2009), while all numerical disphenomenologicrete errors are effectively grouped into τinumjcally. Numerical discrete error is very hard to analyze term byterm in finite-volume framework, since the errors result fromthe spatial reconstruction procedure, time discretization procedure, the design of flux, and the averaging process when updating conservative variables. Another bad news is that the numerical error is grid resolution-dependent, numerical schemedependent and turbulence type-dependent, thus determining its
quantitative expression seems mission impossible.
In view of this dilemma, we adopt to deal with the total SGS dissipation, which is a classical route to understandthe eLES models in LES community. In practical simulations,corresponding to Eq.(15), the pointwise total SGS dissipationprate εsgs contains two parts

ε(Ek ) = −iLESS-ModelV-Model

Time

iLES

S-ModelV-Model

2.5

1.5 × 10−5

2.8 × 10−5

2.6 × 10−55

1.4 × 10−4

3.5 × 10−4

3.2 × 10−4

10

6.7 × 10−4

1.4 × 10−3

1.3 × 10−3

15

1.8 × 10−5

2.5 × 10−4

1.8 × 10−4

Figure 7 shows that ε(Ek ) is well predicted at all grid levels by iLES and eLES, indicating that the resolved physicaldissipation and total SGS dissipation work together consistently in the calculation. Thence, the numerical dissipationin iLES indeed can be treated as the built-in SGS dissipation.
As presented in Figure 1 - 3, the built-in numerical SGS dissipation accounts for the reasonable performance of iLES incurrent low-Reynolds number compressible turbulence. Figure 8 shows that the ensemble total SGS dissipation rate εsgs

(17)with second-order central difference method in computing

5

12th International Symposium on Turbulence and Shear Flow Phenomena (TSFP12)
Osaka, Japan (Online), July 19-22, 2022
based on eLES is approximate 2 times larger than that of iLES,
because the static eddy viscosity model always provides the
positive modeling SGS dissipation. Additionally, we observethat the posteriori ensemble total SGS dissipation rate εsgs inTable 2 is larger
than
E the priori ensemble filtered SGS producD
f ftion term −τi j Sei j in Table 1, except for the late decaying
stage at t = 15. We revisit the time history of ensemble enstrophy dissipation rate ε(ζ ) in Figure 2. We can conclude thatthe additional explicit models in eLES pollutes the resolved
turbulent structures, i.e., blurs the resolution of resolved vorticity. To strengthen above conclusion, effects of static eddy
viscosity models on density, momentum and energy transport
will be explored, beyond grouping the explicit modeling effects into the effective ensemble modeling SGS dissipation.

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CONCLUSION AND DISCUSSION
To address the better performance of iLES, current research conducts the quantitative comparisons between iLES
and eLES, especially focusing on the built-in numerical dissipation and explicit modeling SGS dissipation. It is concluded
that the numerical dissipation in iLES can act as the intrinsic
SGS dissipation, and explicit modeling SGS dissipation is not
required in low-Reynolds number turbulence. In addition, theadditional explicit SGS models in eLES even pollute the resolved turbulent structures. An important consequence is thatthe improvement of the reliability of LES results requires workon both the numerical methods and the subgrid models.
In following studies, more detailed analysis in Eq.(15) deserves to be explored. iLES for complex turbulent flows is still
under debate, current quantitative analysis of total SGS dissipation gives the specific hints on this issue. More importantly,
the LES community should pay special attention to the utilization of static explicit eddy viscosity models, instead of regarding the eLES as holy grail in LES on unresolved grids.

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