Natural convection heat transfer and fluid dynamics for a
pair of vertically aligned isothermal horizontal cylinders
Tim Persoonsa,*, Ian M. O’Gormana, David B. Donoghuea, Gerry Byrnea, Darina B. Murraya
a
Department of Mechanical and Manufacturing Engineering, Parsons Building, Trinity College, Dublin
2, Ireland
Abstract
This paper discusses the close interaction between fluid dynamics and local natural convection heat
transfer rates from a pair of isothermally heated horizontal cylinders submerged in water. The
presence of a second heated cylinder induces heat transfer enhancements of up to 10%, and strong
fluctuations in local heat transfer rate. Therefore specific attention is focused on how the local heat
transfer characteristics of the upper cylinder are affected by buoyancy induced fluid flow from the
lower cylinder. The paper investigates a range of Rayleigh number between 1.8x106 and 5.5x106, and
a vertical cylinder spacing between 2D and 4D. Simultaneous local heat flux measurements and flow
velocity measurements using particle image velocimetry reveal oscillatory behaviour of the thermal
plume, depending on operating conditions. A joint temporal analysis of the data has provided new
insights into the governing mechanisms, which enables further optimisation of the heat transfer
performance.
Keywords: Tubular heat exchangers, flow oscillations, PIV, thermal plume interaction, vortex sheet.
*
Corresponding author. Tel.: +1 765 494 5638; fax: +1 765 494 0539. E-mail: tim.persoons@tcd.ie (T. Persoons).
1
Nomenclature
D
cylinder diameter (m)
U, V
horizontal and vertical fluid velocity (m/s)
DR
dynamic measurement range
Vt
velocity tangential to cylinder (m/s)
Gr
Grashof number (g (Ts - T)D3/2)
Vref
reference plume velocity (Eq. (1)) (m/s)
f
frequency (Hz)
u’, v’
fluctuation intensity of horizontal and
vertical fluid velocity (m/s)
2
g
gravitational acceleration (m/s )
H
immersion depth (m)
x, y
horizontal and vertical coordinate (m)
h
local convective heat transfer coefficient
z
coordinate along cylinder axis (m)
(q/(Ts - T)) (W/(m2K))
Greek symbols
k
thermal conductivity of fluid (W/(mK))
Nu
Nusselt number (hD/k)
Pr
Prandtl number (/)
q
local convective surface heat flux (W/m2)
Ra
Rayleigh number (GrPr)
r
radial coordinate from cylinder centre (m)
Sr
Strouhal number (fD/Vref)
Subscripts
S
cylinder centre spacing (m)
s
cylinder surface
Ts, T
cylinder surface and bulk fluid temperature

bulk fluid conditions
(K)
U
velocity
time (s)
u’
turbulence intensity
t
1

thermal diffusivity of fluid (m2/s)

volumetric expansion coefficient (K-1)

kinematic viscosity of fluid (m2/s)

density of fluid (kg/m3)

circumferential coordinate ()
Introduction
Tubular heat exchangers are among the most common devices for exchanging heat between
two fluid flows. The time-averaged heat transfer characteristics are well known for forced convection
from arrays of horizontal cylinders. However natural convection heat transfer has received less
attention and there remain questions regarding the natural convection heat transfer mechanisms,
especially for closely packed tube arrays, where thermal plumes interact with the thermal boundary
layer around nearby cylinders. Most studies have focused on averaged heat transfer characteristics
either for single cylinders [1] or cylinder arrays [2]. The interaction between adjacent cylinders is
generally described in terms of an overall heat transfer enhancement. Only a very limited amount of
knowledge is available on the influence of plume oscillations from one (upstream) cylinder on the
heat transfer from another (downstream) cylinder; this is mainly due to the difficulty in the
numerical modelling of the transient flow in the plume region.
Some papers discuss the swaying motion caused by a thermal plume from single cylinders as a
result of natural convection [3,4]. Low Rayleigh numbers are characterized by stationary twodimensional laminar plumes. With increasing Ra, the plume starts to oscillate in very irregular flow
patterns, especially near the top of the cylinder. At high Rayleigh number (Ra > 1010), the plume
transitions to the turbulent regime [4]. A far greater amount of research has been carried out into
2
studying plume swaying motion from thin horizontal heated wires, which can be regarded as line
heat sources [5,6,7]. Relationships for the velocity and temperature of a plume from a single wire
have been derived numerically [5]. These numerical predictions agree with experimental data for
thin wires at low Rayleigh number, yet at higher Rayleigh number plume swaying is observed
experimentally, causing substantial temperature oscillations in the plume centreline [6]. Desrayaud
and Lauriat [8] numerically investigated buoyancy induced flow from a horizontal line heat source
inside rectangular vessels with adiabatic sidewalls and isothermal top and bottom walls. For
rectangular vessels two destabilizing mechanisms lead to low frequency motion due to instability of
the buoyant plume. These mechanisms depend on the ratio of depth of immersion to vessel width.
