Rayleigh-Benard Natural Convection Heat Transfer: Pattern Formation, Complexity and Predictability

Abstract

This paper is concerned with fundamental issues in predicting the onset and pattern formation in Rayleigh-Benard thermal convection in a fluid layer heated from below. Theoretically, thermal convection has long been a challenging subject in physics because of the complexity embedded in such unstably stratified flows. Practically, Rayleigh-Benard convection is central to many technologies and situations. In particular, core melt progression in a hypothetical severe accident in a nuclear reactor is governed by heat transfer (and therefore energy splitting) in molten metal/oxide layers. In previous studies, we conducted large-scale experiments and numerical simulations to quantify melt pool heat transfer models. In the present study, we developed a novel experimental approach and used an advanced diagnostic method to obtain first-of-a-kind thermometry data on transient natural convection heat transfer. The data are used to assess whether and to what extent Navier-Stokes and energy equations and their numerical methods are capable of preserving and correctly predicting complexity in thermal convection, i.e. onset and pattern formation. Uncertainty of physical measurements and numerical solutions are discussed. (authors)

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Proceedings of ICAPP ’04
Pittsburgh, PA USA, June 13-17, 2004
Paper 4241
1
Rayleigh-Bénard Natural Convection Heat Transfer:
Pattern Formation, Complexity and Predictability
T.N. Dinh, Y.Z. Yang, J.P. Tu, R.R. Nourgaliev, and T.G. Theofanous
Center for Risk Studies and Safety, University of California, Santa Barbara
6740 Cortona Drive, Goleta CA 93117
Tel: 1-805-893-4942, Fax: 1-805-893-4927, Email: nam@engr.ucsb.edu
Abstract – This paper is concerned with fundamental issues in predicting the onset and pattern
formation in Rayleigh-Benard thermal convection in a fluid layer heated from below.
Theoretically, thermal convection has long been a challenging subject in physics because of the
complexity embedded in such unstably stratified flows. Practically, Rayleigh-Benard convection is
central to many technologies and situations. In particular, core melt progression in a hypothetical
severe accident in a nuclear reactor is governed by heat transfer (and therefore energy splitting)
in molten metal/oxide layers. In previous studies, we conducted large-scale experiments and
numerical simulations to quantify melt pool heat transfer models. In the present study, we
developed a novel experimental approach and used an advanced diagnostic method to obtain
first-of-a-kind thermometry data on transient natural convection heat transfer. The data are used
to assess whether and to what extent Navier-Stokes and energy equations and their numerical
methods are capable of preserving and correctly predicting complexity in thermal convection, i.e.
onset and pattern formation. Uncertainty of physical measurements and numerical solutions are
discussed.
I. INTRODUCTION
During the course of a hypothetical severe accident in
a nuclear power plant, the reactor core may degrade, melt
down and relocate to the reactor lower plenum. Given no
measures in such situations, the reactor pressure vessel
(RPV) will be heated up, and upon the RPV failure, the
molten core will be ejected to the containment. During the
1980s and 1990s, research was initiated and conducted at
the University of California-Santa Barbara (UCSB) to
develop and validate a severe accident management (SAM)
strategy based on in-vessel retention (IVR); see
Theofanous et al, 1996 [1]. In essence, the IVR scheme
involves flooding of the reactor cavity by water from
storage tanks and uses boiling and two-phase natural
circulation to remove the core decay heat from the RPV in
severe accident scenarios. An important element in the
assessment of the IVR performance is thermal loading,
determined by natural convection in the volumetrically
heater oxidic (UO2-ZrO2-Zr) melt pool and a so-called
focusing effect due to a molten metal (Fe-Zr) layer above
the oxide pool; for more information, see Proceedings of
OECD/CSNI Workshop on this topic (Grenoble, 1994). In
both the oxide and metal pools, thermal flow is governed,
to a significant extent, by unstably stratification, at the
upper cooled surface in one case and at the bottom heated
surface in the other. Theofanous and Angelini (2000)
demonstrate that whether internally (volumetric heat
source) or externally (cooling of the outer boundary)
driven, thermal convection is quantitatively the same. They
demonstrated the case in ACOPO experiments in which
thermal energy stored in the liquid is used to drive the
natural convection when a cooled boundary is applied. In
fact, dimensionless groups, named external (RaE) and
internal (RaE) Rayleigh numbers to reflect the driving
sources for thermal convection, are inter-exchangeable.
