Development of laminar mixed convection in a horizontal squarechannel with heated side walls

J.J.M. Sillekens, C.C.M. Rindt *, A.A. van Steenhoven

Faculty of Mechanical Engineering, Eindhoven University of Technology, WH-3.129, P.O. Box 513, 5600 MB Eindhoven, The Netherlands

Received 21 September 1996; accepted 15 August 1997

Abstract

A ®nite element code is used for detailed analysis of mixed-convection ¯ow in a horizontal channel heated from the side walls.

The Reynolds number typically is Re � 500, and the Grashof number varies around Gr � 105. Firstly, the numerical code is val-

idated by a quantitative comparison with results of particle-tracking experiments for the velocity and liquid crystal measurements

for the temperature. It is concluded that the agreement between the numerical and experimental data is satisfactory and that the

®nite element approximation employed can be used in analyzing mixed-convection ¯ow problems in complex geometries. Secondly,

the velocity and temperature ®elds are further described by means of isovelocity and isotemperature lines. The results are quanti®ed

by the relative kinetic energy of secondary ¯ow, the Fanning friction factor, and the Nusselt number as a function of the streamwise

position. It seems that the resulting secondary ¯ow induced by buoyancy forces causes a substantial increase in heat transfer, know-

ledge of which is of importance for the design of compact heat exchangers. Ó 1998 Elsevier Science Inc. All rights reserved.

Keywords: Mixed convection; Liquid crystal thermography; Particle-tracking velocimetry; Channel ¯ow; Laminar ¯ow

1. Introduction

The present study on mixed convection in a horizontalsquare channel is initiated by the research on the complicated¯ow patterns occurring in the coiled heat exchanger of a solardomestic hot water system (SDHWS). When such a system isoperated according to the low-¯ow principle [the Reynoldsnumber being Re � O�103� and the Grashof number beingGr � O�105�], the ¯ow is laminar, and both natural and forcedconvection must be taken into account. Buoyancy and centri-fugal forces give rise to a secondary ¯ow perpendicular to themain axial ¯ow. The relative magnitude of the secondary ¯owdue to buoyancy forces can be estimated by

UGr

U� O

������Grp

Re

!�1�

It is expected that the secondary ¯ow due to buoyancy forcesfor high Prandtl number ¯uids is lower than for low Prandtlnumber ¯uids. This e�ect on secondary ¯ow caused by buoy-ancy forces may be appreciated when the study of Le Fevre(1956) is considered, where buoyancy-induced ¯ow at a verti-cal plate is described. From the similarity formulation forthe boundary-layer equations governing this ¯ow, it can be

International Journal of Heat and Fluid Flow 19 (1998) 270±281

Notation

cp speci®c heat capacityf Fanning friction factorg gravity accelerationGr Grashof numberH height of channel or cavityK relative kinetic energyNu Nusselt numberp pressurePr Prandtl numberRe Reynolds numberT temperatureU mean axial velocityuax axial velocityUGr buoyant secondary velocityusec secondary velocity~u velocity vectoru; v;w velocity components~x position vectorx; y; z Cartesian coordinates

Greekb cubic expansion coe�cientDt time intervalDT characteristic temperature di�erencek thermal conductivity

l dynamic viscosityq densityrxx; rxy ; rxz stress components�sw mean wall shear stress

* Corresponding author. E-mail: c.c.m.rindt@wtb.tue.nl.

0142-727X/98/$19.00 Ó 1998 Elsevier Science Inc. All rights reserved.

PII: S 0 1 4 2 - 7 2 7 X ( 9 7 ) 1 0 0 0 4 - 2

deduced that the in¯uence of the Prandtl number on secondary¯ow is as follows:

UGr

U� O

������Grp

Re������������������1� Pr�p" #

�2�

The e�ciency of helically coiled heat exchangers is largely in-¯uenced by this secondary ¯ow through which temperatureboundary layers remain thin, and heat transfer increases.

In this investigation, attention is focussed on mixed-convec-tion ¯ow in a horizontal square channel heated from the sidewalls. In such a con®guration, the physical phenomena occur-ring can be studied relatively easily using experimental methods,making it possible to validate the numerical methods developed.The literature on this subject up to 1989 has been reviewed byHartnett and Kostic (1989) and Shah and Joshi (1987). The re-sulting ¯ow ®eld is determined by a relatively large set of param-eters: the Grashof, Reynolds, and Prandtl numbers, thetemperature boundary conditions on the walls, and the aspectratio of the channel. Buoyancy-driven ¯ows cause the heattransfer rate to increase, an e�ect more pronounced when theduct is ¯atter (larger width than height). It is shown that for suf-®ciently high Grashof numbers, buoyancy e�ects may give riseto the occurrence of two or more steady longitudinal vortices(Nandakumar et al., 1985) and recirculation zones (Kleijn,1991). The temporal stability of the ¯ow for several boundaryconditions at higher Grashof number �Gr > 105� has been in-vestigated recently by Hosokawa et al. (1993) and Evans andGreif (1993), and Huang and Lin (1995). Air ¯ow heated frombelow becomes unsteady at Gr > 5� 105 for Re � 220 (Hoso-kawa et al., 1993). Evans and Greif (1993) and Huang and Lin(1995) show that the critical Grashof number (above which the¯ow becomes unsteady) is smaller when the Reynolds numberdecreases. In most studies, the case of air �Pr � 0:7� is consid-ered. The case of water in a horizontal channel heated from be-low was investigated numerically by Biswas et al. (1990), whoshowed the existence of several longitudinal vortices near thelower wall. Water in a horizontal channel heated both fromabove and below was studied experimentally by Incropera etal. (1987). It was shown that at the top wall a stably strati®edlayer occurred, which was hardly in¯uenced by the buoyancy-induced ¯ow from the lower wall.

In the present study, water ¯ow in a horizontal square chan-nel heated from the side walls is investigated both experimental-ly, using liquid crystal thermography and particle-trackingvelocimetry, and numerically, using a ®nite element approach.The goal of the experiments was twofold. Firstly, from the ex-periments conducted at various Grashof numbers it could beconcluded that in the range of Grashof numbers under consid-eration (Prandtl and Reynolds were kept constant) no temporal¯ow instabilities are expected to occur, knowledge which is in-dispensable for the development of a proper numerical solutionprocedure. Secondly, the experimentally determined tempera-tures and velocities at one speci®c Grashof number are usedfor validation of the numerical code, because substantial gridre®nement was not possible on the computers available becauseof computation times needed and memory limitations. Because,as we show, the agreement is quite satisfactory, the numericalmodel can be used to analyze in more detail mixed-convective¯ow for various Grashof numbers with emphasis on the in¯u-ence of secondary ¯ow on heat transfer.

