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Original Article
Large eddy simulation study of turbulentflow around smooth and
rough domes
N Kharoua and L Khezzar
Date received: 4 September 2012; accepted: 17 December 2012
Abstract
Large eddy simulation of turbulent flow around smooth and rough
hemispherical domes was conducted. The roughness
of the rough dome was generated by a special approach using
quadrilateral solid blocks placed alternately on the dome
surface. It was shown that this approach is capable of
generating the roughness effect with a relative success. The
subgrid-scale model based on the transport of the subgrid
turbulent kinetic energy was used to account for the small
scales effect not resolved by large eddy simulation. The
turbulent flow was simulated at a subcritical Reynolds number
based on the approach free stream velocity, air properties, and
dome diameter of 1.4 105. Profiles of mean pressurecoefficient,
mean velocity, and its root mean square were predicted with good
accuracy. The comparison between the
two domes showed different flow behavior around them. A
flattened horseshoe vortex was observed to develop around
the rough dome at larger distance compared with the smooth dome.
The separation phenomenon occurs before the
apex of the rough dome while for the smooth dome it is shifted
forward. The turbulence-affected region in the wake was
larger for the rough dome.
Keywords
Computational fluid dynamics, large eddy simulation, domes, wind
load
Introduction
Flow around hemispherical domed structures is rele-vant to a
variety of practical engineering applicationsnotably domed
buildings, roofs, and sub-ocean struc-tures. Domes were also used
in hydraulic channels tostudy the shedding of hairpin vortices in
their wake ina vein to understand the development of the
near-wallregion of turbulent boundary layers.1 For
robustengineering design of such structures, detailed
under-standing and knowledge of the flow structure arounddomes is
necessary.
The flow around domes is three-dimensional andcontains large
scale highly unsteady motions with sep-aration. Complex vortical
structures with sheddingand multiple reattachment and separation
areas char-acterize this flow. Several parameters affect the
flowbehavior around domes; they include the dome shape,the Reynolds
number (based on the approaching freestream velocity and dome
diameter), the inflow con-ditions such as the approaching boundary
layer shapeand turbulence content, the upstream-floor and
domesurface, and the surroundings topology.
A number of relevant experimental studies wereconducted on
domes. Taniguchi et al.2 conductedan experiment to identify the
effects of the domesize and the characteristics of the approaching
bound-ary layer on the pressure coefficient and integral
properties such as drag and lift coefficient. The pres-sure
distributions along the symmetry plane of thehemisphere were found
to be highly similar in theregion of 050, but show a marked
dependency onthe Reynolds number in the region from 50 to
120.Savory and Toy3 conducted experiments to study theeffect of
three different approaching boundary layersand dome surface
roughness on the mean pressuredistribution and on the critical
Reynolds numberbeyond which the pressure distribution
becomesinvariable. For the artificially roughened dome sub-jected
to a thin boundary layer, the reattachmentlength of the downstream
recirculation zone wasequal to 1.25, the dome diameter from its
axis.Tamai et al.4 explored the formation and sheddingof vortices
from a dome in a water tunnel and deter-mined the frequencies
characterizing each phenom-enon for a rather low Reynolds number of
104.Approaching the front side, successive recirculationzones,
increasing in size and forming the well-known
Department of Mechanical Engineering, Petroleum Institute,
United Arab Emirates
Corresponding author:
L Khezzar, Department of Mechanical Engineering, Petroleum
Institute,
PO Box 2533, Abu Dhabi, United Arab Emirates.
Email: [email protected]
Proc IMechE Part C:
J Mechanical Engineering Science
0(0) 115
! IMechE 2013
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DOI: 10.1177/0954406212474211
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horseshoe vortices, were observed. The number ofthese structures
increases with increasing Reynoldsnumber until a critical value of
about 3000 beyondwhich only one single recirculation zone
persistswith a constant size. Cheng and Fu5 studied theeffect of
Reynolds number on pressure distribution.This study identified
three modes of flow separationin the apex region: for low Reynolds
below 1.8 105,the free shear layer which develops in the wake
regionseparates without reattachment on the dome surfacethis
corresponded to one single peak of the fluctuatingpressure
coefficient. At a Reynolds number of1.8 1053 105, a second peak was
observed corres-ponding, to the reattachment point caused by a
sep-aration bubble which has formed on the domesurface. Beyond a
value of 3 105 of the Reynoldsnumber, only one peak was observed
meaning thatthe separation bubble has disappeared.
Using Reynolds averaged NavierStokes (RANS)based turbulence
models, Meroney et al.6 found thatthe three turbulence models,
SpalartAllmaras, k",and Reynolds stress model, gave similar
pressure dis-tribution results with minor variations. Tavakol et
al.7
used the RNG k" turbulence model to investigate theeffects of
thin and thick approaching boundary layerson the flow aerodynamics.