Extensive numerical studies of natural convection heat
transfer in IVR conditions have been performed by Dinh
and Nourgaliev (1997), Nourgaliev and Dinh (1997). The
focus has been placed on high Rayleigh number regimes,
on performance of different turbulence models for
anisotropic flows that occur in such liquid pools. It has
been found from the numerical analysis that the pool’s
aspect ratio may have affect the heat transfer and energy
splitting (Dinh et al, 1997).
Broadly speaking, the physics of natural convection
has long been a subject of experimental, theoretical, and

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computational studies. The challenge lies in the complexity
of natural convection flows, with various patterns
emerging from seemingly chaotic dynamics. It has been
argued that the Navier-Stokes equations, added by the
energy conservation equation (convective heat transfer), is
a proper model for natural convection flow. However, to
date, such Navier-Stokes equations-based models fail to
reproduce flow patterns characteristic of Rayleigh-Benard
convection. Interestingly, while these models are used at
relatively high Rayleigh number, they seem to be able to
produce heat transfer rates at the pool boundary in a
reasonable agreement with experimental data. The
predictability of heat transfer means that the mixing in the
pool is predictable. This is contrasted with the failure of
the model to predict organized flow patterns even at much
lower Rayleigh numbers. In fact, in order to reproduce the
structure in thermal convection, physicists use an order
equation, similar to Carl-Hillard equation for spinodal
decomposition (Bodenschatz et al, 2000; Oh and Ahler,
2003). The dilemma thus persists on whether and to what
extent the Navier-Stokes-equations-based model provides
an adequate representation of the complex physics in
thermal convection.
The objective of this paper, and the work presented
therein, is to examine the ability of the Navier-Stokes
equations to capture the emergence of complex patterns in
thermal convection, and explore factors of physical and
numerical nature that may have contributed to the
deterioration of the model performance when compared to
experimental measurements.
Our approach in this work is integrated experimental
and computational. The motivation for a new experimental
study is the lack of transient data that may reveal how
complex flow structures emerge in thermal convection.
Furthermore, previous velocimetry data were qualitative,
rather than quantitative, making it very difficult for direct
comparison between images and results of direct numerical
simulation (DNS) by solving the Navier-Stokes and energy
equations. The focus in this work is therefore placed on the
instability and onset of pattern formation.
The paper is organized as follows. In section II, we
describe a novel experimental approach to thermal
convection, and present data that were obtained on
transient heatup, onset of thermal convection and thermal
cells. Section III is devoted to a numerical study, where
numerical methods and results are shown, and compared to
the experimental data of section II. The focus is to examine
the ability of a numerical model and method to capture the
onset of thermal convection and pattern formation.
Uncertainties, both experimental and numerical origins, are
discussed. Major findings from the analysis are
summarized as concluding remarks in section IV.
II. EXPERIMENTAL STUDY
In order to study Rayleigh-Benard natural convection,
we use an infrared camera to obtain thermal images of the
heated surface. The main idea is to characterize the onset
and formation of patterns in unstably stratified flow. The
experimental approach for natural convection study stems
from our own work developed under a NASA grant to
investigate pool boiling. Uniform heat flux on the heated
surface is achieved by passing a direct electrical current
through a 460nm-thin Titanium films vapor-deposited on
130 µm borosilicate glass substrate. In this paper, two
heaters “I” and “H” from the same manufacturing series
were used in experiments to assure that behaviors observed
in the experiments are not heater-specific. The heaters’
surface as manufactured is fresh and examined by Atomic
Force Microscopy and Electron Scanning Microscopy. For
more details about the heater manufacturing and
characterization, see Theofanous et al, 2002. Because of
the high accuracy of the vapor deposition process used in
microelectronics industry, the Titanium film is expected to
be of high uniformity in thickness. Also, because of the
small thickness of the Titanium film and the glass
substrate, temperature on and heat removal from the heater
surface reflect nearly in a characteristic time (ms)
significantly shorter than that of thermal convection
(second).
II.A. Experimental Arrangement
The experimental arrangement named BETA-NC
included a test section, a computerized power supply
system, an infrared camera and data acquisition system
(DAS). This system has been developed under a NASA
grant to study boiling heat transfer and boiling crisis, and
reported in detail in Theofanous et al (2002). The infrared
camera is a high-speed (1 KHz) high-resolution (30 µm)
thermometry system from Santa Barbara Focalplane (a
Lockheed-Martin company). The camera is specially
designed and calibrated to provide high-accurate thermal
imaging of the heater surface of the Titanium film-glass
substrate assembly over the temperature range of interest.