2. Numerical procedure

2.1. Equations

The conservation equations of mass, momentum, and ener-gy are simpli®ed using the Oberbeck±Boussinesq approxima-

tions (Oberbeck, 1888). In Gray and Giorgini (1976) thelimitations to those approximations are discussed. For water,only minor errors are introduced when, as in the present study,the temperature di�erences are of O�100�. The following di-mensionless quantities are de®ned as

~u0 � ~uU; T 0 � T ÿ Tinlet

Twall ÿ Tinlet

; p0 � pqU 2

; ~g0 �~gg;

~r0 � H~r; �3�where the characteristic velocity U equals the mean axial veloc-ity and the characteristic length scale the height H of the chan-nel. In dimensionless form, the equations for steady ¯ow thenread (with omission of the primes):

Continuity:

~r �~u � 0 �4�Momentum:

~u � ~r~u� ~rp ÿ 1

Re~r � ~ru� Gr

Re2T~g � 0 �5�

Energy:

~u � ~rT ÿ 1

RePr~r � ~rT � 0 �6�

with Re � qUH=l denoting the Reynolds number, Pr � lcp=kdenoting the Prandtl number, and Gr � gbq2DTH 3=l2 denot-ing the Grashof number �DT � Twall ÿ Tinlet�. All ¯uid proper-ties in the above mentioned equations are constant andtaken at the reference temperature Tinlet. In the present study,channel ¯ow is solved numerically by discretization of thoseequations by a ®nite element formulation. For details aboutthe ®nite element technique one is referred to the standard textbooks (Cuvelier et al., 1986), for details about the implementa-tion used in the present study one is referred to Sillekens et al.(1994a, b). Here only a brief description is given.

2.2. Solution procedure

The non-linear convective term is linearized using Picard'smethod. The penalty function method is used to decouplethe momentum and continuity equations. In order to minimizethe size of the matrices to be handled, the velocity and temper-ature ®elds are solved as separate problems in an iterative fash-ion. To avoid spatial oscillations in the temperature ®eld thestreamline upwind/Petrov±Galerkin (SUPG) method (Brooksand Hughes, 1982) is used for discretization of the energyequation. For O�h3� accurate quadratic elements (which willbe used throughout this study) the SUPG-method showed tobe O�h3� accurate for smooth solutions, even when the Pecletnumber is high (Segal, 1993).

The calculation domain is presented in Fig. 1. From the ex-perimental results, it is seen that the ¯ow in the channel is sym-metric with respect to the plane y � 0. For the computations,therefore, only half of the channel is discretized. In the axialdirection, the calculation domain equals 11 H�ÿ1 <x=H < 10�. For the momentum equation 31� 5� 10 triqua-dratic hexahedral Crouzeix±Raviart elements are used in thex-, y- and z- direction, respectively. The element distributionin a cross section is shown in Fig. 1. The energy equationwas solved on a 50� 8� 12 mesh of triquadratic hexahedrals,which is re®ned about 1.5 times as compared to the mesh forthe momentum equation because of the thinner boundary lay-ers expected �Pr > 1�. To evaluate the temperature in the nodalpoints of the mesh for the momentum equation and to evaluatethe velocity values in the nodal points for the energy equation,a triquadratic interpolation between the meshes is employed.

The problem is solved numerically on a Silicon Graphics Su-per Challenge (MIPS R8000 processor). When using a solution

J.J.M. Sillekens et al. / Int. J. Heat and Fluid Flow 19 (1998) 270±281 271

at a lower Grashof number as a starting solution, typically 25iterations were necessary to obtain a converged solution. Eachiteration took 33 min CPU-time. Both computation timesneeded and memory limitations prevent signi®cant mesh re-®nement studies on the computers available, making experi-ments indispensable.

2.3. Boundary conditions

The boundary conditions are given in Table 1 and in Fig. 1.The prescribed inlet pro®le u�y; z� at x � ÿH (plane IN inFig. 1) is an approximation to within �2% accuracy of the ex-act pro®le for a fully developed channel ¯ow (Shah and Joshi,1987). At the walls (SW-, SW+, TW, BW) the no-slip condi-tion is applied; whereas, at the midplane y � 0 (SY), the sym-metry condition holds. At x � 10 H (OUT) for the velocity®eld, the stresses are set equal to zero; i.e.,

rxx � ÿp � 2

Re

@u@x� 0; rxy � 1

Re

@u@y� @u@x

� �� 0;

rxz � 1

Re

@u@z� @w@x

� �� 0:

For the temperature, all the boundaries act as adiabatic wallsexcept the in¯ow boundary (IN: Tinlet � 0) and the vertical sidewalls for x P 0 (SW+: Twall � 1).

Out¯ow boundary conditions still receive a lot of attentionin literature (Sani and Gresho, 1994) and are often based onintuition, experience, asymptotic behavior, and numerical ex-perimentation (Papanastasiou et al., 1992). The out¯ow boun-dary conditions for the stresses used here are natural boundaryconditions in the Galerkin weak formulation and result in awell-posed problem (Sani and Gresho, 1994). When a ¯ow isfully developed the condition rxx � 0 sets the dynamic pressurep at the out¯ow boundary to zero. A secondary ¯ow, however,generally gives rise to a non-uniform pressure distribution over

Table 1

Boundary conditions prescribed. For a de®nition of the planes, see Fig. 1

IN: x � ÿH u�y; z� � f1ÿ �y=�12H��2:2g � f1ÿ �z=�1

2�H �2:2g In¯ow condition

v � w � 0

Tinlet � 0

OUT: x � 10H 2Reÿ1@u=@xÿ p � 0 Out¯ow condition

@u=@y � @v=@x � 0

@u=@z� @w=@x � 0

@T=@x � 0

SY: y � 0 v � 0 Symmetry condition

@u=@y � @w=@y � 0

@T=@y � 0

SW ) : y � 12H ; x < 0 u � v � w � 0 No-slip adiabatic

@T=@y � 0

SW+: y � 12H ; x P 0 u � v � w � 0 No-slip wall-temp.

