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The 11th International Topical Meeting on Nuclear Reactor Thermal-Hydraulics (NURETH-11)
Popes’ Palace Conference Center, Avignon, France, October 2-6, 2005.
Paper: 405
LARGE EDDY SIMULATION OF FLOW ACROSS IN-LINE TUBE
BUNDLES
Sofiane, Benhamadouche1, Dominique, Laurence,
Electricité de France R&D, MFTT, 6 quai Watier, 78400 Chatou, France.
[email protected]
Nicolas, Jarrin, Imran, Afgan, Charles, Moulinec.
Manchester University, MAME Dept, George Begg Bldg, Sackville St. PO box 88, Manchester
M601QD, UK
ABSTRACT
The cross-flow in a 3D square in-line tube bundle is computed using Large Eddy Simulation (LES)
with periodic boundary conditions. Both the in-house EDF code Saturne and CD-Adapco code STAR
CCM are tested in the present work. The numerical method in both codes is based on a finite volume
approach on unstructured grids with a collocated arrangement for all the unknowns. Two aspect ratios
are tested, P/D=1.44 and P/D=1.75. The influence of two parameters is explored herein; the accuracy
of the pressure gradient calculation and the spanwise extrusion length (Lz). Drag and Lift coefficients
and their rms values are calculated and compared to available experimental data. It is shown that a
high accuracy is needed for pressure gradient calculation to obtain a physical solution. In addition, one
needs at least Lz = 2D in the homogeneous direction, where D is the diameter of the tubes, to
accommodate the use of periodic boundary conditions. Drag and lift coefficients have reasonable
values, but their rms values are very sensitive to the extrusion length. Although, rms values are
satisfactory for P/D=1.44, they are much larger than experimental data for P/D=1.75. All the cases
presented are computed with Saturne, and Star CCM is used on some cases to confirm asymmetric
results and sensitivity for the narrow gap case.
KEYWORDS
Large Eddy Simulation, In-line Tube Bundle, Drag and Lift Coefficients
1.
INTRODUCTION
Heat exchangers are vital components of power generation systems, which present many issues that
are not understood in detail and others that need to be optimized. These include heat transfer
enhancement vs. head loss, hot spots, vibrations, fluid structure coupling, in line / staggered / oblique
flow direction, two phase flows and dust deposition …
A major parametric CFD study of tube bundles would seem timely and worthwhile in view of the
billion dollars per year potential savings worldwide, for coal power plants alone (Bouris et al., 2001).
However, Reynolds averaged Navier Stokes (RANS) simulations have so far failed to produce reliable
predictions of flows across tube bundles.
This provocative statement is based on numerous reported RANS simulations on staggered tube
bundles: the ERCOFTAC workshops on refined flow modeling of 1993 and 1994, (Rollet-Miet et al.,
1999), (Sebag et al., 1991), (Meyer, 1994), and (Bouris and Bergeles, 1999). The general conclusion is
that even advanced RANS models such as non-linear, realizable and RNG types of k-epsilon models
severely underestimate the high turbulent kinetic energy levels observed in densely packed tube
bundles. Paradoxically the standard k-epsilon model returns reasonable predictions of mean velocities
1
Corresponding author
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The 11th International Topical Meeting on Nuclear Reactor Thermal-Hydraulics (NURETH-11)
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and global level of turbulent kinetic energy, but this is purely by chance thanks to the erroneous
overproduction of kinetic energy on the impinging side of the tubes. This artificially raises the overall
turbulence intensity but the locations of the maximum and minimum are erroneous. These
discrepancies have led (Hassan and Ibrahim 1997) and (Bouris et al. 1999 and 2001), to resort to «two
dimensional LES» with some success including the prediction of turbulence levels. However debatable
this approach may be, it yielded better results for the mean velocities and turbulent kinetic energy than
RANS models, thus allowing some analysis of stress loading, heat transfer and deposition rates which
are highly dependent on the turbulence intensity. (Rollet-Miet et al. 1999) and later (Benhamadouche
and Laurence 2003) applied respectively an industrial finite element code, N3S, and an industrial finite
volume code, Saturne, to 3D LES of the flow in a staggered tube bundle and obtained good results and
observed that the subgrid scale model (whether Smagorinsky or dynamic) had very little effect. The
present simulations use the LES implemented in Code_Saturne to compute another configuration of
tube bundle, the square in-line one. The commercial code, STAR CCM is then used to confirm the
present results.
