Sue Ellen Haupt1
e-mail: haupts2@asme.org
Frank J. Zajaczkowski
e-mail: fxz101@psu.edu
L. Joel Peltier2
e-mail: ljpeltie@bechtel.com
Applied Research Laboratory,
The Pennsylvania State University,
P.O. Box 30,
State College, PA 16804-0030
1
Detached Eddy Simulation of
Atmospheric Flow About a
Surface Mounted Cube at High
Reynolds Number
Modeling high Reynolds number (Re) flow is important for understanding wind loading
on structures, transport and dispersion of airborne contaminants, and turbulence patterns in urban areas. This study reports a high fidelity computational fluid dynamics
simulation of flow about a surface mounted cube for a Reynolds number sufficiently high
to represent atmospheric flow conditions. Results from detached eddy simulations (DES)
and zonal DES that compare well with field experiment data are presented. A study of
reducing grid resolution indicates that further grid refinement would not make a significant difference in the flow field, adding confidence in the accuracy of the results. We
additionally consider what features are captured by coarser grids. The conclusion is that
these methods can produce high fidelity simulations of high Reynolds number atmospheric flow conditions with a modest grid resolution. 关DOI: 10.1115/1.4003649兴
Introduction
Flow about a surface mounted blunt body has been a classic
testbed for computational fluid dynamics 共CFD兲 关1–9兴. It demonstrates separation, vortex shedding, and stationary vortical structures, all of which are difficult to model well. There has been
recent interest in pushing such CFD computations to higher Reynold’s numbers 共Re兲 that more closely match atmospheric flows
for studies involving wind loading on structures, dispersion of
contaminant around buildings, and characterizing flow in an urban
environment 关7兴. To that end, this study strives to carefully model
flow at atmospheric Re using high fidelity, but readily achievable,
grid resolution and state-of-the-art techniques and then to systematically relax the grid and modeling assumptions to characterize
the fidelity necessary to reproduce certain features of the high Re
flow. This information is valuable since modern atmospheric
simulations often require modeling multiple buildings in close
proximity in urban areas. It is useful to determine which features
require a very fine grid resolution and which will be captured with
a coarser grid.
The case of a surface mounted square cylinder at lower Re has
been widely studied. Several experimental studies, in particular,
are important for verifying numerical simulation results. One
physical modeling study widely used for the validation of numerical results is the experimental characterization of three dimensional flow around surface mounted prismatic obstacles performed
by Martinuzzi and Tropea 关10兴. They investigated flow around
obstacles in both water and air channels. Static pressure measurements, laser light sheet, oil film, and crystal violet visualization
techniques were used to record results. Comparison CFD studies
showed the ability of Reynolds averaged Navier–Stokes 共RANS兲
models to match the major features 关11–13兴. Although those studies were important steps toward simulating the details of flow
features at high Re, they are insufficient for realistic atmospheric
1
Present address: National Center for Atmospheric Research/Research Applications Laboratory, Boulder, CO.
2
Present address: Bechtel Corporation, Frederick, MD.
Contributed by the Fluids Engineering Division of ASME for publication in the
JOURNAL OF FLUIDS ENGINEERING. Manuscript received December 22, 2006; final
manuscript received July 10, 2010; published online March 15, 2011. Assoc. Editor:
Rajat Mittal.
Journal of Fluids Engineering
conditions because those experiments were performed for fully
developed channel flow at lower Reynolds numbers 共Re= 4
⫻ 104 and 1.2⫻ 105兲.
Richards et al. 关14–18兴 investigated full scale, high Reynolds
number 共Re= 4 ⫻ 106兲 flow around a 6 m cube at the Silsoe Research Institute, producing pressure data for streamwise and crossstream centerlines around the cube. Further investigation produced unsteady flow velocity data along the sides of the cube 关16兴.
These studies provided the case examined in the present paper and
are referred to as the “full scale” and “field measurement” data
throughout the remainder of this paper. Previously, Wright and
Easom 关19兴 used the Silsoe case for a RANS simulation using a
nonlinear k-␧ turbulence model. They were able to reproduce the
pressure distribution on the windward face of the cube but had
difficulty on the top, side, and leeward faces. It was determined
that isotropic turbulence models were inadequate in this case.
