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

EXPLICIT ALGEBRAIC REYNOLDS STRESS MODELLING IN
SCALE-RESOLVED SIMULATIONS OF TURBULENCE

Matteo Montecchia
Department of Engineering Mechanics
KTH, Royal Institute of Technology
Osquars Backe 18, SE-10044 Stockholm, Sweden
matteomo@mech.kth.se

Stefan Wallin
Department of Engineering Mechanics
KTH, Royal Institute of Technology
Osquars Backe 18, SE-10044 Stockholm, Sweden
stefanw@mech.kth.se

ABSTRACT

where many of the problems associated with eddy-viscosity
models are mitigated without major increase in computational
effort or degeneration of numerical stability.

New hybrid RANS/LES methods are proposed where explicit algebraic Reynolds stress modelling is introduced bothin the RANS and LES regions. Both in the form of an IDDES
method allowing for wall-modelled LES simulations and in
the form of an DDES methods where attached boundary layers
are shielded within RANS. The methods are extensions of the
SST-(I)DDES metods by Gritskevich, et al., Flow, turbulence
and combustion 88 (3), 431?449, 2012. In channel flow the log
region is well predicted for different Reynolds numbers similar as with the baseline SST-IDDES, but the EARSM-IDDES
give a more rapid transition to turbulence closer to the wall.
The model gives reasonable results on very coarse meshes,basically similar with the baseline SST-IDDES, for periodic
hill flow at Re = 10.000 and 37.000 Also for the 3D Stanford
diffuser, the EARSM-IDDES gives good results, here superior to the SST-IDDES, which has problems in fully resolving the turbulence in the inlet channel. The model in DDES
mode is tested on the developing shear layer. The so-called
grey area problem is substantially mitigated compared with the
SST-DDES resulting in a more physically correct and fully developed resolved turbulence.

Application of EARSM for scale-resolved simulations
cannot be strictly based on the weak-equilibrium assumption,
which is not well established on the resolved time-dependent
scales. Equilibrium is only reached in an averaged sense,
which to some degree can motivate the use of EARSM for subgrid modelling, though. Nevertheless, SGS models for LES
containing non-linear terms inspired by EARSM have beenshown to give improvements where the SGS anisotropies areof importance, e.g. in the near-wall regions of wall-resolved
LES, see Marstorp et al. (2009), Montecchia et al. (2017) andMontecchia et al. (2018).
Scale-resolved simulations with unified modelling of theunresolved turbulence with variable resolution ranging fromRANS to LES, or even DNS if sufficiently resolved. Such ”hybrid RANS/LES” modelling is an active research field since
the work of Spalart (2009). One of the fundamental problems with such modelling is the transition regions with rapidchanges of resolution. Formulation of the filter commutation
terms and corresponding energy-conserving modelling is proposed by Girimaji & Wallin (2013), Kamble et al. (2022).

INTRODUCTION
Standard eddy-viscosity models (EVMs) are widely used
for RANS models as well as for modelling the unresolved
scales in scale-resolved simulations (e.g. LES) due to theirsimplicity and stability. These models have, however, a number of well known problems associated with the linear relation
between the velocity strain rate and the turbulence stresses resulting in inadequate representation of the stress anisotropy
and turbulence production insensitive of local rotation, e.g.
swirl, and mean-flow curvature. Physical realisability is easily violated due to the fact that the modelled Reynolds stresscomponents are not restricted by the turbulence kinetic energy.
Explicit algebraic Reynolds stress models (EARSM) formally derived from applying the weak-equilibrium assumption
on the full Reynolds stress transport models are now established for RANS modelling, e.g. Wallin & Johansson (2000),

Introduction of EARSM for hybrid RANS/LES models in
the RANS region can be motivated by the need for better representation of attached boundary layers and separation onset
from smooth surfaces, which to a large extent is driven by the
RANS modelling, see e.g. Jaffrézic & Breuer (2008). For consistency between RANS and LES modelling and for utilisingpossible benefits in the LES region, we will introduce a unified
EARSM in both RANS and LES regions as well as in the transition in between. This will be further elaborated in the paper
for flows in simple and more complex geometries. The extension of the modelling accounting for the commutation terms in
the transition regions will be left for later studies.

