-
ce
ing
Received in revised form
Keywords:Drag forceConvective heat transfer
o-dric
nt rol
et al. [7].An analysis of a large range of literature in this
area shows that
considerable efforts have been focused on the effect of
roughness inthe turbulent ow regime. Applied to a ow past a
cylinder, in
critical Reynolds number corresponds to the Reynolds numberwhere
the drag coefcient exhibits a minimum. In the follow-upexperiment
Achenbach [3] carried out an investigation into theeffect of
surface roughness on heat transfer between a cylinderand a gas ow.
The roughness was reproduced using regulararrangements of pyramids,
each with a rhomboidal base. Achen-bachs experiments showed that,
similar to the isothermal case, theroughness parameter did not play
a signicant role in the total heattransfer coefcient under
subcritical ow conditions. However,
* Corresponding author. Tel.: 49 3731394202.E-mail addresses:
[email protected] (F. Dierich), petr.nikrityuk@
Contents lists available at SciVerse ScienceDirect
International Journal
w.e
International Journal of Thermal Sciences 65 (2013)
92e103vtc.tu-freiberg.de (P.A. Nikrityuk).branches, e.g. from
chemical engineering to aerospace engineering,due to their
signicant role in the heat and mass transfer betweena uid and the
surface of a solid. In particular, the effect of surfaceroughness
on the total heat transfer coefcient and the boundarylayer
characteristics has been studied in various experimental [1e4] and
numerical works [5,6], respectively. It should be noted thatthese
works are related to the inuence of surface roughness onheat
transfer. A recent review of pioneering works accounting forthe
surface roughness effect on hydrodynamic characteristics,
e.g.pressure drop and drag coefcient, can be found in work by
Taylor
numbers. In the isothermal experiments described in [1]
theroughness was represented by emery paper covering the
cylinder.To characterize the roughness the so-called roughness
coefcientks/D was utilized, where ks is the height of the sand
grain (Nikur-adse roughness) and D is the cylinder diameter. In
Achenbachswork the roughness coefcient was varied between 1.1 103
and9 103. Experiments showed that the subcritical ow regime wasnot
inuenced by the surface roughness. However, Achenbachfound out that
increasing the roughness parameter causesa decrease in critical
Reynolds number. Here, following [1], theRoughnessImmersed boundary
method
1. Introduction
Rough surfaces plays an importa1290-0729 2012 Elsevier Masson
SAS.http://dx.doi.org/10.1016/j.ijthermalsci.2012.08.009
Open access undeforcing (Khadra et al. Int. J. Numer. Meth.
Fluids 34, 2000) was used to simulate heat and gas ow pasta
cylindrical particle with a complex geometry. A polygon and the
SutherlandeHodgman clippingalgorithm were used to immerse the rough
cylindrical particle into a Cartesian grid. The inuence of
theroughness on the drag coefcient and the surface-averaged Nusselt
number was studied numericallyover the range of Reynolds numbers 10
Re 200. Analyzing the numerical simulations showed thatthe impact
of the roughness on the drag coefcient is negligible in comparison
to the surface-averagedNusselt number. In particular, the Nusselt
number decreases rapidly as the degree of roughness increases.A
universal relationship was found between the efciency factor Ef,
which is the ratio between Nusseltnumbers predicted for rough and
smooth surfaces, and the surface enlargement coefcient Sef.
2012 Elsevier Masson SAS.
e in many engineering
a series of works [1,3] Achenbach carried out experiments
inves-tigating the inuence of surface roughness on the cross-ow
andheat transfer around a circular cylinder for a range of
Reynolds
Open access under CC BY-NC-ND license. Accepted 16 August
2012Available online 2 November 2012 Method (FVM) onto a xed
Cartesian grid, and the Immersed Boundary Method (IBM) with
continuous15 August 2012distributed on the cylinder surface. The
roughness was varied using different notch shapes and heights.The
NaviereStokes equation and conservation of energy were discretized
using the Finite VolumeA numerical study of the impact of surfaa
cylindrical particle
F. Dierich*, P.A. NikrityukCentre for Innovation Competence
VIRTUHCON, Department of Energy Process EngineerFuchsmhlenweg 9,
09596 Freiberg, Germany
a r t i c l e i n f o
Article history:Received 14 October 2010
a b s t r a c t
This work is devoted to a twand uid ow past a cylin
journal homepage: wwr CC BY-NC-ND license. roughness on heat and
uid ow past
and Chemical Engineering, Technische Universitt Bergakademie
Freiberg,
dimensional numerical study of the inuence of surface roughness
on heatal particle. The surface roughness consists of radial
notches periodically
of Thermal Sciences
lsevier .com/locate/ i j ts
-
urnafor the transcritical ow range, the increase in the
roughnessparameter led to an increase in the heat transfer by a
factor of about2.5 [3].