Twall � 1

TW: z � � 12H u � v � w � 0 No-slip adiabatic

BW: z � ÿ 12H @T=@z � 0

Fig. 1. Left: calculation domain; right: element division in half a cross-section with nodal points marked by a circle; x indicates the main axial ¯ow

direction, y � 0 refers to the plane of symmetry (SY), x � ÿH refers to the inlet (IN), x � 10 H refers to the outlet (OUT), y � 12H and x P 0 (SW+)

indicate the side walls and z � � 12H indicates the horizontal top wall (TW) and z � ÿ 1

2H indicates the horizontal bottom wall (BW).

272 J.J.M. Sillekens et al. / Int. J. Heat and Fluid Flow 19 (1998) 270±281

a cross-section. Therefore, the prescription of a zero normalstress generally leads to errors in the induced secondary ¯ow®eld near the out¯ow boundary, referred to as arti®cial out-¯ow boundary layers by Sani and Gresho (1994) and also ob-served by Evans and Greif (1993). By similar arguments, it canbe deduced also that the other boundary conditionsrxy � rxz � 0 are physically not correct. However, from nu-merical experience, it is known that these boundary conditionsonly a�ect the solution a few elements upstream of the out¯owboundary (van de Vosse, 1987; Rindt et al., 1991). This willalso be made plausible in the present paper when discussingthe in¯uence of the Grashof number on the temperature andvelocity ®elds. The numerical results in the out¯ow region in-dicated as shaded areas in Figs. 8±10, should, because of theaforementioned, be handled with care.

3. Experimental procedure

3.1. Experimental setup

The considered con®guration is sketched in Fig. 2 (top).Water ¯ows through a horizontal square channel of 3.1cm cross-section. The medium is symmetrically heated at theside walls by water ¯owing through the heat exchangers. The¯ow through the heat exchangers is separated from the ¯owthrough the channel by 0.25 mm thick polycarbonate (Lexan)

sheets. Because of the high ¯ow rate through the heat exchang-ers and the thin dividing walls, the temperature of the sidewalls can be kept at a relatively uniform temperature ofTwall � 0:15�C, as compared to a typical temperature di�erenceof Twall ÿ Tinlet � 1:5�C. Fig. 2 (bottom) shows a scheme of thetotal experimental setup. The setup consists of two separatecircuits. With the main circuit, the axial ¯ow through the chan-nel is controlled. Conditioned water ¯ows from basin b1

through an in¯ow section, which is su�ciently long (65 H )for the ¯ow to become hydrodynamically fully developed.The developed ¯ow with temperature Tinlet � 19:7�C entersthe measurement section where it is heated by the heat ex-changers (temperature Twall � 21:2�C� of the secondary circuit.The ¯ow through the main circuit is measured by ¯ow meter Vand is pumped by a peristaltic pump p1 from basin b2 to basinb1. The ¯ow rate can be adjusted using a valve at V . The tem-perature of the water in the main circuit is controlled by coolerc and by a Julabo P1 thermostatic heater h1. The ¯ow throughthe channel is heated by two heat exchangers in the measure-ment section. Water is pumped by centrifugal pump p2 fromthe lower basin to the higher basin from which it ¯ows throughthe heat exchangers. The temperature of this circuit is con-trolled by a Julabo P1 thermostatic heater h2.

Although the main circuit was very well insulated by 20 cmthick insulation plates, a heat ¯ux of about 1 Wmÿ2 from themedium in the in¯ow section to the surrounding air could notbe avoided. As a result of this small heat ¯ux the medium in

Fig. 2. Top: 3 D-view of the measurement section; water ¯ows through the channel along the x-axis and is heated at the side walls; the arrows indicate

the ¯ow of the heating medium. Bottom: the experimental setup; the main circuit (dark gray) deals with the conditioning of the axial ¯ow through the

measurement section; the secondary circuit (light gray) maintains the heat exchangers of the measurement section at a constant temperature.

J.J.M. Sillekens et al. / Int. J. Heat and Fluid Flow 19 (1998) 270±281 273

the in¯ow section experienced a buoyancy e�ect with a Gra-shof number of Grq00 � gbq2q00H 4=l2k � 5� 103. It will beshown that, as a result of this e�ect, the velocity pro®le atthe entrance of the measurement section, �x � 0� was some-what shifted with respect to the fully developed velocity pro®lewithout this buoyancy e�ect.

The total accuracy of the setup in terms of Reynolds andGrashof numbers approximately equals 3% �Re � 500� and15% �Gr � 105�, respectively. The accuracy of the latter canbe improved by using a channel of smaller cross-section (tothe cost of worse optical accessibility) and thinner dividingwalls between the main channel and the heat exchangers (tothe cost of rigidity).

3.2. Particle-tracking velocimetry

Particle-tracking (PT) is a method for measuring the veloc-ity ®eld in a ¯uid by means of keeping track of individual par-ticles suspended in the ¯uid (Adrian, 1991). The ¯ow is hardlyin¯uenced by the particle seeding, because the concentration ofparticles can be kept very low. In this study, the PT facility ofthe computer package DigImage is used. Details of the methodas implemented in DigImage can be found in Dalziel(1992a, b).

The ¯ow con®guration is intersected by a thin light sheet.The perpendicular re¯ection of the particles is recorded on S-VHS tape. The video system is capable of grabbing each ®eldof the recording separately. Suppose that two subsequent®elds are grabbed, say ®eld P and ®eld Q at time tP and tQ,tQ � tP � Dt, where Dt � 0:04 s, the time interval betweentwo complete video ®elds. When the distance between the in-dividual particles in ®eld P is larger than the displacement ofthe particle during Dt, then every particle in ®eld P can bematched exactly to a particle in ®eld Q. The average velocityin the light sheet of a particle at position ~xP in P then can becomputed using the Lagrangian de®nition of velocity:~u � �~xP ÿ~xQ�=Dt, ~xQ being the position of the particle in Q.For details about the matching algorithm one is referred toDalziel (1992b).

In the experiments Optimage particles were used with a typ-ical diameter of 100 lm and in a concentration of about 0.001vol%. A cross-section of the channel equal to 6 cm in the axialdirection and 3.1 cm in the transverse direction was visualizedby a 3 mm thick light sheet. The acquired resolution was 85pixels/cm in the axial direction and 120 pixels/cm in the trans-verse direction, respectively. To avoid interlace e�ects, only theodd video ®elds were used, and the e�ective transverse resolu-tion is reduced to 60 pixels/cm. Because the volume centroid ofa particle was used to determine its position, the particles couldbe located with subpixel accuracy (Dalziel, 1992a). The resolu-tion of particle location approximately equals �2 lm and thevelocity resolution then is equal to �0:01 cm=s.