The reattachment pointbehind the obstacle which is of the order
of1.0851.17 the dome diameter was located at a shorterdistance for
the thick boundary layer. All these studieshave simulated the flow
using the steady-state equa-tions of motion. In general steady and
to some extentunsteady RANS simulate such flows with
acceptablequalitative accuracy. Manhart8 used the large
eddysimulation (LES) turbulence model based on theSmagorinsky
subgrid-stress model with the law ofthe wall. In this study, the
dome roughness was gen-erated by a stepwise approximation of the
curvedwall. The study was based on the case reported inSavory and
Toy,3 of an artificially roughened domeat a Reynolds number of 1.4
105 and two grid sizesof 650,000 and 1,863,680 computational cells
wereused with an integration time step of 0.00250.005 s.Two
mechanisms of vortex generation are mentioned,one from the top face
due to the separation of thefree shear layer from the region
surrounding thedome apex causing symmetric vortex shedding andan
another one from the side faces engendering asym-metric vortex
shedding. Tavakol et al.9 conducted acomparison between the RNG k"
and severalLES subgrid turbulence models over smooth domes.Although
their comparison was limited to few meanstreamwise velocity
profiles, the RNG k" model wasoverestimating the streamwise
velocity over the hemi-sphere. The results showed that LES
simulation usingthe single equation transport model for the
subgridkinetic energy turbulence model (KETM) withproper grid
resolution greater than 4 million cells,are able to capture the
mean velocity profiles accur-ately in the apex and wake
regions.
From the review of previous work, it is clear thatRANS-based
approaches to simulate flows over hemi-spherical domes performed
with relative success andthat relatively very few LES-based
numerical simula-tions were conducted on such flows. LES-based
simu-lations offer the advantage of simulating explicitly thelarge
scales while modeling the small subgrid scales.This approach is
hence appropriate for aerodynamicflows around domes. In this study,
a numerical simu-lation based on the LES model using the KETM
sub-grid-scale model is used to calculate and benchmarkthe flow
over two hemispherical domes of Savory andToy3,10,11 and examine in
detail the features of the sur-rounding flow field. Available,
experimental mean sur-face pressure coefficient and velocity
profiles withcorresponding root mean square (RMS) are used inthe
validation of the numerical results. This particulartest case
consists of turbulent flow over an (a) artifi-cially roughened
surface which can exist in practicalapplications and, hence,
presents an additional levelof complexity in modeling the surface
details in con-trast to flows over smooth surfaces for the LES
frame-work and (b) over a smooth dome of the same size.Taking
advantage of the visualization of the flowthat can be provided by
LES the effect of the surfaceroughness is examined in detail and
contrasted to thesmooth dome case. In particular, this article
presents anovel methodology of how to model and account forsurface
roughness in structures such as domes withinthe LES approach for
industrial applications. Thestandard ke model was used to generate
the startingflow for the LES simulation over the rough surface
andis also used in the discussion of the results.
Mathematical model
LES is used in this study. The fluid is assumed incom-pressible
and the filtered continuity and momentumtime dependent equations
solved are given by
@ Ui@xi
0 1
@ Ui@t
@Ui Uj@xj
@ij@xj
@ p@xj
@ij@xj
2
where the variables with an overbar, Ui and p, repre-sent the
filtered (locally averaged) values of the vel-ocity and pressure,
respectively. The laminar stresstensor is given by
ij @ Ui@xj
@Uj
@xi
3
is the molecular viscosity.According to Tavakol et al.,9 where
three subgrid-
scale models were contrasted for flows around domes,the
subgrid-scale model based on the transport of thesubgrid-scale
turbulent kinetic energy performs betterfor flows of this type and
it is hence used in this study.
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This model developed independently by a number ofauthors, is not
based on the local equilibrium hypoth-esis and belongs to the class
of model based onsubgrid scales in contrast to the
dynamicSmagorinskyLilly model which is based on theresolved scales
properties.12 The grid scale cutoff canbe in the inertial range and
thus allowing the usage ofrelatively coarse grids.13
The subgrid stress accounting for the unresolvedscales
contribution defined by
ij UiUj Ui Uj 4
is modeled according to the following equation
ij 2
3ksgsij 2Ckk
12sgsf Sij 5
where ksgs is the subgrid-scale turbulent kineticenergy, f V1=3
the filter size or characteristiclength, based on the average
dimension of a compu-tational finite volume, and Sij the usual
resolved meanstrain rate tensor given by
Sij 1
2
@ Ui@xj
@Uj
@xi
6
and the kinetic energy of the subgrid modes is given by
ksgs 1
2U2k U
2k
7
From equation (5), the subgrid-scale turbulenteddy viscosity is
then implicitly defined from kineticenergy of the subgrid modes
by
t Ckk12sgsf 8
The subgrid-scale kinetic energy adds anotherunknown to the
problem and is evaluated by solvinga modeled transport equation
which takes the formgiven by equation (9).14
@ksgs@t
@Ujksgs@xj
ij@ Ui@xj
C"k
32sgs
f
Ck@
@xj
k12sgsf
k
@ksgs@xj
! 9
The constants Ck and C" are computed dynamic-ally, based on the
least-square method proposed byLilly,15 usually applied to the
standard Smagorinskymodel to compute its constant dynamically,
while thePrandtl number k is equal to unity.