The oscillation frequency collapses well to f = 0.0657Ra0.433 (0.3106< Ra < 8106) [8] which agrees
with earlier results of Noto [7].
These findings for single cylinders may serve as a reference case. Eckert and Soehngen [9]
investigated heat transfer from a vertical cylinder pair and found that the induced temperature and
velocity fields due to the buoyant plume from downstream cylinders have opposite effects. The heat
transfer rate from the upper cylinder was found to decrease with decreasing cylinder spacing due to
a decrease in the local temperature difference, while at larger spacing an increase in heat transfer
was found to occur due to the higher local fluid velocity having a forced convection effect on the
upper cylinder. For the case of a pair of horizontal isothermally heated cylinders, previous studies by
the authors involving measurements of the local heat transfer coefficient around the cylinder
circumference have revealed a range of cylinder spacings and Rayleigh numbers where beneficial
interaction occurs [10,11]. Based on spectral analysis of the local surface heat flux, it was
hypothesized that when the plume from the bottom cylinder oscillates out of phase with the plume
from the top cylinder, beneficial mixing occurs which explains the heat transfer enhancement.
To elucidate these previously obtained results, the objective of this paper is to experimentally
investigate the coupled fluid dynamics and heat transfer of natural convection from a pair of
isothermal horizontal cylinders. The Rayleigh number ranges between 1.8106 and 5.5106,
corresponding to the earlier investigations [10,11] and the cylinder centre spacing ranges between 2
and 4 cylinder diameters.
2
Experimental approach
2.1 Natural convection test facility
The experimental test facility is designed to approximate the testing of a tubular section of
infinite length contained within an infinite fluid medium. The facility uses a pair of isothermally
heated copper cylinders with a diameter D = 30mm. Horizontal and vertical confinement effects are
minimized by choosing the end plate spacing greater than 3D [12] and the depth of immersion H
greater than 3D [13]. Figure 1 shows a schematic diagram of the facility built with these
considerations in mind, measuring 900mm high, 900mm long and 300mm (10D) wide. Deoxygenated
water was chosen as the working fluid for this analysis. Air was deemed unsuitable due to the high
temperatures required for the targeted Rayleigh number range, and due to the difficulty of properly
3
insulating an air control volume from the surroundings. The large vessel volume (about 200 litres)
prevents excessive bulk water temperature drift during testing.
water level
 = 180
y
z
x

 = 0
heat flux
sensor
S
D
(a)
H
y
10D
(b)
Figure 1. (a) Diagram of the natural convection test facility and (b) definition of the coordinate system,
separation distance S and depth of immersion H
2.2 Local heat transfer measurements
The instrumented cylinders are carefully machined and measure D = 30mm in diameter and 10D
long to mitigate end effects (see Fig. 1b). Each cylinder contains two 500W cartridge heaters
embedded along the cylinder axis, using thermal grease to minimise the thermal contact resistance.
The minimum distance between the bottom cylinder and the vessel floor is 10D. Each cylinder is
instrumented with a flush mounted thermopile heat flux sensor (RdF Micro-FoilTM 27036-2-RdF) and
an internally mounted T-type thermocouple. The cylinders can be rotated about their axis to
measure the local surface heat flux around the circumference, as shown in Fig. 1b. To minimize
radiation losses the cylinder surfaces are polished to reduce the surface emissivity to around 0.10;
the emissivity of the Kapton coating on the heat flux sensor is 0.70.
The properties of water are evaluated at the film temperature, (Ts + T)/2. The most accurate
reading of the surface temperature Ts is obtained using a surface-mounted 0.3mm fine wire T-type
thermocouple, mounted at the same circumferential position  as the heat flux sensor yet at an axial
distance of z = 10mm from the sensor. This proved to be a more reliable approach than predicting
the surface temperature based on one-dimensional heat conduction through the heat flux sensor
and the thermocouple embedded in the sensor. For a given test, the Rayleigh number (Ra = GrPr =
g(Ts – T)D3/()) is set by adjusting the difference (Ts – T) between the surface temperature and
the bulk water temperature, which is measured at the same elevation y as the test cylinder.
