Jtam.dvi

JOURNAL OF THEORETICAL

AND APPLIED MECHANICS

49, 4, pp. 1101-1114, Warsaw 2011

EVALUATION OF HIGH ORDER SMOOTH FILTER IN

LARGE EDDY SIMULATION OF FULLY DEVELOPED

TURBULENT CHANNEL FLOW WITH EXPLICIT FILTERING

Norouz M. Nouri

Saber Yekani Motlagh

Ehsan Yasari

Iran University of Science and Technology, Department of Mechanical Engineering, Tehran, Iran

e-mail: mnouri@iust.ac.ir

Explicit filtering with a smooth shape is one of approaches adopted in
large eddy simulations (LES).The presentwork investigates the applica-
tion of an explicit high order smooth (HOS) filter for the LES of a fully
developed turbulent channel flow. The Crank-Nicolson scheme for time
marching and second-order finite-volume schemes for spatial derivatives
were implemented in this investigation. Implicit filtering, together with
the Smagorinsky sub-grid scale (SGS)model, and explicit filtering, along
with a HOS filter were studied in a fully turbulent channel flow. In this
study, explicitHOSfilteringwith an explicit filterwidth to grid size ratio
of 2.0 was in agreement with the available direct numerical simulation
(DNS) data. However, themean velocity profile in the streamwise direc-
tionwas underestimated, and the turbulence intensity in the streamwise
direction improved compared to other directions. Moreover, turbulence
stresses were well predicted using the mixed SGS and sub-filter stress
(SFS) models and applying the HOS filter as an explicit filter.

Key words: large eddy simulation, explicit filtering, implicit filtering

1. Introduction

Large eddy simulation (LES) is one of the best methods for computational
modeling of turbulent flows. In contrast with the direct numerical simulation
(DNS), the LES does not seek to resolve all the length scales of the flow. The
LES is based on the decomposition of flow scales into a resolved sub-grid scale
(SGS) where large scales are directly resolved. In thismethod, the SGS terms
and interactions with the resolved scales aremodeled.Many differentmethods

1102 N.M. Nouri et al.

are used to achieve the desired accuracy in the LES.The eddy viscosity-based
model proposed by Smagorinsky (1963) and Germano et al. (1991) generally
dissipates enough energy but cannot predict the stress value. The classical
Smagorinsky model is obtained by applying a sharp cut-off filter and is valid
only when the cut-off frequency is in the inertial sub-range of the energy
spectrum. The scales in the inertial sub-range are larger than the dissipative
scales and smaller than the scales that contain most of the energy (Piomelli
et al., 1991).

Researchers have conducted extensive studies on fully developed turbulent
channel flows.Deardorff (1970) conducted the first numerical large eddy simu-
lation of a channel flowusing the Smagorinskymodel and applying the central
difference discretization. In the LES, discrete equations act as a low-pass filter
which, by having a width equal to the cell width, produces a smooth shape
in the frequency space. Therefore, there is no control on the filter shape and
width. The use of a secondary filter as an explicit filter, where the shape and
width are somewhat controllable, improves the performance of the solution.
Thewidths of the explicit filter in a turbulent channel flow are 1.5 and 3 times
the grid spacing. By using explicit filtering, the errors are transferred to hi-
gherwave numbers and thus, their effects on the results are small (Lund, 1997;
Lund and Kaltenbach, 2003). Many methods have been proposed for explicit
filtering using sharp cut-off filters. For instance, the explicit filtering of the
whole velocity field at the end of each time step and the explicit filtering of
the convection terms, etc. have been investigated byTellervo (2005a,b, 2008).

Using a finite-volumemethod, Elhami et al. (2005) recently investigated a
fully turbulent channel flowbymeans of a dynamic filteredmodel inwhich the
explicit cut-off filter widthwas twice the grid spacing. The explicit filter shape
can either be smooth or close to a sharp cut-off. Consequently, applying each
of these filters will have different effects on the turbulencemodel (De Stefano
andVasilyev, 2002).When the explicit filter with the smooth shape is applied
to the Navier-Stokes equations, the sub-filter stress (SFS) model is added to
the SGSmodel to indicate the interactions between the small and large scales
(Tellervo, 2005a,b, 2008). The scale-similarity model of Bardina et al.) (1980)
is a well-known SFS model. To explicitly calculate the magnitude of the SFS
term, the filter with a specific function is necessary.