Some of the previous RANS CFD modeling results comparing the
Silsoe field experiments to RANS simulations are summarized by
Richards et al. 关20兴. Those computational solutions were generated as part of the Computational Wind Engineering 共CWE兲 2000
Conference Competition. Here, we wish to determine whether detached eddy simulation 共DES兲 approaches can improve on those
RANS approaches.
Our CFD methodology, including grid and model details, is
explained in Sec. 2. The results are described, compared with the
full scale field data of Richards et al. 关14兴, and evaluated for grid
independence in Sec. 3. Section 4 contains a discussion of the
results and suggestions for further research.
2
CFD Methodology
The experiments performed by Richards et al. 关14–18兴 are modeled here with CFD using both the standard DES methodology
with a Spalart–Allmaras 共SA兲 turbulence model 关21兴 and a recently proposed modification known as zonal DES 共ZDES兲 关22兴.
The computational domain, shown in Fig. 1, features a 6 m surface mounted cube situated within a domain that is 100 m high.
This domain height accommodates the experimentally measured
atmospheric boundary layer 共ABL兲 profile described in detail by
Richards et al. 关15兴.
2.1
LVE™,
ACUSOLVE. We use the commercial flow code, ACUSOas our computational engine. This solver is an incompress-
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and the functional relationship of ␯T to ˜␯ is
f v1 =
␹3
,
␹3 + cv31
␯T = ˜␯ f v1
f v2 = 1 −
˜␯
where ␹ = ,
␯
␹
,
1 + ␹ f v1
fw = g
冉
6
1 + cw3
6
g6 + cw3
g = r + cw2共r6 − r兲,
r=
冊
共1/6兲
˜␯
S̃共␬d兲2
共4c兲
S = 冑2⍀ij⍀ij,
S̃ = S +
˜␯
f v2,
共␬d兲2
⍀ij =
ible finite element code that offers several turbulence modeling
options 关23兴. ACUSOLVE™ allows various implementations and the
ability to customize the modeling strategy. It is based on the standard Navier–Stokes equations as reported in detail by Lyons et al.
关24兴. The filtered equations can be written as
⳵2ũir
⳵ ũir
⳵ ũir
1 ⳵ p̃
⳵˜␶ij
=−
+␯
−
+ ũrj
⳵xj
⳵xj ⳵ xj ⳵xj
⳵t
␳ ⳵ xi
共1兲
2.2 DES. For the present study, we use the DES subgrid
model to compute the flow. DES is a hybrid statistical/eddyresolving technique that harnesses the fidelity of large eddy simulation 共LES兲 in regions of massive separation, like the separated
flow downstream of the cube, while retaining much of the computational efficiency of RANS near boundaries and away from
regions of interest. DES discriminates the LES and RANS regions
by choosing a characteristic model length scale, d̃, that is smaller
than the characteristic grid scale, CDES⌬, and the local RANS
length scale, d,
d̃ ⬅ min共d,CDES⌬兲
⳵ ũir
=0
⳵ xi
共2兲
where the overset tilde denotes a full variable, ũi, one that has
both a statistical mean value, Ui, and a fluctuating component,
ui : ũi = Ui + ui, and where the “filtered”/”resolved” variables 共superscript “r,” ũri and p̃r兲 are computed and the effects of the “subfilter” motions 共superscript “s,” usi 兲 on the resolved field are collected in the subfilter stress, ␶ij, ˜␶ij ⬅ 共uri usj + usi urj + usi usj 兲r + 共uri urj 兲r
− 共uri urj 兲. Introducing an eddy-diffusion closure for the subfilter
stress, followed by collecting terms, yields
⳵t
+
⳵ ũir
ũrj
⳵xj
=−
1 ⳵ p̃
⳵
+ 2 关共␯ + ␯T兲S̃ijr兴
␳ ⳵ xi
⳵xj
共3兲
where ␯T, the eddy diffusivity, must be modeled and S̃rij = 共1 / 2兲
⫻关共⳵ũri / ⳵x j兲 + 共⳵ũrj / ⳵xi兲兴 is the strain rate tensor.