Measurements are only taken once a pseudo steady state is reached. Once the targeted
operating parameters are met the cylinder is rotated in 10 intervals for half a revolution. At each
interval the heat flux, surface temperature, and bulk fluid temperature are recorded. Preliminary
testing has revealed very low frequency oscillations in heat transfer for multiple cylinder
configurations. To capture this oscillatory behaviour a sampling time of approximately 220s and
sampling frequency of 40Hz are applied. For each measurement condition, tests are repeated 5 times
to ensure repeatability.
4
The thermopile heat flux sensor signal is conditioned using a 1000:1 Fylde 351UA amplifier and
read into a National Instruments 9172 data acquisition system (NI 9215, 16bit, 0-10V).
Thermocouples are read into the data acquisition system (NI 9219, 24bit, 0-80mV) with internal cold
junction compensation. The thermocouples were calibrated in situ against a factory-calibrated
resistance temperature detector (RTD) probe with an Omega CL 26 digital meter (uncertainty of
0.3% at 50C). An in-situ calibration was carried out for the heat flux sensors mounted on the
instrumented cylinders, following the procedures outlined by Reymond et al. [11] and Atmane et al.
[13]. Thus, for each sensor, the measured heat transfer was referenced against the Churchill and Chu
[14] correlation for natural convection from a single horizontal cylinder. In addition, an energy
balance was conducted to equate the measured convective heat flux to the electrical power supplied
to the internal cartridge heaters. These independent calibration procedures differed by a maximum
of 8%, giving confidence in the results obtained.
The heat flux measurement is the main contribution to the uncertainty in the Nusselt number.
The maximum measurement uncertainty occurs at the highest Rayleigh number of 5.5106. Based on
a 95% confidence level, the overall estimated uncertainties of the Rayleigh and Nusselt numbers are
around 1% and 16%, respectively. Repeatability across multiple tests is very good, with maximum
rms fluctuations in Nusselt number below 0.5% for the single cylinder tests. More details of the
calibration procedures and error analysis are given by O’Gorman [15].
2.3 Fluid flow measurements
2.3.1 Particle image velocimetry approach
Particle image velocimetry (PIV) is used to quantify the time-varying flow field. These
measurements are performed simultaneously to the heat transfer measurements described above.
Synchronisation of the start of both measurements is achieved using a common trigger signal.
Figure 1a depicts the arrangement of the laser light sheet optics (intersecting the cylinders in
the mid-plane of the vessel) and the double-frame camera (Photron Fastcam SA1, 1024x1024 px,
5400Hz at full frame rate, 12bit, Sigma 105mm f/2.8 lens). The light source is a Quantronix DarwinDuo Nd:YLF twin cavity pulsed laser (15mJ at 1000Hz, 527nm), and the mean light sheet thickness
throughout the field of view is approximately 1mm. With the light sheet entering from the left, the
cylinder casts a shadow where no PIV measurements are possible (e.g. see Fig. 2).
The camera is aligned nearly perpendicular to the light sheet, so as to minimize light reflections
from the cylinder surface and maximize the visible region near the top cylinder surface. Because of
the slightly off-perpendicular position (viewing angle of a few degrees), a calibration target was
positioned in the water tank prior to the measurements to perform an optical distortion correction
using Lavision’s DavisTM 7.2.2 software. Details of the calibration process are given by O’Gorman [15].
Polyamid particles with diameters between 30 and 70μm and density of 1.03g/cm 3 are used as
seeding. The maximum Stokes number (i.e. dimensionless relaxation time) never exceeds 0.2,
indicating that the particles closely follow the water streamlines. After optical calibration of the
camera setup in ambient conditions, the heating power to the instrumented cylinders is adjusted to
meet the required thermal conditions for the given test. As slow frequency oscillations within the
5
flow are expected, the camera frame rate is reduced to its minimum of 50Hz, providing a maximum
test duration of 26s. Local heat transfer data from the instrumented test cylinders is recorded
simultaneously with the PIV images.
The tests were restricted to Rayleigh numbers below 6106, since at higher temperatures the
quality of the vectors obtained close to the cylinder surface deteriorated due to differential optical
refraction within the thermal plume, causing an elongation of the seeding particle images. This
optical aberration is minimized by positioning the camera at a slightly oblique angle (about 5),
thereby minimizing the overall thermal gradients in the light path incident to the camera lens.
2.3.2 High dynamic range PIV measurements
Since the flow field is characterized by strong velocity gradients and differences in velocity
magnitude between the thermal plume and the entrainment region, a novel technique for increasing
the dynamic range of standard PIV is applied. The multiple pulse separation (MPS) technique
described by Persoons and O’Donovan [16] combines information from sets of images acquired at
multiple pulse separations, to automatically determine the locally optimal pulse separation
depending on local flow field characteristics. By increasing the dynamic velocity range, MPS PIV
obtains more accurate mean flow and turbulence measurements compared to conventional PIV.