The density qp of the particles used is only slightly higherthan that of water, i.e. qP � 1000 kg=m

3. From the experi-

ments performed, it was estimated that the vertical velocityof the particles attributable to this density di�erence is lessthan 10ÿ3 cm/s.

The timescale on which suspended particles are able to fol-low rapid changes in velocity can be estimated by equating in-ertia forces and viscous forces acting on a particle. For thechannel ¯ow considered, this timescale is about 5� 10ÿ4 s. Avelocity increase of 1 cm/s will be caught up with within 5 lm.

Furthermore it is known from the experiments of Segre andSilberberg (1962) that small particles suspended in a Hagen±Poiseuille ¯ow have a tendency to move to a speci®c radial po-sition. From these experiments and from the analysis of Hoand Leal (1974), it can be concluded that this inertia-induced

migration (also referred to as the Magnus e�ect) results in atransverse velocity of less than 0.01 cm/s.

3.3. Liquid crystal thermography

Liquid crystal thermography (LCT) is used for quantitativevisualization of the temperature ®elds. The method is based onthe optical properties of a cholesteric liquid crystal. The chole-steric liquid crystal is characterized by its typical layered struc-ture. Because of this layered structure, they show constructiveinterference as a result of optical path di�erences of incominglight. There are two parameters in¯uencing the resulting re¯ec-tion of cholesteric liquid crystal: the angle of incidence h; andthe pitch pLC . When the temperature of cholesteric liquid crys-tal is changed, the orientation of the molecules is changed sothat the pitch decreases with increasing temperature. This phe-nomenon makes it possible to use the cholesteric liquid crystalphase for temperature visualization purposes: at a higher tem-perature shorter wavelengths are re¯ected. In order to avoidmechanical stress in¯uences and chemical in¯uences, the chole-steric material can be encapsulated in gum arabic. Encapsula-tion further prevents liquids crystals from contamination,which is important for use in water for instance. A major dis-advantage of encapsulation is that the re¯ection of encapsulat-ed liquid crystal is less intense than the re¯ection ofunencapsulated liquid crystal. Cholesteric liquid crystal is com-mercially available with red start temperatures ranging fromÿ30 to 115�C and with bandwidths from 1 to 50�C (Parsley,1991).

The time response of encapsulated cholesteric liquid crystalis dependent on the intrinsic time response of the liquid crystaland on the time response of the capsule material. According toFergason (1968), the intrinsic time response (the time the mol-ecules need to rearrange as a result of a sudden increase in tem-perature) is of the order of 0.1 s. The response time of thecapsule material is estimated to be less than 0.1 s.

For calibration of a certain type of liquid crystals, a suspen-sion in demineralized water of approximately 0.04 vol% ismade. To calibrate the re¯ected colours of liquid crystalsagainst temperature, a small transparent container is used inwhich the suspension is cooled slowly, starting from the highertemperature in the temperature bandwidth. The 90� re¯ectionof a 3 mm halogene light sheet crossing the container, is re-corded on an S-VHS video tape, and the temperature of thesuspension is registered simultaneously using a thermocouple.During calibration, the suspension is stirred by a mixer.

4. Comparison between experimental and numerical results

For a channel ¯ow with heated side walls, the density of themedium in the temperature boundary layers near those wallsdecreases as a result of the heat transferred from the side wallsto the medium in the channel. Because this gives rise to a bodyforce, the medium ¯ows upwards near the side walls and re-turns in the core of the channel. The relative magnitude ofthe secondary ¯ow compared to the primary ¯ow can be esti-mated to be

������Grp

=�Re������������������1� Pr�p � � 0:24 for the case of

Re� 500, Gr� 105, and Pr� 6 as described in this section. Be-cause the onset of these secondary vortices, the dynamic pres-sure in the upper region of the channel increases, and the axialvelocity pro®le will change along the channel. For validationof the code, in this section a comparison is made betweenthe experimentally measured and numerically calculated veloc-ities and temperatures. Before doing so, the method of measur-ing temperatures was ®rst tested in a di�erentially heatedcavity.

274 J.J.M. Sillekens et al. / Int. J. Heat and Fluid Flow 19 (1998) 270±281

4.1. An initial test: Di�erentially heated cavity

Liquid crystal thermography was tested in a di�erentiallyheated cavity with an aspect ratio equal to H=B � 2:33. The in-stalled Grashof number (based on the height of the cavity) wasequal to Gr � 1:3� 106, based on the height of the cavity. Thetemperature measurements were compared to ®nite elementcalculations. Fig. 3 shows a satisfactory agreement betweenthe experimental and numerical results, although it seems thatthe measured temperature ®eld is somewhat shifted towardshigher temperatures (more red and less blue), which may wellbe due to experimental uncertainties in the boundary condi-tions. The temperature boundary layers are resolved quite wellby LCT, showing the good spatial resolution of the method. Inthe lower temperature range, the agreement between the exper-imental and the numerical results is fair. In the higher range,the agreement is somewhat worse. In global terms, one cansay that the mean relative error is about 5% but that the max-imal relative error can go up to 10%. For more detailed infor-mation about this experiment, the processing of the liquidcrystal images, and the system calibration, one is referred toSillekens (1995).

4.2. Channel ¯ow: Midplane velocities

As mentioned above, because of the onset of secondary vor-tices in a channel ¯ow with heated side walls, the dynamicpressure in the upper region of the channel increases and theaxial velocity pro®le will change along the channel. This clearlyis observed in Fig. 4 where the measured velocity ®eld at thesymmetry plane of the channel for Re� 500, Gr� 105, andPr� 6 is presented.

In Fig. 4 (top), it is seen that the velocity ®eld (indicated bythe arrows) is still rather symmetric with respect to the planez� 0. From the gray background ®eld, indicating the magnitudeof the shear @u=@z, it can be observed that the maximum in theaxial velocity (indicated by the white band) remains at aboutz� 0. Fig. 4 (bottom) shows that the maximal axial velocityhas shifted towards lower values of z, and a remarkable changein the velocity pro®le is observed, which is caused by the convec-tive transport of the medium by the secondary motion.

The results in Fig. 4 show that particle tracking is a verypowerful tool for quantitative visualization of this ¯ow ®eld.It would, however, be interesting to measure the secondary¯ow ®eld in a cross-section of the channel at a particular axialposition. Unfortunately, this is not feasible: as will be shown,the axial velocity is an order of magnitude larger than the sec-ondary velocity, and, therefore, particles would pass throughthe light sheet too fast to be tracked.