Geometry and computational approach
Geometrical configuration
Figure 1 shows the geometry used for this study andtaken from
Savory and Toy.3,10,11 The hemispherical
dome had a diameter of d 190mm and wasimmersed in a thin
turbulent boundary layer of86mm thickness as in the experiment. In
the experi-ments, two dome models were considered one with asmooth
surface and the other with an artificiallyroughened surface using a
random coat of sphericalbeads to give an equivalent roughness ratio
ofk0/d 0.01, where k0 is the diameter of the sphericalbeads. The
size of the computational domain, in thespanwise y-direction, was
equal to 4d. The experi-ments were conducted at a subcritical
Reynoldsnumber based on the approach free stream velocity,air
properties, and dome diameter of 1.4 105.However, the surface
roughness induced flow separ-ation resulted in an effectively
supercritical flow.
Computational approach
In the present LES approach, the near-wall flow fieldis resolved
rather than modeled. To simulate the sur-face roughness of the
rough dome of Savory andToy,11 Manhart and Wengle16 adopted a
techniqueof blocking out the body-filled grid cells of thedome
within a Cartesian grid. In this study, a ratherdifferent and more
elaborate approach is adopted.Alternate rectangular solid blocks
(Figure 2) wereextruded from the curved wall of the dome. Thesehad
a perpendicular depth to the dome surfaceequal to 1.5mm and edge
sides less than 6mm. Inthis way, the artificial roughness
introduced bySavory and Toy11 which corresponds to an averagebead
size of 1.5mm was modeled. The ideal represen-tation of the dome
roughness would use surface pro-trusions based on solid cubic cells
with an average sizeof 1.5 mm or even smaller. However, the
generationof the corresponding mesh, and the implementation ofthe
boundary conditions would, in this case, be highlyprohibitive. The
approach used represents an accept-able compromise between a
reasonable representationof the surface details and time resources.
The distri-bution of the solid blocks in Figure 2 is uniform in
theazimuthal direction but is in some places non-uniformin the
latitude direction. These would have a minorimpact since real
surface roughness non-uniformity israndom. This simple modeling
strategy will be shownto account for the main flow features.
A hybrid mesh (Figure 3) of hexahedral and tetra-hedral cells
was generated. The computationaldomain was decomposed into several
blocks. A firstblock of clustered grids having the shape of a
hemi-sphere surrounding the dome was created to facilitatethe
generation of a structured hexahedral mesh in thezones where most
of the relevant flow phenomenawere expected to occur namely; the
front side, theapex region, and the wake. Further out, this
hemi-spherical mesh was surrounded by a cube of tetrahe-dral cells
due to the difficulty of generating ahexahedral mesh of good
quality within the outercube truncated by the hemisphere
surrounding
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the dome. Finally, a relatively coarse mesh was gen-erated in
the rest of the domain. The refined-mesh sizein the hemispherical
block surrounding the dome is1.5 mm and approximately the same size
was kept forthe tetrahedral mesh. The computational domain
wasdivided into about 9 million cells for the two casesstudied.
Studies on optimization of computationalresources based on LES
requirements, such asAddad et al.17 for example, showed that an
efficientgrid should have a cell size around the Taylor
microscales. Based on an a priori RANS simulation,the
corresponding scale for this problem, was,approximately, equal to
0.2mm. The cell size in thisproblem was chosen, due to the
computationalresources limitations, to be 0.4 mm inside the
regionsurrounding the dome where smaller scales areexpected to
reside. The values of y near the wall ofthe dome was in the range
114. This mesh resolutionnecessitated the use of a near-wall
treatment modelwhich is discussed below.
Air is then working fluid at ambient temperature,having a
density of 1.225 kg/m3 and dynamic viscosityof 1.7894 105kg/ms.
The boundary conditions were imposed in suchway to mimic the
conditions under which the experi-ments of Savory and Toy3,10,11
were conducted. At theinlet, measured profiles of the mean
streamwise vel-ocity component in addition to the turbulent
param-eters required (k, ", and Reynolds stresses) wereimposed. The
turbulent parameters were calculatedbased on the profile of the
streamwise turbulenceintensity measured in Savory and Toy.3 The
spectralsynthesizer technique18,19 was employed for the gen-eration
of the fluctuating velocity components.
In this method, fluctuating velocity componentsare computed by
synthesizing a divergence-free velo-city-vector field from the
summation of Fourier har-monics. In ANSYS FLUENT, the number of
Fourierharmonics is fixed to 100.
The turbulent kinetic energy and its dissipationrate profiles
were calculated from the measured tur-bulence intensity
profiles.
k 32
UavgIt 2 10
" C3=4k3=2
l11
where It is the turbulence intensity, C the constantequal to
0.09, and l the turbulence length scale, and isequal to 0.07 the
hydraulic diameter based on fullydeveloped duct flow empirical
results.
