unsteady natural convection heat transfer from a pair of vertically aligned horizontal...

8/17/2019 Unsteady Natural Convection Heat Transfer From a Pair of Vertically Aligned Horizontal Cylinders-pelletier2016

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Unsteady natural convection heat transfer from a pair of verticallyaligned horizontal cylinders

Quentin Pelletier ⇑ , Darina B. Murray, Tim PersoonsDepartment of Mechanical and Manufacturing Engineering, Parsons Building, Trinity College, Dublin 2, Ireland

a r t i c l e i n f o

Article history:Received 3 August 2015Received in revised form 18 December 2015Accepted 18 December 2015

Keywords:Natural convectionUnsteady RANSSSTExperimental validationOscillating plumeHeat transfer enhancementPower spectral densityCoherence spectrumVortex shedding

a b s t r a c t

This paper presents the results of a numerical study of unsteady natural convection heat transfer from apair of isothermally heated horizontal cylinders in water. Particular attention is paid to the interactionbetween the two buoyant plumes and the relationship between unsteady uid ow and heat transferenhancement. When the cylinders are vertically aligned, the heat transfer effectiveness of the uppercylinder is affected by buoyancy-induced uid ow induced by the lower cylinder. Moreover, strongoscillations appear in the local heat transfer rate and uid ow eld. A computational uid dynamicsmodel (CFD) is established and validated against experimental results for both averaged and localNusselt number, for a range of Rayleigh numbers between 1 :7 10 6 and 5 :3 10 6 and a centre-to-centre cylinder spacing between 2 and 4 diameters. The validated CFD model is used to identify 3.5diameters as the spacing which maximises the averaged Nusselt number on the upper cylinder amongthe values of Rayleigh number and spacing which were presently investigated. The transient CFD modelis used to characterise the peak frequencies of the oscillating thermal plume, as well as analyse Fouriertransforms and spectral coherence of the local Nusselt number and uid velocity. This detailed numericalanalysis has complemented previous experimental measurements, conrming some hypotheses on themechanisms of heat transfer enhancement in this conguration.

2015 Elsevier Ltd. All rights reserved.

1. Introduction

Convective heat transfer from pipes and tubular arrays occurs inmany technical applications including heat exchangers, boilers, aircooling systems, etc. Although the time-averaged characteristics of external forced convection from arrays of cylinders are well under-stood, the understanding of natural convection heat transfer is lesscomplete, especially for conned geometries with strong interac-tions between the developing thermal plumes and the thermalboundary layer of nearby cylinders. Most studies have focused onthe overall heat transfer from a single cylinder [1] or an array of cylinders [2] , but not many studies have described the effect of thermal plume oscillations on the transient, local Nusselt numberand ow velocity.

Some publications have discussed the swaying motion causedby the thermal plume rising from a single cylinder [3,4] and havecharacterised its behaviour as a function of Rayleigh number.Numerical studies provide a good body of data for single horizontalheated wires, which can be considered as line heat sources [5–8] .

However, only a few studies have investigated the phenomenonfor several cylinders.

Eckert and Soehngen [9] studied the heat transfer from a verti-cally aligned cylinder pair and discovered that the lower cylinder’sbuoyant plume could either have a positive or a negative effect onthe upper cylinder’s heat transfer capacity depending the spacingbetween the two cylinders. The heat transfer rate was found todecrease with decreasing cylinder spacing. Previous experimentalstudies by some of the co-authors revealed a range of cylinderspacing and Rayleigh numbers where benecial interaction occurs[10,11] . Based on experimental measurements of local time-varying Nusselt number, spectral analysis suggested that whenthe plume from the lower cylinder oscillates out of phase withthe plume from the upper cylinder, the mixing around the uppercylinder is enhanced, which in turn increases the heat transfer rate.Positive enhancement has also been reported by Grafsronningenand Jensen [12,13] . They investigated the change in heat transferof the upper of a pair of cylinders for Rayleigh numbers from1:82 10 7 to 2 :55 10 8 and spacing 1 :5D 6 S 6 5D [12] and fora conguration of three vertically aligned cylinders for Rayleighnumbers Ra ¼ 1:96 10 7 and Ra ¼ 5:35 10 7 and spacingbetween the lowermost and middle cylinders and between the

http://dx.doi.org/10.1016/j.ijheatmasstransfer.2015.12.041

0017-9310/ 2015 Elsevier Ltd. All rights reserved.

⇑ Corresponding author. Tel.: +353 1 896 1034.E-mail address: [email protected] (Q. Pelletier).

International Journal of Heat and Mass Transfer 95 (2016) 693–708

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International Journal of Heat and Mass Transfer

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the facility, measuring 900 mm (30 D) high, 900 mm (30 D) long and300 mm (10 D) wide, containing about 200 litres of deoxygenatedwater as working uid.

2.2. Local heat transfer measurements

Each cylinder contains two internal 500 W cartridge heatersembedded along the cylinder axis, using thermal paste to minimisethe thermal contact resistance. Each cylinder is instrumented witha ush mounted thermopile heat ux sensor (RdF Micro-Foil TM

27036-2-RdF) and an internally mounted T -type thermocouple.The cylinders are rotated around their axis to measure the localsurface heat ux around the circumference. The properties of water were evaluated at the lm temperature ðT s þ T 1 Þ=2, withthe surface temperature T s being obtained using a surface-mounted ne wire T -type thermocouple (0.3 mm wire diameter),mounted at the same circumferential position h yet at an axial dis-tance z ¼ 10 mm from the heat ux sensor. For a given test, the

Rayleigh number (Eq. (1) ) was set by adjusting the differencebetween the surface temperature and the bulk water temperatureðT s T 1 Þ, where T 1 was measured at the same elevation y as thecentre of the top cylinder [21] .

Ra ¼ Gr :Pr ¼ g bðT s T 1 ÞD

3

ma ð1 Þ

Measurements were only taken once a pseudo steady state wasreached. Once the targeted operating parameters were met, thecylinder was rotated in 10 intervals over half a revolution(0 6 h 6 180 ). At each position, the heat ux, surface tempera-ture, and bulk uid temperature were recorded once equilibriumwas reached to within 1 % of the time-averaged Nusselt number.Since preliminary testing revealed low frequency oscillations inheat transfer for multiple cylinder congurations, in order to cap-

ture this oscillatory behaviour a sampling time of approximately220 s and sampling frequency of 40 Hz were employed. For eachmeasurement condition, tests were repeated 5 times to ensurerepeatability [21] .

The heat ux measurement is the main contribution to theuncertainty in the Nusselt number. The maximum measurementuncertainty occurred at the highest Rayleigh number of 5:5 10 6 . Based on a 95% condence level, the overall estimateduncertainties of the Rayleigh and Nusselt numbers are around 1%and 16%, respectively [21] . More details of the calibration proce-dures and error analysis are given by O’Gorman [22] .