Numerical efforts to reproduce the effect of roughness on a
owpast a rough cylinder were reported by Kawamura et al. [5]
andLakehal [6]. In particular, Kawamura et al. [5] carried out
directnumerical simulations of the ow around a circular cylinder
witha roughness parameter of about 5 103. The Reynolds numberwas
varied between 103 and 105. The total number of mesh pointswas 80
80. Following Kawamura et al. [5] reasonable qualitativeagreement
was achieved between numerically predicated resultsand results by
Ashebach et al. [4]. Lakehal [6] performed two-
andthree-dimensional RANS simulations of turbulent ows past
rough-
Nomenclature
Roman symbolsAs area of the polygon (m2)AA area of the nite
volume ()cp heat capacity (J kg1 K1)CD drag coefcient (N kg1 s2)cu,
cT constantsd dimensionless height of the notch ()D characteristic
size, diameter (m)R radius (m)Ef heat transfer efciency factor ()ks
height of the sand-grain (Nikuradse roughness) (m)KR roughness
coefcient ()K permeability coefcientF!
IBM IBM forces (N m3)
FD drag force (N)g gravitational constant (m s2)n! surface
normal ()Nu Nusselt number()Pi polygon
F. Dierich, P.A. Nikrityuk / International Jowalled circular
cylinders. A rough-wall model was utilized withinthe k RANS model.
The calculations provided close agreementwith experimental data
published. However, in the twoworks citedabove no heat transfer was
included into considerations.
An analysis of the literature indicates that the basic issue
inearly investigations concerned the effect of roughness on
rela-tively high Re ows. In the laminar ow region, the roughness
wasshown to have very little effect on the drag coefcient. In spite
ofextensive research on the role of the laminar ow in heat
transfernear the cylinder, e.g. see Lange et al. [8], Shi et al.
[9], Juncu [10],so far, however, there has been little discussion
about the inu-ence of surface roughness on heat transfer on bluff
body wakes forlaminar ow regimes. At the same time it should be
noted that,recently, with considerable development in microuidic
devices,where the ow is laminar due to the small scale of the
geometry,researchers have shown an increased interest in the role
of surfaceroughness on the heat transfer in laminar ow regimes,
e.g. seethe works [11e14]. Basically in these works the surface
roughnessis modeled directly using blocks of different shapes
periodicallydistributed on the plane walls. From this point of view
the work byAbu-Hijleh [15], who carried out numerical investigation
into theinuence of radial ns around the cylinder on the enhancement
ofheat transfer, has some similarities to studies on the effect
ofroughness. Abu-Hijleh [15] reported that short ns reduce the
heattransfer from the cylinder surface. This effect is reversed for
longns, where the enlargement of the surface can compensate for
theeffect.Parallel to the direct modeling of roughness, various
modelshave been proposed to account for the effect of roughness
onlaminar ows. In particular, Koo and Kleinstreuer [16]
introducedthe concept of an equivalent porous medium layer to model
therough near-wall region. Using a similar approach,
Bhattacharyyaand Singh [17] carried out numerical investigation
into the inu-ence of a porous layer around the cylinder on the
enhancement ofheat transfer. In particular, Bhattacharyya and Singh
[17] showedthat a thin porous wrapper which has the same thermal
conduc-tivity as the cylinder can signicantly reduce the heat
transferbetween the cylinder and ow. To model the gas ow inside
theporous layer they used the DupuiteForchheimer relationship,which
states that the velocity inside the porous medium is
p pressure (N m2)Pr Prandtl number ()Sr Strouhal number ()Sef
surface enlargement ()_Q IBM IBM source term for energy equation
(W)t time (s)T temperature (K)Ts cylinder surface temperature (K)TN
free stream temperature (K)DT Ts TN temperature difference (K)u!
velocity vector (m s1)
Greek symbols volume fraction of gas ()l heat conductivity (W K1
m1)n kinematic viscosity (kg m1 s1)r density (kg m3)
Subscriptsav averageds surfacein inow
l of Thermal Sciences 65 (2013) 92e103 93proportional to the
bulk velocity multiplied by the porosity. The useof this or the
Darcy ow assumption when modeling particleroughness is questionable
due to the fact that the convection maynot be negligible within the
roughness region.