A zero gradient boundary condition was imposedat the outlet of
the domain relying on the fact that it
x y z
Inlet Wall
Symmetry
Outlet
d8d4
3d
Figure 1. Boundary conditions and dimensions of the domain (flow
from left to right).
Figure 2. Roughened dome model (the blue or dark blocks
represent the solid protrusions).
Figure 3. A view of the computational mesh.
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was sufficiently far from the dome. At the top of thedomain, a
symmetry condition was used at a locationwhere no effect of the
dome on the flow is noticeable.Manhart8 invoked the fact that, with
such boundaryconditions, the mass flux arising from the growth
ofthe approaching boundary layer is omitted and mightresult in some
discrepancies compared with theexperiments which were conducted
inside a wall-bounded test section. In this study, the
computationaldomain was truncated in the free stream, at z/d
3,relying on preliminary simulations conducted with thereal test
section dimensions and the truncateddomain. The two lateral sides
were taken as periodicboundaries, placed at 2d from the central
verticalplane of symmetry. No-slip wall boundary conditionswere
imposed at solid surfaces. The Werner andWengle20 wall function was
used in regions wherethe mesh was not fine enough to resolve the
viscoussub-layer. Fluent documentation states that there areno
restrictions on the grid size near the wall for LESsince the wall
boundary conditions were implementedusing the law of the wall
approach although it isadvisable to refine the mesh near the wall
to a valueof y 1 for a better accuracy of the results.
The finite volume technique implemented in thefluent code
version 12.1 is used to integrate the con-servation equations. The
computations were per-formed using the unsteady segregated solver.
Abounded central differencing scheme was used to dis-cretize the
convective term in the filtered NavierStokes and subgrid-scale
turbulent kinetic energyequations. It is a composition of the pure
second-order central differencing scheme, the second-orderupwind
scheme and the first-order upwind scheme,see the documentation of
Fluent version 12.121 refer-ring to the Normalized Variable Diagram
approachproposed by Leonard.22 The pressure and velocitywere
decoupled using the SIMPLE algorithm. Thepressure interpolation was
performed by thePRESTO scheme.
The implicit temporal discretization was of second-order
accuracy. The steady mean flow was computedusing the k" model to
provide reasonable initial con-ditions for the LES simulation.
Subsequently, an LESsimulation was conducted during about 1 s
corres-ponding roughly to five times the residence time ofthe flow
based on an average velocity of 10.76m/sand the domain length. When
the flow developed,the simulation was run during another 1 s for
the col-lection of statistics. The local time-averaged
variableswere calculated using
; 1N
XNi1
; x, y, z, ti 12
where ; x, y, z, ti is the local and instantaneous vari-able at
the position (x, y, z) of the computationaldomain at time ti and N
the number of samples
collected to compute the statistical averages, the sam-pling
interval time was chosen equal to the integrationtime step.
The integration time step was 104 s and insured acell Courant
number in the domain less than 2. Recentstudies on optimization of
LES computational effort(e.g. Kornhaas et al.23) showed that a
Courantnumber equal to 2 would be sufficient for an accept-able
solution, using an implicit discretization scheme,in terms of
accuracy and stability for shear flows. Thecalculations were
conducted with fluent running inparallel mode using 32 processors
of a high perform-ance cluster HPC during about 60 days of
continuousrun characterized by a total wall-clock time/time
stepequal to 296 s and a total CPU time per time stepequal to 9480
s with a total of 19,597 time steps.
Results
The results of the simulations are presented in thissection for
the smooth and rough domes of Savoryand Toy.3,10,11 The mean flow
field, represented by thepressure coefficient, the average
streamwise velocitycomponent, and the dividing streamline showing
thesize of the wake recirculation zone, are discussed first.A
qualitative and quantitative assessment of the flowfeatures, in
different regions surrounding the dome, isthen presented. Finally,
the development of theReynolds stress profiles, in the wake region
arediscussed.
