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Two different types of instantaneous wall boundary conditions have been proposed for resolved large scale simulations that extend inside the viscous sublayer. These conditions transfer the physical no-slip and impermeability/permeability information,

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A new large-eddy simulation near-wall treatment
M. Iovieno,G. Passoni, andD. Tordella
Citation:Phys. Fluids
16
, 3935 (2004); doi: 10.1063/1.1783371
View online:http://dx.doi.org/10.1063/1.1783371
View Table of Contents:http://pof.aip.org/resource/1/PHFLE6/v16/i11
Published by the American Institute of Physics.
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A new large-eddy simulation near-wall treatment
M. Iovieno
Dipartimento di Ingegneria Aeronautica e Spaziale, Politecnico di Torino, Corso Duca degli Abruzzi 24,10129 Torino, Italy
G. Passoni
Dipartimento di Ingegneria Idraulica, Politecnico di Milano, Ambientale, Rilevamento Piazza Leonardo da Vinci 32, 20133 Milano, Italy
D. Tordella
a
)
Dipartimento di Ingegneria Aeronautica e Spaziale, Politecnico di Torino, Corso Duca degli Abruzzi 24,10129 Torino, Italy
(
Received 24 February 2004; accepted 4 June 2004; published online 5 October 2004
)
Two different types of instantaneous wall boundary conditions have been proposed for resolvedlarge scale simulations that extend inside the viscous sublayer. These conditions transfer thephysical no-slip and impermeability/permeability information, which can only be rigorously appliedto the unﬁltered variables, to the ﬁltered variables. The ﬁrst condition is universal, while the secondone speciﬁes the wall stress and relevant distribution and can be used to treat inverse ﬂow problems.The ﬁlter scale close to the wall is a function which varies according to its position and thus theproblem of the noncommutation of the ﬁlter and differentiation operators arises. Used together withthe explicit noncommutation procedure by Iovieno and Tordella, these boundary conditionsconstitute a wall treatment which could improve the use of the large-eddy methodology in relationto aspects that are independent of the modeling of the subgrid scale motion. When applied in the testcase of the plane periodic channel, intentionally using the most crude subgrid scale model
(
Smagorinsky, with no dynamic procedure or wall damping function
)
to prove its efﬁcacy, theproposed near-wall treatment yielded resolved large-eddy simulations which compare well withboth direct numerical simulations and with experimental data. The effects of the Reynolds numberon the structure of the ﬂow are retained. Distributions of the noncommutation error on the turbulentsolution are also reported. ©
2004 American Institute of Physics
.
[
DOI: 10.1063/1.1783371
]
I. INTRODUCTION
The large-eddy simulation
(
LES
)
method is probably go-ing to be one of the most frequently used tools to predict thebehavior of turbulent ﬂows for many different physical andengineering applications. Among these applications, wallﬂows constitute a separate class, due to the peculiarities of the near-wall dynamics that are related to important applica-tions in geophysics, hydrodynamics, and gasdynamics. Theturbulence near the wall is very unhomogeneous and not inequilibrium. The diffusive vorticity generation is coupledand is of the same order as the unsteadiness and nonlinearity.Such a complex situation is not easily synthesized in amodel, because, close to the wall, the categories on whichthe turbulence modeling of homogeneous or nearly homoge-neous ﬂows relies are not valid, the conceptual separationbetween the large and small scales is not possible, and theasymptotics similarity is not observed in practical problems.It is crucial for physicists and engineers, who neverthelessmust produce approximate but reliable forecasts to improvethe use of the method as much as possible independently of the physical features of the subgrid scale model that isadopted. For this purpose it is important to consider the fol-lowing:
(
1
)
the transfer of the wall physical conditions,which can only be rigorously applied to the unﬁltered vari-ables, to the ﬁltered variables and
(
2
)
the noncommutationproperty loss between the ﬁlter and differentiation opera-tions, which affects the simulation of unhomogeneous ﬁelds,such as the wall ﬂows, in which the ﬁlter scale varies greatlyaccording to the position
͓
␦
=
␦
͑
x
i
͔͒
.