All the studies reviewed so far relating to the effect of
roughnesson heat transfer, however, suffer from the fact that they
directlymodel the inuence of surface roughness on heat transfer
betweenthe cylinder and gas ow. Motivated by this fact the present
workinvestigates the ow and heat transfer from a rough, solid
cylinderplaced horizontally in a cross-ow with an uniform stream of
air.Themainmotivation of this study is to estimate the inuence of
thethickness of the roughness layer on the heat transfer and on
thedrag coefcient for a cylindrical particle. The practical context
ofthis study is to contribute to understanding and developing
closurerelations for the drag coefcient and the Nusselt number,
which canbe used in the so-called subgrid models whenmodeling
particulateows in chemical reactors or coal gasiers.
2. Problem formulation and governing equations
2.1. What is roughness?
Before we proceed with a description of the setup
underinvestigation and the model we use, let us specify what
roughnessis. Following recent work by Taylor et al. [7] the term
roughness isshort for the ner irregularities of surface texture
that are inherent inthe materials or production process, i.e. the
cutting tool, spark, grit
-
size, etc. Difculties arise, however, when an attempt is made
toimplement this denition into a mathematical and numericalmodels.
An analysis of recent works devoted to the modeling ofsurface
roughness shows that basically in numerical investigationsthe
roughness is approximated using two-dimensional or
three-dimensional blocks with different shapes periodically
distributedon a smooth surface, e.g. see [11e13]. Fig. 1 shows an
example ofthis concept. However, this representation has a number
of limi-tations. In particular, periodic structures can not model
nerirregularities. One of the ways to escape this limitation is to
usea fractal geometry to characterize adequately rough surfaces,
e.g.see [14].
To characterize the roughness the so-called roughness
coefcientis utilized [1]:
KR ksD
(1)
where ks is the height of the irregular surface and D is the
char-acteristic size, e.g. diameter.
2.2. Problem setup
It is awell-known fact that a spherical particle shape is
themost-used approximation in subgrid models. In this work,
however, asa rst step we consider a single cylindrical particle
with a radius R
real rough surface
approximation of a real rough surface
k s
Fig. 1. Principal schemes of roughness.
40R 40R
40R
100R
2R
u in
a
b
c
Fig. 2. Size of the domain (a), a zoomed view of the particle
under investigation withroughness parameter d (b) and a zoomed view
of the particle used in Dierich andNikrityuk [39] (c).
F. Dierich, P.A. Nikrityuk / International Journal of Thermal
Sciences 65 (2013) 92e10394c
e f
d
Fig. 3. Clipping of a polygon at one control volume,
SutherlandeHodgman algorithm.(a) Polygon P1, (b) Polygon P2 clipped
against left edge, (c) Polygon P3 clipped againstplaced in a
stationary position, with the main gas ow passingaround it. The
principal scheme of the domain is shown in Fig. 2(a).The inow
velocity, uin, was assumed to be uniform and wasdetermined bymeans
of the Reynolds number calculated as follows:
Re 2Ruinn
(2)
where n is the kinematic viscosity. The particle roughness
ismodeled by 10 notches with the depth d R as shown in Fig.
2(b).
a bbottom edge, (d) Polygon P4 clipped against right edge, (e)
Polygon P5 clipped againsttop edge and (f) Clipped polygon P5.
-
The depth of the notches d R is varied from 0.01 to 0.5 R
usingsteps of 0.01. Inserting these values into eq. (1) gives the
roughnesscoefcient in the range between 5 103 and 0.25.
The cylindrical particle is placed in the center of the
domainwith a total length of 140 R and a total width of 80 R. We
considerthe roughness layer to be made from the same material as
thecylinder.
To proceed with the governing equations the following
basicassumptions have been made:
1. The gas ow is treated as an incompressible medium.2. The
viscous heating effect is neglected.3. The thermophysical
properties are constant, giving a Prandtl
number Pr of 0.7486.4. The buoyancy effect is neglected.