3. Numerical methodology

A numerical model has been developed using ANSYS Fluent 14.0that aims to provide an estimation of the enhancement or

diminishment of the natural convection heat transfer from a pairof horizontal cylinders. Validation of the results is obtained bycomparison with the experimental results for the ranges of Rayleigh numbers and cylinder spacing that were available fromprevious work [21] . Both a (steady) Reynolds-Averaged Navier–Stokes (RANS) and Unsteady RANS (URANS) approach are usedon a two-dimensional numerical domain. The two-dimensional(2D) approach was used in order to save computational time; a sin-gle simulation with the 2D mesh, desired time-step and durationtakes about 10 h on a quad-core desktop workstation. As explainedin the introduction, the uctuations of the third velocity compo-nent are expected to be negligible compared to the two velocitycomponents in the plane considered. This approximation is vali-dated with the results, as it will be shown later in Fig. 22 , in the lastsection of this paper, that a good agreement is found between thepresent numerical unsteady results for the velocity ow eld and2D PIV measurements carried out in the mid-plane of the test facil-ity by Persoons et al. [21] . The analysis is based on a second order

nite volume spatial discretization method for the momentumequation, the PISO velocity–pressure coupling technique (derivedfrom the SIMPLE algorithm [25,26] ) and second order implicit timediscretization for the URANS case. The eddy viscosity is obtainedusing the Shear Stress Transport (SST) k x model with a low-Reynolds model approach to avoid the use of wall functions[25,27] . This model behaves like a classical k x model in theinner parts of the boundary layer, down to the wall through theviscous sub-layer, and it switches to the k model in the free-stream regions to avoid the problem of high sensitivity of thek x model in these regions [25,28] . The relaxation factors havebeen increased from 0.3 to 0.8 for the pressure and from 0.7 to 1(i.e., no relaxation) for the momentum equation, which increasesthe sensitivity of the solver to physical oscillations, in an effort to

better capture the experimentally observed variations in velocityand temperature. Since a low-Reynolds model is used, a 2D meshwith 8600 grid cells is designed to ensure the size of the rst cellsat the cylinder wall does not exceed yþ ¼ 1. The inner part of themesh around the cylinders is discretized using quadrilateral cellswhile the outer part of the numerical domain is discretized withtriangular cells as shown in Fig. 2. The skewness and aspect ratioof the cells (triangles as well as quadrilaterals) are minimised (withan average skewness of 0.03 and an average aspect ratio of 1.29) inorder to avoid any non-orthogonality related issues.

The uid density is described by the Boussinesq approximation(Eq. (2) ). This is a rst order approximation obtained by linearisingthe state equation. It is valid for small changes in density, that is tosay for bD T 1. Given the temperature differences involved in this

simulation, the Boussinesq approximation can be used in the com-putation of the buoyancy force term q g in the momentum equa-

Fig. 1. (a) Diagram of the natural convection test facility, and (b) close-up view of the cylinder pair.

Q. Pelletier et al. / International Journal of Heat and Mass Transfer 95 (2016) 693–708 695

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tion [25] . Eq. (2) expresses a change in temperature as aproportional change in density, and thus enables the simulationof thermally induced gravity driven buoyant motion.

qðT Þ ¼q 1 ½1 bðT T 1 Þ ð2 Þ

where q1 is the bulk uid density, T 1 is the bulk uid temperatureand the thermal expansion coefcient b ¼ 1q1 ð

@ q@ T Þ p. The thermal

boundary conditions on the bottom and side walls of the tank aredened as convective boundary conditions, representing the smallamount of heat loss through the walls of the tank in the experimen-tal test facility [21] . An overall heat transfer coefcient U is com-puted as

1

U ¼ 1

hwater þ

t wallkwall

þ 1

hair ð3 Þ

where t wall and kwall are the thickness and thermal conductivity of the tank walls, respectively. hwater and hair are the estimated averageheat transfer coefcients on the inside (water) and outside (air) of the tank, estimated using established empirical correlations [29] .The top side of the numerical domain is dened as free slip surfaceto represent the free water surface in the experimental test facility[21] .

The mesh size and the time step renement have been studiedfor a single cylinder, to ensure that the solution is independent of both mesh size and time step, as shown in Fig. 3. The characteristic

mesh size is represented by D r in Fig. 3b, which corresponds to theradial thickness of the rst layer of structured cells adjacent to thecylinder surface (see Fig. 2b). For the URANS simulations, a time-step of D t ¼ 50 ms has been used. A grid size of D r ¼ 0:5 mmwas taken for both steady and URANS simulations. Numericaltime-averaged Nusselt number results (presented in Section 4)are evaluated from steady RANS simulation, whereas time-resolved data (presented in Section 5) are obtained from URANSsimulations. The consistency of the RANS results with the averageover time of the URANS results has been veried for the singlecylinder case. In the gures shown further in this paper, some of the time-averaged results have been obtained directly from aver-aging the URANS results and some have been obtained as (steady)RANS solution. Where appropriate, the results shown in subse-

quent gures have been labelled as ‘‘RANS” or ‘‘URANS” resultsfor clarity.

4. Time-averaged results and validation of the CFD model

This section presents a detailed validation of the CFD model fornatural convection heat transfer from one or two isothermallyheated horizontal circular cylinders, using empirical data availablein the literature as well as from experimental data from Persoonset al. [21] .

4.1. Time-averaged heat transfer from a single cylinder

The time-averaged mean surface heat transfer coefcientresults have rst been compared for a single cylinder test case toa widely used empirical correlation [30] (in the form of Eq. (4) )which is valid for a range of Rayleigh numbers going from

0:5 106

to 6 106

as shown in Fig. 4Num ¼ CRa

1 =4 ð4 Þ

where C depends on the Prandtl number ( C ¼ 0:436 ; 0:456 ;0:520 ; 0:523 for Pr ¼ 0:7 ; 1; 10 ; 100 respectively, [30] ) andNum ¼ 12p R

2p0 NuðhÞdh is the mean time-averaged Nusselt number

over the cylinder surface. For the same range of Rayleigh number(0 :5 10 6 6 Ra 6 6 10 6 ), the results of our CFD model with a sin-gle cylinder (yet otherwise the same tank dimensions as describedin the preceding sections) are in good agreement with the abovecorrelation from Merk and Prins [30] (for Pr ¼ 7 ; C ¼ 0:515 in Eq.(4) ). The mean Nusselt number results have been least square ttedwith the power law correlation

Num ¼ 0 :551 Ra0 :245 ðR2 ¼ 0 :9991 Þ ð5 Þ

As the main objective of this paper is to study oscillations of thethermal plume arising from the bottom cylinder, a study of thelocal effects of this plume on the top cylinder is required. There-fore, in addition to only comparing the time-averaged heat trans-fer, a more detailed validation of the local heat transfer ratearound a single cylinder is carried out.

For the single cylinder CFD model results, the time-averagedlocal Nusselt number has been evaluated at small surface incre-ments every 10 from the bottom ( h ¼ 0 ) to the top ( h ¼ 180 ) of the cylinder. The results for different values of Ra have been com-pared to the results obtained by Kuehn and Goldstein [31] and areshown in Fig. 5. It can be noted that our CFD results t the results

by Kuehn and Goldstein for the investigated Rayleigh numberrange (10 4 6 Ra 6 5:3 10 6 ) although a small difference is seen

Fig. 2. (a) Overview of the two-dimensional mesh, and (b) close-up view of the pair of cylinders showing the transition from structured mesh around the cylinders tounstructured mesh in the surrounding region.