Mean flow field
Figure 4 illustrates the distribution of the pressurecoefficient
on the dome centerline, in the streamwisex-direction, for the
smooth and rough domes sub-jected to an approaching thin boundary
layer.Savory and Toy3 stated that for the Reynoldsnumber
considered, 1.4 105, the flow regime overthe rough model was
supercritical in which case thepressure distribution over the dome
surface becomesindependent of the Reynolds number while it was
stillsubcritical for the smooth dome. The calculated
andexperimental values of the mean pressure coefficientfor the
smooth dome are shown on Figure 4(a). BothLES and k"models
predictions are good on the frontside until the apex region at
about 85, where the peakcorresponds to the separation of the
boundary layerdeveloping on the dome surface. A slight decrease
isobserved between 0 and 7. On the front side, a smallrecirculation
zone was generated in the corner formedby the dome curved surface
and the bottom wall. Thesecond peak at about 21 corresponds to the
stagna-tion point caused by the impact of the approachingflow on
the dome surface. The incoming flow impactsfirst on a recirculation
zone generated upstream of thedome which acts as an obstacle. The
approaching flowis, then, deviated upward impacting the dome
surfaceat the position of 21. Discrepancies are observed
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starting from 93 for the LES model and 100 for thek" model. The
LES exhibits a local jump in the pres-sure coefficient profile due
probably to a reattachmentof an existing separation bubble causing
the secondnegative peak at about 97 but during recoverymatches the
experimental values. In the wakeregion, the LES model slightly
under-predicts thepressure coefficient, whereas the k" model
showsbetter agreement with experiments. For the roughdome and in
the front stagnation region, the max-imum value of the pressure
coefficient is slightlyover-predicted by the ke and LES
calculations until40. This behavior might be due to a lack of
resolutionof the flow in this region. This was also observed
byManhart8 who attributed it to the different shapes ofthe oncoming
experimental and calculated boundarylayers. The decrease from the
maximum value is cap-tured correctly by the calculations until the
peak suc-tion value around 80. Both the LES and the kemodels
exhibit a first local negative maximum at 65
followed by a decreasing trend until 75, where asecond sharp
increase is observed. The roughnesseffect has moved the separation
point backward
compared with the smooth dome, i.e. there is an ear-lier
separation. The separation and reattachment phe-nomena are more
pronounced on the surface of theroughened dome due to the nature of
the surface. Therecovery of the ke model appears to be more
rapidthan the LES results.
Savory and Toy3 referred to the local centerlinedrag coefficient
as a useful parameter to predict thetrend of the pressure forces
acting on the dome sur-face for different Reynolds numbers. The
local dragcoefficient exhibits a minimum which designates
thecritical flow regime (critical Reynolds number). In thisstudy,
the local drag centerline drag coefficient wasequal to 0.5 for the
rough dome and 0.63 for thesmooth dome with a deviation from the
experimentalresults equal to 4.6% and 40%, respectively.
The fluctuating pressure coefficient is presented inFigure 5
along the centerline of the dome in the sym-metry plane y/d 0. The
distribution along the roughdome is characterized by three peaks at
10, 90, and109, respectively, while, for the smooth dome, thepeaks
are observed at 2, 14, and 112, respectively.Cheng and Fu5
explained that the fluctuating pressure
1.0
0.8
0.6
0.4
0.2
0.0
0.2
0.4
0.6
0.8
0 20
Mea
n pr
essu
re c
oeff
icie
ntM
ean
pres
sure
coe
ffic
ient
Mea
n pr
essu
re c
oeff
icie
nt
40 60 80 100 120 140 160 180
Angle ()
(a)
[3]
LES
k
1.0
0.8
0.6
0.4
0.2
0.0
0.2
0.4
0.6
0.8
0 20 40 60 80 100 120 140 160 180
Angle ()
(b)
[3]
LES
k
Figure 4. Mean pressure coefficient distribution on the dome
centerline: (a) smooth dome and (b) rough dome.
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coefficient peaks are related to transition of separationbubbles
and reattachment phenomena. ForRe< 1.8 105, as in the present
case, they observeda first peak near the apex region (8090) and
abroader lump at 140. In the present simulation witha smooth dome,
only one peak, near the separationregion, is observed at about 112,
the second peak isnot detected probably because the separation
bubbleis not resolved. Two other peaks are generated in thefront
side under the effect of the impingement of theincoming flow on the
dome surface at 14 and thereattachment of a small recirculation
zone, generatedin the corner formed by the dome and the floor, at
2.The peak at 112 corresponds approximately to theseparation of the
boundary layer developing along theupper surface of the dome. The
related flow behavioris illustrated in Figure 6(a) on which was
superim-posed the surface contours of the RMS of the
pressurecoefficient, and the mean velocity vectors plotted inthe
plane of symmetry. The position of the RMS
Cp peak is indicated by the thick horizontal line.The
abovementioned small recirculation zone andimpact point are clearly
seen. The situation is differentfor the rough dome case where only
one peak isobserved on the front side due to the absence of
thesmall recirculation zone (Figure 6(b)). Near the apex,a first
peak is observed at 90 and second one at 109
which, according to Cheng and Fu,5 are due to sep-aration and
reattachment phenomena, respectively.
Figure 7 shows the dividing streamline in the wakeregion. Taking
the axis of the dome as a reference inthe streamwise x-direction,
the position of the meanreattachment is 0.8d and 1d for the smooth
and roughdomes, respectively. An underestimation of thereattachment
distance is clearly seen compared withthe values measured by Savory
and Toy3 for the samecase or the value of Tavakol et al.7 which was
1.17 fora Reynolds number equal to 6.4 104, based on a freestream
velocity of 8.5m/s and a diameter of the domeof 0.12m.
0.04
0
0.04
0.08
0.12
0.16
0 20 40 60 80 100 120 140 160 180
rms
pres
sure
coe
ffic
ient
Angle ()
LES rough
LES smooth
Figure 5. RMS surface pressure coefficient distribution on the
dome centerline.
RMS: root mean square.
(a) (b)
Figure 6. Mean velocity vectors (plane of symmetry) superimposed
with the contours of RMS of Cp (dome surface).