1–5
In this situation, thegoverning equations change structure, because a noncommu-tation term must be introduced in correspondence to eachspatial differential term. The change in the ﬁltered governingequations introduces variations to their numerical solution.For high Reynolds number ﬂows, the problem of theboundary conditions for the ﬁltered ﬁeld can be treated byadopting one of the classical approximated conditions thatrelies on the introduction of special wall models, which rep-resent the inner layer dynamics
[
usually in a Reynolds-averaged sense, see the review by Piomelli and Balaras, Sec.2
(
Ref. 6
)]
, and by putting the ﬁrst grid point used by thelarge-eddy simulation inside the logarithmic layer
(
see Ref.7, the wall-stress models by Schumann
8
and Piomelli
et al.
,
9
the two-layer models by Balaras
et al.
,
10
the detached eddysimulation approach by Baggett
11
and Nikitin
et al.
12
)
. Thesemodels were conceived to avoid the prohibitively expensivecomputational cost of resolving the wall layer in high Rey-nolds number environmental and engineering applications.However, at a fundamental level, with regard to the LES
a
)
Author to whom correspondence should be addressed. Telephone: 0039011 564 6812; fax: 0039 011 564 6899; electronic mail:daniela.tordella@polito.itPHYSICS OF FLUIDS VOLUME 16, NUMBER 11 NOVEMBER 2004
1070-6631/2004/16
(
11
)
/3935/10/$22.00 © 2004 American Institute of Physics3935
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methodology, and to resolve the near-wall dynamics, it isalso acceptable to place grid points inside the viscous sub-layer.The boundary conditions for the ﬁltered variables shouldbe different from those that are canonical for the unﬁlteredvariables, i.e., the no-slip and impermeability conditions
u
i
=0 at the wall. First, the ﬁltering operation
(
e.g., the volumeaverage
)
is ill deﬁned for grid points placed on the wallbecause, in this case, the ﬁlter width extends beyond the wallboundary
(
i.e., outside the ﬂow domain
)
; second, a ﬁlteringvolume in contact with the wall, but entirely merged withinthe domain, will give averaged velocities that are differentfrom zero and which are placed in the dynamical center of the average volume which will always be located at a ﬁnitedistance from the wall.On the other hand, the alternative option of the grid re-ﬁnement
(
i.e., the ﬁlter width that goes to zero as the wall isapproached
)
is not clearly deﬁned. In this case in fact it is notpossible to automatically determine where the shift fromLES to DNS takes place. This shift would necessitate thechange of the evolution equations from the ﬁltered NS ver-sion
(
LES
)
to the unﬁltered NS
(
DNS
)
. Since this change isnot carried out, which would inevitably imply the introduc-tion of a domain decomposition, the simulation cannot beconsidered as being based on a consistent problem form.Furthermore, even in the hypothesis of having consistentlysplit the domain to carry out the hybrid LES-DNS, it wouldmean adopting a time step which must ﬁt the DNS require-ments close to the wall. The temporal integration scale forthe DNS is faster than that required by the LES, but since itis not possible, advancing on time, to differentiate the tem-poral steps into different regions of the computational do-main, the DNS requirement would take precedence over theLES one, and this is not convenient.It is here proposed to shift the boundary conditions forthe ﬁltered ﬁeld onto a surface that lies on a ﬁrst level of gridpoints and is parallel to the wall, at a distance of the sameorder as the viscous length. The transfer of the informationthat is relevant to the physical properties of the wall is ac-complished by considering a series expansion in
␦
for theﬁltered variable, at the ﬁrst layer of points. If associated to aTaylor expansion of the unﬁltered variable at the wall, thisyields a ﬁrst kind of condition that is universal in character.If the
␦
expansion is instead related to a Mac Laurin expan-sion of the unﬁltered variable at the wall, a second kind of boundary condition is obtained which is suitable to impose aknown distribution of wall stresses, as normally asked in thecontext of inverse mathematical problems. The boundarycondition formulations are described in Sec. II. The related
a priori
tests,
13
which showed a correlation with DNS data
14,15
as high as 0.97 for a boundary shift of ﬁve wall units, aredescribed in Sec. II A.As previously explained, the other feature that has beenimplemented in the simulations is the noncommutationprocedure.