(a)
(b)
2
2 1/2
Figure 2. Comparison of turbulence intensity distribution (u’ +v’ ) /Vref around the upper of two heated
cylinders at Ra = 3.6106 and S = 3D, using (a) conventional PIV with a pulse separation of 40ms and (b) multipulse separation (MPS) PIV based on three pulse separation values (40, 80 and 160ms)
Table 1. Comparison of the dynamic measurement range of several quantities using conventional and MPS PIV,
for Ra = 3.6106 and S = 3D (see Fig. 2)
Quantity
Measurable range
for conventional PIV
for MPS PIV [16]
time-averaged
velocity (U2+V2)1/2
0.028 to 2.06mm/s
0.005 to 1.86mm/s
DRU = 70:1
DRU = 410:1
rms turbulence
intensity (u’2+v’2)1/2
0.063 to 1.38mm/s
0.012 to 1.36mm/s
DRu’ = 20:1
DRu’ = 110:1
6
Figure 2 shows a sample result comparing (a) conventional PIV and (b) MPS PIV using three
pulse separation multiples (1, 2 and 4 the minimum pulse separation of 40ms). Conventional PIV
tends to overestimate turbulence levels in low velocity regions, which has also been noted in other
flow fields with a wide velocity range, such as impinging steady and synthetic jet flows [16-18]. This is
due to the deterioration of vector quality when the particle displacement magnitude reduces to the
minimum resolvable displacement which is about 0.1 pixels in typical laboratory conditions [19],
even using advanced multi-grid correlation techniques.
Table 1 compares the dynamic measurement range of conventional and MPS PIV. The MPS
technique increases the dynamic range for mean velocity and turbulence intensity with a factor of
5.5x compared to conventional PIV, resulting in a considerably higher accuracy. The effect is more
pronounced in flows with a wider velocity range [16-18].
3
Experimental results
A range of Rayleigh numbers between 1.84106 and 5.33106 were investigated, meaning that
the flow is nominally laminar. Simultaneous PIV and heat transfer data were recorded at heat flux
sensor position angles of  = 0, 90, 135 and 180. Four horizontal cylinder configurations were
tested: a single cylinder and a pair of vertically aligned cylinders with centre spacings 2D, 3D and 4D.
3.1 Heat transfer from a single cylinder
Figure 3. Effect of Rayleigh number on the time-averaged local Nusselt number along the circumference of a
single cylinder (0 = bottom, 180 = top)
Firstly, local heat transfer results from single cylinder measurements are presented. Figure 3
shows the time-averaged local Nusselt number around the circumference of the cylinder, where 0
and 180 represent bottom and top of the cylinder, respectively (see definition in Fig. 1(b)). Figure 3
shows that the maximum heat transfer rate occurs at the bottom stagnation point (0) of the cylinder
and steadily decreases with increasing angle until approximately 160, with a sharper rate of
decrease occurring over the final 20 of the cylinder circumference. This result is consistent with
results for the local Nusselt number distribution about a single cylinder reported by Kuehn and
Goldstein [20], Merkin [21] and Reymond et al. [11]. A power law scaling of Nu with Ra is observed
7
with an exponent of 0.32 (R2 > 0.99), which is broadly consistent with the findings reported by
Morgan [1] and Churchill & Chu [14].
Figure 4. Time-averaged contours of velocity magnitude (U2 + V2)1/2/Vref and velocity vectors showing the
buoyant plume above a single cylinder at Ra = 1.92106 (Vref = 5.1 mm/s, plume width 2b = 0.35D at y = 2D)
The accelerated decrease in the local Nusselt number over the final 20 of the circumference is
generally attributed to an insulating effect of the buoyant plume forming near top of the cylinder. To
verify this, PIV flow visualisation was carried out for the case of a single cylinder and within the target
Rayleigh number range of this experiment. For a selected case of Ra = 1.92106, Fig. 4 shows the
time-averaged flow field over a single cylinder as contours of velocity magnitude (U2 + V2)1/2/Vref,
where the reference plume velocity Vref is defined as
Vref 
 V ( x , y ) dx
(1)
x
b
The plume width b is defined as the distance between the locations where the upward velocity drops
to 25% of the peak velocity. Although the plume width slightly increases with y, the value of Vref
according to Eq. (1) is reasonably independent of y. Within the Rayleigh number range investigated
the plume width b remained constant at 0.35D (= 10.5mm; evaluated at y = 2D). The plume velocity
according to Eq. (1) is typically about ¼ of the characteristic velocity defined based on the square
root of the Grashof number, (Ra/Pr)1/2/D.