Thus, the aim of this study is to investigate the use of an explicit HOS
filter for theLESof a fullydeveloped turbulent channel flow.Thefinite-volume
method and PISO algorithm are utilized for the numerical simulations. The
secondorderdiscretizationmethod for convection anddiffusionand theCrank-
Nicolson method for time integration are used. The effects of explicit and

Evaluation of high order smooth filter... 1103

implicit filtering on the prediction of the velocity profile, Reynolds stresses,
and turbulence intensities are considered.

2. Governing equations in large eddy simulation

The closed form of the filtered Navier-Stokes equation is the fundamental
governing equation for simulating the turbulent flow. The following equations
are formally derived by applying a low-pass filter to the continuity andNavier-
Stokes equations and separating the non-linear terms in three-dimensional
forms

∇· Ũ =0

∂Ũ

∂t
+∇· (Ũ ⊗ Ũ)+

ρ
∇P̃ =∇· (2νD̃)−∇· (BSGS)

(2.1)

where D̃ is defined as D̃=(∇Ũ+∇⊤Ũ)/2. The term BSGS = Ũ⊗U−Ũ⊗Ũ
is the sub-grid scale (SGS) stress and represents the interactions between the
small scales and the large ones. This term originates from the nonlinearity of
the convection term ∇· (Ũ ⊗ Ũ). Different models, including the well-known
eddyviscosity Smagorinskymodel (Smagorinsky, 1963), are introduced for the
sub-grid term. There are twomain assumptionsmade in the derivation of this
model: (1) local energy equilibrium and (2) applying the sharp cut-off filter.
By considering these assumptions, the relation for the νt (eddy viscosity) is
obtained as follows

νt =(Cs∆grid)
2|D̃| (2.2)

where Cs is the Smagorinsky coefficient, which Cs = 0.2 is used in this si-
mulation, and ∆grid = (∆x∆y∆z)

1/3 is the grid filter width. The classical
Smagorinsky model with the wall damping function is used in this work for
comparison.

3. Explicit filtering

In the LES, discrete equations act as a low-pass filter in which the cell size
functions as the filter width resulting in a smooth shape in frequency space.
This is referred to as the implicit filter. The limitations of this filter type are
as follows.

1104 N.M. Nouri et al.

There is no control on the filter shape, width, and energy spectrum. As a
result, numerical errors are uncontrollable (Lund, 1997; LundandKaltenbach,
2003). Althoughmostmodels, including Smagorinsky, are derived by applying
a sharp cut-off filter (Smagorinsky, 1963), usually in practice, a smooth grid
filter is utilized.Therefore, there is no consistency between the applied and the
modeled filter. However, using a secondary filter as an explicit filter yields a
controllable shape and width. This improves the performance of the solution.

Consequently, the numerical errors diminish, and the coherency between
the LES model filter and the applied filter for the numerical simulation is
improved. The shape of the explicit filter has different effects on the energy
spectrum and LES turbulence modeling.

4. The effect of filter shape in frequency space on explicit

filtering

In contrast to the sharp cut-off filter, the Gaussian and the top-hat filters
are smooth in frequency space. For this reason, they are called smooth filters.
Because the filter width is larger than the grid spacing in explicit filtering,
additional stresses can be defined according to the available turbulence scales.
For instance, the stresses related to scales smaller than the explicit filterwidth
are called the sub-filter scale (SFS) stresses. The SFS stresses are divided into
two components: the resolved SFS (RSFS) stresses, in which scales are betwe-
en the grid filter width and the explicit filter width, and the unresolved SFS
(URSFS) stresses (i.e. SGS stresses), which are more closely related to the
length scale rather than to the grid spacing, Fig.1. By applying the explicit

Fig. 1. Schematic profile of energy spectrum

Evaluation of high order smooth filter... 1105

smooth filter in frequency space, the energy spectrum can be separated into
three parts. The low wave number portion is filtered and well resolved on the
grid. This portion contains the velocity, which is filtered once to represent the
implicit filter or discretization, anda second time, to represent an explicit smo-
oth filter. The middle portion represents SFS motions that can theoretically
be reconstructed by an inverse filter operation. However, this reconstruction
is impeded by numerical errors which will increase near the cut-off frequen-
cy. The last portion contains SGS motions that cannot be resolved on the
grid and must be modeled. In smooth explicit filtering, SGS modeling is not
sufficient; the SFS term must be added to take into account the turbulence
stresses. There is no need for this improvement when applying a sharp filter
as an explicit filter.Whenever sharp filters are used, the SFS term disappears
(Chow and Street, 2005; Nouri et al., 2008). The main concern is that when
the filter is discretized on the grid, the filter acts as a smooth filter. It fol-
lows that methods to obtain filters in a physical space with shapes close to
the sharp cut-off in the frequency space must be developed (De Stefano and
Vasilyev, 2002; Nouri et al., 2007; Tellervo, 2008).