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for 135 6 h 6 180 , especially for Ra ¼ 10 4 . The discrepancyappearing at the top of the cylinder (for h close to 180 ) remains

unexplained. The main difference between the present simulationand that of Kuehn and Goldstein [31] is the outer domain. For their

simulation, Kuehn and Goldstein assumed that the single cylinderwas placed in an innite medium and they designed the meshaccordingly while in the present work, the cylinder is placed atthe centre of a tank whose dimensions are based on the experi-mental facility used by Persoons et al. [21] .

4.2. Time-averaged heat transfer from a pair of cylinders

The validation of the CFD model with experimental results hasalso been conducted for natural convection from a pair of isother-mally heated horizontal cylinders in water. The two horizontalcylinders are vertically aligned, separated by a centre-to-centredistance S , as shown in Fig. 1 and heated at the same temperatureT s. The surface temperature T s is set to different values correspond-ing to the targeted Rayleigh numbers (see Eq. (1) ). The results forthe investigated ranges of Rayleigh numbers Ra and dimensionlessspacing S =D are presented in the following sections.

4.2.1. Effect of Rayleigh number Fig. 6 shows the (a) numerical and (b) experimental results for a

spacing S ¼ 2D. A good qualitative agreement is found, with curvesshowing similar trends and inection points, marking three dis-tinct heat transfer regions (I, II, III). The Nusselt number decreasessharply from the bottom of the cylinder for 0 6 h < 40 (region I)followed by a more gradual decrease for 40 6 h < 150 (region II)and again a sharper decrease towards the top of the cylinder(150 6 h 6 180 , region III). For each tested Rayleigh number, amonotonic increase of the local Nusselt number with Rayleighnumber is observed. The numerical and experimental results exhi-bit a reasonable quantitative agreement as this region of graduallydecreasing Nu (region II) is reached at around h ¼ 40 with a sim-ilar value of Nu 20 in both sets of results for all Rayleigh num-bers. Region I shows a good quantitative agreement betweennumerical and experimental data. Region II shows a reasonable

agreement, although the experimental local Nusselt number dataexhibit a more pronounced plateau at higher Rayleigh numbers,whereas the numerical data retain a monotonic decreasing trendfor increasing angle. Region III (near the top of the cylinder) showsdecreasing local Nusselt number values in both numerical andexperimental results, yet there remains a discrepancy nearh ¼ 180 . The discrepancy between the experimental and numeri-cal results consistently appears near the top of the cylinder asshown in Fig. 6. Two plausible reasons why the numerical resultsunderpredict the experimental results relate to the measurementmethod. Firstly, the differential thermopile sensor (RdF Micro-Foil 27036–2) has an approximate spatial resolution of 15 andits reading would therefore overestimate the true local value. Sec-ondly, the sensor is mounted on a polymer substrate which inevi-

tably causes an additional thermal resistance on the surface of thecylinder, thereby disturbing the thermal boundary condition. By

Fig. 3. Sensitivity study of the solution to (a) the time step and (b) mesh spacing, for a single cylinder at Ra ¼ 1:70 10 6 .

Fig. 5. Normalised local Nusselt number. RANS CFD results from the present workas solid lines (from top to bottom; Ra ¼ 10 4 ; 10 5 ; 1:7 10 6 ; 5:3 10 6 ) and numer-ical results from Kuehn and Goldstein [31] as markers (from top to bottom;Ra ¼ 10 4 ; 10 5 ; 10 6 ).

Fig. 4. Mean time-averaged Nusselt number for a single cylinder. Validation of numerical results from the present work (RANS) by comparison with empiricalcorrelation from Merk and Prins [30] (Eq. (4) with C ¼ 0:515). Eq. (5) is not shown

for clarity.


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contrast, the CFD model assumes a perfect isothermal boundarycondition at the cylinder surface. The observed deviation betweennumerical and experimental results near the top of the cylinder islikely due to a combination of these two effects: (i) limited spatialresolution and (ii) a disturbance to the thermal boundary conditionin the experimental measurements. However, for the purpose of validating the CFD code, the observed agreement in the trendsand locations of these three regions is found satisfactory.

A numerical simulation has also been conducted for S ¼ 3D andfor the same Rayleigh numbers, as shown in Fig. 7. A similar beha-viour with 3 regions along the circumference (I, II, III) is observed.From the numerical results shown in Figs. 6a and 7, it can be con-cluded that the Rayleigh number mainly inuences the averagedNusselt number. In other words, a change in Rayleigh number doesnot signicantly alter the relative local Nusselt number at a givenposition h when normalised by the averaged Nusselt number. Aglobal trend appears in the distribution of the local Nusselt num-ber, which is scaled up or down as a whole with an increase ordecrease in Rayleigh number, respectively, as shown in Figs. 6and 7 . This scaling however only slightly affects the local Nusseltnumber around 180 (the top of the upper cylinder). The sensitivityof the Nusselt number curve to a change in Rayleigh number isexpressed by a similar power law relationship to Eq. (5) .

4.2.2. Effect of cylinder spacing S =DThe effect of the spacing on the distribution of the local Nusselt

number has been investigated in more detail for a Rayleigh numberRa ¼ 5:33 10 6 . Fig. 8 shows the local time-averaged Nusseltnumber distributions for (a) the numerical model and (b) experi-mental measurements for three cylinder spacings and the singlecylinder case. Each set of results shows a satisfactory agreementin terms of absolute values and trends, with the exception of theregion near the top of the cylinder where the experimental datadiverge somewhat.

Unlike the Rayleigh number, Fig. 8 demonstrates that the cylin-der spacing S =D has a strong inuence on the local Nusselt numberdistribution. The sharp decrease in local Nusselt number near thebottom of the cylinder is more pronounced for smaller spacings,and tends to a more gradual trend at larger spacing S =D > 3. ForS ¼ 2D, the Nusselt number drops by 54% (from 59 to 27) between

h ¼ 0 and 40 whereas for S ¼ 4D, the Nusselt number drops byonly 13% from h ¼ 0 to 40 .

The greater the dimensionless spacing S =D, the more closely theNusselt number distribution agrees with the single cylinderresults, as shown in Fig. 9. Indeed when the two cylinders are farfrom each other, the interaction between them is weak and if thedistance is great enough, this interaction is expected to become

negligible. For the investigated range (2 6 S =D 6 4), Fig. 9 showsthat the increase in local heat transfer coefcient near the bottomof the cylinder reduces from a maximum of 90% for S =D ¼ 2 to 23%for S =D ¼ 4. A reduction in local heat transfer (albeit smaller inmagnitude) is observed in the upper half of the cylinder; howeverthe reduction magnitude in this region is less affected by the cylin-der spacing.

Fig. 10 compares the numerical against experimental values forthe local Nusselt number for the same three spacings atRa ¼ 5:33 10 6 . The highest values of Nusselt number representthe region near the bottom of the upper cylinder. A good agree-ment can be noted for most values within approximately ±15%,except for the smallest values representing the heat transfer nearthe top of the cylinder. In this region, the numerical results consis-

tently underpredict the experimental values as explained inSection 4.2.1 .

Fig. 7. Local Nusselt number along the circumference of the upper cylinder forS ¼ 3D (numerical results); numerical results (RANS).

Fig. 6. Local Nusselt number along the circumference of the upper cylinder for S ¼ 2D. (a) Numerical results (RANS) and (b) experimental results [21] .



