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

NEAR-AUTONOMOUS LARGE EDDY SIMULATIONS OF TURBULENCEBASED ON INTERSCALE ENERGY TRANSFER AMONG RESOLVEDSCALES

J.A. Domaradzki
Department of Aerospace and Mechanical Engineering
University of Southern California
Los Angeles, CA 90089-1191, USA
jad@usc.edu

ABSTRACT

1

Spectral eddy viscosity models proposed by Kraichnan(1976) and by Chollet & Lesieur (1981) account explicitly forphysical properties of interscale interactions deduced from analytical theories of turbulence. In the present work we showthat the main features of the subgrid-scale (SGS) energy transfer can also be obtained directly from the evolving large eddy
simulations (LESs), allowing a near autonomous LES where
the SGS model is provided by the analysis of the LES fields
at each time step in simulations. Specifically, the method
computes the SGS energy transfer among resolved scales andits wave number distribution from the evolving LES velocity
fields. This information is supplemented by well establishedasymptotic properties of the energy flux in the inertial range:
its locality scaling exponent 4/3 and the value of the spectral eddy viscosity in the limit k → 0 . The resulting SGS
energy transfer, when cast in the form of a spectral eddy viscosity, allows self-contained simulations without use of extraneous SGS models. The method is tested in LES of isotropic
turbulence at high Reynolds number where the inertial range
dynamics is expected and for lower Reynolds number decaying turbulence under conditions of the classical experiments ofComte-Bellot & Corrsin (1971).

f LES(k|(kc)

0.8

0.6

0.4

0.2

0
0

0.2

0.4

0.6

0.8

1

k/kc

Figure 1. Spectral eddy viscosity shape functions. Solid linewith symbols ◦: analytical theory of turbulence (EDQNM);
horizontal solid line: asymptotic plateau value from the
EDQNM theory; shape functions computed from LES datafor cases ”ceddy” (solid line), ”pconst” (broken line), ”pvar”(broken-dotted line).

proximation (EDQNM), with both approaches leading to similar eddy viscosities. For the EDQNM formulation the authors
subsequently provided an analytical fit to the computed eddy
viscosity and used it as a SGS model in large eddy simulationsof Navier-Stokes equations (see, e.g., Lesieur (1997); Lesieur
et al. (2005)). In such an approach to SGS modeling the primary physical quantity is the energy transfer across a wave
number cutoff kc between the resolved scales (k < kc ) andthe subgrid-scales (k > kc ) and the eddy viscosity is a derived
quantity. This is different from a more common approach topostulate first a functional form of the eddy viscosity and thenobtain values of model constants that best match known theoretical and experimental results for a given turbulent flow. The
former approach can be advantageous if information about the
SGS energy transfer is directly available for a given flow. In
previous work by Domaradzki (2021a,b) it was shown that the
subgrid-scale energy transfer among resolved scales in large
eddy simulations (LES) as well as its wave number distribution can be obtained directly from the evolving LES velocity
fields. This information, supplemented by known asymptotic
properties of energy flux in the inertial range, when cast in theform of a spectral eddy viscosity, allows SGS modeling without need for explicit expressions of the analytical theories or

INTRODUCTION
Analytical theories of isotropic turbulence as originatedby Kraichnan’s Direct Interaction Approximation (Kraichnan(1959)) provide closure expressions for the energy transfer
term T (k) in the spectral kinetic energy equation in terms of
the energy spectrum E(k). Modern, exhaustive review of analytical theories of turbulence and closures has been recently
provided by Zhou (2021). Kraichnan (1976) employed suchclosure expressions to compute the subgrid-scale (SGS) energy transfer TSGS (k|kc ) from a range of resolved scales k ≤ kc
caused by nonlinear interactions involving subgrid scales k >kc , where kc is a cutoff wavenumber of a sharp spectral filter.
The SGS energy transfer, when normalized by 2k2 E(k), gives
a spectral eddy viscosity νeddy (k|kc ). Such an eddy viscosity,
computed for the infinite inertial range spectrum E(k) ∼ k−5/3 ,has a relatively simple form with a constant plateau for wave
numbers k less than approximately 0.4kc and rising in a formof a cusp to the maximum value at k = kc (see Fig. 1). Kraichnan (1976) used a particular analytical theory, the Test Field
Model (TFM), while Chollet & Lesieur (1981) used anotherformulation, the Eddy Damped Quasi-Normal Markovian Ap-