Thick lines indicate the maximum RMS of Cp. (a) Smooth and (b)
rough.
RMS: root mean square.
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To investigate the causes of the underestimation ofthe
reattachment point, profiles of the average stream-wise velocity
component, obtained by LES for bothrough and smooth domes and
experiments ofSavory24 for the rough dome, are plotted at
differentlongitudinal positions in the vertical z-direction
inFigure 8. At x/d 0.8 and approaching the floor sur-face, the
reversed flow starts to become underestimatedcausing the location
of zero-streamwise velocity tooccur at a lower position z/d. At x/d
1, the simulated
flow has already reattached while the experiments stillshow a
small backflow in the region below z/d 0.33.At x/d 1.2, which is
approximately the measuredreattachment position, the streamwise
velocity compo-nent, predicted by LES, is positive everywhere in
thevertical direction while it is equal to zero belowz/d 0.3. From
the mean velocity profiles, the maindiscrepancy between a rough and
smooth dome existwithin the separating shear layer, with the rough
domeprofiles showing a higher momentum deficit.
0
0.1
0.2
0.3
0.4
0.5
z/d
0.6
0.7
0.8
0.5 0.5 1.5
x/d=0.4
0.5 0.5 1.5
x/d=0.6
0.5 0.5 1.5
x/d=0.8
0.5 0.5 1.5
x/d=1
0.5 0.5 1.5
x/d=1.2
Figure 8. Profiles of the mean streamwise velocity component in
the vertical direction at different x/d positions: , Savory;24 ,
LESrough; and - - -, LES smooth.
LES: large eddy simulation.
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
z/d
0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 1.1 1.2 1.3
x/d
[3]: rough
LES rough
LES smooth
DomeSurface
Figure 7. Dividing stream line.
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Flow structures and patterns
Isosurfaces of vorticity and mean path lines are usedto
illustrate features of the flow. In addition, streamtraces obtained
from the mean velocity field are alsoused. A comparison between the
smooth and roughdomes is conducted in the light of the qualitative
andquantitative results presented.
Figure 9 shows isosurface vorticity contours whichillustrate the
main flow features previously described.An intense turbulent
activity close to the floor isobserved in the front side of the
dome and developsaround it. This turbulent behavior corresponds to
thehorseshoe vortex and is more pronounced for thesmooth dome. For
the rough dome, it is not clearlyshown due to the value of the
vorticity, which waschosen to capture the wake turbulence rather.
Theflow separation occurs earlier for the rough than forthe smooth
dome as shown before with the wall pres-sure coefficient results.
The turbulent structures gen-erated by the separated boundary layer
from the topface of the smooth dome, tend to be entrained in
adirection slopping downward, while for the roughmodel, they move
parallel to the longitudinaldirection.
Figure 10 shows the streamlines projected on sev-eral horizontal
planes. Close to the bottom atz/d 0.005, the horseshoe vortex is
present with a lat-eral axis at about x/d0.9 for the smooth dome
andx/d1.1 for the rough dome. Two counter-rotatingrecirculation
zones, generated in the wake region,extend to a longitudinal
position of x/d 0.8 for thesmooth dome and x/d 1 for the rough
dome. Atz/d 0.05, the horseshoe vortex for the rough
dome,disappears while, in the wake, the recirculation zonesare
still present with a larger size for the rough dome.
Once the horseshoe, for the rough dome, disappears,the
streamlines tend to converge in a more pro-nounced way in the far
wake compared with thesmooth dome case. For both geometries, it
seemsthat, in the positive lateral wake side, some of
thestreamlines circumventing the recirculation zone areentrained by
the opposite recirculation zonerather than the streamwise flow. The
counter-rotatingrecirculation zones vanish at a vertical position
ofz/d 0.25 for the smooth dome, while for the roughdome, they
disappear at z/d 0.375.
The most important flow feature seen in the verti-cal planes
perpendicular to the main flow direction(Figure 11) is the rolled
up vortices in the lateralsides of the dome belonging to the
horseshoe vortex.At x/d 0 (apex), trailing vortices, with a
quasi-circu-lar cross section, are seen to be closer to the dome
forthe smooth model while flattened vortices are locatedfarther
from the rough dome, at about y/d 1.25. Inthe wake region at x/d 1,
the trailing longitudinalvortices are flattened for both cases
under the effectof diffusion and decay. In the vertical streamwise
dir-ection (Figure 12), the main observed features are thehorseshoe
vortex upstream of the dome and the recir-culation zone in the
wake. In the plane y/d 0, thehorseshoe vortex axis is located at
x/d0.65 andz/d 0.05 for the smooth dome. A flattened
thinnerhorseshoe vortex is generated at x/d0.98 andz/d 0.025. In
the wake of the smooth dome, thecenter of the recirculation zone
has the coordinatesx/d 0.58 and z/d 0.27 while a larger one is
gener-ated behind the rough dome at x/d 0.59 and at ahigher
vertical position z/d 0.355. The center ofthe recirculation zone
obtained in the experiments ofSavory and Toy,3 was at approximately
x/d 0.77and z/d 0.38. The recirculation zone axis predicted
(a1) smooth (b1) rough
(a2) smooth (b2) rough
Figure 9. Isosurface vorticity contours from side and isometric
views: (a) smooth (1000 s1) and (b) rough (1200 s1).