Our RANS closure is the SA one-equation turbulence model
关21兴. The SA model relates the eddy diffusivity, ␯T, to a computed
diffusivity, ˜␯, that satisfies the transport equation,
冉冊
˜
˜
˜␯
⳵␯
⳵␯
= cb1S̃˜␯ − cw1 f w
+ Uj
⳵xj
⳵t
d
+
2
+
冋
˜
⳵␯
1 ⳵
共␯ + ˜␯兲
␴ ⳵ xk
⳵ xk
册
˜ ⳵␯
˜
cb2 ⳵␯
␴ ⳵ xk ⳵ xk
共4a兲
where d is the distance to the nearest no-slip surface. The model
constants are
cb1 = 0.1355,
cb2 = 0.622,
cb1 共1 + cb2兲
,
cw1 = 2 +
␬
␴
cv1 = 7.1,
␴=
031002-2 / Vol. 133, MARCH 2011
cw3 = 2,
␬ = 0.41
共5兲
where CDES is an adjustable constant and ⌬ ⬅ max共⌬x , ⌬y , ⌬z兲.
When ⌬ Ⰶ d, the DES subgrid model becomes a Smagorinskytype LES. Likewise, when ⌬ Ⰷ d, the model remains RANS. Otherwise, the DES model blends the RANS and LES behaviors.
Strelets 关25兴 described how DES can be implemented for an arbitrary turbulence model and provided examples showing improved
turbulence statistics over RANS predictions in regions of massive
separation. ACUSOLVE™ implements the one-equation Spalart–
Allmaras turbulence model with curvature corrections in its
RANS mode. Its DES implementation, therefore, follows the
original prescription 关21兴.
2.3 Zonal DES. In addition to the standard DES technique, a
modification known as ZDES 关25兴 is implemented in order to
improve the agreement with the measurements. This zonal approach adds a discriminator function, ⌿, designed to retain RANS
modeling in boundary layers regardless of the local grid density.
The modification is implemented to correct an observed deficiency of DES—that it is allowed to transition from RANS to
LES in boundary layers, losing the statistical effect of turbulence
stresses without developing resolved turbulence stresses to compensate. Menter et al. 关13兴 described this aberration as gridinduced separation. Therefore, it is necessary to force the use of
RANS in the attached regions of the flow.
Specifically, we define a discriminator function, ⌿, which is
used in a black/white form,
␺ = min
冉
1 C ␮⍀ d 2⍀
d
,
2 a1k1/2 500␯
冊
共6a兲
␺ ⱕ 1 identifies boundary layers. The model constants, a1 and C␮,
are
a1 = 0.31,
2
3
共4b兲
cw2 = 0.3,
冊
The SA model diagnoses the time scale of the turbulence from the
mean field vorticity and chooses the characteristic length as the
maximum distance to the wall. The model constants and functions
are tuned to the data.
Fig. 1 Computational domain
⳵ ũir
冉
1 ⳵ Ui ⳵ U j
−
2 ⳵ x j ⳵ xi
C␮ = 0.09
共6b兲
Access to the turbulent kinetic energy, k, is required to compute ␺
for Eq. 共6a兲. However, k is not a variable directly available from
the Spalart–Allmaras model. Slimon 关22兴 deduced k from the
strain rate and eddy diffusivity,
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Table 1 Grid details
Total no. of elements
Near-wall spacing, m
Total no. of prism layers
Maximum element:
surface, m
Maximum element:
cube sides, m
Maximum element:
cube top, m
Maximum element:
density region, m
Growth ratio
No. of processors
Fig. 2 Computational mesh for the BASE case runs. „a… Top
view showing the refinement region, „b… side view through the
cube, and „c… close-up of the side view indicating the prism
layers near the wall.