1

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

employed from a dynamic procedure, in the same spirit ofGermano et al. (1991) (Marstorp et al. (2009), Montecchia(2019)). A possible further development could be to adopt the
dynamic procedure also in the LES regions of the (I)DDES.
The k − ω model equations are adopted as defined in
Gritskevich et al. (2012), with the important difference thatthe turbulence production is not modelled through the eddyviscosity assumption, but defined without further modellingthrough the EARSM as

Explicit Algebraic Reynolds stress model (EARSM) formulation for the Reynolds stress tensor can be written as:2
u0i u0j = kδi j + β1 kτ ∗ Si j + β4 kτ ∗2 (Sik Ωk j − Ωik Sk j ) + hot 0 s.
3(1)
where Si j and Ωi j are the dimensional resolved symmetricstrain and antisymmetric rotations rate tensors, respectively,
such as ∇U = S+Ω and τ ∗ the modelled turbulence time scale.
The formulation and derivation follow the work by Wallin &Johansson (2000), but here without the normalisation.
The β ’s depend on invariants of Si j and Ωi j resulting
from the formal solution of the differential Reynolds stressmodel equations in the weak-equilibrium limit where variationof Reynolds stress anisotropy is neglected. The second term on
the right-hand-side of (1) is an eddy-viscosity term while thethird non-linear term aims to improve the modelling of stressanisotropy. High-order terms account for 3-D effects.
The second, eddy-viscosity, term will have an effective
viscosity1νTeff = − β1 kτ ∗2

Pk = −u0i u0j

∂Ui
.
∂xj

(3)

The coefficients are all following Wallin & Johansson(2000), except the recalibration of A1 for a precise consistency
with the Menter BSL/SST model. Then, c1,RANS = 1.8 and
A1 = 1.245. The Rotta constant c1 in LES requires further
considerations involving local flow and Re dependencies, seeMarstorp et al. (2009). Moreover, we are utilising that the SGS
k is estimated from the transport equation, resulting in

c1,LES = c01(2)

with a non-constant effective Cµ = −β1 /2, where β1 is given
from the EARSM solution of Wallin & Johansson (2000).
The adaptation of expression (1) as a sub-grid scale (SGS)model for scale-resolved simulations is not strictly motivatedfrom the weak-equilibrium. Hence, it can be considered as atruncation of the full EARSM retaining terms up to the third
term in (1) and does not consider higher order terms. The second, eddy-viscosity, term is dissipative while the third term isbasically redistributive and non-dissipative contributing to theanisotropy.
A further step in the modelling explored in this study isthe adoption of the EARSM relation (1) for hybrid RANS-LES
computations. For this purpose we will extend the existing
SST k − ω IDDES and DDES methodologies of Gritskevich
et al. (2012), by replacing the eddy-viscosity relation with the
EARSM formulation. In the RANS region the full 3D formulation including high order terms will be used, while in the
LES region only terms up to second order are kept.
The blending between LES and RANS stress is made by
the use of the fed and fd blending functions, respectively, as
defined in Gritskevich et al. (2012) without further adaption
or tuning. The blending functions are functions of an eddy
viscosity and here the SST viscosity is used, fed (νTSST ) andfd (νTSST ), for closest consistency with the original definition.
Consistently with the original model, the DDES mode isused for detached eddy simulations where the whole boundary layer is shielded to be purely within the RANS zone. The
IDDES mode is an extension where the RANS-LES interface
will be located well inside of the boundary layer if the grid
resolution is fine enough to allow wall-modelled LES.
When EARSM is adopted in RANS modelling, the estimation of kinetic energy and turbulence time scale is performed by employing model transport equations for the turbulent kinetic energy k and a length-scale determining quantity, e.g. ε or ω, the latter applied in this work. For consistency with the SST k − ω (I)DDES modelling of Gritskevich
et al. (2012), we are using the model transport equations for kand ω for determining the resolved scales in both RANS andLES modes. This is contrary of our earlier work on EARSM
for WRLES, where the subgrid turbulence kinetic energy was

q

c03

k1.25
.
(2Cs ∆S)2.5(4)

with c01 = 4.2, c03 = 2.4, Cs = 0.1 and A1 = 1.65.
The time scale τ = k/ε is further approximated by
Marstorp et al. (2009). By replacing the dynamic procedure
with the transport equation for k we deduceτLES = c03 1.5C1.5√k,
2Cs ∆S2(5)

where C = 0.5 is the Kolmogorov constant.
All coefficients and model details that differ between
RANS and LES are blended by fed and fd blending functions before the EARSM solution is obtained for the mixed
anisotropies. Also, N and β3,6,9 are blended.