5. Explicit filtering with smooth filter

By filtering the Navier-Stokes equations for a second time by a smooth filter
and separating the non-linear term, we obtain the closed form of the Navier-
Stokes as the equation below

∂Ũ

∂t
+∇· (Ũ⊗ Ũ)+

ρ
∇P̃ =∇· (2νD̃)−∇· (B) (5.1)

where

B=BSGS +BSFS =(Ũ ⊗U− Ũ⊗ Ũ)+(Ũ ⊗ Ũ− Ũ⊗ Ũ) (5.2)

(∼) and (−) represent the primary implicit and the secondary explicit filter,
respectively. BSGS is the sub-grid scale (SGS) stress, whichmust bemodeled
using a model such as Smagorinsky. Moreover, BSFS is the sub-filter scale
stress and should be reconstructed by the resolved velocity field (Tellervo,
2005b).The scale-similaritymodel proposedbyBardina et al. (1980) is defined
as

BSFS = Ũ⊗ Ũ− Ũ⊗ Ũ (5.3)

1106 N.M. Nouri et al.

TheBardinamodelmakes theapproximation U = Ũ (i.e., that the full velocity
is equal to the filtered velocity) to close the SFS stress term, resulting in

BSFS = Ũ⊗ Ũ− Ũ⊗ Ũ. It allows backscatter of energy from the small to the
large scales, making it a desirable component of any SFS model. The model
does not dissipate enough energy, however. Thus, Bardina also introduced the
mixedmodel,which combined theSmagorinskyand the scale-similaritymodel.
This gavemuch improved correlations while allowing for adequate dissipation.
In this study, a mixed model consisting of Smagorinsky’s SGS model and

Bardina’s SFS is used for turbulence stress modeling. In the numerical algo-
rithm, the value of sub-filter scale stress is calculated by applying the HOS
filter.Moreover, the turbulence viscosity is obtained fromSmagorinsky’s SGS
model and added to the physical viscosity.

6. Derivation of HOS filter

The expansion of the multi-dimensional velocity series at the point xj =
=(x,y,z) is written as

Ũi(x
′

j)= Ũi(xj)+(x
′

k−xk)
∂Ũi(xj)

∂xk
+
1

2
(x′k−xk)(x

′

l−xl)
∂2Ũi(xj)

∂xk∂xl
+. . . (6.1)

and the smooth Gaussian filter function is defined as

G(x−x′′,y−y′′,z−z′′,∆x,∆y,∆z)

=
(√6
π

)3
2 1

∆x∆y∆z
exp
(
−
6(x−x′′)

∆2x
−
6(y−y′′)

∆2y
−
6(z−z′′)

∆2z

) (6.2)

Using the following convolution integration

Ũ(x,y,z) =

+∞∫

−∞

+∞∫

−∞

+∞∫

−∞

G(x−x′,y−y′,z−z′)Ũ(x′,y′,z′) dx′dy′dz′ (6.3)

the high order smooth filter for the smooth Gaussian filter is achieved as

Ũ(x,y,z)≈ Ũi−
∆2

24
∇2Ũi+

∆4

1152

(∂4Ũi
∂x4
+
∂4Ũi
∂y4
+
∂4Ũi
∂z4

)

+
5∆4

1728

( ∂4Ũi
∂x2∂y2

+
∂4Ũi
∂y2∂z2

+
∂4Ũi
∂z2∂x2

)
+O(∆6)

(6.4)

In this work, we consider four terms of the equation (10) as a HOS filter.

Evaluation of high order smooth filter... 1107

7. Numerical considerations in flow simulation

Thepresent studyuses the second-order central difference scheme for thediffu-
sion term, the interpolation of the convection termon the collocating grid, and
the finite-volume PISO algorithm. For time integration, the Crank-Nicolson
method is performed. Themain steps of the numerical procedure of the PISO
algorithm along with explicit filtering are defined as follows:

1. Numerical calculations begin by discretizing and solving the linearized
Navier-Stokes equation consideringa specificpressurefield andassuming
definite flux to obtain the mean velocity (i.e. prediction process).

2. Next, the Poisson equation of pressure is used to achieve a corrected
pressure field. The terms of this equation are also discretized with the
second-order central difference method.

3. The velocity field is also corrected by using the pressure field obtained
from Step 2. Steps 2 and 3 can be iterated several times.