冉
␯T r
k ⬇ max
S̃ ,k0
a1
冊
共6c兲
where k0 is a freestream value and S̃r ⬅ 共2S̃rijS̃rij兲1/2. Slimon 关22兴
also reported that numerical experiments showed that best results
were obtained when f v1 = f w = 1 and f v2 = 0, values also adopted
here. ZDES redefines d̃ such that
d̃ =
再
d
for ␺ ⱕ 1
min共d,CDES⌬兲 for ␺ ⬎ 1
冎
共7兲
The method sharply delineates RANS and LES regions but ensures a continuous, smooth solution across the ZDES interface.
Thus, DES is active only for ⌿ greater than 1. In this sense, the
algorithm is zonal. Otherwise, the governing equations are continuous throughout the flow. The ZDES algorithm was described
in detail by Slimon 关22兴.
2.4 Grids. The base case grid 共BASE兲 is a high resolution,
hybrid tetrahedral-prism grid. The unstructured tetrahedral-prism
mesh was generated with ANSYS ICEM CFD. Prism layers were extruded from solid walls to provide better modeling of near-wall
physics than tetrahedrons. A near-wall spacing of 0.003 m 共about
300 wall units兲 was used in grid construction, and a grid expansion rate of 1.2 was enforced in the wall normal direction, yielding a total of 22 prism layers. Tetrahedral elements surround the
prism layers. The size of the elements is limited to no larger than
0.85 m in the density region surrounding the cube. The far field
elements are allowed to grow to a maximum of 2.35 m. This high
resolution BASE grid contains 3.4⫻ 106 elements. Figure 2 displays the grid. A top view 共Fig. 2共a兲兲 indicates a refinement region
surrounding the cube. A side view 共Fig. 2共b兲兲 shows a cut plane
through the center of the cube, and a blow-up 共Fig. 2共c兲兲 indicates
the refinement of the prism layers.
Three progressively coarser grids were generated to perform a
grid study and are one-half, one-quarter, and one-eighth the linear
resolution of the base case, respectively. Care was taken to mainJournal of Fluids Engineering
BASE
BASE/2
BASE/4
BASE/8
3,387,676
0.003
22
2.35
916,663
0.006
19
4.70
222,709
0.012
15
9.40
66,666
0.024
12
18.8
0.20
0.40
0.80
1.60
0.05
0.10
0.20
0.40
0.85
1.70
3.40
6.80
1.2
11
1.2
4
1.2
1
1.2
1
tain consistency in grid quality across several resolutions. Our
goal is to determine what features become under-represented with
grid coarsening. For the remainder of this paper, these grids will
be referred to as BASE, BASE/2, BASE/4, and BASE/8. Table 1
compares the details of each.
Finally, an extremely fine near-wall grid, beginning at one wall
unit from the cube, was constructed to study the effect of a very
fine resolution. We found that there was no improvement with
such fine resolution. We believe that this observation is due to the
fact that wall functions are employed, which smoothly transition
from the turbulent region above to the viscous sublayer. Since
these functions act to produce a smooth vertical velocity profile, a
finer resolution is not helpful, and those results are not shown
explicitly here.
2.5 Boundary Conditions and Implementation. Inflow conditions were extracted from the Silsoe full scale experimental
data. Profiles of the inflow velocity and eddy viscosity are shown
in Fig. 3. The eddy viscosity was diagnosed from the measured
turbulence intensities and length scales 关15兴. Thus, both velocity
and turbulence profiles are matched to the full scale data. In the
experimental case, a reference velocity at a cube height was found
to be ⬃10 m / s. The Reynolds number based on this reference
and the cube height is Re= 4 ⫻ 106. An exit condition is used at the
downstream boundary that allows vortical structures to advect out
of the domain without unphysical pressure reflections. Symmetry
conditions are imposed on the upper and side boundaries.
Fig. 3 Inflow conditions from experiment
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Fig. 4 Time averaged horizontal streamlines near bottom wall
for BASE DES case as viewed from above
The ACUSOLVE model with DES implemented was run in unsteady mode. Selecting a time step of 0.1 s gives a maximum
Courant–Friedrichs–Lewy 共CFL兲 number of 1 in the wake for the
BASE grid. The BASE model was run for over 8000 time steps.
The first 1400 steps are discarded from the analysis. To derive
pressure coefficients, we select pressure profiles along the streamwise and cross-stream lines corresponding to the same distances
measured in the Silsoe full scale field experiments. Such profiles
are derived at 1000 step intervals and averaged for inclusion in the
plots presented below.
