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: frank.dierich@vtc.tu-freiberg.de (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.

References

[1] E. Achenbach, Inuence of surface roughness on the cross-ow arounda circular cylinder, J. Fluid Mech. (1971) 321e335.

[2] T. Fujii, M. Fujii, M. Takeuchi, Inuence of various surface roughness on thenatural convection, Int. J. Heat Mass. Transf. 16 (1973) 629e640.

[3] E. Achenbach, The effect of surface roughness on the heat transfer froma circular cylinder to the cross ow of air, Int. J. Heat Mass. Transf. 20 (1977)359e369.

[4] E. Achenbach, E. Heinecke, On vortex shedding from smooth and roughcylinders in the range of Reynolds numbers 6 103 e 5 106, J. Fluid Mech.109 (1981) 239e251.

[5] T. Kawamura, H. Takami, K. Kuwahara, Computation of high Reynolds numberow around a circular cylinder with surface roughness, Fluid Dyn. Res. 1(1986) 145e162.

[6] D. Lakehal, Computation of turbulent shear ows over rough-walled circularcylinders, J. Wind Eng. Ind. Aerodyn. 80 (1999) 47e68.

[7] J. Taylor, A. Carrano, S. Kandlikar, Characterization of the effect of surfaceroughness and texture on uid ow - past, present, and future, Int. J. Therm.Sci. 45 (2006) 962e968.

[8] C. Lange, F. Durst, M. Breuer, Momentum and heat transfer from cylinders inlaminarow at 104 Re 200, Int. J. HeatMass. Transf. 41 (1998) 3409e3430.

[9] J.-M. Shi, D. Gerlach, M. Breuer, G. Biswas, F. Durst, Heating effect on steadyand unsteady horizontal laminar ow of air past a circular cylinder, Phys.Fluids 16 (2004) 4331e4345.

[10] G. Juncu, Unsteady conjugate heat/mass transfer from a circular cylinder inlaminar crossow at low Reynolds numbers, Int. J. Heat Mass. Transf. 47(2004) 2469e2480.

[11] Y. Ji, K. Yuan, J. Chung,Numerical simulationofwall roughnessongaseousowandheat transfer in a microchannel, Int. J. Heat Mass. Transf. 49 (2006) 1329e1339.

[12] G. Gamrat, M. Favre-Marinet, S. Le Person, Modelling of roughness effect onheat transfer in thermally fully-developed laminar ows through micro-channels, Int. J. Therm. Sci. 48 (2009) 2203e2214.

[13] C. Zhang, Y. Chen, M. Shi, Effect of roughness elements on laminar ow andheat transfer in microchanels, Chem. Eng. Proc. 49 (2010) 1188e1192.

[14] Y. Chen, P. Fu, C. Zhang, M. Shi, Numerical simulation of laminar heat transferin microchannels with rough surfaces characterized by fractal Cantor struc-tures, Int. J. Heat Fluid Flow 31 (2010) 622e629.

[15] B.A. Abu-Hijleh, Numerical simulation of forced convection heat transfer froma cylinder with high conductivity radial ns in cross-ow, Int. J. Therm. Sci. 42(2003) 741e748.

[16] J. Koo, C. Kleinstreuer, Analysis of surface roughness effects on heat transfer inmicro-conduits, Int. J. Heat Mass. Transf. 48 (2005) 2625e2634.

[17] S. Bhattacharyya, A. Singh, Augmentation of heat transfer from a solid cylinderwrapped with a porous layer, Int. J. Heat Mass. Transf. 52 (2009) 1991e2001.

[18] C. Peskin, Flow pattern around heart valves: a numerical method, J. Comput.Phys. 10 (1972) 252e271.

l of Thermal Sciences 65 (2013) 92e103[19] D. Goldstein, R. Handler, L. Sirovich, Modeling a no-slip owboundary with anexternal force eld, J. Comput. Phys. 105 (1993) 354e366.

[20] W. Bennon, F. Incropera, A continuummodel for momentum, heat and speciestransport in binary solid-liquid phase change systems i. model formulation,Int. J. Heat Mass. Transf. 30 (1987) 2161e2170.

[21] K. Khadra, P. Angot, S. Parneix, J.-P. Caltagirone, Fictitious domain approachfor numerical modelling of NaviereStokes equations, Int. J. Numer. MethodsFluids 34 (2000) 651e684.

[22] J. Ravoux, A. Nadim, H. Haj-Hariri, An embedding method for bluff body ows:interactions of two side-by-side cylinder wakes, Theor. Comput. Fluid Dyn. 16(2003) 433e466.