2.
NUMERICAL METHOD
A finite volume code for complex geometries, Code_Saturne (Archambeau et al. 2004), is used to
solve 3D Navier-Stokes equations on unstructured grids. The flow is assumed incompressible and
Newtonian and the density ρ is constant. Let u be the filtered velocity when using LES. The filtered
Navier-Stokes equations can be written for LES:
∂ τ ij
⎧∂ u i ∂ u i u j
∂ 2 ui
1 ∂ p
ν
, i ∈ {1,2,3}
+
=
−
+
−
⎪
∂ xj
ρ ∂ xi
∂ xi ∂ x i ∂ x j
⎪∂ t
(1)
⎨
⎪∂ u i
⎪∂ x = 0
i
⎩
As (Rollet-Miet et al. 1999) have shown that the subgrid scale model is not crucial in the tube
bundle case, only the standard Smagorinsky model is used herein:
1
3
τ ij − τ kk δ ij = −2ν t S ij = −2(C S ∆) 2 S S ij
(2)
S ij is the filtered strain rate tensor ( S = 2 S ij S ij ), νt the turbulent viscosity and ∆ the filter length.
As the cells used in the present work are hexahedral (even if Saturne or STAR CCM accept cells of
1
3
any shape), one can take ∆ = 2Ω , where Ω is the volume of a cell. The Smagorinsky constant, CS,
is set to 0.065 .
In the collocated finite volume approach used here, all variables are located at the centers of
gravity of the cells (which can be of any shape). The momentum equations are solved by considering
an explicit mass flux (the three components of the velocity are thus uncoupled). Velocity and pressure
coupling is insured by a prediction/correction method with a SIMPLEC algorithm (Ferziger & Peric,
1999). The Poisson equation is solved with a conjugate gradient method. The collocated discretization
requires a (Rhie and Chow 1982) interpolation in the correction step to avoid oscillatory solutions. In
Saturne, a second order centered scheme (in space and time) is used. It is Crank-Nicolson in time with
a linearized convection, and the second order Adams-Bashforth method is used for the part of the
diffusion involving the transposed gradient operator, that couples the velocity components. The nonorthogonalities are taken into account with a reconstruction technique explained in (Archambeau et al.
2004) and (Ferziger and Peric 1999). When a non-orthogonal grid is used, the matrix contains the
orthogonal contribution only and the non-orthogonal part is included in the right hand side of the
equation. This is known as the deferred correction (the non-orthogonal part is explicit), however one
can iterate on the system to make it implicit. Typically 5 iterations are performed in Saturne and the
CPU cost is double compared to the deferred correction. The effect of these two different methods will
be shown later.
The 11th International Topical Meeting on Nuclear Reactor Thermal-Hydraulics (NURETH-11)
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3.
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CASE DESCRIPTION
The present test-cases require only periodic and wall boundary conditions. All the meshes used in the
present work do not require wall functions, thus no-slip boundary conditions are used.
In-line configurations of tube bundles are computed in the present work. The bundle is assumed
infinite in all directions, thus one needs to choose a periodic cell (as shown in figure 1) to compute the
whole bundle. Although several tests have been performed with several cells, one will focus on the
cell with two tubes to minimize the computational costs. As periodic boundary conditions are used in
the three directions of space, the mass flow is imposed by the mean pressure gradient. This is done in
Saturne by adding a source term to the momentum equations as shown in equation (3).