3
The DES Solutions
3.1 Qualitative Description. We first describe our time averaged results from the high resolution DES BASE case and then
compare the pressure and velocity profiles to the full scale experimental data and the CWE 2000 Competition RANS simulations
关20兴 for model validation. The BASE case averaging period is 7
min and omits a nonconverged spin-up period of 1400 time steps,
equivalent to 140 s.
Time averaged streamlines from the BASE DES case are
shown, looking down from above on a slice near the surface 共Fig.
4兲 and along the vertical center plane of the cube 共Fig. 5兲. These
results display the symmetry expected of a converged time averaged solution and capture the same observed flow features obtained via time averaged LES results at lower Re described by
Ferziger and Peric 关26兴. Incoming flow reaches a stagnation point
near the ground upstream from the cube and flows around the
sides of the cube, as seen in our DES simulation in Fig. 4. Further
above the ground, the flow impinges on the front face of the cube
and separates, and some of it descends into a region of reversed
flow. Figure 5 shows the stagnation point above the halfway point
in agreement with the full scale Silsoe field measurements. Just
upstream from the front face of the cube and along the lower
surface, there is a separation zone, which is the head of a horseshoe vortex. This horseshoe vortex extends along the sides of the
cube, as is evident in both Figs. 4 and 5. The top view 共Fig. 4兲
shows the horseshoe vortex in roughly the correct location. It
shows up as a vortex recirculation zone in Fig. 5. Of particular
note for the present results is the lack of reattachment for the
recirculation zone along the top face. While this agrees with lower
Fig. 5 Time averaged streamlines along a vertical slice
through the centerline for BASE DES case
031002-4 / Vol. 133, MARCH 2011
Fig. 6
Q-criteria isosurface colored by helicity
Re LES results 关11兴, it does not match the full scale results, which
do reattach. A single step of the transient solution is shown in Fig.
6. This figure plots the Q-criterion as a way to visualize curvature
in the flow. The Q-criterion is defined by 关27兴 as
冉
冊 冉
冊
⳵ Ui 1 ⳵ Ui ⳵ Ui
1 ⳵ Ui ⳵ Ui
=
+
+
−
= Sij + ⍀ij
2 ⳵xj ⳵xj
⳵xj 2 ⳵xj ⳵xj
Q = ⍀ij⍀ij − SijSij
共8兲
Here, 共⳵Ui / ⳵x j兲 is the gradient of the ith velocity component in
the j-coordinate direction, where i and j range from 1 to 3. Sij is
the strain tensor and ⍀ij denotes vorticity. Q is the computed
value of the Q-criterion, which is meant to help identify turbulent
features. The horseshoe vortex is the dominant feature in Fig. 6,
beginning upstream of the cube and extending around the cube.
The turbulent features form in the separation regions and extend
into the wake.
3.2 Impact of Grid Resolution. We investigate grid dependence by running the DES simulation on each of the four grids
described in Sec. 2 and detailed in Table 1. In addition, we wish to
assess which features are lost as the grid coarsens. Our rationale is
based on the fact that for various modern applications, such as
transport and dispersion or wind loading in urban areas, one must
compute flow around multiple buildings in an area, precluding
high resolution around each of the structures. Thus, it is useful to
assess which features are maintained in the coarsening and which
may be lost. The following discussion details the changes in features, and in particular, the coefficient of pressure, between the
differing resolutions as well as with the Silsoe full scale field data.
Figure 7 compares the coefficient of pressure 共Cp兲 between the
coarsened grids to the BASE case, both over the top and crossstream of the cube 共Fig. 7共a兲兲 and along stream of the cube 共Fig.
7共b兲兲. Over the top of the cube, none of the resolutions of DES
sufficiently capture the pressure drop. In general, the BASE/2 provides similar results to the BASE case. In fact, over the top of the
cube 共Fig. 7共a兲兲, the pressure drop seen using the BASE/2 grid
actually agrees better with the full scale field data than the BASE
grid simulations, although it is too low on the sides. Along stream,
however, the BASE/2 grid produces pressure profiles quite close
to those of the BASE grid although neither profile drops as deep
in the lee of the cube as the full scale field measurements. We
suspect that stochastic averaging issues impact these results. One
must be careful in interpreting these profiles—they are not steady
state values but rather represent an average over selected times in
an unsteady simulation.