The plume detaches from the cylinder surface at an angle of approximately 159, confirming
that the sharp decrease in the local Nusselt number from a single cylinder is a direct result of the
formation of the buoyant plume at this location.
3.2 Heat transfer from two vertically aligned cylinders
3.2.1 Heat transfer characteristics
Previous work by the authors [10, 11] has shown that the heat transfer characteristics of a pair
of closely spaced horizontal cylinders are significantly different from those of a single cylinder.
Reymond et al. [11] have demonstrated that only when both cylinders are heated a considerable
degree of interaction occurs in the surface heat transfer rate from the upper cylinder. For a spacing
8
of 1.5D or greater, the lower cylinder is unaffected by the upper cylinder and the upper cylinder is
unaffected by an unheated lower cylinder.
Table 2. Average Nusselt number of a single cylinder and the upper of a pair of vertically aligned cylinders, with
values in brackets representing the relative deviation to a single cylinder case, (Nu – Nu0)/Nu0  100%
Ra = 1.8106
Ra = 3.6106
Ra = 5.3106
Nu = 16.9
Nu = 20.7
Nu = 23.8
Upper cylinder
(S = 4D)
Nu = 18.0
Nu = 22.8
Nu = 24.3
(+6.1%)
(+9.2%)
(+2.0%)
Upper cylinder
(S = 3D)
Nu =18.2
Nu = 22.4
Nu = 26.5
(+7.2%)
(+7.6%)
(+10.2%)
Upper cylinder
(S = 2D)
Nu = 16.1
Nu = 22.3
Nu = 24.5
(–5.1%)
(+7.2%)
(+3.1%)
Single cylinder
(S = )
Table 2 gives an overview of the circumferentially averaged heat transfer coefficient, both for
the single cylinder case and for the upper cylinder at inter-cylinder spacings of 2D, 3D and 4D. For the
2D spacing, a slight decrease in mean Nusselt number can be observed at low Rayleigh number.
However for all other cases, an overall heat transfer enhancement is noted. This is in agreement with
previous results by Reymond et al. [11].
For a fixed Rayleigh number (Ra = 5.33106), Fig. 5 shows the effect on the local heat transfer
coefficient of the presence of a second heated cylinder at spacings of 2D, 3D and 4D below the
primary cylinder. Higher local heat transfer levels are evident, especially near the bottom (0) and to
a lesser extent near the top (180) of the upper cylinder. For the smallest spacing (S = 2D), local
increases of 60% to 100% are observed near the bottom and top of the upper cylinder. A reduction in
heat transfer is observed along the sides of the cylinder, which is insignificant for larger spacings. For
the largest spacing investigated (S = 4D), the heat transfer profile is similar to that of a single cylinder
except near the top. For the parameters of the current investigation, the heat transfer characteristics
of the lower cylinder were unaffected by the presence of the upper heated cylinder and are
therefore not reported here.
Figure 5a shows that the local Nusselt profile of the upper cylinder is strongly dependent on the
cylinder spacing. Although only shown for Ra = 5.3106, this is true for the entire investigated
Rayleigh range (1.8106  Ra  5.3106). Figure 5b shows that the shape of the Nusselt profile at a
fixed spacing is very similar under different Rayleigh number conditions.
For the cylinder pair, strong fluctuations in heat transfer were noted, in particular near the
bottom and top of the upper cylinder. Near the bottom, the thermal plume from the lower cylinder
joins the upper cylinder’s boundary layer, whereas near the top (180) the combined plume
detaches. Figure 6(a,b) shows the fluctuations in local Nusselt number at the bottom of the upper
cylinder ( = 0) for an cylinder spacing of (a) S = 3D and (b) S = 4D. At a moderate spacing (S = 3D),
Fig. 6(a) shows a stable periodicity in the fluctuations induced by the presence of the lower heated
cylinder. At a large spacing (S = 4D), the time trace in Fig. 6(b) shows evidence of two distinct
9
regimes, one at a nearly constant Nusselt number and one exhibiting high amplitude and high
frequency fluctuations.
(a)
(b)
Figure 5. Time-averaged local Nusselt number along the circumference of the upper cylinder: (a) effect of
spacing 2D, 3D and 4D ( represents the single cylinder case) at Ra = 5.33106 and (b) effect of Rayleigh
number at a fixed spacing S = 2D
(a)
(b)
Figure 6. Time-resolved local Nusselt number on the bottom of the upper cylinder (  = 0) at (a) S = 3D and (b)
S = 4D, for low and high Rayleigh numbers (1.7106 and 3.3106)
3.2.2 Simultaneous flow field and heat transfer measurements
The observation of this oscillatory phenomenon in the heat transfer justified a detailed study of
the fluid dynamics. Due to camera memory limitations, the duration of these measurements is
restricted to 26 seconds. Figure 7 shows a single period of a buoyant plume oscillation in the region
between a pair of vertically aligned cylinders at a Rayleigh number of 3.7x106 and a spacing of 3D.