In order to compare the numerical results to the experimen-tal results, the axial and vertical velocities at the symmetryplane y � 0 will be considered at three axial positions:x � ÿH , where a fully developed velocity pro®le is expected;x � 4:1 H , where the magnitude of the secondary velocityhas reached its maximum; and further downstream atx � 8 H . From Fig. 5 (top), it is seen that the measured in¯owvelocity pro®le di�ers somewhat from the analytical pro®le forfully developed channel ¯ow. As has been mentioned before,this is attributable to buoyancy e�ects in the in¯ow section,where the ¯ow is cooled slightly by the environment. The re-sulting secondary velocity causes the in¯ow pro®le to be shift-ed upwards slightly. The right graph of Fig. 5 (top), however,shows that the magnitude of the secondary ¯ow is very lowand is of the order of the uncertainty of the measurement.

Fig. 5 (middle) shows the axial and vertical velocity compo-nents at axial position x � 4:1 H . The computed vertical veloc-ity is seen to be in a good agreement with the experimentalresult, the measured axial velocity, however, still seems to suf-fer from the shifted in¯ow e�ect. A fair comparison betweenthe numerical and experimental results is, therefore, not possi-ble. It can be observed, however, that the downward move-ment of the maximum in the axial velocity pro®le from thein¯ow pro®le (Fig. 5, top) is about 0.07 H, both for the mea-sured and for the computed pro®le.

At x � 8 H , the maximum in the axial velocity pro®le isseen to be shifted by about 0.16 H. Although the measuredpro®le still seems to su�er from the shifted in¯ow, the agree-ment between the measured and the computed pro®le is good.Again, there is good agreement between the measured andcomputed vertical velocity, although the measured vertical ve-locity su�ers from the fact that in the upper region of the chan-nel �z > 0:3�, hardly any particles were observed. The reason

Fig. 3. Temperatures in a di�erentially heated cavity with aspect ratio 2.33; left: using LCT; right: computed temperatures.

J.J.M. Sillekens et al. / Int. J. Heat and Fluid Flow 19 (1998) 270±281 275

for this is that the particles used were slightly more dense thanwater, resulting in a particle-free upper layer in the channel.Because this particle-free layer is conveyed to the symmetryplane by the secondary ¯ow (see Fig. 7), in the upper regionof the channel e�ects of noise in the video system become im-portant, resulting in spurious mappings.

4.3. Channel ¯ow: Temperatures

To visualize the temperature ®eld in the channel, LCT isemployed. As explained before, the medium in the channel¯ows upwards near the side walls of the channel and ¯owsdownward in the core region. In the symmetry plane y� 0the medium thus becomes strati®ed: the warm medium, com-ing from the heated side walls, pushing away the cold medium.In Fig. 6 (left) the build-up of the warmer layer in the symme-try plane is clearly visualized. The experimentally visualizedtemperature ®eld is indicated by the color ®eld, whereas, thenumerical result is given by means of contours of equal tem-perature. Both from the experimental and the numerical re-sults, it is seen that the size of the hot layer grows slightly inthe axial direction, indicating the downward transport in thesymmetry plane. Fig. 6 (right) shows that this downwardtransport is the largest near the center of the channel. Boththe experimental and numerical results exhibit a typical M-shaped temperature boundary layer, showing that, althoughthe magnitude of the secondary ¯ow is small, its in¯uence onthe temperature ®eld (and thus on the heat transfer) is large.From this ®gure, it is also seen that the physical situation, in-deed, shows symmetry in the plane y� 0. The blue spots at the

left and right upper corners in Fig. 6 (right) are the result ofre¯ections at the wall of the channel.

4.4. Conclusion

From the above, it can be concluded that agreement be-tween the experimental and numerical velocity data is quitesatisfactory. The di�erences that occur in the axial compo-nent of the velocity ®eld are probably mainly due to di�eren-ces in the in¯ow conditions. Even the quite small secondarycomponent of the velocity ®eld detected in the experimentsagrees quite well with the numerically calculated one. Di�er-ences are smaller then 1% of the mainstream velocity compo-nent. The agreement between the experimental and numericaltemperature data is less satisfactory and is of the order of10% of the installed temperature di�erence. This is mainlydue to the inaccuracies in the temperatures determined withthe liquid crystal technique. Also in the cavity problem, dif-ferences were found in the order of 5% to 10%. On the basisof these ®ndings, it is concluded that the numerical model inthe range of Grashof numbers under consideration in thepresent study, produces reliable results and can be used forfurther investigation.

5. In¯uence of Grashof number

5.1. Temperature and velocity ®elds

In Fig. 7 both the temperature (left half of each picture)and the velocity ®elds (right half) are displayed at two axial po-sitions x � 4 H (left-hand side of the page) and x � 8 H (right-hand side of the page) for the three di�erent values of the Gra-shof number: top: Gr � 5� 104, middle: Gr � 105, and bot-tom: Gr � 2� 105. The temperature ®eld is given bycontours of constant temperature, the di�erence between thecontours is DT=10. The velocity ®eld is given by contours ofconstant axial velocity (with a di�erence of U=2:5) and by vec-tors indicating the direction and the magnitude of the second-ary ¯ow ®eld. All pictures show that the temperature boundarylayer at the side wall remains small. The heated medium ¯owsupwards near the side wall, along which the temperature boun-dary layer grows. This is quite similar to the ¯ow along a heat-ed vertical plate. At the top wall, the medium has to bend andinitially a ¯ow develops, similar to a density current (¯ow thatoccurs when a ¯uid with a density di�erent from the bulk ¯uidpenetrates along a wall) as can be observed in Fig. 7 (top), leftside. Farther downstream, the warmer medium starts to ®ll upthe upper part of the channel, and in the core, a stably strati-®ed ¯ow develops. At higher Grashof numbers, the warm ¯uidlayer in the core is thicker than at a lower Grashof number, in-dicating that the secondary ¯ow is more intense at a higherGrashof number.

Comparing the velocity ®elds at di�erent Grashof numbers,it is indeed seen that the secondary ¯ow ®eld intensi®es forhigher Grashof numbers. Near the side wall, the secondary ve-locities clearly are larger than in the core region. The second-ary ¯ow, however, remains an order of magnitude lowerthan the primary ¯ow. The secondary ¯ow consists of one vor-tex in either side of the channel. The centre of this vortex is ob-served to shift downward along the channel, an e�ect that ismore pronounced when then Grashof number is higher.