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12th International Symposium on Turbulence and Shear Flow Phenomena (TSFP12)
Osaka, Japan, July 19–22, 2022
written for cutoff 12 kc , over all wave numbers less than thiscutoff. However, as shown in Domaradzki (2021a) it can also
be computed using standard LES formulas for SGS dissipationin the physical space based on the SGS tensor and the resolved
rate-of-strain tensor.
A wave number distribution of the resolved SGS energyres (k| 1 k ) can be computed from LES data durtransfer TSGS
2 c
ing an actual run and cast in the form of the k-dependent
res (k| 1 k ) =eddy viscosity (4), which is normalized to unity fLES
2 c
1
1
1
res
res
νeddy (k| 2 kc )/νeddy ( 2 kc | 2 kc ). That last quantity, the eddy viscosity shape function, is rescaled from the test cutoff (1/2)kc
to LES cutoff kc , using the similarity variable 0 ≤ k/kcuto f f ≤1. Such computed eddy viscosities for several LES cases areshown in Fig. 1. Finally, the values of the eddy viscosity at
low k are modified to make them consistent with the asymptotic value provided by the analytical theories for the inertial
range at k/kc → 0. Based on results from the EDQNM theory the plateau asymptotic value p was determined as 0.37 of
the peak value at the cusp, i.e., p = 0.37 for the eddy viscosity shape function normalized to unity at kc (horizontal line in
Fig. 1). The final shape function fLES (k|kc ) comprise a constant plateau up to an intersection with a cusp of the resolvedres (k|k ), followed by the unmodified cusp
shape function fLEScpart from the intersection point to k = kc .
The complete procedure is implemented in several steps.
At each time step in simulations the eddy viscosity is

any other classical SGS models. Effectively, the procedure allows self-contained LES without use of extraneous SGS models, or equivalently, at each time step the model is obtained
from a simulated field itself and two asymptotic properties ofthe energy flux in the inertial range.

DESCRIPTION OF THE METHOD
The spectral LES energy equation for scales k ≤ kc is obtained by defining first energy transfer T < (k|kc ) among resolved modes, where the notation signifies that only modes
satisfying the inequality k ≤ kc , i.e. scales that are fully known
in LES with the cutoff kc , are retained in computing T < (k|kc ).
The complete spectral energy equation can then be rewritten
for LES scales k ≤ kc as follows∂ <E (k|kc ) = T < (k|kc )+TSGS (k|kc )−2νk2 E < (k|kc ), k ≤ kc ,∂t(1)where the SGS energy transfer term is

TSGS (k|kc ) = T (k) − T < (k|kc ), k ≤ kc ,(2)

where T (k) is the full nonlinear energy transfer computed using all modes, resolved and subgrid scale.
Following Kraichnan (1976), the SGS spectral energy
equation can be formally rewritten asνeddy (k|kc ) = Cm fLES (k|kc ),∂ <E (k|kc ) = T < (k|kc )−2νeddy (k|kc )k2 E < (k|kc )−2νk2 E < (k|kc )∂t(3)
where the SGS energy transfer is expressed in the same functional form as the molecular dissipation term by introducing
the theoretical, effective eddy viscosity

where Cm is a model constant and fLES (k|kc ) is a shape function, determined as described above. The model constant Cm is
computed using known total SGS energy transfer as an integral
constraint

Z kc

TSGS (kc ) =νeddy (k|kc ) = −TSGS (k|kc ).
2k2 E < (k|kc )

(4)