RESULTSThe Explicit Algebraic subgrid-stress model (EAM) was
previously extensively validated and tested for wall-resolved
large-eddy simulations (WRLES) of turbulent channel flow ata range of Reynolds numbers up to Reτ = 5200. Both a specificsolver with spectral accuracy and the general-purpose OpenFOAM solver have been used. The resolution requirement forwall-resolved LES scales with the near-wall viscous scales andis almost as high as for DNS. We found that EARSM as a
SGS model could relax the resolution requirements comparedwith the dynamic Smagorinsky model. One example of mean
velocity profiles are shown in figures 1 at different Reynolds
numbers. As throughly described in (Montecchia et al. (2017),
Montecchia et al. (2019)) the EAM has also improved the estimation of Reynolds stresses, in particular of the diagonal components. These improvements have been consistently observed
for all Reynolds numbers and for solvers with high numericalaccuracy as well as standard general purpose CFD solvers.
IDDES RESULTS
The development of the EARSM-IDDES has been implemented into OpenFOAM and was validated and tested for turbulent channel flows at Reτ = 934 and 5200. The number of2

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

25

30

20

25

< U >+< U >+

grid points in the stream- and spanwise directions are the same
60 × 60 for both Re resulting in that the grid resolutions inwall units are (∆x+ , ∆z+ ) = (98, 49) and (544, 272), respectively for the two Reynolds numbers, resulting in less than ahalf million grid points. The wall-normal resolution is set suchthat y+ ∼ 1 at the first cell.
Mean velocity profiles, shown in figure 2 are reasonably computed by EARSM-IDDES and the k − ω SST IDDES for both Reynolds numbers. The log-layer mismatch isquite limited for both models. However, the transition fromthe near-wall RANS to LES is located closer to the wall andis more rapid for the EARSM-IDDES, resulting in a considerably larger amount of resolved turbulence near the wall seenin the figure. Although, the more rapid transition for the
EARSM-IDDES induces slightly less accurate log profiles but
might be of importance in mitigating the so called grey-area
problems in more complex cases.
The EARSM-IDDES was further tested by the computation of the periodic hill flow Mellen et al. (2000) at bulk
Reynolds numbers of Reb = 10595 and Reb = 37000. The
wall-normal resolution is y+ ∼ 1 at the first cell, while thenumber of grid points along the streamwise and spanwise directions are about 70 × 46. The grids for both cases are, hence,very coarse with less than half million grid points to be able
to observe influence of modelling. The flow as well as the
Reynolds number dependency is reasonably captured by the
new model on these coarse meshes, see figure 3. SST-IDDES
performs slightly better in the separated region and capturing
the peak at the hill crest, while EARSM-IDDES captures thereattachment point slightly better.
The 3-D diffuser case by Cherry et al. (2008), also simulated by DNS by Ohlsson et al. (2010) at the lower Re, isa more difficult case to compute where the corner vortices
in the inflow duct play a major role in the development of
the asymmetrical separation. Two different Reynolds numbers
were computed, at Reh = 10000 and Reh = 30000. Two different grids are used, a ”coarse” one of 1.4 and a ”fine” oneof 2.8 million grid points. Pressure coefficient as a functionof the streamwise direction at Reh = 10000 and Reh = 30000
is shown in figure 5. The coarse resolution could not well
capture the flow and will not be further considered. For thefine resolution, both models provide a similar prediction of
the pressure coefficient at the lower Reynolds number. At
the higher Reynolds number, EARSM-IDDES gives a significant improved C p compared to SST-IDDES. No DNS is yet