[23] K. Ng, A collocated nite volume embedding method for simulation of owpast a stationary and moving body, Comput. Fluids 38 (2009) 347e357.

[24] I.E. Sutherland, G.W. Hodgman, Reentrant polygon clipping, Commun. ACM(CACM) 17 (1974) 32e42.

[25] A. Watt, S. Maddock, Mainstream Rendering Techniques, in: A.B. Tucker (Ed.),Computer Science. Handboook, Taylor & Francis Group, LLC, 2004 (p. SectionIV: Graphics and Visual Computing).

[26] S.V. Patankar, Numerical Heat Transfer and Fluid Flow, Taylor & Francis, 1980.[27] J.H. Ferziger, M. Peric, Computational Methods for Fluid Dynamics, third ed.,

Springer, Berlin, 2002.[28] C. Rhie, W. Chow, Numerical study of the turbulent ow past an airfoil with

trailing edge separation, AIAA J. 21 (1983) 1525e1532.[29] H.L. Stone, Iterative solution of implicit approximations of multidimensional

partial differential equations, SIAM J. Numer. Anal. 5 (1968) 530e558.[30] ANSYS, Inc, ANSYS-FLUENT V 13.0 e Commercially Available CFD Software

Package Based on the Finite Volume Method. Southpointe, 275 TechnologyDrive, Canonsburg, PA 15317, U.S.A. (2011) www.ansys.com

[31] C. Jackson, A nite-element study of the onset of vortex shedding in ow pastvariously shaped bodies, J. Fluid Mech. 182 (1987) 23e45.

[32] M. Moryznski, K. Afanasiev, F. Thiele, Solution of the eigenvalue problemsresulting from global non-parallel ow stability analysis, Comput. MethodsAppl. Mech. Eng. 169 (1999) 161e176.

[33] M. Provansal, C. Mathis, L. Boyer, Bnard-von Krmn instability: transientand forced regimes, J. Fluid Mech. 182 (1987) 1e22.

[34] C. Norberg, An experimental investigation of the ow around a circularcylinder: inuence of aspect ratio, J. Fluid Mech. 258 (1994) 287e316.

[35] D. Barkley, R.D. Henderson, Three-dimensional Floquet stability analysis of thewake of a circular cylinder, J. Fluid Mech. 322 (1996) 215e241.

[36] C. Williamson, Vortex dynamics in the cylinder wake, Annu. Rev. Fluid Mech.28 (1996) 477e539.

[37] M.S. Bloor, The transition to turbulence in the wake of a circular cylinder,J. Fluid Mech. 19 (1964) 290e304.

[38] G. Miller, G. Williamson, Control of three-dimensional phase dynamics ina cylinder wake, Exp. Fluids 18 (1994) 26e35.

[39] F. Dierich, P. Nikrityuk, A numerical study of the inuence of surfaceroughness on the convective heat transfer in a gas ow, CMES 64 (2010)251e266.

[40] L. Baranyi, Computation of unsteady momentum and heat transfer froma xed circular cylinder in laminar form, J. Comput. Appl. Mech. 4 (2003)13e25.

[41] H. Takami, H.B. Keller, Steady two-dimensional viscous ow of an incom-pressible uid past a circular cylinder, Phys. Fluids 12 (Suppl. II) (1969)51e56.

[42] S. Dennis, G.-Z. Chang, Numerical solution for steady ow past a circularcylinder at Reynolds numbers up to 100, J. Fluid Mech. 42 (1970) 471e489.

[43] G. Biswas, S. Sarkar, Effect of thermal buoyancy on vortex shedding pasta circular cylinder in cross-ow at low Reynolds numbers, Int. J. Heat Mass.Transfer 52 (2009) 1897e1912.

[44] O. Posdziech, R. Grundmann, A systematic approach to the numerical calcu-lation of fundamental quantities of the two-dimensional ow over a circularcylinder, J. Fluid Struct. 23 (2007) 479e499.

[45] H. Persillon, M. Braza, Physical analysis of the transition to turbulence in thewake of a circular cylinder by three-dimensional NaviereStokes simulation,J. Fluid Mech. 365 (1998) 23e88.

[46] A.-B. Wang, Z. Trvncek, On the linear heat transfer correlation of a heatedcircular cylinder in laminar crossow using a new representative temperatureconcept, Int. J. Heat Mass. Transfer 44 (2001) 4635e4647.

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