1
1
T res ( kc ),1 − b SGS 2

0

dkTSGS (k|kc ) = −Z kc
0

dk νeddy (k|kc )2k2 E(k),(7)

which gives

As stressed before, the eddy viscosity is obtained from the primary physical quantity which is the energy transfer across awave number cutoff kc between resolved scales (k < kc ) andsubgrid-scales (k > kc ).
It was shown by Domaradzki (2021a) that the task ofmodeling TSGS (k|kc ) can be productively split into finding thetotal SGS transfer/dissipation, integrated over 0 < k < kc and,separately, its distribution in wave numbers k. The total SGS
energy transfer across the cutoff kc is determined by the formula derived in Domaradzki (2021a) using the Germano identity (Germano et al. (1991)),

TSGS (kc ) =(6)

Cm = R kc

0−TSGS (kc )fLES (k|kc )2k2 E(k)dk.

(8)

In LES runs the eddy viscosity (6) is determined at each time
step in simulations and used in the eddy viscosity term added
to the Navier-Stokes spectral solver as a SGS modeling term.
In (8) TSGS (kc ) is expressed in terms of SGS transfer
res ( 1 k ), Eq. (5), computed at eachamong resolved scales TSGS
2 ctime step in LES with the spectral eddy viscosity given by (6).
Similarly, the shape function fLES (k|kc ) is computed at eachres (k| 1 k ),time step from the resolved SGS energy transfer TSGS
2 ci.e., both factors in the formula (6) are computed from information available in LES. In effect, the SGS model is not prescribed but obtained from the resolved SGS energy transfer
res (k| 1 k ) in a given LES and two well-established properTSGS
2 cties of the energy flux for the inertial range in the asymptotic
res (k| 1 k ) and E(k) inlimit k/kc → 0. Note also that since TSGS
2 cgeneral are time dependent, both factors in (6) are also functions of time, Cm (t) and fLES (k,t|kc ).
The purpose of this paper is to revisit derivations ofparameters b and p and to assess the performance of the(5)res ( 1 k ) is the energy transfer
where b is a constant and TSGS
2 ccomputed for the resolved LES modes (k < kc ) and the cutoff (1/2)kc . Using multiple theoretical and DNS results forinertial range turbulence the constant b was determined to beapproximately b ≈ 0.40. The total resolved SGS energy transres ( 1 k ) can be computed by integrating expression (2),fer TSGS
2 c

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12th International Symposium on Turbulence and Shear Flow Phenomena (TSFP12)
Osaka, Japan, July 19–22, 2022
present method, however, uses only the value of the eddy viscosity in the infrared limit k/kc → 0. In that limit the eddy
viscosity has a form

method for allowable choices of these parameters. In particular, derivation of constant b using the Germano identity
requires assumptions about a form of the shape function for
the final spectral eddy viscosity. We show that these assumptions are not necessary because the constant b can be deduced
from the scaling properties of the energy flux for k/kc → ∞,without reference to the Germano identity. Similarly, we showthat the plateau value p does not need to be set to a constantvalue but can be obtained in course of simulations solely fromthe asymptotic properties of the spectral eddy viscosity fork/kc → 0.

νeddy (0|kc ) =The constant b was computed in Domaradzki (2021a) using the Germano identity. However, its value can be obtainedusing solely asymptotic properties of the energy flux in the ultraviolet limit k/k0 → ∞. Kraichnan (1959) introduced a scale
locality function Πuv (k0 |k), k > k0 that measures the amount ofenergy flux across k0 caused by interactions involving at least
one wave number mode with a wave number greater than k.
Analytical theories of turbulence consistent with the Kolmogoroff inertial range (Kraichnan (1971a,b)) produce the scaling
result(9)where K is a constant. This result was reviewed and reinforcedby theoretical analyses of Navier-Stokes solutions by Eyink(2005) and numerical results of Zhou (1993) and Domaradzki
et al. (2009). The theoretical analysis predicts the scaling exponent but not the constant K. However, if modes from theforcing band and the band adjacent to the mesh cutoff are removed from the analysis of DNS data one observes that K ≈ 1
in the entire range of wavenumbers, down to k0 /k = 1, as longas k0 is firmly in the inertial range (Domaradzki et al. (2009)).
Assuming K = 1 and k0 = ak, a < 1, Eq. (9) allows to split theenergy flux across k0 as follows

0

4/3

Π(k ) = a

0

res

0

Π(k ) + Π (k ),TSGS (kc ) = −2ν̄eddy (kc )1
T res (akc ),1 − a4/3 SGS

Z kc

k2 E(k)dk.