Taking into account the assumptions made above, the
conser-vation equations for mass, momentum and energy
transportwritten for the gas phase take the following form:
V$ u! 0; (3)
v u!vt
u!$V u! Vpr nV2 u! FIBM (4)
vTvt u!$VT l
rcpV2T QIBM (5)
Here u! is the velocity vector, p is the pressure, n is the
kinematicviscosity, l is the thermal conductivity, r is the
density, cp is the heatcapacity and Ts is the temperature of the
particle.
On the bottom we set an inow boundary condition withconstant
temperature. We treat the top as having an outletboundary condition
and on the sides we apply the Neumannboundary condition.
To enforce no-slip boundary conditions on the particle surfacewe
introduce a body force to the momentum equation F
!IBM and
a source term to the temperature equation _Q IBM using the
so-calledcontinuous-forcing approach, e.g. see [18,19]. In this
work weutilize a modication of this approach, the so-called
porous-medium approach, which is used extensively when
modelingsolidication [20]. This method was summarized by Khadra et
al.[21] for the case of moving bodies including heat transfer
modeling
a
b
c
F. Dierich, P.A. Nikrityuk / International Journal of Thermal
Sciences 65 (2013) 92e103 95dFig. 4. Different variants of clipping
(a) pi inside pi1 outside, (b) pi outside, pi1 inside,(c) both
points outside and (d) both inside.with Dirichlet, Neumeann and
Robin boundary conditions on themoving surface. In this method, the
grid region occupied by thesolid body is assumed to be a Birkman
porous medium, charac-terized by its permeability K (t, x, y),
which can be variable in timeand space. A mathematical description
of the source terms appliedto our problem is given in Section
4.
3. Immersed surface reconstruction
Before we move on to the description of source terms in eqs.(4)
and (5) a short explanation of the immersed interfaceapproximation
is necessary. One of the most important steps inusing IBM methods
is the approximation of the interface locationand the identication
of the interface cells where appropriateboundary conditions have to
be set up. Here it should be notedthat interface cells are those
control volume cells that are crossedby the immersed surface. In
this work we use the so-called
0.00
0.00
0.00
0.00
0.00
0.00
0.73
0.00
0.00
0.00
0.00
0.00
0.06
0.97
0.00
0.00
0.00
0.00
0.00
0.37
1.00
0.00
0.00
0.00
0.00
0.00
0.74
1.00
0.73
0.24
0.00
0.00
0.15
0.99
1.00
1.00
1.00
0.75
0.26
0.58
1.00
1.00
1.00
1.00
1.00
1.00
0.99
1.00
1.00
1.00
1.00
1.00
1.00
1.00
1.00
1.00
1.00
1.00
1.00
1.00
1.00
1.00
1.00Fig. 5. Zoomed view of the spatial distribution of the
volume fraction of uid near theparticle notch.
-
nite-volume embedding methodology introduced by [22,23]
torepresent the solid immersed in the Cartesian grid. However,
incontrast to these works, we use the SutherlandeHodgman clip-ping
algorithm, which is well-known from computer graphicstheory
[24,25], to calculate the volume fraction of uid in eachcontrol
volume.
In particular, to calculate the volume fraction of gas in
eachcontrol volume we describe the surface of the particle by means
ofthe polygon P1. A polygon is a closed path consisting of a
nitesequence of straight line segments. The coordinates of all
polygonvertices are stored in the circular list LP . Using the
polygons theSutherlandeHodgman clipping algorithm can be applied,
tocalculate the volume fraction of gas. The
SutherlandeHodgmanclipping algorithm is applied separately in each
control volume.This algorithm starts with the innite extension of
the left edge ofthe control volume in both directions. The
algorithm clips thepolygon against this edge. This clipping process
is explained in
shows zoomed view of the spatial distribution of the
volumefraction of uid in the area around the particle notch
surface. Theparticle surface is indicated with the polygon (black
line). It can beseen that the volume fraction of the uid calculated
in each celltake the following values:
8:n u
! u!scu$MIN 1:;1 2
3
!0 < 1
0; 1(7)
QIBM
8>:
1rcp
T TscT$MIN 1:;1 2
3
!0 < 1
0; 1(8)
where cu and cT are constants whose dimensions make F!