Thermal images obtained at high speeds and resolutions
are transferred to DAS for storage and post-test analysis.
The accuracy of temperature measurement is achieved
through a calibration using the camera and the actual
heater kept under fixed temperatures. Specifically, for the
temperature range experimented in this work (20-100oC),
the accuracy is ±0.3K.
The power supply provides a current- and voltage-
controlled power input to the Titanium film for Joule
heating. A step-change heating scheme is used through the
BETA-NC test program. Temperature profile of the heater
surface upon a sudden application of electric current over a

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(heat-conduction) phase is used as an indicator of the
uniformity of the applied heat flux. We will discuss results
of relevant tests and measurement shortly below.
The test section used is a rectangular glass vessel,
closed at the bottom by the heater element, it occupying
the entire 27x40 mm2 cross section of the vessel (Figure
1). Special techniques allow gasket-free sealing and hence
avoidance of contamination. Neither the vertical glass
walls nor the glass substrate are isolated in experiments
shown in this paper. The heat losses were estimated to be
insignificant to the onset and pattern formation phase
studied in this paper.
Figure 1. BETA-NC test section for the study of thermal
convection.
II.B. BETA-NC Experimental Program
The BETA-NC experiments were performed using
super-clean high-quality (HPLC) water as the working
fluid. In future, other fluids will be used to examine the
effect of fluid viscosity and Prandtl number on the thermal
convection patterns. Before the test starts, water was kept
at room temperature. Selected tests were conducted with
disturbed water temperature field, and their results are
discussed separately.
Fluid layers with different thickness (from 1 to 55
mm) were also tested. It was found that for water layer
thicker than 5 mm, the onset of thermal convection and
thermal patterns are found to be insensitive to the water
layer thickness. In this paper, results obtained in
experiments with water thickness larger than 5 mm are
shown. A fixed water layer thickness H=10 mm is used in
all numerical simulations.
Also, the fluid layer upper surface was kept as free in
most tests. For comparison, a wall boundary condition was
applied in selected test runs, and results of comparison
between non-slip and free upper surfaces did not reveal
any effect on the infrared thermal images of thermal
convection onset and development. The free surface
condition is then chosen as the base test case shown in this
paper.
Before proceeding further, it is worth noting that the
“classical” Rayleigh-Benard thermal convection is
characterized by the Rayleigh number defined as
αν
β
3
)( HTTg
Ra topbot −
=
where H is height of fluid layer, m; Tbot and Ttop are
temperatures of the bottom and upper surfaces,
α
is
thermal diffusivity, m2/s, and
ν
is kinematic viscosity,
m2/s. In the BETA-NC experiments, temperatures on the
bottom and upper surfaces are not fixed. Since in this case,
q = cons, and H ~ (
α
t)1/2, we have
ν
αβ
2
tqg
Raq=
It can be seen that the Rayleigh number, Raq, increases
with time.
In total, 34 test runs were conducted, with 9 runs on
“I” heater and 25 on “H” heater. The two heaters are from
the same manufacturing series and used to The results are
similar under the same given conditions. The heat flux was
in the range from 16 to 62 kW/m2, and the typical run
duration recorded by IR camera is 8 s (up to 10 s). During
an initial conduction phase, the heater surface is reported
to heat up from room temperature (20oC) to about 80oC,
followed by the onset of thermal convection that cause the
heater to cool down.
II.C. Infrared Thermometry
First, we analyze the infrared thermometry data over
an initial heatup period upon the application of electric
current to the heater. We observe a general trend of
temperature increase, which is consistent with conduction-
controlled process. Quantitatively, temperature
measurements agree reasonably well with predictions
using the Fluent code (see section III below), in which
water’s physical properties are temperature-dependent,
heat generation is localized within the thin heater and
uniform over the heated area, and heat conduction in the
heater-glass substrate assembly is included in a conjugate
fashion.

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Figure 2. IR images of onset and patterns of thermal
convection in BETA-NC experiments. Test BETA-NC I9
(Heat flux 44 kW/m2, 5 mm water layer, covered surface);
t= 4.538,4.970, 5.402, 5.834, 6.266, 6.698, 7.131, 7.563,
7.995, 8.427, 8.859, 9.291, 9.727 s.