1
This is based on an approximation of the differ-ent noncommutation terms in the governing equations asfunctions of the
␦
gradient and of the
␦
derivatives of theﬁltered variables. The anisotropic noncommutation approxi-mating terms, of the fourth order of accuracy in the ﬁlterscale, are obtained using series expansion in
␦
of approxima-tions based on ﬁnite differences and introducing two succes-sive levels of ﬁltering. A brief outline of this procedure isgiven in Sec. III. The distribution of the noncommutationerrors on the Reynolds stresses is given in Sec. IV.The results that were obtained when using the presentwall treatment applied to the LES of the channel ﬂow andobtained by utilizing the most crude SGS model
(
Smagorin-sky, with no dynamic procedure or wall damping function
)
are discussed in Sec. IV. The simulations compare well withthe direct numerical simulations
14,15
and with laboratoryobservations.
16–18
The simulations show the correct Rey-nolds number dependency. Given this, much greater progresscan be expected if the dynamic procedure
19,20
and modelswhich allow for signiﬁcant nonlocal and nonequilibrium ef-fects are used.
21–24
The concluding remarks are given in Sec.V.
II. NEAR-WALL TREATMENTA. Wall conditions for the ﬁltered variables
The shifting of the boundary conditions for the ﬁlteredﬁeld on a surface that lies beside the wall at a distance
y
គ
of the same order as the viscous length
ᐉ
͓
y
=
O
͑
ᐉ
͒
=
O
͑
␦
min
͔͒
, see Fig. 1, is here proposed. The ﬁrst level of thegrid points should in turn be positioned on the shifted bound-ary. This shift offers a twofold advantage—ﬁrst, the depen-dent variables are correctly determined, as it is possible to seta local volume of integration which does not cut the physicalboundary; and second, since the shifted boundary is close tothe wall, it is possible to transfer the physical informationthat corresponds to the no-slip and impermeability/ permeability conditions to the shifted condition through aseries expansion. An expansion in series of
␦
of the ﬁlteredvariable can in fact be associated, at the ﬁrst layer of points,to a Taylor or a Mac Laurin expansion of the unﬁltered vari-able at the wall, where the ﬁlter length reaches the minimumvalue normal to the wall
␦
min
. While using this boundaryformulation, one is aiming at simulating the inner viscouslayer in the region
y
+
Ͻ
50, which requires
␦
min
=
y
គ
ϳ
1–5.
FIG. 1. A schematic view of the shifted boundary conditions, the ﬁlter, andthe grid.