The BASE/4 results show a sudden drop in accuracy in both
plots. Values across the top face lose the pressure drop in the
vortex as the grid resolution is relaxed, but the profile maintains
the same basic shape as the higher resolution DES runs. Along the
sides, however, the solution exhibits the rapid dip near the windward face characteristic of RANS results 共shown below兲. Likewise, the BASE/8 results fail to capture the pressure drops on both
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Fig. 7 Coefficient of pressure profiles at different grid resolutions
the side and top faces. Overall accuracy declines everywhere, and
even the windward face is no longer well matched 共Fig. 7共b兲兲.
Therefore, we conclude that the grid spacing of BASE/2 is sufficient to generate high fidelity Cp profiles. For a good match of Cp
to the field test data, the coarser BASE/4 and BASE/8 are not
adequate since they do not show the expected pressure drops observed in the center of the vortices.
We also compared the velocities at the points monitored in the
full scale field test data. The results confirm our conclusions above
and are not shown here. Once again, the BASE/2 solution comes
close to matching the velocity values of the BASE case. The
coarser BASE/4 and BASE/8 cases, however, in the regions closest to the cube where the flow is irregular and detached vary
significantly from the finer resolutions. For these regions, the finer
resolution of the BASE/2 grid is necessary to get closer matches
to the field test data. The proximity of the BASE/2 solution to the
BASE solution provides confidence that the grid resolution is adequate to provide a high fidelity solution. The DES solution alone,
however, does not show a sufficient pressure drop in the center of
the cube. Thus, ZDES results are included in the results discussed
below.
3.3 Comparison to Full Scale Silsoe Field Data and Prior
RANS Simulations. Figure 8 compares pressure coefficient 共Cp兲
results for the high resolution BASE case DES and ZDES runs to
Journal of Fluids Engineering
the full scale experimental field data taken at Silsoe and to the
CWE 2000 Competition RANS results reported in Richards et al.
关20兴. The RANS results include a standard k-␧ turbulence model
关28,29兴 and two k-␧ modifications, MMK and renormalized group
共RNG兲 theory based on the models developed by Tsuchiya et al.
关4兴 and Yakhot and Orszag 关30兴 respectively. Figure 8共a兲 shows
Cp profiles along a streamwise vertical centerline that moves up
the windward face of the cube, along the top face, and down the
leeward face. Generally, the comparison indicates that all models
match the windward face reasonably well. Standard k-␧ predicts
pressures that are too low on the front of the top face and premature reattachment. The RNG modification improves the profile but
underpredicts the pressure drop in the vortex along the entire top
face, while MMK fails to capture the low pressure over the top.
Our DES simulation, although performing better than MMK, also
exhibits a relatively flat profile, resulting from an oversized and
slow recirculation zone. The ZDES implementation, however,
shows improved agreement with the experimental data, particularly along the top face. It captures the reattachment much better
than standard DES or k-␧ models.
Figure 8共b兲 displays vertical transverse Cp plots that move up
the center of one side of the cube, across the top in the crossstream direction, and down the other side, as indicated in the inset.
The field experimental data exhibit minimum pressure in the cenMARCH 2011, Vol. 133 / 031002-5
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Fig. 8 Coefficient of pressure profiles comparing DES and
ZDES to full scale and RANS. „a… Along the centerline of the
cube, „b… along the transverse line of the cube, and „c… along
the horizontal mid-height line of the cube.
ter of the cube. The RANS solutions, as expected from the streamwise plots, underpredict the pressure drop across the top face.