15
10
5

15
10
5

0

10

0

2

1

10

10

y+

1.0

1.0

0.8

0.8

0.6

+
Ruv

+
Ruv

0

20

0.4

100

0.2
0.0
0.0

0.6

0.8

1.0

0.2

0.4

y/δ
1.50

1.50

1.25

1.25

0.8

1.0

0.6

0.8

1.0

1.00

f˜d

1.00

f˜d

0.6

y/δ

0.75

0.75

0.50

0.50

0.25

0.25

0.00
0.0

0.2

0.4

0.6

0.8

0.00
0.0

1.0

0.2

0.4

y/δ

y/δ

Figure 2. Channel flow simulation using IDDES showing
inner-scaled mean velocity profile, Reynolds shear stress andthe blending function fed for Reτ ≈ 950 (left) and Reτ ≈ 5200
: k − ω SST-IDDES,: EARSM-IDDES,(right).
: DNS. The turbulence shear stress is divided intomodelled (− − −) and resolved (− · −) contributions.

0.03

0.004
0.002
0.000

Cf

0.02

−0.002
0

1

2

3

4

5

6

7

8

−0.004

0.01

0.00
0

2

4

6

8

x/h
0.004

0.020

30

0.002

cLES-DSM
vcLES-DSM
cLES-EAM
vcLES-EAM
DNS

0.000

0.015

−0.002

Cf

20

103

0.6

0.2
0.4

102

y+

0.4

0.0
0.0

0.2

101

0.010 0

1

2

3

4

5

6

7

8

−0.004

0.005

10
0.000
0

0
10

2

4

6

8

x/h

0

Figure 3. Periodic hill simulation using IDDES showing
lower wall skin friction for Reb = 10595 (top) and Reb =37000 (bottom).
: k − ω SST-IDDES,: EARSMIDDES,: DNS.

Figure 1. Inner-scaled mean velocity profile computed fromWRLES at Reτ ≈ 5200, with two different resolutions. vcLES:(∆x+ , ∆z+ ) = (255, 102), cLES: (∆x+ , ∆z+ ) = (159, 64).
EAM compared with dynamic Smagorinsky as SGS model.

312th International Symposium on Turbulence and Shear Flow Phenomena (TSFP12)
Osaka, Japan (Online), July 19-22, 2022

Colorplot of the streamwise velocity along the mid-span plane at Reh = 10000.

Figure 4.

0.7

We will here use a developing free shear layer as a test
case. This case was previously described in the EU project
Go4Hybrid Mockett et al. (2018) and is of particular interestfor hybrid RANS-LES methods due to that fully developed
thick boundary layers are advected into a free shear layer witha transition from RANS to LES. Standard DES methods haveshown severe suppression and delay of the developing of resolved turbulence in the shear layer, the so-called grey-area
problem.
The analysis of the turbulent structures, depicted in figure 6, shows that EARSM-DDES, differently from k − ω SSTDDES, has a much faster RANS to LES transition, and a much
shorter extent of the grey area. The blending function fd shows
similar transition from RANS to LES for both models shown
as iso-lines of fd in figure 7 very near the flat-plate trailingedge. Instead, SST-DDES shows a locally higher SGS shear
stress in this region while EARSM-DDES predicts lower SGS
shear stress. The SGS shear stress is closely connected to the
SGS dissipation. This plot clearly illustrates that the grey area
reduction by EARSM is mostly due to the refined and more
physical Reynolds stress modelling and not due to differencesin the blending function. It is known that the EARSM in
RANS mode is better responding to the rapid shear avoiding
the severe unphysical over-prediction of modelled turbulence
observed from eddy-viscosity models. Our hypothesis is that
EARSM for the same reason will preserve a more physicalSGS stress in the transition region compared with SST-DDES.
On the other hand, the SST k-ω model does have an eddyviscosity limiter which is supposed to mitigate this particular
problem, and it is not clear why this is not reflected in the results shown here.
The resulting velocity profiles are shown in figure 8 at two
different x positions. The first position is in the later initialtransient while the second position is where the shear layer
can be considered to be fully developed. At both positions
the EARSM-DDES is superior to the SST-DDES, which is areflection of the very damped and delayed transition seen in
the SST-DDES results. The corresponding Reynolds stresses(or u0i u0j ) are shown in figure 9 for the later fully developed
position. EARSM-DDES here shows a good representation of
the different components.