3936 Phys. Fluids, Vol. 16, No. 11, November 2004 Iovieno, Passoni, and Tordella
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Let us consider the class of integration volumes
V
␦
=
ͭ
R
3
:
Ͱ
ͩ
1
␦
1
,
2
␦
2
,
3
␦
3
ͪ
Ͱ
Ͻ
1
ͮ
,
͑
1
͒
where
␦
͑
x
͒
=
(
␦
1
͑
x
͒
,
␦
2
͑
x
͒
,
␦
3
͑
x
͒
)
and the transformation
j
=
␦
j
j
[
with det
͑
i
/
k
͒
=
␦
1
␦
2
␦
3
and where no summationis implied
]
has been introduced. Let us consider the averageoperation for the variable
f
͑
x
͒
=
f
͑
x
j
+
␦
j
j
͒
:
͗
f
͘
␦
=1
V
␦
͵
V
␦
f
͑
x
+
͒
d
=1
V
1
͵
V
1
f
͑
x
j
+
␦
j
j
͒
d
,
͑
2
͒
where
1
=
͑
1,1,1
͒
and
V
1
=
V
␦
/
␦
1
␦
2
␦
3
.For the sake of simplicity, let us now consider the caseof a ﬂat wall ﬂow. Here the ﬁlter can be opportunely repre-sented by the widely used notation
␦
͑
x
͒
=
(
⌬
x
,
͑
y
͒
⌬
y
,
⌬
z
)
with constants
⌬
x
,
⌬
y
, and
⌬
z
and where
y
is the coordinatenormal to the wall. The
␦
expansion for a general variable
f
,after setting
␦
͑
y
គ
͒Х
␦
min
=
min
⌬
, yields
͗
f
͑͘
y
គ
͒
=
f
͑
y
គ
͒
+
L
͓
f
͔
␦
min2
+
O
͑
␦
min4
͒
,
͑
3
͒
where
L
͓
·
͔
=
12
a
˜
2
,
˜
=
y
2
2
+
ͩ
⌬
x
␦
min
ͪ
2
x
2
2
+
ͩ
⌬
z
␦
min
ͪ
2
z
2
2
,
͑
4
͒
a
=1
V
1
͵
V
1
i
2
d
,
͑
5
͒
and where it should be recalled that, due to
(
1
)
and
(
2
)
, thecoefﬁcients of the odd powers of
␦
min
are zero. By applyingthe operator
L
to
(
3
)
one obtains
L
͓͗
f
͔͘
=
L
͓
f
͔
+
L
†
␦
min2
L
͓
f
͔
‡
+
O
͑
␦
min4
͒
=
L
͓
f
͔
+
O
͑
␦
min2
͒
.
͑
6
͒
As a consequence,
(
3
)
can be written as
f
͑
y
គ
͒
=
͗
f
͑͘
y
គ
͒
−
␦
min2
L
͓͗
f
͔͘
+
O
͑
␦
min4
͒
.
͑
7
͒
In turn, to transfer the no-slip and impermeability informa-tion, which applies at
y
=0, let us consider the Taylor expan-sion along the normal to the wall,
f
͑
0
͒
=
f
͑
y
គ
͒
−
y
គ
y
f
͑
y
គ
͒
+
y
គ
2
2
y
f
͑
y
គ
͒
+
O
͑
y
គ
3
͒
.
͑
8
͒
Since relation
(
6
)
can be generalized as
d
m
dx
m
f
͑
x
j
͒
=
d
m
dx
m
͗
f
͑͘
x
j
͒
+
O
͑
␦
2
͒
,
m
= 1,2,
…
,
͑
9
͒
expansion
(
8
)
can be written as
f
͑
y
គ
͒
=
f
͑
0
͒
+
y
គ
y
͗
f
͘
−
y
គ
2
2
y
2
͗
f
͘
+
O
͑
␦
min3
͒
.