(13)

0

The ratio of asymptotic eddy viscosity from the EDQNM theory and the averaged eddy viscosity with the same energy fluxis (Lesieur (1997))

νeddy (0|kc ) 0.441= 0.6615.
=ν̄eddy (kc )
(2/3)

(14)

Note that the averaged eddy viscosity is not dependent
on any specific analytical theory so the equation (14) allows
to determine the plateau value of the eddy viscosity entirely
from the integral relation (13) rather than from the ratio ofpoint values. Parameter p refers to eddy viscosity normalized to unity at k/kc = 1, with the normalization factor being
the maximum value of the resolved eddy viscosity at the cusp
res = ν res ( 1 k | 1 k ), giving for pνmax
eddy 2 c 2 c(10)where Πres (k0 ) is contribution to the energy flux across k0 due
to interactions with all modes below wave number k. If k
is a cutoff wave number in LES, k = kc , the second term isthe energy flux across k0 < kc that is resolved using only LES
res (k0 ).
data and is equal to the resolved SGS energy transfer TSGS
Also, in the inertial range the total flux is independent of the
wave number and equal to the total SGS energy transfer, i.e.,Π(k0 ) = Π(kc ) = TSGS (kc ). Using this observation Eq. (10)leads to the relation

TSGS (kc ) =∂ E(q,t)θ0qq 5E(q,t) + q
dq, (12)
∂q
kc

Z ∞

where θ0qq is a triad interaction time in that limit, where q > kc
(Kraichnan (1976); Lesieur (1997)). Despite differences indefinitions of θ for different analytical theories they all leadto essentially same value of νeddy (0|kc ) for the inertial range
spectrum E(q,t) ∼ q−5/3 . In Domaradzki (2021b) νeddy (0|kc )was used to constrain the plateau of the eddy viscosity computed from LES data to p = 0.37 of the peak value at the LES
cutoff kc , consistent with the prediction of the EDQNM theory. This is a pointwise constraint in a sense that it is based on
the ratio of the eddy viscosity at two points k = 0 and k = kc .
Since the cusp value at kc results from local interactions ofmodes with wave numbers in the vicinity of kc the parameter p
is not strictly dependent only on the asymptotic properties fork/kc → 0. To retain a dependence of the plateau level only on
asymptotic values at k/kc → 0 we have explored replacing thepointwise constraint by an integral constraint, based on knowledge of the total SGS energy transfer (11). The total transfer
TSGS (kc ) allows to define the average constant eddy viscosity
for LES with cutoff kc , ν̄eddy (kc ), through the relation

THE USE OF ASYMPTOTIC PROPERTIES OF
ENERGY FLUX

Πuv (k0 |k) = K(k0 /k)4/3 , k  k0 .

1
15

p = fLES (0|kc ) = 0.6615

ν̄eddy (kc )
,
res
νmax

(15)

which is a time-dependent quantity because both ν̄eddy andres are functions of time.
νmax
(11)

RESULTS
which is equation (5) with b = a4/3 . Specifically, for a = 1/2
the value of b ≈ 0.4.
Analytical theories of turbulence predict dependence of aspectral eddy viscosity on a wave number, νeddy (k|kc ). The

To test these concepts and the proposed modification of
the method we have performed several forced large eddy simulations initialized with k−5/3 energy spectrum. Details of
the numerical method and parameters in the simulations are312th International Symposium on Turbulence and Shear Flow Phenomena (TSFP12)
Osaka, Japan, July 19–22, 2022

10

0

0.5

0.4
10

-1

10

-2

E

E(k)