It is worth noting that there exists uncertainty in
characterizing uniformity of input heat flux. We have
examined the uncertainty by varying input heat flux and
compare the predicted temperature rise during the
conduction phase with the measured temperature and
found typical variations of input heat flux within ±3%, but
locally the variation can go as high as 10% even in time-
averaged sense (2 seconds of conduction phase). We
performed a detail examination of IR images during a
period of 100 ms immediately following the current’s
application, and found that the heater’s response is not
simultaneous over the whole heater. Certain delay (10 ms)
may be observed in some areas of the heater indicating
Joule heating is first “nucleated” in the weakest locations
and then spread broadly. The high-speed high resolution
infrared thermometry is instrumental to the present
observation of Joule heating behavior that has not
previously been reported in the literature.
Besides experimental runs with water, we have also
performed tests with air. The “delayed nucleation”
behavior discussed above repeats itself in air as well. The
delay is the longest and most distinct when the heater and
test section was kept overnight in an undisturbed (cold)
state. We found that thermal-state non-uniformities (seen
through infrared signal fluctuations before power
application) in the heater or in test section was able to
cause the delay to disappear. Again, the observations
indicate the “phase-change”-like nature of the Joule
heating. We devote a separate study to this phenomenon. It
suffices to mention here that the onset and formation of
convection patterns will likely depend on the heating
process.
II.D. Experimental database and observations
Figure 2 depicts typical images obtained from infrared
camera. The images are shown for different time moments
after onset on natural convection, in order to convey the
evolution of pattern formation. The effect of side walls is
apparent, because of the colder side walls confine the
thermal convection field, and contribute to the heat transfer
in the near-wall region. It can also be seen that the onset is
most rapid at certain locations (near “south pole” in the
images shown) while quite uniformly over the whole
heater surface. Interestingly, the pattern structure formed
early on persisted until neighboring cells merge. The
thermal patterns obtained from IR thermometry images can
be analyzed to derive the cell’s length scale evolution. It
can be seen that the cell size and cell form may vary,
particularly in an initial period after the onset. The cells
become more “rectangular”-like as they merge.
Besides the pattern information, quantitative data were
also obtained. In fact, it is a large and first-of-its-kind
database when direct characterization of transient thermal
convection heat transfer is made. Figure 3 depicts an
example of the temperature data at a latter time moment
when thermal convection has established. It can be seen
that temperature variations are within 20K. Near the ends,
temperature drops are visible. Figure 4 depicts temperature
histories at selected locations. The graphs of Figure 4 show
a parabolic temperature rise, characteristic of heat
conduction, until the convection onset, manifested by a

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rapid cooling, followed by a level-off. The onset occurred
in about 4 seconds after the power start.
Figure 3. Heater surface temperature on different lines
across the heater (taken from IR images) 1s after the onset
of thermal convection (measurements BETA-NC H14 run
are shown in pixels of IR image).
Figure 4. Temperature transient in BETA-NC H14 run.
On the one hand, data obtained in the BETA-NC
experiments are consistent with our knowledge about
thermal convection, that under a bottom heating, the fluid
layer is heated up in a conduction mode, rendering the
thermal boundary layer unstable due to the lower density
of the heated fluid (thermal boundary layer grows to 1mm
in the conduction phase, 4 s). The physical picture is
similar, in principle, to Rayleigh instability when lighter
fluid is accelerated into heavy one. The distinction, and
difficulty, here is in the absence of any interface and sharp
change in fluid density. It is also not straightforward to
apply the instability criterion for classical Rayleigh-Benard
thermal convection, in which a conductive fluid layer is
motionless and under a temperature gradient between its
upper and bottom walls (Ra = 1200…2400 ~ RaCR =1700).
In interpreting the BETA-NC experimental runs, the term
“conduction” is conditional as the fluid expends during the
heatup phase. The fluid velocity is upward, and we define
the onset of thermal convection when the colder fluid
descends toward the heater and initiate cooling.
On the other hand, because no such thermometry data
were available previously, the BETA-NC data bring out
several new insights. They show that the side walls, and
therefore aspect ratio, may have a significant effect on
thermal stability and convection pattern. The input heat
non-uniformity is another factor that has not been
addressed. Detail analysis of the BETA-NC data, by
themselves as well as in conjunction with numerical
simulations (next section) is instrumental to the
quantification of these effects.
III. NUMERICAL SIMULATION
Thermal convection considered in this study is at the
incompressible limit. Navier-Stokes equations added by an
energy transport equation are the governing equations for
such flow.