0.6
0.5

Cp

0.4
0.3
0.2
0.1
0.0
−0.1
−0.2
0.0

0.2

0.4

0.6

0.8

1.0

x/L
0.009
0.008
0.007

Cf

0.006
0.005
0.004
0.003
0.002
0.001
0.000
−0.4

−0.2

0.0

0.2

0.4

0.6

0.8

1.0

x/L

0.8
0.7
0.6

Cp

0.5
0.4
0.3
0.2
0.1
0.0
−0.1
0.0

0.2

0.4

0.6

0.8

1.0

x/L

Figure 5. 3D diffuser simulations. Pressure and friction
coefficients, C p and C f , along the streamwise direction atz/B = 0.5 for Reh = 10000 (top, middle) and 30000 (bottom)
computed using EARSM-IDDES (red) and SST-IDDES (blue)
for fine (solid lines) and coarse (dashed-dotted lines) meshes.
Compared with reference DNS (black lines) and experimental(black symbols) data.

available, so C f cannot be compared for the high Re case. At
Reh = 10000 the friction coefficient is better predicted by using EARSM-IDDES, the inlet duct is affected by the largest
improvements, where SST-IDDES fails to resolve the fully developed turbulence at the inlet channel. The improvements
related to the inlet channel for EARSM-IDDES might be related to the more near-wall transition from RANS to LES seenin the channel flow above.

CONCLUDING REMARKSThe introduction of EARSM for hybrid RANS/LES simulations is primarily expected to give improvements for attachedboundary layers and initial separation from smooth surfaces
when 3D effects and curvature/rotation is present. Incorporating EARSM in both RANS and LES regions is mainly motivated for consistency reasons, with less expected incompatibilities in the interface regions. Also our previous studies on
WRLES indicate that some improvements could be expectedalso in the pure LES regions.
It was found that the interface regions were particularly
influenced by the novel modelling, which was somewhat unexpected. That can partly be explained by that the blending
function reacts differently when used for EARSM even though
the formulation is kept as close as possible with the baseline

DDES RESULTS
Also the development of the EARSM-DDES was implemented into OpenFOAM. Here, the boundary layers areshielded and completely in RANS mode. Validation of DDES
models are difficult since it will involve cases with both thin
attached boundary layers and free turbulence, which are cases
more complex than typical generic benchmark cases.

4

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

Figure 6.
tom).

Vorticity
Magnitude
Q-criterion isosurfaces coloured with vorticity
magnitude,
computed by EARSM-DDES (top) and k − ω SST-DDES (bot0.0e+00

200

300 400 500

600

7.5e+02

SST-(I)DDES. However, the blending function is non linear
and might be influenced through the coupling with the accompanying k-ω model equations where the production term is differently modelled. The other explanation is that the EARSM
might introduce a more physically correct modelling with, inparticular, a different response to non-equilibrium conditions
typically for these transition regions.
Some recent studies on mitigating the grey-area problem
are focused on directly reducing the length-scale definition bymeasuring the amount of resolved turbulence, e.g. the sigma
model, see e.g. Fuchs et al. (2020). Moreover, the commutation term emerging from the rapid change of resolution, heremeasured by the transition from RANS to LES length scales,
might be of large importance. Further studies in this direction should include the tentative modelling of this phenomenon
proposed by Girimaji & Wallin (2013). These phenomena arebasically independent and should be combined for a more general and consistent approach.

Figure 7. Mixing layer simulation using EARSM-DDES(top) and SST-DDES (bottom) showing SGS shear stress τxy
and iso-lines of the blending function fd = 0.1 and 0.9.

This work has been funded by the Swedish Governmental
Agency for Innovation Systems (VINNOVA) in the Swedish
National Flight Research Program (NFFP-CIAO, Contract
No. 2017-04887). Computer time has been provided by the
Swedish National Infrastructure for Computing (SNIC).
Figure 8. Mixing layer simulation using DDES. Computed
velocity profiles at x = 200 (left) and 800 (right) using SSTDDES (blue) and EARSM-DDES (red) compared with experiments (symbols).