Further it is interesting to observe that for the case ofGr � 2� 105; at x � 8 H (Fig. 7) the medium ``bounces''against the upper wall, and a corner structure similar to thatobserved by Lankhorst (1991) forms: ®rst the medium re¯ectsfrom the upper wall, then ¯ows slightly upwards, but ¯owsdownward again near the symmetry plane.

Fig. 4. Measured velocity ®eld in symmetry plane y � 0 for Re � 500,

Gr � 105, Pr � 6; the arrows indicate the velocity vectors; gray-scaled

background indicates the absolute magnitude of the shear: a high shear

is represented by a dark background, zero shear is represented by

white; top: 3:3 < x=H < 4:9; bottom: 7:2 < x=H < 8:8.

276 J.J.M. Sillekens et al. / Int. J. Heat and Fluid Flow 19 (1998) 270±281

Finally, as discussed before, the axial velocity pro®le is seento be shifted downward. The upper left picture resembles theprescribed (fully developed) inlet velocity pro®le the most,the lower right picture shows how this pro®le has changed be-cause of buoyancy.

5.2. Flow and heat transfer parameters

The analysis of the mixed convective ¯ow ®eld in the chan-nel will be completed with a description of the heat transferfrom the side walls to the medium, in relation to the magnitudeof the secondary ¯ow, followed by a description of the frictionexperienced by the ¯ow in the channel.

Heat transfer will be expressed as function of the axial po-sition x=H in terms of the local Nusselt number. The localNusselt number is based on the height of the channel and

the heat transfer coe�cient averaged over the perimeter ata particular cross-section. Heat transfer is thought to be ina close relation with the magnitude of the secondary ¯ow:when the heat transfer is high, the buoyancy forces will berelatively large, resulting in a higher magnitude of the second-ary ¯ow; when a strong secondary ¯ow is present, the tem-perature boundary layer at the side wall will be small, andthus the heat transfer will be high. The magnitude of the sec-ondary ¯ow is expressed as a ratio of the kinetic energy ofthe secondary ¯ow to the kinetic energy of the primary ¯owat a cross-section and will be referred to as the relative kineticenergy K.

K �R

A u2sec dy dzR

A u2ax dy dz

; �7�

Fig. 5. Measured and computed axial (u) and vertical (w) velocity components at various axial distances for Re � 500, Gr � 105 and Pr � 6; top:

x � ÿH ; middle: x � 4:1 H ; bottom: x � 8H . (±) Analytical (top) or numerical (middle and bottom) data; (±±) experimental data.

J.J.M. Sillekens et al. / Int. J. Heat and Fluid Flow 19 (1998) 270±281 277

where usec ����������������v2 � w2p

; uax is the axial velocity, and A is thearea of the cross-section.

Fig. 8 gives the computed value of the Nusselt number as afunction of the axial position in the channel for ®ve di�erentvalues of the Grashof number and for Re� 500 and Pr� 6.Near the entrance of the channel, the Nusselt number is thehighest and approximately equal for all Grashof numbers con-sidered. Because the magnitude of the secondary ¯ow here issmall (see Fig. 9), conduction of heat from the side wall tothe medium is equal at all Grashof numbers. The temperatureboundary layer still is very thin, and therefore the relative heattransfer is high. From position x � 0:5 H , e�ects of the second-ary ¯ow become important. For x > 8 H , oscillations occur inthe solution that are caused by the incorrect boundary condi-tions prescribed. For all cases, in the fully developed situation,a Nusselt number of Nu1 � 3:72 will be reached (Shah andJoshi, 1987). Comparing the Nusselt number for the di�erentGrashof numbers considered, it is clear that the heat transferis much better at high Grashof numbers than at low Grashofnumbers. The Nusselt number for Gr � 2� 105 is up to threetimes as large as the Nusselt number for Gr� 0. These di�er-ences in the heat transfer coe�cient lead to di�erences in thebulk temperature. Because the Nusselt numbers for the Gra-shof numbers under consideration are more or less constantfor x > 1 H , the bulk temperatures show almost an exponen-tial decrease. The values of the bulk temperatures at the outlet�x � 10 H� are presented in Table 2.

In Fig. 9 it is observed that for all situations considered (ex-cept for Gr� 0), the relative kinetic energy initially grows,reaches a maximum, and then decreases. The sudden increaseat the end of the channel �x > 9 H� is a result of the physicallyincorrect out¯ow boundary conditions and will be discussedbelow. In the ®rst part of the channel, the secondary ¯ow gainsmomentum because of buoyancy forces. Because of the rise intemperature of the medium in the downstream direction, thebuoyancy forces in the channel gradually decrease. Besides,¯uid is heated at the side walls, ¯ows upwards, and results ina thermal strati®cation (see Fig. 7). This strati®cation leadsto damping of the secondary ¯ow ®eld. The in¯uence of the de-

creasing forcing rate and the strati®cation on the relative kinet-ic energy is clearly visible in Fig. 9. It can be seen that therelative kinetic energy shows a more or less exponential de-crease for the Grashof numbers under consideration. Note thatbecause the buoyancy forces become zero when the ¯ow is inthermal equilibrium with the side walls, the relative kinetic en-ergy ®nally will stabilize at a value of zero. The behavior of thesecondary ¯ow ®eld is also re¯ected in the Nusselt number.The Nusselt number reaches a local maximum (atx � 5:5 H for Gr � 105) and then decreases slightly. Note thatthe occurrence of the maximum is somewhat lagging behindthe occurrence of the maximum in the kinetic energy ratio, in-dicating that at this stage the heat transfer is increased by thesecondary ¯ow.

In Fig. 10 the Fanning friction factor is shown as a functionof the axial position for the di�erent values of the Grashofnumbers. The Fanning friction factor f is de®ned as:

f �x� � �sw

12qU 2

; �8�

where �sw is the average wall shear stress at a certain axial po-sition:

�sw�x� � 1

4 ReH

I�

ÿ @u@n

ds: �9�

Here, � denotes the perimeter of the channel, and n denotesthe local normal to the perimeter. The Fanning friction factoris a measure of the local skin friction experienced by the ¯ow.For fully developed ¯ow in a square channel a value off1 �Re � 14:2271 will be reached (Shah and Joshi, 1987).Comparing the di�erent curves, it is seen that friction is in-creased by buoyancy forces. As has been shown before, the ax-ial velocity pro®le changes because of buoyancy, and especiallyin the lower half of the channel, the normal axial velocity gra-dient substantially increases (see Fig. 7). Surprisingly, also thefriction for Gr� 0 changes with axial position. Although theinlet velocity pro®le prescribed only di�ers from the analyticalpro®le by about 2% (Shah and Joshi, 1987), this di�erence isenough to observe a development towards the analytical ex-

Fig. 6. Measured temperatures using LCT and computed temperature contours for Re � 500, Gr � 105 and Pr � 6; left: in the symmetry plane

8 H < x < 9 H , y � 0; right: in cross-section at x � 8 H ; DT � Twall ÿ Tinlet � 1:5�C and di�erence between the contours is DT=10.