0)( =⋅∇+
∂
∂
U
U
ρ
ρ
t
(1)
)()( UUU
U∇+−⋅∇+=⊗⋅∇+
∂
∂
µδρρ
ρ
Pg
t
(2)
T
P
Tkh
t
h
∂
∂
=∆⋅∇−⋅∇+
∂
∂
)()( U
ρ
ρ
(3)
where h is enthalpy,
ρ
is density,
µ
is viscosity, and U is
the velocity vector,
δ
: Kroenecker’s delta. For thermal
convection, Boussinesq approximation has often been used
for the case with a constant thermal expansion coefficient.
For working fluids (such as water in the BETA-NC
experiments), the thermal expansion coefficient varies
more than 4 time from 1.47x10-3 1/K to 6.2x10-3 1/K over
the temperature range in every run (20 to 80oC). As a
result, a full Navier-Stokes equations formulation is

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needed for the numerical simulation. In fact, viscosity and
other physical properties are also temperature dependent in
water, and these dependencies must be accounted for in the
solution of the Navier-Stokes equations.
There are two major issues in using the above model
to compute and analyze inherently transient processes in
thermal convection. First of all, in order to predict heat
transfer in thermal convection, one must be able to
correctly describe the mixing in unstably stratified regions.
Since every scheme for numerical discretization of the
Navier-Stokes and energy equations involves numerical
diffusion, it is important that the numerical diffusion does
not interplay with physical mixing to the level that it
causes the mixing to diminish. Often, physical models are
added in an ad hoc manner so to recover the lost mixing
source. However, development and validation of such
models have been a challenge.
In a series of previous publications (Dinh and
Nourgaliev, 1997; Dinh et al, 1997; Nourgaliev and Dinh,
1997), we have examined, numerically, natural convection
heat transfer in unstably stratified flow. We demonstrated
that anisotropy is the utmost important factor in
understanding and modeling of high Rayleigh-number
thermal convection. We went on to suggest that large eddy
simulation (LES) is the most promising and efficient
approach to numerical simulations in both Rayleigh-
Benard fluid layers and volumetrically heated liquid pools.
In fact, our numerical simulations for different unstably
stratified thermal flows using a “no model” LES showed
that the method works quite well in predicting mixing and
heat transfer in natural convection in volumetrically heated
liquid pools and for transient cooldown liquid pools
Nourgaliev et al. (1997a, 1997b). Based on these works,
we recommended the QUICK-modified third-order
bounded CCCT upwinded scheme for the simulation tasks
(Nourgaliev and Dinh, 1997).
A critical analysis of past works as well as a review of
recent advances in turbulence modeling and simulation
revealed that previously intuitive, empirical selection of
numerical schemes for use in “no model” LES model of
turbulent natural convection in internally-heated liquid
pools has a root in Monotonically Integrated LES (or
MILES) method (see Boris et al, 1992). The MILES
approach has recently received a substantial body of
supporting results as well as theoretical basis (Grinstein
and Fureby, 2002). The chief idea in MILES is that
numerical diffusion in an appropriately constructed
numerical scheme plays the role of sub-grid scale mixing,
so that no additional sub-grid scale treatment is required.
The second issue is of integrated theoretical/
computational origin; it is built upon the long-standing
question to whether the incompressible Navier-Stokes
equations are capable of correctly describing complexity
(e.g., pattern formation) in fluid dynamics. It has been
argued that hyperbolicity is key to preserve (and describe)
shocks, discontinuities and their interactions. While
numerical diffusion expectedly contributes to smearing
(parabolization), incompressible Navier-Stokes equations
are parabolic in the first place.
Given the above discussion, it is clear while the onset
of convection and formation of patterns such as those
observed in the BETA-NC experiments are such an
invaluable test bed for addressing issues.
III.A. Numerical methods and performance
Figure 5. Onset and development of thermal
convection cells in Rayleigh-Benard convection. Note the
regularity of initial “bubbles” and their coalescence to
form larger loops.
In the present study, the Fluent code is used for
solving the Navier-Stokes and energy equations (1-3). We
have examined first-order (SIMPLE), second-order
(SIMPLEC, QUICK) accurate numerical schemes on a
standard Rayleigh-Benard (Figure 5), for which heat
transfer measurements are available. We found, again, the
QUICK scheme be applicable. Parameters of spatial and
time discretization were also tested. The new results
confirmed previous experience in Nourgaliev and Dinh
and Nourgaliev et al (1997). In all calculations presented
for analysis of the BETA-NC experiments, non-uniform
grids are used so that thermal/fluid boundary layers be

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resolved accurately. The grid refinement effect was also
verified.