dp
k +1
= dp + α
k
2(U k − U g ) − (U k −1 − U g )
2 ∆t
(3)
dp k is the source term (homogeneous to a pressure gradient) at the k th time iteration, U k the bulk
velocity calculated in a chosen plane of the computational domain (the first periodic plane in the
streamwise direction for example), ∆t the time step and α a relaxation coefficient as this formulation
may be unstable at the beginning of the calculation. In STAR CCM, a constant pressure gradient that
gives the right bulk velocity is added to the right hand side of the momentum equations. Boundary
conditions are shown in figure 2. No particular initial field is used; the initial value of the velocity is
zero. The relaxation coefficient has been taken to 10% to stabilize the imposed flux at the beginning of
the calculation. Drag and Lift coefficients are calculated for the central tube at each time step. When
frequencies of oscillations are given, this is done by using Power Density Spectra (PDS) and localizing
the peak values (the most energetic frequency). All the experimental data used in the present paper are
taken from (Chen, 1987).
P and D stand respectively for the horizontal or vertical distance between two centers of tubes and
the diameter of the tubes. The given Reynolds numbers (Re) are based on the diameter of the tubes, the
kinematic viscosity of the fluid and the gap velocity. Two configurations are studied with Saturne:
P
P
= 1.44 , Re = 70 000) and ( = 1.75 , Re= 20 000). STAR CCM calculations are done for
D
D
P
( = 1.5 , Re = 45 000). This aspect ratio is close to 1.44 and the two configurations can be compared
D
(
as they are both at high Reynolds numbers.
The same mesh is used for the two configurations with Saturne (figure 3a); whereas mesh for
STAR CCM is shown in figure 3b. Table 1 summarizes the different parameters of the main
calculations. The number of cells in 2D indicates the number of 2D cells obtained by cutting the
computational domain with a plane orthogonal to the spanwise direction (this 2D mesh is then
extruded in the z direction to obtain the 3D mesh).
Case 1
Case 2
Case 3
Code
Saturne
Saturne
STAR CCM
P/D
1.44
1.75
1.50
Re
68 420
20 000
45 000
Table 1: Test-case configurations
Number of cells in 2D
11 000
11 000
12 800
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Popes’ Palace Conference Center, Avignon, France, October 2-6, 2005.
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4.
Figure 1: Possible computational cells
Figure 2: Boundary conditions
Figure 3a: Mesh used with code Saturne
Figure 3b: Mesh used with code STAR CCM
RESULTS
4.1 Pressure accuracy,
P
= 1.44
D
The need of high accuracy for pressure is shown in this section. As the mesh is non-orthogonal, a
reconstruction method of gradients has to be used to take into account the non-orthogonalities that
occur particularly with the present meshes. One can either use a deferred non-orthogonality correction
(deferred n-o.c.) where the reconstruction is put explicitly on the right hand side of the Poisson
equation or iterate on the reconstruction to make it implicit (implicit n-o.c.). The two methods have
been tested with Saturne for
P
= 1.44 . The extrusion length in the spanwise direction is Lz = D .
D
Figures 4 and 5 show an instantaneous pressure field for the two methods. It appears that the
pressure with an explicit n-o.c. exhibits a staggered mode as the peak of the pressure is alternatively on
the top or on the bottom of the tubes. According to (Chen, 1987) this pattern occurs for wider crossstream spacing.
When implicit n-o.c. is used, the pressure field has the same behavior on all the tubes and shows a
high pressure region on the bottom of the tubes, which is believed to be the correct solution according
to the experimental lift coefficient. Figure 6 shows instantaneous evolution of the lift coefficient
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Popes’ Palace Conference Center, Avignon, France, October 2-6, 2005.
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computed for the two cases. With deferred n-o.c., the lift coefficient has a non-zero mean value. The
mean value of the lift is zero when implicit n-o.c. is used although a non-symmetrical solution is
obtained for the pressure. There is no numerical reason to obtain a high-pressure region on the bottom
or on the top as the mesh is perfectly symmetrical. The solution is stable and the phenomenon is
similar to the Coanda effect obtained with diaphragms. It is mentioned in (Chen, 1987, p. 322) that
Heinecke and Mohr observe experimentally a non-symmetrical solution for the pressure field. It is
observed that the high-pressure region is compensated with a low-pressure one just downstream (see
figure 5), what gives a zero mean lift coefficient.