IBM and_Q IBM consistent with the units of the rest of the terms
in themomentum and energy equations, respectively. Basically,
theconstants cu and cT are grid-dependent and must be chosen
care-fully. For example, if cu takes a too-small value, the
velocity insidethe particle is not zero and the particle is treated
as porous. On theother hand, if cu has a too-large value, the
solution does notconverge normally. Based on the numerous tests, in
our case wefound out that the choice:
cu 2$104Dx1min (9)Fig. 7. Computational grid in the simulations
using the IBM (lscheme was used to stabilize the pressure-velocity
coupling. Thediscretization of the time derivatives uses an
implicit three-time-level scheme. The matrix solver SIP developed
by Stone [29] isused to solve the system of linear equations. Time
marching withxed time steps was used. A pseudo-unsteady approach
was usedfor the steady cases at Reynolds numbers of 10e40.
Thecalculations were performed using the unsteady approach butonly
one outer iteration was carried out per time step. The timestep was
equal to 0.05 s, which corresponds to a non-dimensional time step
of 4.59 103 to 1.84 102. The compu-tations were stopped when the
normalized maximal residual of allequations was less than 1010. In
the unsteady cases at Reynoldsnumbers of 100 and 200 the time step
was equal to 0.01 s and0.005 s, respectively, which both
corresponds to a nonedimensional time step of 9.18 103. In all
simulations a gridwith 400 600 control volumes was used. The size
of a controlvolume (CV) inside the solid particle is about one
hundredth of theparticle diameter. This is achieved by local
renement of the gridinside the particle.
5. Code validation
To validate the code and the IBM model implemented wereproduced
the results of the ow around a circular cylinder atseveral Reynolds
numbers. It is a well-known fact that at Rey-nolds numbers of 1
< Re < 47, the ow past a cylinder islaminar, where a steady
recirculation region with toroidalvortex occurs behind the
cylinder. The size of the recirculationregion grows as the Reynolds
number increases. At Reynoldseft) and in the simulations using
Ansys Fluent 13 (right).
-
6. Results
Before starting with a description of the results the next
twoparagraphs gives a brief overview of the ow behavior for
differentRe. For very low Re (Re(5) the ow is attached to the
cylinder and
gh cylinder calculated with the IBM (left) and with Ansys Fluent
13 (right).
F. Dierich, P.A. Nikrityuk / International Journal of Thermal
Sciences 65 (2013) 92e10398numbers of Re 47, the ow becomes
unsteady with vortexshedding (von Karman vortex shedding) in the
near wakebehind the cylinder.
For the Reynolds numbers 10, 20 and 40 we compare the
dragcoefcient CD, the angle of separation qs and the vortex length
L/Rwith other published data in Table 1. The denition of the
vortexlength is shown in Fig. 6, where L is the length of the
recirculatingvortexes and R is the radius of the cylinder.
For higher Reynolds numbers of 100 and 200 we validate thedrag
coefcient CD and the Strouhal number St in Table 2. Theresults show
a good agreement with the published data.
To validate the implementation of the heat transfer model vs.the
immersed boundary method we carried out calculations of the
Fig. 8. Contour plot of the nondimensional temperature prole of
a rouNusselt number Nu for a circular cylinder for different
Reynoldsnumbers. The comparison of our predictions with data
published inthe literature is given in Table 3. Good consistency
with publishedresults can be seen.
The nal validation case represents a calculation of the heat
anduid ow past a rough cylinder using the commercial CFD
softwareAnsys Fluent 13 [30], which utilizes a conventional CFD
approach,and the IBMmodel implemented in our code. A setupwith a
dimpledepth of d 0.2 was simulated using a body-tted mesh
withFluent software and using our immersed boundary method.
Thegrids used in both simulations are shown in Fig. 7. The
body-ttedgrid of the Ansys Fluent 13 simulations is only exists in
the uidphase. In contrast, in the IBM the grid also exists in the
solid phase.A contour plot of the temperature of both simulations
at a Reynoldsnumber of 20 is shown in Fig. 8. The comparison of the
drag coef-cient CD and the Nusselt number given in Table 4 shows
that thetwo simulations are consistent.
Table 4Comparison for CD and Nu from the present simulations
with values calculated withAnsys Fluent 13 for a dimple depth of d
0.2.
CD Nu CD Nu CD Nu
Re 10 20 40Ansys Fluent 13 2.857 1.069 2.056 1.402 1.534
1.849Present results 2.874 1.081 2.084 1.428 1.571 1.898 Fig. 9.
Spatial distribution of non-dimensional vectors u
!=uin (a) and (b) zoomed viewof
(a), and the nondimensional temperature (c) predicted for Re 40
and d 0.5, Sef 2.12.