From left to right and top to bottom, the plots represent instantaneous flow fields at 2.5 second
intervals.
10
Figure 7. Instantaneous streamline and velocity magnitude plots at t = 0s to 12.5s in steps of 2.5s (left to right,
top to bottom) showing one plume oscillation period around the lower stagnation point of the upper cylinder
at Ra = 3.7x106 and S = 3D (Vref = 4.2 mm/s, plume width 2b = 0.42D at y = -1D)
Figure 8. Simultaneously acquired time-resolved () local Nu on left side of upper cylinder ( = 90) and ()
near-cylinder tangential velocity Vt (in mm/s) just above the sensor position, at Ra = 3.7106 and S = 3D (Vref =
4.2 mm/s, plume width 2b = 0.42D at y = -1D)
Figure 7 shows that the plume from the lower cylinder does not split and rise symmetrically
around the upper cylinder, but instead oscillates back and forth across the upper cylinder. As a result
of the swaying motion, the cylinder experiences alternating pockets of high velocity fluid passing
along its sides. Studies by Sadeghipour and Asheghi [22] and Sparrow and Niethammer [23] have
attributed enhancement of the heat transfer from the upper cylinder to the forced convection effect
caused by the plume from the lower cylinder, however these flow field measurements suggest that
11
lateral oscillation of the thermal plume from the lower cylinder is an additional mechanism of heat
transfer enhancement from the upper cylinder.
Figure 8 shows the local Nusselt number time trace at a position of 90°, on the left side of the
upper cylinder. The circular markers represent the simultaneously measured tangential velocity Vt
just above the heat flux sensor. This velocity is determined from the time-resolved PIV data, by
averaging the tangential velocity in the boundary layer region up to 0.2D from the cylinder surface:
Vt (90, t )  0.21 D 
0.7 D
V (r ,  90, t ) dr
(2)
r  0.5D t
As shown in Fig. 7, the plume alternates between the left and right side of the upper cylinder.
Figure 8 demonstrates that the time-resolved Nusselt number Nu(90) and the tangential velocity on
the same location Vt (90) correlate well. The spectrum of both signals exhibits a peak fluctuation
frequency of 0.065 ( 0.003) Hz, with only a minor phase difference between the local velocity and
Nusselt number.
This coherence between the local Nusselt number and local plume velocity highlights the impact
of the swaying motion of the plume from the lower cylinder on the heat transfer performance of the
upper cylinder. Figure 8 does show some higher order components in both heat transfer and plume
velocities. As shown in Fig. 7, the flow field features a wide range of larger vortices down to smaller
eddies. Stochastic phenomena including shedding, merging and dissipation of vortices into small
scale turbulence can explain these higher order perturbations observed in Fig. 8.
4
Discussion
4.1 Positive heat transfer enhancement and plume instability
At a spacing of 4D, Fig. 6b shows two distinct regimes occurring in the Nusselt number time
trace at the bottom of the upper cylinder, one of high frequency fluctuations and one of relatively
constant Nusselt number. The preceding discussion suggests that the highly fluctuating Nusselt
number is a consequence of oscillation of the thermal plume from the lower cylinder. However this
does not explain the steady regime in which the mean Nusselt number is comparable with that of a
single cylinder at the same location, as shown in Fig. 9a. These two regimes are indicated with letters
A and B in Fig. 9. No fixed periodicity is observed in the alternation between both regimes. Their
occurrence is stochastic, with regimes A and B occurring respectively in 17% and 83% of the total
measurement time covering 12 separate tests.
Figure 9b gives a schematic representation of the flow field in both regimes. In regime A, no
oscillations of the buoyant plume from the lower cylinder are observed and the plume from the
bottom cylinder rises steadily along one side of the upper cylinder, maintaining a constant separation
distance from the surface of the upper cylinder. This separation leads to two independent plumes
rising above the upper cylinder. In regime B, oscillatory plume motion is observed and a single
intertwined plume rises above the upper cylinder.
In regime A the flow field around one side of the upper cylinder is similar to that around a single
cylinder, thus explaining why the heat transfer rate in regime A is comparable to that of a single
12
cylinder (see Fig. 9a). The oscillatory regime B is clearly beneficial, with an averaged heat transfer
level of about 50% higher than regime A.
A
B
(a)
(b)
Figure 9. (a) Comparison between the time-resolved local Nusselt number at  = 0 (i.e. bottom stagnation
point) of the upper of two cylinders at S = 4D (black line) to a single cylinder (grey line), both at Ra = 3.3106.