278 J.J.M. Sillekens et al. / Int. J. Heat and Fluid Flow 19 (1998) 270±281

pected value for the friction factor. The Fanning friction factordoes not show a local maximum like the relative kinetic energyor the Nusselt number. It is interesting, however, to observethat the rate of increase of the friction factor in axial directionis closely related to the magnitude of the relative kinetic energy:

the higher the magnitude of the secondary ¯ow, the morethe axial velocity pro®le changes, and thus the higher theincrease in shear. It is expected that a maximum in shearwill be reached when the axial velocity pro®le is maximallyshifted.

Fig. 8. Calculated Nusselt number for Re � 500, Pr � 6; I : Gr � 0; II:

Gr � 5� 104, III : Gr � 105, IV : Gr � 1:5� 105, V : Gr � 2� 105.Fig. 9. Calculated relative kinetic energy for Re � 500, Pr � 6; I:

Gr � 0; II: Gr � 5� 104; III: Gr � 105, IV : Gr � 1:5� 105; V :

Gr � 2� 105.

Fig. 7. Calculated temperature and velocity ®elds for Re � 500, Pr � 6 and various Grashof numbers; left: x � 4 H ; right: x � 8 H ; isotherms (left

side of each ®gure) and isotachs and secondary velocity vectors (right side of each ®gure); a vector length equal to the height of the channel corre-

sponds to the mean axial velocity; top: Gr � 5� 104; middle: Gr � 105; bottom: Gr � 2� 105.

J.J.M. Sillekens et al. / Int. J. Heat and Fluid Flow 19 (1998) 270±281 279

5.3. In¯uence out¯ow boundary conditions

As mentioned before the results beyond position x � 8 Hshould be handled with care. In Figs. 8±10, the numerical re-sults in the out¯ow region are, therefore, indicated as shadedareas. The physically incorrect out¯ow boundary conditionslead to wiggles in the Nusselt number and Fanning friction fac-tor and to a sudden increase of the relative kinetic energy in theout¯ow region. For the case where no buoyancy forces arepresent (Gr� 0; line I in Figs. 8±10), the ¯ow should be fullydeveloped over the full length of the calculation domain, im-plying no changes in the axial ¯ow ®eld and zero secondary ve-locities. For this case, setting the normal stress equal to zero�rxx � 0� is a proper boundary condition. As already men-tioned, the change in the Fanning friction factor for this case(see Fig. 10) is attributed to the small di�erences between theinlet pro®le prescribed and the exact pro®le for fully developed¯ow in a square channel. The secondary velocities for this case(see Fig. 9) are almost zero over the entire domain. In the out-¯ow region, a sudden increase is observed which is attributedto the physically incorrect boundary conditions for the tangen-tial stresses �rxy � rxz � 0�. For the cases where buoyancyforces are present (lines II to IV in Figs. 8±10) also, the zeronormal stress boundary condition is physically incorrect be-cause of pressure di�erences over a cross-section. This resultsin an `extra' secondary ¯ow in the out¯ow region, the e�ectof which on the relative kinetic energy is clearly seen inFig. 9. For higher Grashof numbers, larger errors are intro-duced because then larger pressure di�erences are present overa cross-section. However, from the results presented it can beconcluded that the upstream in¯uence of the out¯ow boundaryconditions is restricted to an axial distance of about 2 H .

6. Conclusions

In the present study, the ®nite element method (FEM) isemployed for detailed analysis of mixed-convection ¯ow in ahorizontal straight channel with heated side walls. The Rey-nolds and Grashof numbers are such that the ¯ow is laminarand steady. Because of the simple geometry, a detailed com-parison could be made with velocity measurements using par-ticle tracking and temperature measurements using liquidcrystals. The agreement is quite satisfactory: di�erences canbe explained by the di�erences in inlet conditions and the inac-curacies of the experimental methods used. It is, therefore,concluded that the numerical method used in the present studyis a very powerful tool for analyzing mixed-convection ¯ows incomplex geometries, such as helically coiled heat exchangers orcorrugated tubes. Besides, using the method employed, de-tailed insight into the physics can be obtained, such as the in-¯uence of the buoyancy-induced secondary ¯ows on the main¯ow characteristics and the heat transfer, knowledge of whichis of great importance for the design of new heat exchangerconcepts.

Particle-tracking velocimetry seems to be a powerful toolfor measuring velocity ®elds. In the present study, only veloc-ities are detected in the midplane (symmetry plane) of thechannel where the ¯ow is quasi two-dimensional (2-D). Atall other positions, the ¯ow is essentially 3-D, leading to veloc-ity components perpendicular to the light sheet. This makesproper matching of particles in successive images a di�cultjob, certainly when axial and secondary velocities are of thesame order. Therefore, 3-D particle-tracking velocimetryseems to be indispensable in analyzing complex ¯ow problems.

In the present study, the steady momentum and energyequations are solved, supposing that no temporal instabilitiesoccur. In most studies on the temporal stability of a channel¯ow, the case of air is considered (Hosokawa et al., 1993;Evans and Greif, 1993; Huang and Lin, 1995), and other boun-dary conditions are applied (mostly heated from below). As faras known, no data are available of the situation under consid-eration in the present study (water and heated side walls).However, from the liquid crystal and particle-tracking experi-ments, it may be deduced that no instabilities were present.Also, for higher Grashof numbers, the aforementioned tech-niques may be extremely useful in detecting the occurrenceof unstable ¯ow phenomena.

Acknowledgements

The authors are grateful to A.J.M. van Beers for his exper-imental assistance and to L.G. van der Schoot for manufactur-ing the experimental setup.

References

Adrian, R.J., 1991. Particle-imaging techniques for experimental ¯uid

mechanics. Ann. Rev. Fluid Mech. 23, 261±304.

Biswas, G., Mitra, N.K., Fiebig, M., Cornelius, J.S., 1990. Numerical

investigation of laminar mixed convection ¯ows in rectangular

horizontal ducts. In: Proceedings of the Ninth International Heat

Transfer Conferences, vol. 2, pp. 435±439.