-
steady. The viscous transport is the dominating phenomenon
inthis ow regime and inuences the ow eld over large distances.For
this regime a very large computational domain must be chosento
achieve accurate results. This was reported by Lange et al.
[8].Therefore, the lowest Re investigated in this paper is 10. The
nextow regime reaches from 5(Re to Re< Rec1z47 and is still
steadyand has two vortices behind the cylinder. The exact value of
thecritical Reynolds number Rec1 has been studied by several
authors.Jackson [31] andMoryznski et al. [32] carried out a
stability analysisand found Rec146.184 and Rec147.00, respectively.
This result isin agreement with the experimental results of
Provansal et al. [33]and Norberg [34].
For Rec1 < Re < Rec2z 190 the ow around a cylinder
becomesunsteady with vortex shedding, the so-called Krmn vortex
street.The vortex shedding is described by the Strouhal number,
denedas Sr 2Rf/uin where f is the frequency of vortex shedding, R
is theradius of the cylinder and uin is the free-stream velocity.
In thisregime the ow is still 2D and 3D effects only occur for Re
> Rec2.Barkley and Henderson [35] used a stability analysis to
calculatea value of Rec2 188.51. In the literature, experimental
results forRec2 can be found in a wide range: Rec2 140e194
(Williamson
[36]). The variations are explained by Bloor [37] as due to
thedifferent free-stream turbulences and by Miller and
Williamson[38] as due to the different end conditions. Therefore,
2D simula-tions with Re> 200 do not describe the ow past a 3D
cylinder. Forthis reason Re 200 is the largest Re in the present
simulation. Atthe same time it should be noted that for the large
values of ks thethree-dimensionality of the ow may occurs at lower
Reynoldsnumber values.
To proceed with our analysis of the results, we briey
describethe main input and output parameters we use to study the
systembehavior shown in Fig. 2. To study the heat transfer
characteristicswe use the Nusselt number. In particular, we
introduce the surface-averaged Nusselt number Nuav given as
follows:
Nuav HSNulocaldsH
S1ds; Nulocal
2RTs TN
vTvn
(11)
where Nulocal is the local Nusselt number, TN is the
free-streamtemperature, Ts is the particle surface temperature and
n is theinward-pointing normal.
F. Dierich, P.A. Nikrityuk / International Journal of Thermal
Sciences 65 (2013) 92e103 99Fig. 10. Contour plots of the isotherms
T TN/Ts TN for different Re and roughnesses (a) R(d) Re 10, d 0.5,
Sef 2.12, (e) Re 40, d 0.5, Sef 2.12 and (f) Re 100, d 0.5, See 10,
d 0.1, Sef 1.08, (b) Re 40, d 0.1, Sef 1.08, (c) Re 100, d 0.1, Sef
1.08,f 2.12.
-
Fig. 11. Snapshot of the isotherms T TN/Ts TN for Re 100 and Sef
2.12.
Fig. 12. Contour plots of the non-dimensional temperature
gradientvTvx2 vTvy
2q 2R
DTfor
Re 40 and Sef 1 (left), and Sef 2.12 (right), respectively. The
maximum in the leftgure is 6.33 and the maximum in the right gure
is 7.27.
Fig. 13. Effect of surface enlargement (Sef) on the efciency
factor (Ef).
Table 5Nusselt number (Nu) in different Re and Sef.
Sef d Re: 10 20 40 100 200
1.00 0.00 1.847 2.451 3.280 5.279 7.6961.08 0.10 1.654 2.188
2.916 4.633 6.7131.29 0.20 1.345 1.779 2.369 3.757 5.4142.12 0.50
0.786 1.039 1.381 2.168 3.129
Table 6Drag coefcient (CD) in different Re and Sef.
Sef d Re: 10 20 40 100 200
1.00 0.00 2.782 2.007 1.502 1.289 1.3001.08 0.10 2.757 1.993
1.496 1.302 1.3271.29 0.20 2.768 2.000 1.501 1.331 1.3592.12 0.50
2.882 2.091 1.577 1.413 1.467
F. Dierich, P.A. Nikrityuk / International Journal of Thermal
Sciences 65 (2013) 92e103100In order to study the inuence of
roughness on the heat transferwe introduce the heat transfer
efciency factor Ef, described byBhattacharyya and Singh [17], given
by:
Ef NuavNu0av
(12)
where Nu0av is the surface-averaged Nusselt number for the
particlewith zero roughness. Thus, Ef measures the ratio between
theaverage rate of heat transfer from a rough particle to the
averagerate of heat transfer from a particle without roughness.