This case is thus sensitive to the pressure resolution. Initially it was believed that this was due to
insufficient representative cells so a test case with four tubes (figure 1) was run. Figure 7 exhibits the
mean value of pressure for this case. The alternating mode of the stagnation pressure is again obtained.
These observations have been confirmed by the code STAR CCM which uses an explicit deferred
correction on the pressure. Figure 8 shows the pressure field obtained for
Figure 5: instantaneous pressure field
Figure 4: instantaneous pressure field
(
P
= 1.5 with STAR CCM.
D
P
= 1.44 , Lz = D , deferred n-o.c)
D
(
P
= 1.44 , Lz = D , implicit n-o.c)
D
0.5
0
−0.5
0
0.5
1
temps (s)
1.5
Figure 6: instantaneous lift coefficient (black: low
accuracy, red: high accuracy)
2
Figure 7: mean pressure field (
P
= 1.44 , Lz = D ,
D
deferred n-o.c, four tubes)
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Popes’ Palace Conference Center, Avignon, France, October 2-6, 2005.
Figure 8: mean pressure field (
4.2 Length extrusion L z ,
P
= 1.5 , Lz = D , deferred n-o.c, STAR CCM code)
D
P
= 1.44
D
All the following cases with Saturne will be carried out with an implict n-o.c. in pressure gradient
calculations.
The same 2D mesh has been used by taking L z = 2 D . The results are qualitatively similar the
ones with L z = D . Figure 9a shows the mean pressure field obtained by Saturne. A non-symmetrical
solution is observed, as was the case with L z = D . A calculation was also done with STAR CCM code
with an explicit deferred correction,
P
= 1.5 and L z = 2 D , shown in Figure 9b. The same behavior
D
for the pressure field is noticed. This is strange as the case was sensitive to the pressure accuracy with
STAR CCM while using L z = D . This is a first indication that one has to take at least L z = 2 D to
compute this case to be insensitive to the periodic boundary conditions in the spanwise direction.
Figures 10 and 11 show the mean velocity vector field for Saturne and STAR CCM codes. Two
recirculation bubbles coexist, a big one downstream the tube in the bottom due to the acceleration of
the fluid and a small one on the top. The shear stress in the bottom of the tube is then higher than in
the top. To illustrate this, figure 12 shows the fluctuations in the streamwise direction. Figure 13 gives
the evolution of normalized pressure coefficient along the curvilinear coordinate around the central
tube. Both codes agree very well with each other, though somewhat shorter time-sampling was
collected with Star-CCM. As one can see, the pressure peak is clearly offset from the axis, in
correspondence with the velocity plot. Moreover the Cp curve show that the stagnation point pressure
is compensated by a low-pressure region (negative values of Cp) on the same side, which gives an
overall zero mean lift coefficient.
The mean values of the drag and lift coefficients for the pressure forces and their rms values are
summarized in table 2. It has been observed that viscous coefficients are small compared to pressure
ones (not more that 10%). Drag and lift coefficients are computed using the gap velocity as a reference
velocity and the reference surface is S = L z D . The drag and lift coefficients remain unchanged and
are close to usual experimental values. The rms values are more sensitive to the extrusion length
particularly for the rms value of the drag coefficient which is two times bigger with L z = D . To
validate our approach, the experimental values found in (Chen, 1987) for
P
T
= 1.42 ,
= 1.42 and
D
D
Re = 1360 are used. T is the vertical spacing between the tubes. The coefficients should not be
strongly affected by variations of the turbulent Reynolds numbers (this is observed in Chen’s book for
The 11th International Topical Meeting on Nuclear Reactor Thermal-Hydraulics (NURETH-11)
Popes’ Palace Conference Center, Avignon, France, October 2-6, 2005.
7/12
P
= 1.75 ). The rms of the lift coefficient is equal to 0.078 which is close to the numerical value
D
found in the present work with Saturne and L z = 2 D . It would have been interesting to test a wider
P
= 1.75 . At this stage one can argue that it is
extrusion. This is done in the next section with
D
mandatory to take an extrusion length at least equal to 2D. It is shown next that this extrusion length is
enough for
P
= 1.75 .