4. The value of sub-filter scale stress BSFS = Ũ⊗ Ũ− Ũ⊗ Ũ is calculated
by applying theHOSfilter. This step is the equivalent of smooth explicit
filtering.

5. The turbulence viscosity (νt) is obtained and added to the physical vi-
scosity.

8. Fully developed turbulent channel flow

In the present paper, implicit and explicit filtering were applied. Based on
the friction velocity and the channel half-width, the Reynolds number Reτ
equals 180. This value is equivalent to Re = 2800, based on the average
velocity. The computational domain is 4πδ × 2δ× 4πδ/3; δ is the channel
half-width; and the x, y and z are the streamwise, wall-normal, and spanwise
directions, respectively. Four mesh resolutions are performed for this simula-
tion which are 303, 403, 503 and 653. Non dimensional streamwise velocity
profiles for different mesh resolution are shown in Fig.2. The results have
shown that there are no significant differences between 653 and 503. The-
refore, the resolution of 653 is selected for all the runs. The grid spacing
in wall coordinates in the x and z directions are ∆x+ = ∆xuτ/υ ≈ 35,
∆z+ = ∆zuτ/υ ≈ 12. A non-uniform mesh generated with the hyperbolic
tangent distribution yj = 0.01(1− tanh(γ(1− 2j/Ny))/tanhλ) was used in

1108 N.M. Nouri et al.

Fig. 2. Non-dimensional streamwise velocity profiles for different mesh resolution

the wall-normal direction, where Ny is the number of whole grids in the y di-
rection, and γ is the mesh distribution factor, which is associated with the
ratio of the last grid cell size in the channel centerline to the first grid cell
size by cosh3γ/sinhγ. The first mesh point away from the wall occurred at
y+ = ∆xuτ/υ ≈ 0.49, and the maximum spacing at the centerline of the
channel was 13.8 wall units. The initial flow field was set at 20% turbulence
intensity along with a mean velocity of 15.63 in the streamwise direction. To
maintain a fixedmass flow rate in each time step, a constant pressure gradient
was added to the Navier-Stokes equation as a definite term. Periodic bounda-
ry conditions were applied in the streamwise and spanwise directions; no slip
condition was considered at the solid walls.

9. Results

Figure 3 shows the dimensionless mean streamwise velocity profile for the
implicitfilter (Smagorinskymodel), and for theexplicit filter applyingtheHOS
filter (mixed model). As seen in this figure, the results improve when using
themixedmodel. Compared to implicit filtering, themean velocity profilewas
underestimated. The results were compared to the DNS results obtained by
Kim et al. (1987).

The shear stress on the wall was the one of the effective parameters in the
near-wall region. This shear stress was obtained via the slope of the velocity
field near the wall. To compare the implicit and the explicit HOS filters, the
magnitude and error of non-dimensional stress on the wall, which is called the
skin friction factor, are compared in Table 1.

Evaluation of high order smooth filter... 1109

Fig. 3. Non-dimensional mean streamwise velocity obtained by the mixedmodel
compared with Kim’s DNS (Kim et al., 1987) and the Smagorinskymodel

Table 1. Magnitude and maximum error of skin friction factor for different
filtering methods

Method
Kim’s DNS Smagorinsky Mixed model
data [7] implicit filter (HOS filter)

Cf,wall 8.18E-3 7.99E-3 8.14E-3

ErrorCf,wall [%] – 2.33 0.04

The results obtained from the explicit filter were better than those from
the implicit filter. Moreover, it is evident that the smooth filter accurately
predicted the near-wall velocity slope. Numerical simulation for turbulence
intensity in the streamwise direction comparing with Kim’s DNS (Kim et
al., 1987) and the Smagorinsky model is shown in Fig.4. Using the sub-filter
model together with the sub-grid model improved the results, though the
difference was obvious. This difference was due to the dissipative property
of the Smagorinsky model. Figures 5 and 6 show the turbulence intensity
profiles in the wall-normal and spanwise directions, respectively. It is evident
that themixedmodel influences turbulence intensity most significantly in the
streamwise direction.

The peak value of the turbulence intensity profiles was one of the parame-
ters related to thebuffer layer.Table 2presents thenon-dimensionalmaximum
error andmagnitude of the peak value of the streamwise turbulence intensity
for the DNS and different filteringmethods.