REFERENCES
Cherry, Erica M, Elkins, Christopher J & Eaton, John K 2008 Geometric sensitivity of three-dimensional separated flows. International
Journal of Heat and Fluid Flow 29 (3), 803–811.
Fuchs, Marian, Mockett, Charles, Sesterhenn, Jörn & Thiele, Frank2020 The grey-area improved σ -ddes approach: Formulation review and application to complex test cases. In Progress in Hybrid
RANS-LES Modelling, pp. 119–130. Springer.
Germano, Massimo, Piomelli, Ugo, Moin, Parviz & Cabot, William H
1991 A dynamic subgrid-scale eddy viscosity model. Physics of
Fluids A: Fluid Dynamics 3 (7), 1760–1765.
Girimaji, Sharath S & Wallin, Stefan 2013 Closure modeling in bridging regions of variable-resolution (vr) turbulence computations.
Journal of Turbulence 14 (1), 72–98.
Gritskevich, Mikhail S, Garbaruk, Andrey V, Schütze, Jochen &Menter, Florian R 2012 Development of DDES and IDDES formulations for the k-ω shear stress transport model. Flow, turbulence
and combustion 88 (3), 431–449.
Jaffrézic, B & Breuer, M 2008 Application of an explicit algebraic
Reynolds stress model within a hybrid LES–RANS method. Flow
Turbul. Combust. 81 (3), 415.
Kamble, Chetna, Girimaji, Sharath, Razi, Pooyan, Tazraei, Pedram& Wallin, Stefan 2022 Closure modeling in near-wall region ofsteep resolution variation for partially averaged navier-stokes simulations. Phys. Rev. Fluids 7, 044608.
Marstorp, Linus, Brethouwer, Geert, Grundestam, Olof & Johansson,
Arne V 2009 Explicit algebraic subgrid stress models with appli-

Figure 9. Mixing layer simulation using DDES. Computed
Reynolds stress profiles at x = 800 using SST-DDES (blue)and EARSM-DDES (red) compared with experiments (symbols). Dashed curves shows the modelled, or SGS, part of thestress.

5

12th International Symposium on Turbulence and Shear Flow Phenomena (TSFP12)
Osaka, Japan (Online), July 19-22, 2022
cation to rotating channel flow. Journal of Fluid Mechanics 639,
403–432.
Mellen, CP, Fröhlich, J & Rodi, W 2000 Large eddy simulation of theflow over periodic hills. In 16th IMACS world congress, pp. 21–25.
Mockett, C, Haase, Werner & Schwamborn, Dieter 2018 Go4Hybrid:
Grey Area Mitigation for Hybrid RANS-LES Methods, , vol. 134.
Springer.
Montecchia, Matteo 2019 Numerical and modelling aspects of largeeddy and hybrid simulations of turbulent flows. PhD thesis, KTH
Royal Institute of Technology.
Montecchia, Matteo, Brethouwer, Geert, Johansson, Arne V & Wallin,
Stefan 2017 Taking large-eddy simulation of wall-bounded flows
to higher reynolds numbers by use of anisotropy-resolving subgrid
models. Physical Review Fluids 2 (3), 034601.
Montecchia, Matteo, Brethouwer, Geert, Wallin, Stefan, Johansson,
Arne V & Knacke, Thilo 2019 Improving LES with OpenFOAM

by minimising numerical dissipation and use of explicit algebraic
SGS stress model. Journal of Turbulence pp. 1–26.
Montecchia, Matteo, Knacke, Thilo, Brethouwer, Geert, Wallin, Stefan & Johansson, Arne V. 2018 Improved LES of turbulent channel
flow by using OpenFOAM with the Explicit Algebraic SGS model.
12th International ERCOFTAC Symposium on Engineering Turbulence Modelling and Measurements .
Ohlsson, Johan, Schlatter, Philipp, Fischer, Paul F & Henningson,
Dan S 2010 Direct numerical simulation of separated flow in athree-dimensional diffuser. Journal of Fluid Mechanics 650, 307–
318.
Spalart, Philippe R 2009 Detached-eddy simulation. Annual review of
fluid mechanics 41, 181–202.
Wallin, Stefan & Johansson, Arne V 2000 An explicit algebraic
Reynolds stress model for incompressible and compressible turbulent flows. Journal of Fluid Mechanics 403, 89–132.

6