Thus, Ef > 1corresponds to heat transfer enhancement and Ef <
1 correspondsto insulation.
The last parameter to characterize the roughness is the
surfaceenlargement Sef given by:
Sef SroughS0
(13)
where S0 and Srough are the geometric surface area of the
particlewithout roughness and with roughness, respectively.
The equation to calculate the drag coefcient takes the
followingform:
CD FD
ru2inRF!
I p n! n
V u! V u!T
$ n!ds (14)
In this work the numerical simulations are carried out for
ve
Reynolds numbers: 10, 20, 40, 100 and 200. For each
Reynoldsnumber we systematically investigate the inuence of the
Fig. 14. Comparison of the present results and the results of
Dierich and Nikrityuk [39].
-
roughness of the cylinder on the surface-averaged Nusselt
number.The roughness of the particle is varied by increasing d, see
Fig. 2.
Fig. 9 shows an example of the velocity vectors and
thetemperature distribution near the particle surface for Re 40
andd 0.50. It can be seen that the velocity is zero in the dimples.
Thus,the air in the dimples plays the role of an isolator, which
decreasesthe convective heat transfer. This effect can be seen
clearly inFig. 10, which depicts the contour plots of the
nondimensionaltemperature T TN=Ts TN for different Re and
roughnessratios. It should be noted that in the case of Re 100 we
useda time-averaging over ve periods to obtain the spatial
distributionof the mean time temperature. Fig. 11 shows a snapshot
of thenondimensional temperature contour plot calculated for Re
100.The increase in the Re number at a constant value of Sef leads
toa decrease in the thermal boundary layer, which is a
well-knownphenomenon. The results show that due to the isolation
effectproduced by the dimples the thermal boundary layer
thicknessincreases in comparison to the cases with less roughness.
Thus, thetemperature gradient in the dimples is decreased. This
effect ispartially explained by the low value of the local Reynolds
number,which takes the following form:
Renotch uindnotch
n(15)
where dnotch is the characteristic size of the dimple with
themaximum value calculated as follows:
-180 -135 -90 -45 0 45 90 135 1800
0.05
0.1
0.15
0.2
0.25
u/u
in
D2 /D 1= 0.0, Sef = 0.00, Re = 40D2 /D 1= 0.5, Sef = 2.12, Re =
40
Fig. 15. Azimuthal prole of the velocity magnitude at a distance
of 0.2 R from the R.Here D2/D1 1 d.
Table 7Strouhal number (Sr) with different Re and Sef.
Re 100 100 200 200Sef 1.00 2.12 1.00 2.12Sr 0.164 0.169 0.196
0.196
0.9996
0.9998
1
1.0002
1.0004
350 352 354 356
0.0001
0.0002
Nu a
v / N
u av,
timem
ean
ux,volav
t*
Nuav, Sef=1.00, d=0.00Nuav, Sef=2.12, d=0.50
0.996 0.997 0.998 0.999
1 1.001 1.002 1.003 1.004
350 352 354 356
Nu a
v / N
u av,
timem
ean
t*
Nuav, Sef=1.00, d=0.00Nuav, Sef=2.12, d=0.50
a
b
Fig. 16. Plot of normalized Nusselt number Nuav and normalized
volum
F. Dierich, P.A. Nikrityuk / International Journal of Thermal
Sciences 65 (2013) 92e103 101358 360 362 364
-0.0002
-0.0001
0
/ uin
ux,volav, Sef=1.00, d=0.00ux,volav, Sef=2.12, d=0.50
358 360 362-2e-05-1.5e-05-1e-05-5e-060 5e-06 1e-05 1.5e-05
2e-05
ux,volav
/ uin
ux,volav, Sef=1.00, d=0.00ux,volav, Sef=2.12, d=0.50e-averaged
cross-ow velocity for (a) Re 100 and (b) Re 200.
-
urnadnotch 2pR20
$R2
r R
p20
r(16)
Thus, in the case of maximum roughness we have the localReynolds
number equals to Renotch
p=20
pRez0:4Re. If
Re 200, we have Renotch z 79, which is not enough to
establishthe convective heat transfer inside the dimple. However,
at thesame timewe have an increase in the temperature gradient in
frontof the stagnation point on the particle surface. This can be
seen inFig. 12, which shows contour plots of the
non-dimensionaltemperature gradient VT:
VT vTvx
2 vTvy
2s
2RDT
(17)
It can be seen that the local heat transfer changes
dramatically.In particular, we have temperature gradients
concentrated on theparticle ledges. This effect can play a very
important role in thecombustion of rough particles leading to the
local speed-up of thecombustion rate on the convex interfaces.