0.3

0.2

0.1

10

-1

10

0

0

0

k/kc

5

10

15

20

25

30

20

25

30

k
0.25

2

0.2

1.5

0.15

E

CK

2.5

1

0.1

0.5

0.05

0
10

-1

10

0

0

0

k/kc

5

10

15

k

Figure 2. Results for forced LES. Solid line: k−independent
eddy viscosity, case ”ceddy”; broken line: case ”pconst” with
p = 0.37; broken-dotted line: case ”pvar” with time dependent
p given by Eq. (15). Thin straight lines show, as appropriate,−5/3 slope, and a boundary of the forcing band at k = 3. For
compensated spectra in the bottom figure horizontal lines markexpected range of values for the Kolmogoroff constant.

Figure 3. Time evolution of energy spectra in LES for a decaying Comte-Bellot and Corrsin case (symbols show experimental data). Top figure for time interval U0 t/M = [42, 98];
run continued in the bottom figure for time interval U0 t/M =[98, 171].

constant and is determined by a ratio of pointwise values oftheoretical eddy viscosity at k = 0 and k = kc . In case ”pvar”
the plateau value varies in time but depends only on theoreticaleddy viscosity at k = 0 through formula (15), as a fraction of
the averaged, k−independent eddy viscosity.
In Fig. 2 we plot energy spectra obtained using all three
implementations. In all cases the spectral energy slopes at latetimes are in an an excellent agreement with −5/3 exponent,
though the case ”ceddy” exhibits slight departure from thatform in the vicinity of kc . The compensated spectra in a formof a k-dependent Kolmogoroff function

provided in Domaradzki (2021a,b) for corresponding LES in
those papers. Specifically, Reynolds numbers Reλ exceed 104 ,indicating that the inertial range theory should apply. LESs
were run with a resolution of 643 modes for 3000 time steps,
corresponding to about 15 large eddy turnover times, and results for plotting were averaged over last 1000 time steps.
Three implementations of the method were employed. In allcases the parameter b = 0.4. Case ”ceddy” corresponds to prescribed shape function independent of k, i.e., f0 (k|kc ) = 1 (seeDomaradzki (2021a)). Effectively it is a constant in k eddy
viscosity enforcing the integral relation (7). The case ”pconst”
implements the method with fixed value of parameter p = 0.37
(see Domaradzki (2021b)). Finally, the case ”pvar” implements the method with value of p varying in time according to
Eq. (15). One can think of these three cases as a progression
in relaxing constraints on the model. Case ”ceddy” prescribes
probably the simplest form of the spectral eddy viscosity, similar to constant molecular viscosity ν. However, that eddy viscosity is time dependent, with the dependence imposed by enforcing the total SGS energy constraint (7) (or (13)). In cases”pconst” and ”pvar” the model shape function fLES is not fullyprescribed but partially recovered from the the eddy viscosity
obtained from the LES fields. Specifically, the eddy viscosity from LES data in the low wavenumber range is replaced
by a constant in k plateau up to point where the plateau intersects the rising cusp in the eddy viscosity curve (Fig. 1). Thatpart of the unmodified cusp is responsible for about 50% of
the total SGS transfer. In case ”pconst” the plateau value p is

CK (k) =E(k),ε 2/3 k−5/3(16)

fall within the expected range 1.4 − 2.1 outside the forcing
wave numbers. However, as the cutoff kc is approached thecase ”ceddy” shows a steep increase in CK . The behaviourof spectra for this case in the vicinity of kc is consistent withinsufficient SGS dissipation in that range. The presence of acusp at kc in the eddy viscosity for two other cases (see Fig. 1)
increases SGS dissipation in the vicinity of kc , leading to better
agreement with the inertial range form. It is quite clear, however, that all three approaches produce overall similar and acceptable spectral results. This implies that the total SGS transfer constraint (7), being the same for all cases, must play the
primary role, while the eddy viscosity wave number distribution plays a secondary role.