D
An analytical formula for the Strouhal number (4) is proposed in Chen’s book. It depends only on
the tube spacing but not on the Reynolds number which is not really realistic but gives a good order of
the Strouhal number.
St =
For
1
(4)
P
2( − 1)
D
P
= 1.44 , one finds S t = 1.13 . By applying PDS to the lift’s signal, two main peaks are obtained
D
around the frequencies 35 Hz and 50 Hz (see figure 14). These two values correspond respectively to
the Strouhal numbers 0.8 and 1.25 which straddles the analytical model. This means that the vortex
shedding detected in the shear regions with LES is realistic. The Strouhal number obtained with STAR
CCM and based on the gap velocity is around 0.9 which is also close to Saturne results.
It is concluded that the length of the extrusion has to be at least 2D to obtain a physical solution
and realistic drag and lift coefficients. The previous configuration corresponds to an industrial one.
Unfortunately, additional experimental data is only available for the
P
= 1.75 case. This is presented
D
in the next section.
case
Saturne, L z = D
CL
0.001
CD
0.339
C’L
0.107
C’D
0.091
Saturne, L z = 2 D
0.013
0.350
0.074
0.049
STAR CCM, L z = 2 D
-0.004
0.351
0.032
0.026
Table 2: Drag and Lift coefficients
Figure 9a: mean pressure field (Saturne,
Figure 9b: mean pressure field (STAR CCM,
P
= 1.44 , L z = 2 D , implicit n-o.c)
D
P
= 1.5 , L z = 2 D , deferred n-o.c)
D
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8/12
Figure 10: mean velocity field (Saturne,
Figure 11: mean velocity field (STAR CCM,
P
= 1.44 , L z = 2 D , implicit n-o.c)
D
P
= 1.5 , L z = 2 D , deferred n-o.c)
D
1.00
SATURNE
STAR CCM
0.75
Normalized Pressure
0.50
0.25
0.00
-0.25
-0.50
-0.75
Azimuthal Angle
-1.00
0
Figure 12: u 'u ' (Saturne,
P
= 1.44 , L z = 2 D ,
D
45
90
135
180
225
3e−04
2e−04
1e−04
0e+00
0
315
Figure 13: Cp around the central tube (Saturne and
STAR CCM, L z = 2 D )
implicit n-o.c)
−1e−04
270
50
100
150
frequence (Hz)
Figure 14: DSP of the lift coefficient (Saturne,
P
= 1.44 , L z = 2 D , implicit n-o.c)
D
360
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5. ASPECT RATIO
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P
= 1.75
D
In this section all the calculations are done with Saturne and with implicit deferred n-o.c. The
Reynolds number is moderate and equal to 20 000. The parameter which varies in the present section
is the extrusion length ( L z = D , L z = 2 D , L z = 2.5 D and L z = 5 D ). All the meshes have the same
2D mesh and exactly the same numerical options.
Figures 15 and 16 show respectively the mean pressure field and the mean velocity vector for the
case with L z = 2 D . A perfectly symmetrical solution is observed with two high-pressure regions
upstream the tubes and two symmetrical recirculation bubbles downstream. This entails naturally a
zero mean lift coefficient. Table 3 gives the drag and lift coefficients and their rms values. The last
line indicates the range of the rms values in an experiment (Chen, 1987) with exactly the same
parameters (as the number of tubes in an experiment is limited, only the middle tubes which are
supposed to be independent from the inlet and outlet conditions have been considered). The mean
drag and lift coefficients are not available for this set of data. These results are also available in
(Zdravkovich, 2003, p. 1137). As it was the case with the previous tube spacing, drag and lift
coefficients are not sensitive to the extrusion length. The lift coefficient has almost a zero mean value
and the drag coefficient has a value around 0.34. The rms values are very sensitive to L z particularly
when this parameter is small. Both the rms values of the drag and the lift have an asymptotic behavior
after L z = 2 D . The rms of the drag is around 0.07 whereas that of lift is around 0.3. Table 3 shows
that the value L z = 2 D is enough to simulate the present case. Although, the rms value of the drag
coefficient is close to the experiment, the rms lift coefficient is still three times higher than the
experimental value. The use of the dynamic model seems mandatory and this will be done in a future
study.