Additionally, the Cp values moving up the side faces increase for
all of the RANS cases, while they decrease for the field study
data. Our DES simulation provides comparable and decreasing Cp
values along each face. The magnitude of these pressures is closer
to experimental values despite being steadier than the experimental values. On the top face 共center of the plot兲, ZDES shows significant improvement over the other simulation methods, reproducing the experiment’s drop in pressure toward the center of the
face. The low pressure zone is too small, however, and on the
sides of the cube, ZDES predicts too severe of a pressure drop. It
also shows an anomalous rise in pressure at the top edge similar to
the RNG and RANS simulations, but more pronounced here.
031002-6 / Vol. 133, MARCH 2011
Fig. 9 Velocities at selected positions/heights comparing DES
to full scale and RANS. Locations 1, 2, and 3 are 600 mm away
from the centers of the windward, side, and leeward faces, respectively, and include measurements at heights of 1 m, 3 m,
and 6 m. Locations 5, 6, and 7 are positioned 9 m away from the
same faces and include measurements at the same heights.
Location 4 is 600 mm above the top face of the cube with measurements at the center of the face, 2 m upstream, and 2 m
downstream of the center. „a…, „b…, and „c… indicate u/Uref,
v/Uref, and w/Uref, respectively.
Figure 8共c兲 is a set of Cp plots along a mid-height horizontal
line that moves along the windward face, back one side, and halfway along the leeward face 共see inset兲. Here, all simulations do a
reasonable job at predicting along the windward and leeward
faces. On the side face, the RNG model performs the best. DES
results continue to produce a steady and flat profile. The ZDES
curve once again shows disagreement along the side. An explanation for this disagreement will be proposed later by considering
the velocities along the side of the cube.
Figure 9 displays measured velocity values from seven points
around the cube at three heights each and compares our BASE
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Table 2 Reattachment and stagnation lengths
From cube center
Wake 共u = 0 at x = 0.01 h兲
Top face 共u = 0 at x = 0.01 h兲
Front face stagnation
Upstream stagnation
Full scale
RNG
MMK
K-E
DES
ZDES
11.44
0.6
3.5
⫺7.5
14
⫺0.1
4.4
⫺6.4
22.1
2.15
4.43
⫺8.26
13.85
None
3.93
⫺6.14
13.8
None
4.3
⫺8.2
13.1
0.18
4.3
ⴚ6.9
DES and ZDES results to the full scale data and the three RANS
results. At the windward locations 共1 and 5兲, all of the models
match the Silsoe field test data favorably. In the wake 共3 and 7兲,
both DES and ZDES perform well at 1 m and 3 m. At 6 m, the
DES u-velocity is too slow, while ZDES matches the Silsoe experimental data quite well. On the top of the cube 共point 4兲, DES
predicts reversed velocities at the center and downstream positions, supporting the observation that it was not able to produce a
reattachment over the cube top. Further, the velocity magnitudes
are relatively small, which is consistent with a large recirculation
zone. The ZDES produces velocities over the cube that compare
favorably with the field measurements. Recall, however, that the
Cp profiles produced by the ZDES model did not match the measurements and showed more spatial variability than observed.
Thus, it is not surprising that additional disagreement is found in
the velocity measurements near the side of the cube at location 2.
At heights of 1 m and 3 m, ZDES predicts reversed flow, while
the Silsoe study observed forward flow. This indicates that the
Silsoe and ZDES measurements were taken on opposite sides of
the side vortex core and could suggest that the time averaged
Silsoe side vortex is narrower than that predicted by ZDES or that
it has reattached by midface. In the far side wake region 共6兲, all of
the models perform well. In this qualitative comparison, however,
the ZDES produces better agreement with the measurements than
the RANS runs reported previously 关20兴.
Table 2 shows a quantitative assessment of the reattachment
and stagnation lengths for the Silsoe full scale field experiment
data and each model. BASE ZDES predicts reattachment most
accurately both in the wake and on the top of the cube. On the
front face of the cube, the CFD models all predict stagnation too
high on the cube but agree well with each other. The stagnation
point upstream of the horseshoe vortex is best predicted by our
BASE ZDES. Generally, BASE ZDES performs the best for predicting stagnation and reattachment for three of the four
measurements.