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Table 1 summarises the time-averaged mean Nusselt numberresults as a function of the Rayleigh number and cylinder spacing.

The normalised heat transfer enhancement compared to the singlecylinder case (at the same Rayleigh number) is represented inTable 1 by percentage values of dNum , dened as

dNum ¼Num Num;single

Num;singleð6 Þ

It is worth noting that, even for the numerical results, the over-all heat transfer does not exhibit a monotonic dependence oneither the spacing or the Rayleigh number. Indeed the relativedeviation from a single cylinder Nusselt number varies betweenslightly negative values (e.g., 1.8% for S ¼ 4D andRa ¼ 5:33 10 6 ) to positive values up to +11% for S ¼ 3:5D at Ra¼ 5:33 10 6 .

Fig. 11 shows a contour plot of the heat transfer enhancementdNum (in percentage values) as a function of cylinder spacing and

Rayleigh number. Apart from the sudden decrease at S ¼ 4D andRa ¼ 5:3 10 6 , the heat transfer enhancement dNum tends to

increase with increasing Ra and decrease with decreasing S =D.The locus of optimum spacing and Rayleigh number, for the speci-c congurations investigated here, is in the range 2 :8D < S < 3:6Dand 4 :7 10 6 6 Ra 6 5:3 10 6 . However the objective of thisstudy was not to provide a denitive optimum and further workwould be needed to ascertain that the 10% contour in Fig. 11 isin fact a global extremum.

5. Unsteady results

5.1. Spectral analysis of local Nusselt number and ow velocity

Results have been presented in the previous section for the local

time-averaged Nusselt number and demonstrate the strong depen-dence of the Nu distribution on the cylinder spacing S /D. Since

Fig. 8. Local time-averaged Nusselt number along the circumference of the upper cylinder for Ra ¼ 5:33 10 6 . (a) Numerical time-averaged URANS results and (b)experimental results [21] .

Fig. 9. Deviation of the local time-averaged Nusselt number from the singlecylinder case along the circumference of the upper cylinder for Ra ¼ 5:33 10 6 ;numerical time-averaged URANS results.

Fig. 10. Numerical time-averaged URANS versus experimental local Nusseltnumber at D h ¼ 10 increments along the circumference for Ra ¼ 5:33 10 6 . Thehighest values of Nusselt number represent the bottom of the upper cylinder.


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experimental measurements for the same conditions have previ-

ously revealed strong oscillatory ow and heat transfer [21] ,unsteady (URANS) simulations have been performed for each casein Table 1 to further investigate the origin of these oscillations, andtheir relationship to heat transfer enhancement. As discussed inSection 3, a time step of D t ¼ 50 ms has been used. The simulationsran for 80,000 iterations or a simulated time of 4000 s.

Fig. 12 identies a number of points where time-resolved localNusselt number and uid velocity data are extracted for furtheranalysis in this section. Points A1, A2 and A3 are dened on the cir-cumference of the upper cylinder at angular locationsh ¼ 0 ; 90 ; 90 , respectively. Points B1 and B2 are dened in theow eld with B1 midway between the two cylinders.

At the location B1, the reference plume velocity V ref is averagedalong a horizontal line,

V ref ¼1

b Z b=2

b=2V ð x; y ¼ yB1 Þdx ð7 Þ

where the plume width b is dened as the distance between thelocations (along the same horizontal line through B1) where theupward time-averaged velocity magnitude drops to 25% of the peakvelocity. Point B2 is somewhat arbitrarily dened at a radial dis-tance from point A2 equal to b/2 As shown in Fig. 21 , the trajectoryof the vortex centre generally passes by the side of the cylinder at alateral distance from point A2 in the range b2 : b . By arbitrarilychoosing the location as b=2, any vortex-related uctuations of the local velocity vector are captured in the analysis. In other loca-

tions adjacent to the side of the upper cylinder, oscillations areobserved at the same frequencies yet different magnitudes.

For the sake of brevity, the spectral analysis is only presented infull for a single representative case in the investigated range,Ra ¼ 3:35 10 6 and S = 3D. Fig. 13 shows the strong oscillationsarising at the bottom of the upper cylinder ( h ¼ 0 , point A1) in

Table 1

Mean time-average Nusselt number Num of a single cylinder and the upper of a pair of vertically aligned cylinders with the values in brackets representing the relative deviationto a single cylinder, dNu 100 % . Experimental results from [21] .

Case Ra ¼ 1:7 10 6 Ra ¼ 3:4 10 6 Ra ¼ 5:3 10 6

Numerical Experimental Numerical Experimental Numerical Experimental

Single cylinder Num ¼ 18 :3 Num ¼ 16 :9 Num ¼ 21 :6 Num ¼ 20 :7 Num ¼ 23 :6 Num ¼ 23 :8Upper cylinder ( S ¼ 4D) Num ¼ 19 :2(+5.0%) Num ¼ 18 :0(+6.1%) Num ¼ 23 :3(+7.7%) Num ¼ 22 :8(+9.2%) Num ¼ 24 :5(+4.1%) Num ¼ 24 :3(+ 2.0%)

Upper cylinder ( S ¼ 3:5D) Num ¼ 26 :2(+11.0%)Upper cylinder ( S ¼ 3D) Num ¼ 19 :5(+6.5%) Num ¼ 18 :2(+ 7.2%) Num ¼ 23 :1(+6.8%) Num ¼ 22 :4(+7.6%) Num ¼ 26 :1(+10.8%) Num ¼ 26 :5(+10.2%)Upper cylinder ( S ¼ 2:5D) Num ¼ 25 :6(+8.6%)Upper cylinder ( S ¼ 2D) Num ¼ 18 :1( 0.9%) Num ¼ 16 :1( 5.1%) Num ¼ 22 :3(+3.1%) Num ¼ 22 :3(+ 7.2%) Num ¼ 24 :7(+4.9%) Num ¼ 24 :5(+3.1%)Upper cylinder ( S ¼ 1:5D) Num ¼ 23 :7(+0.7%)

Fig. 11. Heat transfer enhancement dNum ¼ ðNum Num;single Þ=Num;single ( 100 % ) asa function of Rayleigh number and cylinder spacing S /D.

Fig. 12. Monitored points where solution data is extracted for FFT spectra andcoherence analysis (see Figs. 13–16 ).

Fig. 13. Time history of the local Nusselt number at h ¼ 0 (point A1 in Fig. 12 ) forRa ¼ 3:35 10 6 , S ¼ 3D. The start-up and initial oscillations from 0 s until 570 s areshown in the supplemental video in appendix ( 1 ).



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Fig. 14. Time-resolved local Nusselt number at h ¼ 0 (point A1 in Fig. 12 ) for Ra ¼ 3:35 10 6 , S ¼ 3D. (a) Numerical and (b) experimental results [21] .

Fig. 15. FFT amplitude spectrum for (a) local Nusselt number at point A1 ( h ¼ 0 ), (b) (normalised FFTs) velocity components V x and V y at point B1, (c) local Nusselt number atpoint A2 ( h ¼ 90 ) and (d) (normalised FFTs) velocity components V x and V y at point B2 (see Fig. 12 ) (Ra ¼ 3:35 10 6 , S ¼ 3D).