The analytical formula (4) indicates that the Strouhal number should be of the order of 0.7. Table
4 gives the Strouhal numbers for the different cases. All the cases exhibit two main frequencies
respectively around 0.7 and 1.2. The first frequency is very close to the analytical result. The case
Lz=D is particular as it exhibits the two frequencies but with more energy at the second one (see figure
17). In the three other cases, the behavior is different as the two main frequencies have a comparable
energy. It is then highly recommended to take at least the value Lz=2D to simulate the present case.
Lz = D
Lz = 2D
L z = 2 .5 D
L z = 5D
Chen exp.
CL
CD
C' L
C' D
-0.011
0.358
0.430
0.138
-0.017
0.331
0.291
0.074
-0.010
0.331
0.313
0.076
-0.015
0.339
0.307
0.070
-
-
0.08-0.10
0.04-0.08
Table 3: Drag and lift coefficients and their rms values (
Lz = D
Lz = 2D
L z = 2 .5 D
L z = 5D
Analytical formula
St1
St2
0.73
1.22
0.74
1.18
0.73
1.19
0.72
1.18
0.7
-
P
Table 4: Strouhal numbers (
= 1.75 )
D
P
= 1.75 )
D
10/12
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Popes’ Palace Conference Center, Avignon, France, October 2-6, 2005.
Figure 15: mean pressure (
P
= 1.75 , L z = 2 D )
D
Figure 16: mean velocity (
Figure 16: DSP of the lift coefficient (
P
= 1.75 , L z = 2 D )
D
P
= 1.75 )
D
6. CONCLUSIONS
The in-line configuration of infinite tube bundle is investigated in the present paper. Two tube spacing
have been computed with collocated finite volumes using Large Eddy Simulation (LES):
and
P
= 1.44
D
P
= 1.75 .
D
For the narrow gap, a strongly asymmetric solution is obtained, with one large and one small
vortex in the mean velocity field, and a stagnation point well off the symmetry plane. The larger
vortex is on the same side as the stagnation point, which deflects the main street flow towards the
same side for the next tube. Hence all tubes in the same line have the same asymmetry. In earlier
simulations, a depth of only 1D had been used in the homogeneous direction. In this case a lower time
accuracy scheme for the pressure yielded a spatially alternating pattern for the stagnation point,
whereas only a higher accuracy pressure scheme seemed to produce the correct results. However,
when L z = 2 D , both schemes yielded the same correct pattern. This dependency on the extrusion
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Popes’ Palace Conference Center, Avignon, France, October 2-6, 2005.
length has been confirmed with
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P
= 1.75 in which the drag and lift values have an asymptotic
D
behavior after L z = 2 D . This is due to the periodic boundary conditions which require a spanwise
length twice larger than the bigger 3D turbulent structures. One assumes that in the case L z = D , two
dimensional structures may be created that entail a non-physical solution and wrong rms values for the
drag and lift coefficient otherwise. The higher accuracy pressure scheme yielded correct results with
L z = D but is not explained. Perhaps it is coincidental, or perhaps higher pressure resolution allows a
breakdown of the 2D solution but smaller 3D perturbations. With
P
= 1.44 and L z = 2 D , the results
D
are very satisfactory and this promising for future calculations which aim to estimate the derivative of
the lift coefficient for a series of fixed displacements of the central tube in the vertical direction. These
results can be exploited to compute a realistic fluid-structure with Quasi Linear Models for example
(QMI) (Price and Païdoussis, 1997).
Unfortunately, the conclusions are not totally optimistic for
P
= 1.75 . In this case, one shows
D
that a symmetrical solution is obtained, which is physical because of the tube spacing, but the rms
values of the lift coefficient are seriously overestimated. The next step in this work is to test the
dynamic Smagorinsky model in which the Smagorinsky constant varies in space and is more
representative of the reality in the boundary layers, which will probably affect considerably the lift
coefficient.
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Popes’ Palace Conference Center, Avignon, France, October 2-6, 2005.
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