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by LES is located at a closer distance to the dome andslightly
lower which explains the shorter reattachmentdistance compared with
the experimental results(Figure 7). At y/d 0.5, the recirculation
zone in thewake has disappeared for the smooth dome while it
isstill present for the rough dome.
The main findings through this comparison are thedifferent
shapes and positions of the features charac-terizing the turbulent
flow around domes especiallythe flattened shape of the horseshoe
vortex for therough dome and its position away from the
obstacle.Bradshaw25 explained that the pressure gradientsmight play
a key role in determining the position ofthe characteristic
phenomena. Longitudinal pressuregradients affect the separation
position along thedome surface while lateral pressure gradients
mightmove the vortex shoe away from the corner formedby the dome
and the floor. Unfortunately, the litera-ture does not contain
enough experimental results for
a detailed comparison between the smooth and therough domes.
Turbulent flow field prediction
Figures 13 and 14 show the profiles of the Reynoldsstresses , ,
and turbulent kinetic energy k inthe wake region in the plane of
symmetry y/d 0 andin a horizontal plane at z/d 0.158. These results
referto the rough dome for which experimental values areavailable
in Savory and Toy.11
In the plane of symmetry, the calculated longitu-dinal
correlation over-predicts the experimentalmaximum peaks within the
shear layers betweenx/d 0.4 to 0.8. The values inside the
recirculationwake zone are, however, nicely replicated.Nonetheless,
the positions of the peaks, indicatingthe approximate location of
the shear layer boundingthe wake region, are well captured. In view
of the
Streamlines at z/d =0.005
x/d
y/d
-1 0 1 2 3
-1
-0.5
0
0.5
1
x/d
y/d
-1 0 1 2 3 4
-1
0
1
a) smooth b) rough
Streamlines at z/d = 0.05
x/d
y/d
-1 0 1 2 3
-1
-0.5
0
0.5
1
x/d
y/d
-1 0 1 2 3-1
-0.5
0
0.5
1
a) smooth b) rough
Streamlines in different planes xy
Figure 10. Time-averaged streamlines plotted in xy planes at
different vertical positions z/d.
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Kharoua and Khezzar
Streamlines at 1Jd = 0.1
0 x/d
a) smooth
Streamlines at d cl =0.25
0.5
~ 0
-0.5
0 2 x/d
a) smooth
Streamlines at d cl =0.375
~ 0
-0.5
x/d a) smooth
nature and structure of the wake for both rough and smooth
domes, the profiles of for the smooth dome case exhibit large
deviations. The region bounded by the shear layer is remarkably
larger for the rough dome. The profiles for the component are,
also, better predicted for the rough geometry. They show that the
velocity fluctuations are Jess important at high z/d positions
corresponding to the shear layer separated from the top surface of
the dome. In contrast , the Reynolds stress component in the
spanwise direction exhibit a highly turbulent behavior at z/d <
0.2 far from the dome at 0.8 :( x/d :( 1.2. This region corresponds
to the zone of impact of the turbulent structures resulting from
the separation phenomenon from the lateral sides of the dome, at
low z/d positions, known as Von Karman vortices. Similarly to the
Reynolds stresses, the turbu-lent kinetic energy, shown in Figure
13(c), is well pre-dicted at x /d = 1.2 while, for the remaining
axial positions, discrepancies, especially for the peaks, can be
observed. It is noteworthy to mention that the
2
11
x/d b) rough
0.5
~ o
-0.5
x/d b) rough
0.5
" ->. 0 -0.5
x/d
b) rough
RNG k- e model, also, predicted, correctly, the loca-tion of the
peaks of turbulent kinetic energy although with a clear
underestimation. This is expected since such model is unable to
simulate correctly flows with strong mean streamline curvature and
anisotropy as exists in the wake of the dome.
Figure 14(a) shows profiles of the component in a horizontal
plane at z/d= 0.158 and shows that the profiles are, relatively,
well predicted for the positions x/d = I and x/d = 1.2 while
remarkable overestimated peaks are observed close to the obstacle
at x /d = 0.6 and x /d = 0.8. This is, probably, due to the strong
turbulent effect induced by the technique used to rep-resent the
surface roughness of the dome which is not entirely perfect.
However, contrary to the predictions on the plane of symmetry, the
location of the peaks is slightly shifted downward compared with
the experi-mental results which means that a shorter
shear-Jayer-bounded region is being predicted by the numerical
simulation and which might , also, explain the shorter reattachment
length observed from Figure 8.