The increase in the thickness of the effective thermal
boundarylayer also leads to a decrease in the surface-averaged
Nusseltnumber with an increase in Sef. This effect is demonstrated
in Fig.13and some data is also presented in Table 5. It can be seen
that theefciency factor Ef is proportional to the surface
enlargementcoefcient Sef as follows:
Ef S5=4ef (18)
We found out that this equation is valid for all Re
numbersconsidered. This is consistent with the results predicted
for anothernotch shape for low Re (Re 40), see Dierich and
Nikrityuk [39]. Inparticular, the shape used in Dierich and
Nikrityuk [39] is shown inFig. 2(c). Fig. 14 conrms that eq. (18)
is valid in both cases.However, in comparison to the behavior of
the Nusselt number, thedrag coefcient CD increases only slightly.
The increase at thehighest roughness d 0.5 depends on the Reynolds
number. Fora Reynolds number of 10 the increase is only 3.6% but
for a Reynoldsnumber of 200 it is 12.8%, see Table 6. This can be
explained by theinuence of the roughness on the hydrodynamic
boundary layer.The thickness of the hydrodynamic boundary layer
decreases as theReynolds number is increased. A small boundary
layer increases theinuence of the roughness and leads to a higher
increase in thedrag coefcient. The inuence of the roughness on the
boundarylayer is demonstrated in Fig. 15, which shows the azimuthal
proleof the velocity magnitude at a distance of 0.2 R from R. The
calcu-lated proles for the rough and smooth particles are almost
iden-tical except for the region at q 135.
This also shows that the inuence of the roughness on theStrouhal
number is low. It changes less than 3%. This is shown inTable 7 and
demonstrated in Fig. 16. Fig. 16 shows the plots of thenormalized
Nusselt number Nuav/Nuav,timemean and normalizedvolume-averaged
cross-ow velocity ux,volav/uin for Re 100 andRe 200. The cross-ow
direction ux,volav is dened asux;volav
RVuxdv and Nuav,timemean is the time average of Nuav over
one period. t* is the dimensionless time dened as t* t$uin/(2R).
Itcan be seen that the oscillation frequency of the Nusselt
numberNuav is twice as large as the oscillation frequency of the
volume-averaged velocity in the cross-ow direction ux,volav. One
cycle ofthe Nusselt number consists in shedding one vortex on one
side ofthe cylinder, while one cycle of the volume-averaged
velocity
F. Dierich, P.A. Nikrityuk / International Jo102consists in
shedding two vortices. A similar result with Nu and liftcoefcient
is reported by Baranyi [40].7. Conclusions
A numerical investigation was carried out of steady laminar
andunsteady ow past a heated cylindrical particle with
differentroughnesses. The effect of the thickness of the roughness
layer onthe ow and heat transfer was systematically investigated.
Basedon the numerical data and discussions presented, several
conclu-sions can be summarized as follows:
1. The roughness has a signicant impact on the
surface-averagedNusselt number. In particular, the Nusselt number
decreasesrapidly as the degree of roughness increases.
2. The dependency of the efciency factor Ef on the
surfaceenlargement coefcient Sef can be approximated using
thefollowing relation EfzS
5=4ef for 10 Re 200.
3. The impact of the roughness on the drag coefcient and
theStrouhal number is small in comparison to the surface-averaged
Nusselt number.
Acknowledgments
This work was nancially supported by the Government ofSaxony and
the Federal Ministry of Education and Science of theFederal
Republic of Germany as a part of CIC VIRTUHCON. Theauthors thank
Prof. F. Durst for his comments concerning the localReynolds
number.
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F. Dierich, P.A. Nikrityuk / International Journal of Thermal
Sciences 65 (2013) 92e103 103
A numerical study of the impact of surface roughness on heat and
fluid flow past a cylindrical particle1. Introduction2. Problem
formulation and governing equations2.1. What is roughness?2.2.
Problem setup
3. Immersed surface reconstruction4. Source terms and numerics5.
Code validation6. Results7.
ConclusionsAcknowledgmentsReferences