412th International Symposium on Turbulence and Shear Flow Phenomena (TSFP12)
Osaka, Japan, July 19–22, 2022
In practice, however, enforcing constant value of p (case”pconst”) was found to result in most robust LES for several
other cases of isotropic turbulence, forced and decaying, atvery high as well as at low Reynolds numbers (Domaradzki(2021b)). As an example we show in Fig. 3 results of LES
with p = 0.37 for the classical experimental dataset of ComteBellot & Corrsin (1971) (for more details see Domaradzki(2021b)). Nevertheless, relaxing extraneous information input by making the parameter p variable through Eq. (15) provides valuable guidance regarding the goal of designing an autonomous LES. We showed that a near autonomous LES procedure is possible, in a sense that beyond actual LES data only
two asymptotic results from theory of turbulence are needed:ultraviolet scaling of the energy flux for k/kc → ∞ and infrared
limit of spectral eddy viscosity for k/kc → 0. It is difficult toanticipate that further limiting this information input could result in acceptable LES methods.

of the spectral eddy viscosity on k can be extracted from evolving LES fields, thus moving us in the direction of autonomous
LES. The original method, however, requires constants b andp, that are not needed in DNS of the same flows, and thus constitute extraneous information input. The purpose of this paper
was to revisit derivations of parameters b and p in order to minimize such extraneous information. In particular, derivation of
constant b using the Germano identity requires assumptions
about a form of the shape function for the final spectral eddy
viscosity. We showed that such assumptions can be entirely
avoided because the constant b can be deduced solely from thescaling properties of the energy flux for k/kc → ∞, without
reference to the Germano identity. Similarly, we showed thatthe plateau value p does not need to be set to a constant value
but can be obtained in course of simulations solely from theasymptotic properties of the spectral eddy viscosity in the limit
k/kc → 0. It may be that such asymptotics constitute minimum
extraneous information required for well behaved LES of homogeneous, isotropic turbulence, especially for high Reynolds
numbers. If that is the case the proposed method can be considered as a near-autonomous LES in a sense that minimizing
further extraneous information input is unlikely to be possible.

CONCLUSIONSA previously proposed subgrid-scale modeling procedureof Domaradzki (2021a,b), based on the interscale energy transfer among resolved scales in LES, has been improved by increasing its reliance on information available directly fromknown LES fields and minimizing information from theories
of turbulence. The original procedure consists of two steps.
In the first step, the total unknown SGS transfer across a fixed
cutoff wave number kc is determined using the computed SGS
transfer within the resolved range for the cutoff akc , a < 1,with a set to 21 in this paper. The main parameter in this stepis a ratio b of the SGS transfer at test cutoff akc due to interactions with scales above the LES cutoff kc . In the second stepa distribution of SGS transfer among resolved wave numbers
k < kc is determined through an eddy viscosity shape function f (k|kc ), normalized to unity at the cutoff kc . The shapefunction is obtained directly from a k-dependent eddy viscosity computed using the actual, resolved SGS transfer at thetest cutoff akc . Such an eddy viscosity is qualitatively similar to the eddy viscosity computed from the analytical theories of turbulence, exhibiting a low wave number plateau anda cusp at akc . However, the low wave number plateau levelis too small because the resolved SGS transfer is lacking contributions from the nonlocal interactions with modes k > kc .
The missing interactions were accounted for by replacing thecomputed plateau by a k-independent value p, representing a
constant asymptotic eddy viscosity acting on large eddies bysmall eddies in the presence of a spectral gap (here between(1/2)kc and kc ). For such a hybrid shape function the cusp
is attributable primarily to local interactions and its values,
greater than the plateau value p, are responsible for about 50%of the total SGS dissipation. This local transfer is not modeled
but is a result of the actual interscale interactions operating at
a given time step in actual LES. This implementation of themethod was very successful in LES of forced, high Reynolds
number turbulence, and for decaying turbulence at both highand low Reynolds numbers.
The secondary motivations behind this research was toexplore what is minimum information input into LES as compared with DNS for the same physical problem. We postulated a target of fully autonomous LES, defined as a simulationthat produces the same quality results within resolved rangeof scales as DNS, and uses only the same information that isavailable to DNS. In the previous work we showed that information about the total SGS transfer and the partial dependence

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