The heat transfer problem is solved in a conjugated
formulation that accounts for the localized heat source in
the film heater and multi-dimension conduction in glass.
4.0s 4.4s
4.8s 5.4s
6.0s 9.0s
10.3s
Figure 6. Onset and development of thermal
convection. Two-dimensional simulation of run BETA-NC
I6 (44 kW/m2, 10 mm water layer). Uniform heat flux.
Instability was initiated near the side walls.
Figure 5 depicts results of numerical simulation during
the startup phase in Rayleigh-Benard convection. Such
flow situations may also occur, for example in a transient
cooldown experiment when a hot pool is subjected
suddenly to a cooled (upper) boundary. Unfortunately,
characterization of flow development in Rayleigh-Benard
convection is not available, so we do not have a direct
evidence of whether the instability, the growth of bubble,
their coalescence were predicted correctly, and what are
possible factors that may govern their behavior.
III.B. Simulation of the BETA-NC experiments:
Uncertainty in heat flux uniformity
We chose to analyze run I6, in which the fluid layer is
10 mm and the upper boundary is non-slip (wall). The
input heat flux as determined from current and voltage
measured values is 44 kW/m2. The BETA-NC run I6
shows that the conduction phase lasted until the heater
surface reached 65-70oC at about 3 seconds. Figure 6
shows the calculated temperature field in the fluid layer.
Generation of “bubbles” in the central region is delayed for
about one second, as compared to first bubbles near the
side walls. The first “bubble” cycle lasts about 10 s, when
the second-generation bubble appeared (see image for t=
10.3 s).
Figure 7 depicts results of calculations and
temperature measurements. The result is shown for
different points across the heated area (over a length of 20
mm). The simulations shown in Figure 7 were two-
dimensional. Three-dimensional simulation with uniform
heating shows similar time moment when the instability
sets in.
02468
20
40
60
80
Tem pera tur e (C )
Point-a
Uniform heating
0.2mm no heating
0.5mm no heating
3d experiment
0 2 4 6 8
20
40
60
80
Tem pera tur e (C )
Point-b
02468
20
40
60
80
Tem pera tur e (C )
Point-c
0 2 4 6 8
20
40
60
80
Tem pera tur e (C )
Point-d
02468
20
40
60
80
Tem pera tur e (C )
Point-e
0 2 4 6 8
20
40
60
80
Tem pera tur e (C )
Point-f
02468
20
40
60
80
time (s)
Tem pera tur e (C )
Point-g
0 2 4 6 8
20
40
60
80
time (s)
Tem pera tur e (C )
Point-h
Figure 7. Numerical simulations and comparison to
experimental run (BETA-NC I6 run, 44 kW/m2, 10 mm
water layer with an upper wall). Dash-dot line (blue):
experiment; solid (black) line: calculation with uniform
heat flux; long-dashed (red) line: calculation with 0.2 mm
non-heating edge; dotted (black) line: calculation with 0.5
mm non-heating edge.
It can be seen that simulation performed with the
uniform heat flux predicted much delayed onset of thermal
convection, from 1 to 1.5 second. Correspondingly, the
predicted onset temperature is 10K higher. It is worth
noting that the heater’s temperature growth during an
initial period (3 s) is predicted correctly, indicating that the
power measurement was adequate. A careful look at the
test section suggested a possible non-uniformity of heating
in edgy regions near the side walls. Calculations were
performed for non-heated 0.2 mm and 0.5 mm slices

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adjacent to the side walls. The result shown in Figure 7
indicates that the thermal boundary layer became unstable
at much early times. The non-heated edge is colder, hence
causing a lateral flow into the heated area. Apparently, the
disturbances are sufficient and propagated into the heated
area.
Figure 8. Results of simulation with disturbed profiles of
surface heat flux. See text for explanations.
We have also performed sensitivity analysis for cases
when the surface heat flux is disturbed within a middle
region (as vs to edgy non-uniformity). Figure 8 depicts
results for two cases, with about +7.5% and –5%, in a
small area of L ~ 40 µm in length. Transient results
showed that the instability is predicted to occur exactly at
the disturbance flux location (point “d”); one “bubble” is
nucleated at this point for case of “+7.5%” (Figure 9, left
column) and by two “bubbles” right next it for the case
with “–5%” reduced flux (Figure 9, right column). From
the timing point of view, the onset is about 1 second early
(compare Figure 9 with Figure 6), bringing it closer to the
experimental observation. However, for the remaining
area, the small disturbance at a center location does not
seem to significantly ease the global thermal convection
onset. Possibly, an increase in length scale L the disturbed
(power) zone or presence of many disturbed zones may
lead to the sooner onset of global thermal convection. The
sensitivity of onset of thermal convection to heat flux
profile found in this study indicates a source for mismatch
between numerical and experimental simulations in
unstably stratified layers. Early instability means the
thermal boundary layer in the unstably stratified layer is
susceptible to small disturbances due to imperfections, and
therefore an enhanced mixing is to be expected.