͑
10
͒
By equating
(
7
)
and
(
10
)
, while recalling deﬁnition
(
4
)
, andtruncating the third-order terms, a new boundary condition isobtained at
y
=
y
គ
,
͗
f
͘
=
f
͑
0
͒
+
y
គ
y
͗
f
͘
+12
a
˜
2
͓͗
f
͔͘
␦
min2
−
y
គ
2
2
y
2
͗
f
͘
,
͑
11
͒
where the ﬁltered variable at each instant depends explicitlyon the position
y
គ
, the values of its ﬁrst and second deriva-tives, and the wall value of the unﬁltered variable, whichintroduces the physical information. This condition—whichin the following is called condition I—is an instantaneouscondition and it is universal because it can be applied to anykind of wall boundary. One should note that, according to thetheory established by Kreiss
25
for Dirichlet differential prob-lems discretized with a ﬁnite-difference scheme with order
O
͑
␦
p
͒
at inner points and
O
͑
␦
p
−1
͒
at points close to theboundary, the error of the discrete solution is
O
͑
␦
p
͒
through-out. Therefore, with such a boundary condition formulationand truncating the terms of order
O
͑
␦
3
͒
, the fourth order of accuracy reached for the approximation of the noncommuta-tion term proposed by Iovieno and Tordella
1
can be expectedto be preserved. In the present simulation, the
y
derivativesof the ﬁltered variables at
y
=
y
គ
have been calculated usingone-sided discrete operators.Another similar type of boundary condition can be writ-ten by equating
(
7
)
to the Mac Laurin series
f
͑
y
គ
͒
=
f
͑
0
͒
+
y
គ
y
f
͉
y
=0
+
͑
y
គ
2
/2
͒
y
2
f
͉
y
=0
+
O
͑
y
គ
3
͒
, which gives the condition
͗
f
͘
=
f
͑
0
͒
+
y
គ
y
͉
f
͉
y
=0
+
y
គ
2
2
y
2
͉
f
͉
y
=0
+
a
2
␦
min2
˜
2
͓͗
f
͔͘
.
͑
12
͒
In the case of a steady
(
in the mean
)
turbulent ﬂow it ispossible, with this formulation, to transfer the informationrelevant to the time average of the wall shear stress distribu-tion and its derivative to the ﬁltered ﬁeld. Condition
(
12
)
—which in the following is called condition II—is an instanta-neous condition. However, it is physically reasonable toinsert the time averaged values of the wall stress only in thecase of ﬂows which are steady in the mean, as is the case of the example that we have considered in this paper. However,this is not the only possibility. In fact, if detailed informationon the temporal and spatial variation of the wall stress areavailable, it would in fact be better to insert them into inﬁeldcondition II. The innovative and srcinal character of condi-tion II is that it allows the ﬂow to be fed with different wallinformation, which, apart from the pure no-slip condition,also includes information on the time and space ﬂuctuations,or possible evolutions of the wall stress along the wall. Allthis can be gathered in one single condition. This is feasiblebecause, having placed the condition relatively close to thewall—at a distance of almost one viscous length—one canuse a Mac Laurin series expansion to transfer the physicalinformation at the wall to the inﬁeld condition. In otherwords, a convenient situation is obtained: instead of usingthe physical boundary—the wall—where only one boundarycondition
(
bc
)
can be placed for each variable
(
on the vari-able itself or on one of its derivatives in the direction normalto the wall
)
, one can give a plurality of information concern-ing the values of the variables and of their relevant wallderivatives to the ﬁeld, with just one condition and withoutoverconstraining the ﬂow.
Phys. Fluids, Vol. 16, No. 11, November 2004 A new large-eddy simulation near-wall treatment 3937
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A solution accuracy of the fourth order could also beexpected when using this boundary condition
(
see the previ-ous comments
)
. It must be noticed that, with this kind of formulation, the shear stress distribution along the wall canbe imposed to the ﬁeld, as can the related characteristics suchas the intensity of the wall roughness. In this case, thisboundary condition can be applied to inverse mathematicalproblems.
B.
A priori
test on the approximate boundaryconditions
The correlation between the ﬁltered values that are ob-tained by the present shifted boundary conditions and theﬁltered values that are obtained by a direct numerical simu-lation can be deﬁned, as a function of the distance of theshifted boundary from the wall, as
C
=
͑͗
u
i
͘
DNS
−
͗
u
i
͘
DNS
͒͑͗
u
i
͘
−
͗
u
i
͒͘
/
ͱ
var
͑͗
u
i
͘
DNS
͒
var
͑͗
u
i
͒͘
,
͑
13
͒
where
͗
u
i
͘
DNS
are the ﬁltered data from the direct simulationdata base
(
Passoni
et al.