Richards et al. 关20兴 compared models with the average magnitude of coefficient of pressure differences. Table 3 reproduces this
comparison and includes our DES and ZDES results. The best
results in each row are indicated in boldface and underlined. We
see that for four of the six criteria, ZDES performs at least as well
as any other model, with DES being the best at predicting the
vertical transverse line. This analysis again supports the conjecture that the DES-type models improve on RANS for modeling
details of detached wake flow.
4
Discussion
The Silsoe field experiments provide an opportunity to evaluate
best practices for modeling flow about buildings at Re appropriate
for the atmospheric boundary layer. Overall, the DES results, particularly the zonal version ZDES, match the Silsoe full scale field
measurement data as well as or better than the RANS solutions,
with the added benefit of predicting turbulence. The agreement of
the reattachments was quite good for the ZDES model, although
the BASE DES did not reattach on the top face of the cube. The
primary observed discrepancy in Cp observations when compared
with experimental data is the disagreement over the top face of the
cube. While the tuned RNG k-␧ model improves the RANS agreement significantly, all of the CFD results underpredict the pressure
drop in the vortex over the bulk of this face. The closest to matching the experimental data is the ZDES simulation, but it comes at
the expense of overpredicting the low pressure anomaly on the
sides of the cube. ZDES also best captures reattachment over the
top of the cube.
Details of the pressure field are expected to be sensitive to the
exact location of the side vortices. A difference between the CFD
and the experiment is the ability to control the direction of the
mean wind. In the CFD, the wind direction is prescribed so that
the oncoming wind is exactly perpendicular to the upstream face
of the cube. This condition was also sought in the experiment;
however, local meteorological conditions cannot be controlled. Instead, the experimental results rely on conditional sampling of the
experimental data to extract instances when the oncoming wind
had the proper orientation. These instances become members of a
conditional ensemble whose mean is presented. In fact, it has been
noted that the characteristics of reattachment are modified when
the experimental cube is pitched slightly into the wind 关28兴. One
expects some effect of the fluctuating wind field to be aliased into
the mean simply because of limited control of the conditional
ensemble. These slight variations in the effective mean wind direction and the associated changes in the mean side vortex positions may account for much of the differences between the CFD
result and the experimental data, particularly on the sidewalls.
There are several other possible reasons for discrepancies between the BASE case and full scale experimental results. Richards
et al. 关20兴 noted that modeling the atmospheric boundary layer
with both full scale velocity and turbulence profiles creates a nonhomogeneous boundary layer with the k-␧ turbulence model. For
the present study, both profiles are based on full scale data and the
Table 3 Average magnitude of coefficient differences
Vertical center line
Vertical transverse line
Horizontal mid-height line
u / Uref
v / Uref
w / Uref
Journal of Fluids Engineering
K-E
RNG
MMK
DES
ZDES
0.18
0.32
0.11
0.11
0.02
0.04
0.14
0.21
0.05
0.25
0.01
0.05
0.19
0.19
0.09
0.13
0.04
0.05
0.13
0.07
0.13
0.20
0.02
0.05
0.06
0.20
0.10
0.11
0.01
0.04
MARCH 2011, Vol. 133 / 031002-7
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Spalart–Allmaras turbulence model is used. Further, the eddy viscosity inlet profile used here is based on turbulence measurements
taken at only four different heights and is linearly interpolated
共Fig. 2兲. As a result, the current inlet boundary layer is not homogeneous and evolves for a short distance downstream. The outflow
boundary layer in our solutions has a similar velocity profile but a
much lower eddy viscosity profile.
In summary, the results of DES and ZDES simulations of flow
around a surface mounted cube have been validated with experimental data. It is shown that ZDES performs at least as well as the
best RANS solutions, and on most metrics, better. Further, there is
promise that future studies of the variability in wind direction will
improve DES agreement with full scale experimental results. In
addition, a study of model performance as a function of grid resolution demonstrates that the BASE/2 mesh has very similar results as the BASE case, establishing an appropriate resolution for
the high fidelity DES-type modeling of atmospheric flow around a
blunt body.
Acknowledgment
Robert P. Wilson performed much of the modeling effort for
this paper, funded by the ARL Undergraduate Honors Program.
The authors would like to thank ACUSIM Software Inc. for their
support.
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