3.0s
3.4s
3.8s
4.2s
4.4s
5.0s
5.6s
3.1s
3.5s
3.9s
4.4s
4.8s
5.3s
6.2s
Figure 9. Simulation of nucleation of “bubbles” upon
the thermal convection onset. Central part (40
µ
m) is with
heat flux increased 7.5% (shown in the left column) and
decreased 5% (shown in the right column). Conditions
simulated are that of BETA-NC I6 run. The flow topology
is very similar to that of Rayleigh-Taylor instability.
III.C. Simulation of the BETA-NC experiments:
Pattern formation
In this section, we turn our attention to the later stage
of instability development that is the formation of patterns
in thermal convection. Figure 2 already exhibits the kind of
thermal patterns observed in the BETA-NC experiments
(run I6 for 44 kW/m2). Most strikingly, the patterns
observed in experiment I9 are nearly identical to patterns

Proceedings of ICAPP ’04
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Paper 4241
9
seen in I1…I8 for heat fluxes from 22 to 44 kW/m2, with
water depth 5 and 10 mm, and with free surface and with
wall upper surface. The two black circles seen in first two
images in Figure 2 repeated in all runs on I heater. Patterns
observed in runs on heater “H” are also similar to each
other. Again, one location near a heater edge cooled first.
4.5s 5s 5.5s 6s
6.5s 7s 7.5s 8s
4.5s 5s 5.5s 6s
6.5s 7s 7.5s 8s
Figure 10. Temperature patterns on the heater surface
as obtained from a three-dimensional numerical
simulation of the onset and development of thermal
convection. Simulation was performed for conditions of the
BETA-NC I6 run. First frame shown is for 4.5 s, with
interval 0.5 s between frames. Calculations were
performed for transient with a free upper surface that
allows for water expansion. Real water properties were
used. Computational grids tested were 50x25x30,
50x25x55, 100x50x55, 100x50x30, and 150x75x55. No
significant differences were found for the last two grids.
The result in this figure was obtained with 150x75x55.
Time step in the numerical simulation was chosen to satisfy
both stability and accuracy check.
Figure 10 shows thermal patterns resulted from a
three-dimensional simulation of thermal convection.
Figures 11 and 12 are line results for two different time
moments, one is 1 s after the onset of thermal convection,
and another is 1.5 second after the onset.
It can be seen from Figures 11-12 that while the
temperature fluctuations increase in amplitude, the length
scale of thermal cells also increases. The calculated
thermal fluctuation behavior resembles that of
experimental measurements shown in Figure 3. It can also
be seen that a lower-temperature corridor exists all around
the heater surface near the size walls, both in numerical
results and in experimental images. This behavior is related
to the asymmetry of the fluid layer near the side wall (due
to non-slip velocity boundary condition applied on the
vertical wall), that caused an earlier onset of convection.
Computationally, rectangular shapes are predicted to have
onset to occur simultaneously around the heater, while in
experiments, the onset appeared in certain locations first,
reflecting the effect of heater conditions on the onset.
Figure 11. Temperature profiles across on the heater
surface as obtained from a three-dimensional numerical
simulation of the onset and development of thermal
convection. Conditions for BETA-NC I6 run, t=5 s.
In general, the heat-transfer pattern predicted by the
CFD code is similar to that observed in the BETA-NC
experiment (Figure 2). Initially, fairly circular cells, with
closely equal size emerged. These cells then coalesced to
form larger cells. There is an apparent randomness in the
coalescence process, so that at a later time, we have cells
with large and small sizes. Averaged cell length scales are
“measured” using a procedure similar to Fourier time
series analysis, but applied to spatial dimension instead. It
can be seen that the length scale developed during an
initial phase, then slowed down and remains unchanged.
This prediction is close to observations in the BETA-NC
experiments (also see Figure 2).

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Figure 12. See caption in Figure 11; t= 5.5 s (see
also Figure 3 for reference).