15
)
,
͗
u
i
͘
have been computed from
(
11
)
and
(
12
)
by introducing the ﬁltered direct simulationdata into the right-hand side, the overbar means the averageover the surface parallel to the wall, and var is the variance.For Re
=180
(
the Reynolds number for which a wide set of instantaneous ﬁelds was available
)
Fig. 2 shows that bothboundary conditions I and II yield a correlation
C
which isover 0.97 for the three velocity components up to a distanceof ﬁve wall units and goes down to 0.8, 0.88, 0.7 for
͗
u
͘
,
͗
v
͘
,
͗
w
͘
, respectively, at seven wall units.
C. Noncommutation treatment
The noncommutation procedure and the new wall condi-tion models together constitute the new treatment for wallturbulence that is here proposed. The main applications thatrequire a highly variable ﬁltering in LES are wall ﬂows.They must be represented through
(
a
)
accessory conditionsthat are consistent with the LES methodology, on the onehand, and
(
b
)
, on the other hand, a possibly explicit noncom-mutation procedure. As explained in the Introduction, apartfrom the matter relevant to the quality of either dynamical ornot subgrid model, the no-slip condition associated to the useof a ﬁlter width which goes to zero as the wall is approachedis not a fully consistent treatment for large-eddy simulationsof near-wall turbulence.A brief description of the noncommutation procedure isgiven in the following. When
␦
, the linear scale of ﬁltering,is not uniform in the ﬂow domain, the averaging and differ-entiation operations no longer commute. This leads to inho-mogeneous terms in the motion equations that act as sourceterms whose intensity depends on the gradient of the ﬁlterscale
␦
and which, if neglected, induce a systematic errorthroughout the solution. One kind of noncommutation termfor each differential term is present in the governing equa-tions. By introducing the subgrid turbulent stresses
R
ij
͑
␦
͒
=
͗
u
i
͘
␦
͗
u
j
͘
␦
−
͗
u
i
u
j
͘
␦
and the noncommutation terms
C
i
Ј
,
C
ii
Љ
, for the ﬁrst and second derivatives, as discussed byIovieno and Tordella
1
(
see there, Sec. II for the isotropicﬁlter conﬁguration and the Appendix for the general aniso-tropic and the wall-bounded ﬂow conﬁgurations
)
the aver-aged governing equations can be written as
i
͗
u
i
͘
␦
= −
C
i
Ј
͑͗
u
i
͘
␦
͒
,
͑
14
͒
t
͗
u
i
͘
␦
+
j
͑͗
u
i
͘
␦
͗
u
j
͘
␦
͒
+
i
͗
p
͘
␦
−
jj
2
͗
u
i
͘
␦
−
j
R
ij
͑
␦
͒
= −
C
j
Ј
͑͗
u
i
͘
␦
͗
u
j
͘
␦
͒
−
C
i
Ј
͑͗
p
͘
␦
͒
+
C
jj
Љ
͑͗
u
i
͘
␦
͒
+
C
j
Ј
͑
R
ij
͑
␦
͒
͒
.
͑
15
͒
The present procedure approximates the noncommutationterms on the right-hand side of
(
14
)
and
(
15
)
by means of series expansion in
␦
of ﬁnite difference approximations of the space derivatives and by introducing a successive level of ﬁltering. An accuracy of the fourth order is reached withrespect to
␦
.Let us recall that the anisotropic noncommutation termof the ﬁrst derivative
C
i
Ј
, deﬁned by
FIG. 2. An
a priori
test on the approximate boundary conditions: correlationlevel as a function of the distance
y
min
=
y
គ
of the shifted boundary from thewall. Top panel condition I, Eq.
(
11
)
; bottom panel condition II, Eq.
(
12
)
.
3938 Phys. Fluids, Vol. 16, No. 11, November 2004 Iovieno, Passoni, and Tordella
Downloaded 13 Mar 2012 to 130.192.25.44. Redistribution subject to AIP license or copyright; see http://pof.aip.org/about/rights_and_permissions

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