Brooks, A.N., Hughes, T.J.R., 1982. Streamline upwind / Petrov±

Galerkin formulations for convection-dominated ¯ows with par-

ticular emphasis on the incompressible Navier±Stokes equations.

Comp. Meth. Appl. Mech. Eng. 32, 199±259.

Table 2

Dimensionless bulk temperature at the outlet for various Grashof

numbers

Gr �Twall ÿ Toutlet�=�Twall ÿ Tinlet�5� 104 0.82

105 0.79

1:5� 105 0.77

2� 105 0.75

Fig. 10. Calculated Fanning friction factor for Re � 500, Pr � 6; I:

Gr � 0; II : Gr � 5� 104, III : Gr � 105; IV : Gr � 1:5� 105, V :

Gr � 2� 105.

280 J.J.M. Sillekens et al. / Int. J. Heat and Fluid Flow 19 (1998) 270±281

Cuvelier, C., Segal, A., Steenhoven, A.A.v., 1986. Finite Element

Methods and Navier±Stokes Equations. Reidel, Dordrecht, The

Netherlands.

Dalziel, S.B., 1992a. Decay of rotating turbulence: Some particle-

tracking experiments. Appl. Sci. Res. 49, 217±244.

Dalziel, S.B., 1992b. DigImage, Image Processing for Fluid Dynamics.

Cambridge Environmental Research Consultants Ltd.

Evans, G., Greif, R., 1993. Thermally unstable convection with

applications to chemical vapor deposition channel reactors. Int. J.

Heat Mass Transfer 36, 2769±2781.

Fergason, J.L., 1968. Liquid crystals in nondestructive testing. Appl.

Opt. 7, 1692±1737.

Gray, D.D., Giorgini, A., 1976. The validity of the Boussineq

approximation for liquid and gases. Int. J. Heat Mass Transfer

19, 545±551.

Hartnett, J.P., Kostic, M., 1989. Heat transfer to Newtonian and non-

Newtonian ¯uids in rectangular ducts. In: Irvine, T.F., Hartnett,

J.P. (Eds.), Advances in Heat Transfer, Supplement 19. Academic

Press, New York, pp. 247±356.

Ho, B.P., Leal, L.G., 1974. Inertial migration of rigid spheres in two-

dimensional unidirectional ¯ows. J. Fluid Mech. 65, 365±400.

Hosokawa, I., Tanaka, Y., Yamamoto, K., 1993. Mixed-convective

¯ow with mass transfer in a horizontal rectangular duct heated

from below simulated by the conditional Fourier spectral analysis.

Int. J. Heat Mass Transfer 36, 3029±3042.

Huang, C.C, Lin, T.F., 1995. Vortex ¯ow and thermal characteristics

in mixed convection of air in a horizontal rectangular duct: E�ects

of the Reynolds and Grashof numbers. Int. J. Heat Mass Transfer

38, 1661±1674.

Incropera, F.P., Knox, A.L., Maughan, J.R., 1987. Mixed-convection

¯ow and heat transfer in the entry region of a horizontal

rectangular duct. J. Heat Transfer 109, 434±439.

Kleijn, C.R., 1991. Transport phenomena in chemical vapor deposi-

tion reactors. Ph.D. Thesis, Delft University of Technology, The

Netherlands.

Lankhorst, A.M., 1991. Laminar and turbulent natural convection in

cavities; Numerical modeling and experimental validation. Ph.D.

Thesis, Delft University of Technology, The Netherlands.

Le Fevre, E.J. 1956. Laminar free convection from a vertical surface.

In: Proceedings of the Ninth International Congress on Applied

Mechanics, vol. 4, pp. 168±173.

Nandakumar, K., Masliyah, J.H., Law, H.S., 1985. Bifurcation in

steady laminar mixed convection ¯ow in horizontal ducts. J. Fluid

Mech. 152, 145±161.

Oberbeck, A., 1888. Uber die Bewegungserscheinungen der Atmospha-

ere. Sitzungsberichte der K�oniglich Preuszischen Akademie der

Wissenschaften, pp. 1129±1138.

Papanastasiou, T.C., Malamataris, N., Ellwood, K., 1992. A new

out¯ow boundary condition. Int. J. Numer. Meth. Fluids 14, 587±

608.

Parsley, M., 1991. The Hallcrest Handbook of Thermochromic Liquid

Crystal Technology. Hallcrest Prod.

Rindt, C.C.M., van Steenhoven, A.A., Janssen, J.D., Vossers, G.,

1991. Unsteady entrance ¯ow in a 90� curved tube. J. Fluid Mech.

226, 445±474.

Sani, R.L., Gresho, P.M., 1994. R�esum�e and remarks on the open

boundary condition minisymposium. Int. J. Numer. Meth. Fluids

18, 983±1008.

Segal, A., 1993. Finite element methods for advection±di�usion

equations. In: Vreugdenhil, C.B., Koren, B. (Eds.), Numerical

Methods for Advection±Di�usion Problems. Vieweg, Germany.

Segr�e, G., Silberberg, A., 1962. Behaviour of macroscopic rigid spheres

in Poiseuille ¯ow. Part 2: Experimental results and interpretation.

J. Fluid Mech. 14, 136±157.

Shah, R.K., Joshi, S.D., 1987. Convective heat transfer in curved

ducts. In: Kaka, S., Shah, R., Aung, W. (Eds.), Handbook of

Single-Phase Convective Heat Transfer. Wiley, New York.

Sillekens, J.J.M., 1995. Laminar Mixed Convection in ducts. Ph.D.

Thesis, Eindhoven University of Technology, The Netherlands.

Sillekens, J.J.M., Rindt, C.C.M., Steenhoven, A.A.v., 1994a. Mixed

convection in a 90� horizontal bend. In: Proceedings of the Tenth

International Heat Transfer Conference, vol. 5, pp. 567±572.

Sillekens, J.J.M., Rindt, C.C.M., Steenhoven, A.A.v., 1994b. Mixed-

convective heat transfer in a horizontal bend. In: Advanced

Concepts and Techniques in Thermal Modelling, Proceedings of

Eurotherm 36.

Van de Vosse, F.N., 1987. Numerical analysis of carotid artery ¯ow.

Ph.D. Thesis, Eindhoven University of Technology, The Nether-

lands.

J.J.M. Sillekens et al. / Int. J. Heat and Fluid Flow 19 (1998) 270±281 281