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the local Nusselt number. A supplemental video of the developingow eld from the start of the simulation until t = 570 s is providedas an electronic appendix. 1 After the start of the simulation ( t = 0 s),the ow initially develops as a stable plume, switching to a quasi-periodic oscillation at t ffi250 s. Once plume oscillation is estab-lished, the ow switches between regimes of almost purely har-monic swaying and more non-periodic uctuations, as previouslyobserved experimentally [21] .

Fig. 14 shows a close-up view of these quasi-periodic oscilla-tions, where the data in Fig. 14 a are obtained from the numericalresults and in Fig. 14 b from experimental measurements at thesame conditions [21] . The strong temporal uctuations in localNusselt number shown in Figs. 13 and 14 a correspond to lateralswaying oscillations of the thermal plume rising from the lowercylinder. Depending on the time-varying position and shape of the plume, the buoyant ow occasionally impinges directly ontothe bottom of the upper cylinder causing a sharp rise in instanta-neous local Nusselt number, or it is deected along either side of the cylinder, causing a reduction in local Nu. The uid temperatureand ow velocity magnitude and direction near the bottom of theupper cylinder (point A1) also vary strongly.

Upon closer inspection of Fig. 13 , the plume oscillations seem toswitch between two regimes, a quasi-periodic pattern alternatedby more non-periodic uctuations. This behaviour agrees withthe previous experimental observations [21] . Moreover, whenlooking at a period of time in which the uctuations are periodic,a remarkable similarity can be seen between the numerical URANSresults ( Fig. 14 a) and the experimental results [21] (Fig. 14 b). Thefact that the numerical model is able to capture both the temporalcharacteristics and Nu amplitude of the oscillation provides strongvalidation evidence for the URANS methodology.

Fig. 14 shows the variations of the Nusselt number at the lowestpoint of the upper cylinder (point A1). The Nusselt number at pointA1 goes approximately from 25 up to 60 with a time average of 1=T R T 0 Nuðh ¼ 0 ; t Þdt ¼ Num ð0 Þ 43as isshown on Fig.7 . Interest-ingly, the Nusselt number achieved by a single cylinder with thesame Rayleigh number is 28 (this value has been calculated fromthe data shown in Fig. 5, with Pr ¼ 7 and C ¼ 0:515,Nu ¼ 1:25 C Ra1=4 ). The deviation of the time-averaged localNusselt number from that for a single cylinder, at the location

h ¼ 0 , is observed, and is remarkably high with dNum ð0 Þ ¼ 53 :6% ,as shown in Fig. 9. The increase or decrease of the global heattransfer characteristics of the upper of a pair of cylinders comparedto the single cylinder case appears to be directly linked to theseoscillations.

For this same dataset ( Ra ¼ 3:35 10 6 and S ¼ 3D), a frequencyanalysis of the local Nusselt number at points A1 and A2, andvelocity at points B1 and B2 has been performed. Fig. 15 a and bshow respectively the Fourier transform of the local Nusselt num-ber at point A1 and the velocity components at point B1, based on asampling frequency of 2 Hz for a 500 s long signal. For this case, adistinct peak frequency f 1 ¼ 0:0195 Hz is found in both Nusseltnumber and vertical velocity, while a peak frequency

f 2 ¼ 0:00977 Hz is found for the horizontal velocity. At locationsA2 and B2, the peak frequency f 1 ¼ 0:0195 Hz is found for the Nus-selt number and the two components of the velocity. It is worthnoting that the frequency of the vertical velocity oscillation istwice the frequency of the horizontal velocity oscillations at pointB1. Miyabe et al. [32] report frequencies around 0.01 Hz, for theswaying mo C A1;B1 tion of the thermal plume rising from a heatedcylinder in spindle oil, for a wide range of immersion depth andfor temperature difference ranging from 10 C to 100 C. Theirresults agree well with the present numerical results. Reymondet al. [11] also reported, for a pair of cylinders, a similar pair of fre-quencies governing the oscillations; they observed peak frequen-cies at 0.016 Hz and 0.008 Hz for a spacing of S ¼ 1:5D and 2 Dand a Rayleigh number of 6 10 6 . This can be explained by the fact

that each time the thermal plume oscillates around the centrelinefrom the left side to the right side, it corresponds to one full periodfor V yðt Þbut only half a period for V xðt Þ: Fig. 15 c and d show a peakfrequency f 2 ¼ 0:00977 Hz for the Nusselt number at point A2 andboth velocity components at point B2.

Fig. 16 a shows the magnitude and phase angle of the coherencespectrum between the Nusselt number at point A1 and the velocitycomponents at point B1. Similarly, Fig. 16 b shows the coherencespectrum C A2;B2 between the Nusselt number at point A2 and thevelocity components at point B2.

The coherence between the Nusselt number Nu at point A1 andthe horizontal velocity V x at point B1 is relatively low with a maxi-mumof 0.3, accordingto Fig.16 a. Nu atA1and V x atB1arenotcoher-ent, which agrees with Fig. 15a and b where the dominant

frequencies of Nu at A1 and V x at B1 are different. Nevertheless,Fig. 16 a shows a broad range of high coherencebetween the Nusselt

Fig. 16. Magnitude and phase angle of the coherence spectrum (a) between local Nusselt number at A1 and velocity components at B1, (b) between local Nusselt number atA2 and velocity components at B2 ( Ra ¼ 3:35 10 6 , S ¼ 3D). Markers indicate two sets of harmonics.

1 The video le can be accessed here: https://drive.google.com/open?id=0B0Q2curze67ORnN3V2VKb08yVTQ .


https://drive.google.com/open?id=0B0Q2curze67ORnN3V2VKb08yVTQhttps://drive.google.com/open?id=0B0Q2curze67ORnN3V2VKb08yVTQhttps://drive.google.com/open?id=0B0Q2curze67ORnN3V2VKb08yVTQhttps://drive.google.com/open?id=0B0Q2curze67ORnN3V2VKb08yVTQ


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number Nu at point A1 and theverticalvelocity V y at point B1, atlowfrequencies up to approximately 0.03 Hz with a peak at f 3 ¼ 0:0234 Hz. This suggests that the oscillations in Nu and V y atpoints A1 and B1 (resp.) follow a coherent periodic pattern with a

base frequency f 3 which is close to f 1 ¼ 0:0195 Hz and agrees wellwith the Fourier transforms of Nu at A1 and V y at B1 shown inFig. 15 a and b. As the frequencies f 1 and f 3 are close in value, it hasbeen assumed that these two results characterise the same

Table 2

Summary of frequency analysis based on URANS results. The dimensionless frequencies Sr 1 and Sr 2 represent peak coherence between Nusselt number and velocity betweenpoints A1 and B1 ( Sr 1 ), and between points A2 and B2 ( Sr 2 ). Heat transfer enhancement is related to the equivalent single cylinder case at the same Rayleigh number.