For a pair of cylinders, alternating heat transfer regimes are observed (A = constant Nusselt number and B =
fluctuating Nusselt number), as schematically depicted in (b).
(a)
(b)
Figure 10. Simultaneously acquired local heat flux at the lower stagnation point ( = 0) of the upper cylinder
and upper stagnation point ( = 180) of the lower cylinder, at Ra = 3.3106 and (a) S = 3D and (b) S = 4D
A further investigation was carried out to determine the mechanism triggering the instability in
the lower cylinder plume, by simultaneously measuring the heat fluxes on the lower and upper
cylinder. Figure 10 presents the local convective heat flux q at the lower stagnation point (0) on the
upper cylinder and the upper stagnation point (180) of the lower cylinder, for a Rayleigh number of
3.3x106 and two cylinder spacings. The heat flux at the top of the lower cylinder remains almost
constant throughout the 400s test period. This steady heat flux from the lower cylinder and the lack
of coherence between the heat flux signals on the upper and lower cylinders suggests that unsteady
heating from the lower cylinder is not the driving mechanism for plume oscillation. Therefore PIV
measurements were employed to analyse the flow surrounding the upper cylinder in further detail.
13
(a)
(b)
(c)
(d)
(e)
(f)
Figure 11. Instantaneous streamline and velocity magnitude plot at the time of plume inflection around the
upper of a pair of cylinders at (a-c) S = 3D and (d-f) S = 4D; (a,d) Ra = 1.8x106, (b,e) Ra = 3.6x106, (c,f) Ra =
5.3x106
Figure 11 shows the velocity distribution around the upper cylinder at the point of plume
inflection (i.e. when the lower plume changes direction), for three Rayleigh numbers and a cylinder
spacing of 3D and 4D. At plume inflection a strong counter-clockwise vortex is formed to the left of
the plume (and vice versa, a clockwise vortex is formed to the right when the plume transitions
towards the other side of the upper cylinder). The formation of these vortices creates an unbalance
in the lateral forces acting on the plume, which suggests that this mechanism may be responsible for
the fluctuating oscillatory motion of the lower cylinder plume. Similar coupled oscillatory plume and
vortex formation was observed by Incropera and Yaghoubi [24] using dye injection methods for the
case of a circular cylinder confined by a free water surface. The observation of vortex rings by these
researchers for the case of a vertically confined cylinder suggests that vertical confinement of the
buoyant plume from the lower cylinder by the upper cylinder surface may be the cause of these
alternate rotating vortices which drive the swaying oscillatory motion of the lower cylinder plume.
A second possible interpretation for the plume swaying is that the vortices formed near the
bottom of the upper cylinder are governed by a mechanism similar to Kelvin-Helmholtz instability in
wake flows. The shear layers in the wake of a single cylinder in cross-flow are known to interact and
form a von Karman vortex street. When expressed as Strouhal number, the dimensionless shedding
frequency is quite constant over a wide range of Reynolds numbers based on approach velocity and
cylinder diameter (300 < Re < 105) [25]. For the case shown in Fig. 8 (Ra = 3.7x106 and S = 3D), the
observed plume fluctuation frequency f = 0.065 ( 0.003) Hz becomes:
Sr 
fD
 0.46 ( 0.02)
Vref
(3)
14
where Vref is determined according to Eq. (1) as the average plume velocity at a distance of 0.5D
below the upper cylinder. Using the peak centreline plume velocity instead as reference velocity in
Eq. (3) yields Sr = 0.30  0.01. Although there is a significant difference between pressure-driven flow
across a single cylinder and buoyancy-driven flow across a pair of cylinders, the Strouhal number
value is of the same order of magnitude as the natural shedding frequency for a cylinder in cross-flow
(Sr = 0.19  0.01) [25].
At the largest separation distance (S/D = 4), a similar value of Sr = 0.54 is found based on Vref (or
Sr = 0.36 based on the maximum plume velocity) at a comparable Rayleigh number (Ra = 3.6106).
Further research is required to validate this finding in a wider range of parameters.
4.2 Heat transfer reduction and plume confinement
Figure 5a shows that for small S/D, the local Nusselt number of the upper cylinder is lower than
that of a single cylinder. As mentioned in Sect. 1, heat transfer from the upper cylinder may be
negatively affected by a reduced local temperature difference between the upper cylinder and the
surrounding fluid, which can counteract any potential enhancement due to plume oscillation.