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Figure 11. Time-averaged streamlines plotted in yz planes at
different longitudinal positions x/d.
x/d
z/d
-1 -0.5 0 0.5 1 1.5 20
0.5
x/d
z/d
-1 -0.5 0 0.5 1 1.5 20
0.5
a) smooth b) rough
x/d
z/d
-1 -0.5 0 0.5 1 1.5 20
0.5
x/d
z/d
-1 -0.5 0 0.5 1 1.5 20
0.5
a) smooth b) rough
Figure 12. Time-averaged streamlines plotted in xz planes at
different lateral positions y/d.
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0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0 0.1
y/d
x/d=0.6
0 0.12
x/d=0.8
0 0.1
x/d=1
0 0.1
x/d=1.2(c)
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0 0.1
y/d
x/d=0.6
0 0.12
x/d=0.8
0 0.1
x/d=1
0 0.1
x/d=1.2(a)
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0 0.05
y/d
x/d=0.6
0 0.052
x/d=0.8
0 0.05
x/d=1
0 0.05
x/d=1.2(b)
Figure 14. Profiles of the Reynolds stresses and the turbulent
kinetic energy in the horizontal plane z/d 0.158 at different
lon-gitudinal positions: , Savory and Toy;11 , LES rough; - - -,
LES smooth; and , RNG k".LES: large eddy simulation.
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0 0.1
z/d
x/d=0.4
0 0.1
x/d=0.6
0 0.1
x/d=0.8
0 0.1
x/d=1
0 0.1
x/d=1.2(a)
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0 0.1
z/d
x/d=0.4
0 0.1
x/d=0.6
0 0.1
x/d=0.8
0 0.1
x/d=1
0 0.1
x/d=1.2(b)
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0 0.1
z/d
x/d=0.4
0 0.1
x/d=0.6
0 0.1
x/d=0.8
0 0.1
x/d=1
0 0.1
x/d=1.2(c)
Figure 13. Profiles of the Reynolds stresses and the turbulent
kinetic energy in the plane of symmetry y/d 0 at different
longi-tudinal positions: , Savory and Toy;11 , LES rough; - - -,
LES smooth; and , RNG k".LES: large eddy simulation.
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The peaks are much lower for the smooth model lead-ing to a
smaller shear-layer-bounded region in thewake and an even shorter
reattachment longitudinallocation. Figure 14(b) illustrates
profiles of the lateralReynolds stress component in the same
hori-zontal plane close to the bottom surface. It can beobserved
that, close to the dome at x/d 0.6, aslight peak exists at
approximately y/d 0.47 whichis not well predicted by LES. Beyond
x/d 0.6, thelateral stress component is overestimated andthe trend
is a monotonic decay of the lateral Reynoldsstress component from
the symmetry plane towardthe outer region until, approximately, y/d
0.5.Profiles of the turbulent kinetic energy are shown inFigure
14(c). At x/d 0.6 and x/d 0.8, the peaks aty/d 0.38 and y/d 0.32,
respectively, represent theeffect of the longitudinal component
while the rela-tively good prediction of k beyond x/d 1 is
consist-ent with the good prediction of the stress componentsat the
same positions.
Conclusions
A LES of a turbulent flow around smooth and roughdomes was
conducted. The subgrid-scale model basedon the transport of the
subgrid-scale turbulent kineticenergy was used. The rough dome
surface modelingpresented a challenge to capture its
geometricaldetails. The surface roughness of the rough dome
sur-face was incorporated into the geometry using solidblocks
extruded with an average size of the glassbeads diameter used in
the experiment and representsa reasonable approach to model surface
details.
The pressure coefficient distribution along the cen-terline of
the dome was predicted with very goodaccuracy although with slight
discrepancies in therecovery region. The LES under-predicted the
pres-sure coefficient slightly for the smooth dome
butover-predicted it for the rough dome although thedifferences
were slight. The predicted rear reattach-ment point downstream of
the dome was locatedcloser to the dome compared to the
experiments.
For the rough dome, the Reynolds stresses werewell predicted in
the vertical plane of symmetryalthough the peaks were
overestimated. In the hori-zontal plane close to the floor,
however, the peakswere underestimated announcing a smaller
turbu-lence-affected region in the wake compared with
theexperiments.
The LES model allowed the visualization of fea-tures of the flow
that are difficult to observe experi-mentally at high Reynolds
numbers. The flow aroundthe rough dome was characterized by a
flattenedhorseshoe vortex shifted away from the obstacle com-pared
to the smooth dome. The horseshoe vortexdevelops around and behind
the dome in the formof trailing vortices, becoming progressively
parallelto the flow direction with a separating lateral
distancebetween either sides being larger for the rough dome.
In the wake, a larger vorticity content region boundedby the
separated shear layers was observed for therough dome.
Funding
This research received no specific grant from anyfunding agency
in the public, commercial, or not-for-profit
sectors.
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Appendix
Notation
d dome diameterIt turbulence intensityk turbulent kinetic
energyp pressureSij strain rate tensort timeu, v, w fluctuating
velocity componentsU, V, W velocity componentsUi filtered velocity
component in the xi
directionx, y, z Cartesian coordinates
f LES filter size laminar dynamic viscosityt turbulent dynamic
viscosityij laminar stress tensorij subgrid-scale stress tensor
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