Rayleigh number Ra Cylinderspacing S =D

Referencevelocity V ref (mm/s)

Plumewidth b=D

Peak Strouhal number Heat t ransferenhancement dNum (%)Sr 1 Sr 2

5:33 10 6 1 (single) 6.2 0.53 – – –4 5.3 0.57 0.094 0.033 +4.13.5 4.2 0.47 0.086 0.049 +11.03 4.1 0.50 0.094 0.058 +10.82.5 3.5 0.40 0.101 0.051 +8.62 2.8 0.43 0.104 0.083 +4.91.5 2.2 0.20 0.145 0.079 +0.8

3:35 10 6 1 (single) 5.5 0.60 – – –4 4.4 0.77 0.092 0.054 +7.73 3.4 0.53 0.171 0.026 +6.82 2.4 0.43 0.122 0.085 +3.1

1:70 10 6 1 (single) 4.0 0.62 – – –4 3.5 0.73 0.093 0.042 +5.03 2.7 0.53 0.150 0.075 +6.52 1.9 0.47 0.137 0.167 0.8

Fig. 17. Instantaneous streamline and velocity magnitude plot at (a) t = 280 s, (b) 300 s, (c) 320 s, (d) 340 s, (e) 360 s, (f) 380 s, showing one plume oscillation period(Ra ¼ 3:35 10 6 , S ¼ 3D).


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phenomenon.From now on, we will refer to f 1 only, when consider-ing theperiodicpattern which presents a peak frequency of approx-imately 0.02 Hz. A higher frequency peak in the coherence betweenNu at A1 and V y at B1 can be observed at f ¼ 0:09375 Hz ¼ 4 f 3 ,which can be assumed to be a harmonic of the base frequency f 3 .Inthelowfrequencyrangeupto0 :03 Hz, Fig. 16 a showsa strongcor-relation between the Nusselt number at A1 and the vertical velocityat B1, even though both locations are separated by a distanceD y A1B1 ¼ ðS DÞ=2 ¼ D. The correlation between Nusselt numberat A1 andthe horizontal velocity at B1 is small throughout theentirefrequency range.

Fig. 16 a shows that the coherence phase angle near the peakfrequency f 1 is slightly negative ( / 1 ffi 25 ), which indicates thatthe Nusselt number at A1 lags the vertical velocity at B1 byð/ 1 =360 Þ= f 1 ffi3:6 s. Considering the distance between locationsA1 and B1 (see Fig. 12 ), one can assume that this lag is mainlydue to the time it takes for the rising uid inside the thermal plumeto be advected from point B1 to the stagnation zone near point A1.For comparison, based on the value of the reference velocity V ref and distance between points A1 and B1, the estimated bulk advec-tion time would be D t ¼ D y A1B1V ref ¼ 8:8 s. However, this can be consid-

ered an overestimation since the reference plume velocity isaveraged over the full plume width, and the peak velocity in the

centre of the plume is about double this average, which means thatthe time lags are indeed comparable. This advection time lag hasbeen veried for other points in the ow eld, and therefore thisseems a reasonable explanation for the observed phase lag inFig. 16 a.

Interestingly, Fig. 16b shows different peak frequencies(marked by triangles and circles) in the coherence spectrumbetween the Nusselt number on the side of the cylinder (at A2)and the velocity components adjacent to that location (at B2, a dis-tance b=2 from A2). Compared to the coherence with the ow

below the cylinder ( Fig. 16 a), the coherence between the Nusseltnumber and the horizontal velocity V x is much more pronouncedalongside the cylinder ( Fig. 16 b). A peak frequency is observed at f 2 ¼ 0:00977 Hz (leftmost triangular marker in Fig. 16b) whichconrms the FFT results shown in Fig. 15 c and d. Alongside thecylinder the Nusselt number and both velocity components followa periodic pattern at the frequency f 2 ¼ 0:00977 Hz. Four distinctpeaks at higher frequencies are assumed to be harmonics of thisfrequency f 2 ; these are marked by triangles in Fig. 16b(0 :08789 Hz ¼ 9 f 2 , 0:1465 Hz ¼ 15 f 2 , 0:2637 Hz ¼ 27 f 2 and0:3223 Hz ¼ 33 f 2 .

However, two other peaks can be discerned at f 4 ¼ 0:06641 Hzand its harmonic at 0 :1992 Hz ¼ 3 f 4 . The peak frequency f 4 whichcan also be observed in the Fourier transform of the horizontal

velocity uctuations, indicates that another periodic pattern is pre-sent in the variations of the Nusselt number at point A2 and the

Fig. 18. Schematics of the stages of (i) plume swaying corresponding to low frequency oscillations of the velocity in between the two cylinders and the local Nusselt numberat point A1 (ii) vortex formation linked to high frequency uctuations of the velocity and the local Nusselt number observed at the side of the upper cylinder.

Fig. 19. Nusselt number variations around a zero mean ( Nu ¼ Nu Nu) at (a) pointA1 (h ¼ 0 ), (b) A2 ( h ¼ 90 ) and A3 ( h ¼ þ 90 ) (Ra ¼ 3:35 10 6 , S ¼ 3D).

Fig. 20. Fluctuations of the Nusselt number at point A2 and the transverse velocity(arbitrarily scaled) at point B2 ( Ra ¼ 3:35 10 6 , S ¼ 3D). Five indicated events in (b)correspond to the ow elds shown in Fig. 21 .



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velocity at point B2, especially its horizontal components V x. Thishigh frequency suggests that rapid variations (the period of theseoscillations is 1 = f 4 ffi15 s) in the instantaneous heat transfer occurin parallel to the slow uctuations due to the swaying of the plume(the period is 1 = f 2 ffi100 s). The phase angle at this frequency isfound to be / 4 ¼ 11 :5 which indicates that the Nusselt numberat point A2 lags the horizontal velocity at point B2 byð 11 :5=360 Þ=0:06641 ffi0:5 s.

The relationship between ow velocity and Nusselt number onthe side of the cylinder can lead to a better understanding of howthe buoyant plume dynamics affect the heat transfer characteris-tics of the upper cylinder. An interpretation of these two periodicphenomena (one characterised by f 1 and f 2 and the other one by

f 4 ) is given in the following section.In order to better characterise the oscillatory nature of this ow

for the entire range of Rayleigh number and cylinder spacing, thesethree peak frequencies f 1 , f 2 and f 4 are represented by dimension-less Strouhal numbers Sr 1 ¼ 0:171, Sr 2 ¼ 0:086 and Sr 4 ¼ 0:58where

Sr ¼ fDV ref

ð8 Þ

Unsteady RANS simulations have been carried out for all testconditions in Table 1 . The results are analysed in a similar wayas described above for the arbitrarily selected dataset(Ra ¼ 3:35 10 6 , S ¼ 3D), and are summarised in Table 2 .

Based on the data in Table 2 , two empirical power law correla-tions are least squares tted to the reference plume velocity and

the secondary peak Strouhal number, representing the peak oscil-lation frequency along the side of the cylinder. The referenceplume velocity increases monotonically with both Rayleigh num-ber and cylinder spacing, towards a maximum velocity for the sin-gle cylinder case (represented by S =D ¼ 1 in Table 2 ). However,the single cylinder case was not taken into account in least squaretting the correlation:

V ref ¼ 8 :77 106 Ra0 :33

S D

0 :85

ðm =s Þ ð9 Þ

This correlation shows an RMS deviation of 2.7% and a coef-cient of determination R2 = 0.98. The Strouhal number at the sideof the cylinder ( Sr 2 ) does not exhibit a strictly monotonic beha-viour, yet as a rst approximation, Sr 2 decreases for both increasingRayleigh number and cylinder spacing. For the single cylindercases, the results for the time history of the Nusselt number ath ¼ 0 (corresponding to point A1) did not exhibit signicant oscil-lation, and thus no values are obtained for those cases. This agreeswith the results of Grafsronningen and Jensen [12] .