However, the local reduction in heat transfer can be quite significant and additional mechanisms may
be at play. The maximum local reduction was found to occur at  = 50 for Ra = 3.5x106 and S = 2D,
which is the focus of the following investigation. Figure 12 compares the local Nusselt number at  =
50 for this case against a spacing of 3D. A low frequency, low amplitude oscillatory pattern is
observed at S = 2D, very different from the behaviour at S = 3D.
Figure 12. Time-resolved local Nusselt number on the upper cylinder at Ra = 3.5106 and S = 2D and 3D, at  =
50 corresponding to the circumferential location of maximum heat transfer reduction compared to a single
cylinder case (see Fig. 5)
Although not shown here, PIV measurements show a regular swaying motion of the plume at S =
2D, however with a much lower amplitude and frequency compared to those found at the same
Rayleigh number and larger spacings, as shown in Fig. 7. Incropera and Yaghoubi [24] found that the
behaviour of a buoyant plume confined by a free water surface is strongly dependent on the
immersion depth H/D, where H is the distance from the cylinder top to the confining surface. They
found that the amplitude of the plume oscillation decreases with decreasing H/D and that the
15
oscillation became negligible for H/D < 0.5. This has been confirmed by Atmane and Murray [13]. The
reduction in oscillation amplitude in Fig. 12 is consistent with these findings [13,24], suggesting that
excessive confinement of the plume by the upper cylinder can be responsible for the smaller
fluctuations in heat transfer on the upper cylinder at a spacing of 2D.
In addition to the weak plume oscillations at S = 2D, the formation and detachment of counterrotating vortices on either side of the plume (as seen in Fig. 7 for S = 3D) was not observed at S = 2D.
Incropera and Yaghoubi [24] reported that no vortex is formed below the air-water interface for H/D
< 0.5, in contrast to larger immersion depths. The lack of vortex formation at this close spacing
directly coincides with a large reduction of the local heat transfer of the upper cylinder compared to
that found at a larger spacing. This supports the contention that vortex formation acts as an
additional enhancement mechanism for increasing heat transfer from the upper cylinder of a pair of
isothermal horizontal cylinders aligned in the same vertical plane.
5
Conclusions
By combining particle image velocimetry (PIV) and local convective heat transfer measurements
using hot-film anemometry, a methodology has been established to simultaneously study fluid
dynamics and heat transfer in natural convection from a pair of horizontal cylinders. A multi-pulse
separation PIV technique has been successfully applied to increase the dynamic range in measuring
the local velocity and turbulence fields. In typical conditions, an increase in dynamic range of about
5.5 times has been obtained, resulting in more accurate flow fields.
A single cylinder does not show any significant fluctuations in heat transfer in the Rayleigh
number range under investigation (1.8106  Ra  5.5106), and agrees well with established
correlations in terms of the average heat transfer rate. However, in the case of two vertically aligned
horizontal cylinders (centre spacing from 2D to 4D), a strong periodicity is observed in the local heat
transfer rate. Fluctuations are highest around the bottom and top of the upper cylinder, where
respectively the lower plume impinges and the upper plume detaches. The average heat transfer
rate of the upper cylinder increases by up to 10% depending on Rayleigh number and separation
distance, although a slight heat transfer reduction of –5% is observed at a small separation distance
and low Rayleigh number (S/D = 2, Ra = 1.8106).
The simultaneous flow field and heat transfer results show the interaction of oscillating thermal
plumes, and have confirmed the existence of different alternating regimes. Related to the plume
swaying, vortex shedding occurs near the bottom of the upper cylinder. After formation, these
vortices travel upwards across alternating sides of the upper cylinder, resulting in pockets of high
velocity fluid thus contributing to break up and mixing of the thermal boundary layer. At a cylinder
spacing of S = 2D, excessive confinement of the buoyant plume from the lower cylinder is believed to
be responsible for minimal fluctuations in heat transfer and regions of reduced local heat transfer on
the upper cylinder. A temporal analysis shows how the plume velocity affects the local heat transfer
rate. Further work is needed to confirm a hypothesis of a constant Strouhal number, reminiscent of
vortex shedding in forced flow across a single cylinder.
16
The established methodology of simultaneously acquiring flow field and heat transfer
measurements has yielded a better insight into the fundamentals of natural convection heat transfer
in this particular case, and enables a further optimisation of the observed heat transfer enhancement
towards larger scale systems and industrial applications.
Acknowledgements
Dr. Tim Persoons is a Marie Curie research fellow of the Irish Research Council for Science,
Engineering and Technology (IRCSET). Ian M. O’Gorman is a postgraduate research scholar of the
Irish Research Council for Science, Engineering and Technology (IRCSET). The authors acknowledge
the work of Antoine Van Noorte and Pauline Vancraenenbroek, and the financial support of Science
Foundation Ireland (SFI 09-RFP-ENM2151).
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