2
3
4
5
6
456789
Time (s)
Averaged length scale (mm)
Figure 13. Thermal cell’s (surface-averaged) length
scale evolution derived from results of 3D numerical
simulation (Figure 10).
While a systematic presentation of data from all 34
BETA-NC runs and their analysis are documented in a
separate publication elsewhere (Dinh et al, 2004), it is
worth noting here the IR-recorded thermal patterns on the
same heater are highly reproducible even for different test
conditions (heat flux, free or non-slip upper surface of the
fluid layer). This reproducibility is interesting and puzzling
in light of sensitivity of the onset to heat-flux non-
uniformity discussed in section III.a above. It is possible
that non-uniformity in the input heat flux pre-determined
locations of “nucleation” where instabilities in thermal
boundary layer occur. The convection that followed the
onset is not a chaotic process, but governed by growth and
coalescence of “bubbles”.
IV. DISCUSSION AND CONCLUSIONS
In Introduction (section I), we asked whether the
Navier-Stokes and energy equations adequately represent
the flow physics under Rayleigh-Benard thermal
convection. Previously, methods of Computational Fluid
Dynamics (CFD) have been used to compute natural
convection and their performance was validated on
surface-averaged, quasi-steady-state heat transfer data. The
question is whether the CFD methods only served as an
effective (averaged), global approximation or they are
indeed able to capture the underpinning complexity. In
thermal convection, complexity is manifested by the
formation of patterns of thermal cells and by the
convection onset being sensitive to a number of factors.
Until now, the question about predictability of these
complexities remains open because of the absence of data
needed for the CFD model qualification. Such a basic
question requires a fundamental approach as offered in the
present work by comparing results of direct numerical
simulations with data from “direct” measurements of
heater surface temperatures. The data obtained provide a
test bed for different numerical methods and computer
codes; such a comparison will help to reveal the suitability
of different numerical approaches and schemes before they
are used in large-scale computations at higher Rayleigh-
number flow of practical and theoretical interest.
Because heat transfer in liquid pool under natural
convection is determined, to a large extent, by flow in
unstable stratification region, it is crucial that formation of
thermal “blobs” (or “bubbles”) is correctly predicted. Our
numerical prediction of the onset of thermal convection
shows that the surface heat flux’s non-uniformity (thermal
boundary condition) may be the source of uncertainty.
Physically, variations in the input heat flux render a
relative motion in fluid layer in the direction normal to the
heated boundary, and this motion initiate the layer’s
destabilization. The effect was detected even when the
disturbance’s length scale and magnitude (disturbed heat
flux) are fairly small.
The numerical result led us to suggest that the
difference between the measured onset moment and the
onset moment computed by solving the Navier-Stokes and
energy equations with ideally uniform input heat flux can
be explained by the effect of boundary condition’s non-
uniformity, rather than by the model’s limitation. The
concept of thermal noise and phase transition may be
theoretically attractive, but physically extraneous. In other
words, we suggest that the Navier-Stokes equations are
capable of capturing the onset of thermal convection
theoretically, but uncertainty remains in the practical
ability to characterize thermal boundary conditions in
experiments as well as in technological processes (e.g.,

Proceedings of ICAPP ’04
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Paper 4241
11
core melt pools). In fact, different heat transfer results
were measured in natural convection experiments
performed with different walls (materials, conductivity);
e.g. Nikolaenko and Ahler (2003). Additional calculations
as well as controlled experiments (with pre-defined non-
uniformities) are needed to quantify the effect of non-
uniformity on surface-averaged heat transfer coefficients.
The 3D numerical simulations performed in this work
provided the basis to conclude that the mathematical model
based on the Navier-Stokes and energy equations is
capable of capturing and predicting the thermal patterns
formed in Rayleigh-Benard convection. The use of a high-
order accurate numerical scheme in this work may have
been a significant factor, helping limit the numerical
diffusion. Excessive numerical diffusion is known to delay
or even suppress physical instabilities, which are essential
for the physics in unstably stratified flow such as in the
thermal Rayleigh-Benard convection.
ACKNOWLEDGMENTS
The work described in this paper was supported by the
U.S. Department of Energy’s International Nuclear Energy
Research Initiative Program (I-NERI) under contract DE-
FG06-02RL14337. Basic guidance for this work derived
from the BETA program carried out for NASA under
grants NAG3-2119 and NAG3-2761. The authors thank
Mr T. Salmassi for his help with BETA test section design.
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