Sr 2 ¼ 219 Ra 0 :47 S D

1 :05

ð10 Þ

This correlation shows an RMS deviation of 24.6% and a coef-cient of determination R2 = 0.75. The primary Strouhal number(Sr 1 ) representing the peak oscillation frequency in the regiondirectly below the cylinder exhibits a maximum of Sr 1 ffi0:17 for

the intermediate condition of Ra ¼ 3:35 106

and S = 3D, andtherefore does not collapse well to a power law correlation such

(a) (b) (c)

(d) (e)

Fig. 21. Instantaneous streamline and velocity magnitude corresponding to ve events indicated in Fig. 20 : (a) t 1 ¼ 426 s, (b) t 2 ¼ 427 s, (c) t 3 ¼ 428 s, (d) t 4 ¼ 429 s, (e)t 5 ¼ 430 s. Ra ¼ 3:35 10 6 , S ¼ 3D, with points A1, A2, B2 and A3.


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as Eqs. (9) and (10) . Similarly, the results for Sr 4 did not collapsewell enough ( R2 < 0:5) and thus it was not deemed relevant toadd this in Table 2 .

As noted by Persoons et al. [21] , the Strouhal number values arecomparable in magnitude to those found for vortex sheddingbehind a cylinder in forced cross-ow with an approach velocityof V ref (Sr ¼ 0:19 0:01) [33] . The following section will examinethe unsteady ow elds in the thermal plume in more detail, withparticular emphasis on the relationship between vortices and heattransfer enhancement.

5.2. Discussion: Effect of vortices in the oscillating plume on the localheat transfer coefcient

Fig. 17 shows instantaneous velocity magnitude distributionsand streamlines around the upper cylinder. Similar results havebeen obtained from experiment and are shown in Fig. 11 of Per-soons et al.’s paper [21] . The colour scale represents velocity mag-nitude from 0 to 18 mm/s. Based on the behaviour shown in Fig. 17and on the spectral analysis described inSection 5.1 , the hypothesishas been developed that the uctuations of the Nusselt number atvarious points on the cylinder circumference are affected by thecombination of (i) the overall swaying of the plume and (ii) smallerscale vortices that periodically form alongside the plume at inec-tion points.

On alternating sides of the cylinder, vortices can be seen toappear, advect, and merge with larger scale recirculating ows sur-rounding the plume and stretching into the rest of the domain.

Fig. 17 shows that a vortex is appearing alternately on each sideof the upper cylinder, prior to the plume motion at an approximateangular position of h ¼ 40 as shown in Fig. 17 a (where the signdepends on whether the vortex forms on the left or on the right

hand side of the cylinder). Consequent to the creation of the vortex,

the plume from the lower cylinder is deected toward the sidewhere the vortex has developed. The plume remains attached tothe side of the cylinder and merges with the plume from the uppercylinder. The plume from the upper cylinder is deected in theopposite direction than the plume from the lower cylinder, asshown in Fig. 17 b. The vortex is seen to travel around the cylinderwith its centre at an approximate distance of b, the plume width,from the cylinder surface, as shown in Fig. 17 b. As the vortex prop-agates (from h ¼ 40 to h ¼ 120 on the right hand side and simi-larly on the left hand side of the cylinder), it is stretchedvertically, following the plume direction, until it eventually dissi-pates. When this vortex has almost disappeared at approximatelyh ¼ 120 , another vortex forms at the opposite side of the cylin-der at a position h ¼ 40 as shown in Fig. 17 c. It is worth notingthat when this new vortex forms ( Fig. 17 c), the thermal plume aris-ing from the lower cylinder is still attached to the cylinder on theopposite side, i.e. on the side where the previous vortex had beencreated. Once the new vortex appears, the plume from the lowercylinder and the plume from the upper cylinder sway towards

the opposite side of the cylinder (as seen in Fig. 17 d and e) so thatthe plumes are always deected in the opposite directions. Thispattern is then repeated.

Fig. 18 shows a schematic diagram which illustrates our inter-pretation of this pattern. In Fig. 18 , the thin grey lines representthe outline of the plume as it sways through one full cycle. Thecurved red arrow represents the two vortices that form on alter-nating sides of the upper cylinder. In the diagram, the rst oneforms at h ¼ 40 and propagates until it dissipates at h ¼ 120 ,at which time the second vortex forms at h ¼ þ 40 . The secondvortex then propagates upwards until it dissipates at h ¼ þ 120and the cycle repeats itself.

Firstly, the plume swaying explains why the dominant fre-quency of the Nusselt number oscillation at points A2 and A3 on

the side of the cylinder (represented by Sr 2 ) is half the peak

Fig. 22. Instantaneous streamline and velocity magnitude (experimental data) corresponding to ve events indicated in Fig. 20 : (a) t 1 s, (b) t 2 ¼ t 1 þ 1 s, (c) t 3 ¼ t 1 þ 2 s, (d)t 4 ¼ t 1 þ 3 s, (e) t 5 ¼ t 1 þ 4 s, from PIV data set also presented in Persoons et al. [21] .



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oscillation frequency at point A1 (represented by Sr 1 ), at least formost conditions listed in Table 2 . Indeed, Fig. 19 shows the Nusseltnumber variations about the mean value, Nu0 ¼ Nu Nu, at pointsA1 (h ¼ 0 ), A2 (h ¼ 90 ) and A3 ( h ¼ 90 ). The oscillations of Nu0

at points A2 and A3 are similar in waveform but are 180 out of phase, or shifted by half a period. At each trough of Nu0(0 ) inFig. 19 a, a peak in the Nusselt number is alternately reached in

Fig. 19 b for the Nusselt number at point A3 (right side of the uppercylinder, Nu0ðþ 90 Þ) and the Nusselt number at point A2 (left sideof the upper cylinder, Nu0ð 90 Þ), which explains the factor of twobetween the peak frequencies Sr 1 and Sr 2 .

Secondly, the higher frequency oscillations represented by Sr 4can be explained by the effect of vortices on local instantaneousheat transfer enhancement and can be illustrated by taking a closerlook at the time evolution of the local Nusselt number at point A2and the adjacent transverse (horizontal) velocity V x at point B2. Asexplained in the previous section, the coherence spectrum revealeda strong dependency between these two signals at a peak fre-quency of f 4 ¼ 0:06641 Hz between the Nusselt number and theV x velocity uctuations which cannot be explained merely by bulkadvection.