The deviation of the peak value of turbulence intensity from the real value
was improved by applying explicit filtering. Moreover, the peak value of the

1110 N.M. Nouri et al.

Fig. 4. Streamwise turbulence intensities for the mixedmodel, Kim et al. (1987)
DNS model, and the Smagorinskymodel

Fig. 5.Wall-normal turbulence intensities for the mixedmodel compared with Kim
et al. (1987) DNS model, and the Smagorinskymodel

Fig. 6. Spanwise turbulence intensities for the mixedmodel, Kim et al. (1987) DNS
model, and the Smagorinskymodel

Evaluation of high order smooth filter... 1111

Table 2. Non-dimensional maximum error and magnitude of pick values of
streamwise turbulence intensity for DNS and different filtering methods

Method
Kim’s DNS Smagorinsky Mixed model
data [7] implicit filter (HOS filter)

Pick valu of urms/Uτ 2.68 3.04 2.74

Error [%] – 13.432 2.23

turbulence intensity, in y+ ≃ 10, was closest to the real value when themixed
model and the HOS filter were used.

Figure 7 shows the profiles of the Reynolds stress component on the
xy-plane. The Sub-filter scale stress predicted the turbulence stress pretty
well. As seen from this Figure, explicit filtering using the mixed model had a
considerable effect on the streamwise Reynolds stress. This concept was veri-
fied for the other component of the Reynolds stress.

Fig. 7. Reynolds stress profiles for the mixedmodel, Kim et al. (1987) DNSmodel,
and the Smagorinskymodel

In the calculations performed in this study, the necessary time for having
a solution independent of the initial condition was considered to be about
10δ/Uτ . The simulation was performed using a Pentium 3.2 GHz processor.
One step in the explicit filtering with the SFS model averaged 70 seconds.
When implicit filteringwas applied, each step took approximately 40 seconds,
but the explicit filter reached the developed conditions faster than the implicit
one.This differencemaybe explained by the fact that explicit filtering is capa-
ble of damping non-physical oscillations caused by incorrect initial conditions
in conjunction with the discretization method or the turbulence modeling.
Consequently, the rate of convergence increases.

1112 N.M. Nouri et al.

10. Conclusion

In this paper, the performance of an explicit HOS filter was investigated for
a fully developed turbulent channel flow as the benchmark problem. For si-
mulations, the second-order finite-volume schemewas applied on a collocating
grid system for spatial discretization. For time integration, theCrank-Nicolson
method was used. The results were improved by using a mixed model along
with aHOS filter; in this case, however, themean velocity profile was undere-
stimated.Using this approach, the turbulence intensity profileswere improved
in three directions, however, the turbulence intensity in the stremwise xdirec-
tion showed the maximum improvement. By considering the Reynolds stress
profiles, it is obvious that the explicit filtering using themixedmodel has the
maximum effect. This is due to the sub-filter stresses, which are accurately
predicted from the turbulence stresses.

References

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subgrid-scale eddy viscosity model,Phys. Fluids, 3, 1760-1765

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Ocena gładkiego filtra wysokiego rzędu przy symulacji wielkowirowej

z filtrowaniem jawnym w pełni rozwiniętego turbulentnego przepływu

w kanale

Streszczenie

Jawne filtrowanie typu gładkiego jest jedną z metod stosowanych w symulacjach
wielkowirowych (tzw. LES). W pracy opisano zastosowanie filtra wysokiego rzędu
o charakterystyce gładkiej (HOS) do symulacji wielkowirowej w pełni rozwinięte-
go turbulentnego przepływu w kanale. W badaniach wykorzystano metodę Cranka-
Nicolsona oraz techniki przyrostów czasowych i objętości skończonychdrugiego rzędu
do wyznaczania pochodnych zmiennych przestrzennych. Przeanalizowano filtrowanie
niejawne łącznie z podsieciowymmodelem skalowymSmagorynskiego (SGS) oraz fil-
trowanie jawne HOS w pełni turbulentnego przepływu w kanale. W badaniach wy-
kazano, że filtrowanie jawne za pomocą filtra HOS, którego stosunek długości do

1114 N.M. Nouri et al.

rozmiaru siatki wynosił 2, dało zgodne wyniki z dostępnymi rezultatami bezpośred-
nich symulacji numerycznych (DNS). Nieco niedoszacowany okazał się średni profil
prędkości w kierunku wzdłużnym przepływu, natomiast poprawiły się wyniki doty-
czące intensywności turbulencji właśnie w tym kierunkuw porównaniu do kierunków
pozostałych.Zaobserwowanoponadto, że zastosowaniekombinacji SGSzużyciemmo-
deli subfiltrów naprężeń (SFS) oraz HOS jako filtru jawnego pozwoliło na poprawne
wyznaczenie wartości naprężeń turbulentnych.

Manuscript received July 21, 2010; accepted for print January 26, 2011