Fig. 20 shows both signals (with subtracted mean and arbitrar-ily scaled) for (a) a time duration of 400 s, and (b) a close-up viewof 30s corresponding to the leftmost encircled interval on Fig. 20 a.The two signals are quite similar which conrms the high coher-ence between them (see Fig. 16 b). The four encircled events inFig. 20 a exhibit a repeating pattern with a period correspondingto Sr 2 (here, a frequency f 2 ffi0:0097 Hz or period 1 = f 2 ffi100 s).As explained above, this long period represents the plume swayingand alternating between both sides of the cylinder. However, uponcloser inspection each encircled event also contains higher fre-quency oscillations. Fig. 20b shows these oscillations correspond-ing to the frequency f 4 ¼ 0:06641 Hz identied in the coherencespectrum between the Nusselt number at point A2 and the velocityat point B2 ( Fig. 16 b) and the Fourier transform of the velocity atpoint B2 ( Fig. 15 d). This ‘high’ frequency oscillation is characterisedby a period of approximately 1 = f 4 ffi15 s. In terms of Strouhalnumber, Sr 4 ¼ f 4 D=V ref ffi0:58. A broadly comparable peak fre-quency was also found experimentally by Persoons et al. [21](Sr ¼ 0:46 0:02) and Grafsronningen and Jensen [12] (1= f ffi7 s,S =D ¼ 2, Ra ¼ 5:2 10 7 corresponding to a Strouhal numberSr ¼ 0:73). As shown in Fig. 20 b, the Nusselt number (solid line)lags the velocity V x (dotted line) by approximately 0.5 s which con-rms the results from the coherence phase lag. Exactly what causesthis peak at f 4 is as yet undetermined. However the fact that itappears in these numerical URANS results as well as in previousexperimental results [21,12] gives credence to the validity of theresults.

Figs. 21 and 22 show the instantaneous ow elds from the pre-sent numerical results and experimental PIV results from Persoonset al. [21] respectively, at the ve time instants identied duringone of the high-frequency oscillations in Fig. 20 b. The numericalresults show a good agreement with the experiment in terms of vortex size, plume width and velocity magnitude. However a dif-ference is noticeable as the vortex centre reaches the positionh ¼ 120 in Fig. 21e whereas the vortex centre seems to belocated at the position h ¼ 90 in Fig. 22 e. The vortex capturednumerically is moving faster than what is seen in the experimentalresults even though the local Nusselt number at point A2, from thenumerical results, is oscillating at the same frequency f 4 ¼ 0:0664 Hz observed in the experiments. At t ¼ t 1 the plumeis still impinging on the bottom of the cylinder although it isalready starting to sway towards the left hand side of the cylinder.The swaying is difcult to see in Figs. 21 and 22 because of the dif-ferent time scales involved, with a period of 1 = f 2 ffi100 s for plume

swaying compared to 1 = f 4 ffi15 s for this high frequency oscilla-tion. Considering the data shown in Fig. 20 b, the horizontal veloc-ity uctuations at point B2 are rst negative (times t 1, t 2 and t 3 ),pass by zero at time t 4 and then reach a positive peak at time t 5 .This shows that the vortex is rotating in an anticlockwise directionon the left hand side of the cylinder. At t ¼ t 1 the anticlockwisevortex shown in Fig. 21 is visible with a centre at an angular posi-

tion h ¼ 40 . Point B2 is at the leading edge of the vortex, withlocal ow directed away from the cylinder, or negative horizontalvelocity uctuations. This corresponds to a decrease from 0 to -10 approximately in the variations of the local Nusselt number atpoint A2 according to the data shown in Fig. 20 b. Over the nextfour seconds ( t 1 through t 5), the vortex travels from an angularposition h ¼ 40 to h ¼ 120 . During that time, the local trans-verse velocity in B2 goes from a negative extremum at t ¼ t 2 (i.e.,ow pulling away from point A2) to a positive extremum att ¼ t 5 (i.e., ow impinging onto point A2). This leads to an increasefrom -10 to +10 in the variations of the local Nusselt number atpoint A2. The URANS simulation enabled us to identify how thevariations in transverse velocity at point B2 correlate with the vari-ations of Nusselt number at point A2. However, the time-averagedlocal Nusselt number at point A2 is 24.1 as is shown in Fig. 7whereas the time-averaged local Nusselt number of a single cylin-der is 23.5 at that same location with the same Rayleigh number,according to the data presented in Fig. 5. The difference betweenthe time-averaged local Nusselt number at this location(h ¼ 90 ) and the single cylinder case is only dNum ð 90 Þ ¼ 2:5% .The change in local Nusselt number at point A2 due to the presenceof another cylinder seems not to contribute as much to the globalenhancement of the heat transfer potential of the upper cylinder asdoes the change in local Nusselt number at point A1.

6. Conclusion

A numerical simulation methodology has been established to

study the time-dependent interaction between two isothermallyheated, vertically aligned horizontal cylinders. The numericalresults obtained have been conrmed with previous experimentalmeasurements by some of the co-authors [21] , both in terms of time-averaged and unsteady quantities. For further validation of the numerical model, the single cylinder natural convection beha-viour is in excellent agreement with well-known empirical correla-tions and experimental local Nusselt number distributionsobtained by other researchers [30,31] .

A number of steady RANS simulations were carried out for arange of Rayleigh numbers (1 :7 10 6 6 Ra 6 5:3 10 6 ) andcentre-to-centre cylinder spacing (1 :5 6 S =D 6 4). The overalltime-averaged heat transfer rate is found to be affected by boththe Rayleigh number and the cylinder spacing, in a similar way

to that observed experimentally [21] . Mean heat transfer enhance-ment results are presented in Fig. 11 and Table 1 , showing a locusof optimum spacing (to achieve maximum enhancement) in therange 3 D 6 S 6 4D and 4 :7 10 6 6 Ra 6 5:3 10 6 , with morepronounced enhancement for increasing Rayleigh number. A max-imum enhancement of dNu ffi þ11 % was observed forRa ¼ 5:33 10 6 and S ¼ 3:5D.

For all conditions described in Table 1 , an unsteady RANSsimulation was also carried out. After an initial non-oscillatingstart-up, quasi-periodic plume oscillations were observed in thenumerical results and two regimes characterised by two differentfrequencies were identied. Having observed remarkably similarunsteadiness in the previous experiments [21] , this agreement fur-ther conrmed the validity of the numerical methodology. Fourier

transforms and coherence spectra were calculated for local Nusseltnumber and ow velocity signals, extracted from different points



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in the domain. A strong periodicity and coherence between heattransfer and ow velocity was observed. Trends were analysed inthe peak oscillation frequencies (expressed as dimensionlessStrouhal numbers) for the entire investigated range of Ra and S /DThe unsteady results were analysed in further detail, yieldinginterpretations for the two main time scales in this ow, resultingfrom (i) slow large scale swaying of the buoyant plume with a per-

iod of approximately 100 s and (ii) faster dynamics of vorticesforming adjacent to the plume with a corresponding time scaleof approximately 15 s. Their effect on local instantaneous heattransfer enhancement has been discussed. Based on the presentstudy, the low frequency (i.e., large time scale) uctuations corre-spond to the swaying of the thermal plume rising from the lowercylinder. This phenomenon affects the overall heat transfer of theupper cylinder to a greater extent than the high frequency uctu-ations associated with the formation and propagation of vortices atthe side of the upper cylinder.

Overall, the availability of detailed time-resolved ow and heattransfer data from the URANS simulations has provided insightinto this ow beyond the practical limitations of experimentalmeasurements [21] . However, the experimental data wasindispensable to validate the numerical model. Therefore, only bycombining the strengths of both approaches was this study(in combination with the previous work [21] ) able to provideexplanations for the intricate dynamics of this natural convectionconguration.

Acknowledgements

Quentin Pelletier is a PhD research student supported by theEngineering, Energy and Environment (E3) Institute at Trinity Col-lege Dublin.

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