A. Ibrahim, D. Saury, D. Lemonnier, “Coupling of turbulent natural convection with radiation heat transfer in a square differentially-heated cavity for Ra = 1.5×109”, Computers and Fluids, Volume 88, December 2013, pp. 115–125, 2013. doi:10.1016/j.compfluid.2013.09.006
Coupling of turbulent natural convection with radiation heat
transfer in an air-filled square differentially-heated cavity at
Ra = 1.5×109
Adel Ibrahim, Didier Saury, Denis Lemonnier1
Institut Pprime, UPR CNRS 3346
CNRS, ISAE-ENSMA, University of Poitiers
1 avenue Clément Ader, BP 40109
F-86961 Futuroscope Chasseneuil CEDEX, France
Abstract
This paper deals with the influence of radiation on natural convection airflows in confined
areas. The investigated configuration is a square differentially-heated cavity at Ra = 1.5×109.
Two dimensional LES computations have been carried out using a specific subgrid model for
the thermal diffusivity and taking into account gas and wall radiation. Radiation within the
fluid (due to the presence of water vapor in humid air) is considered using the SLW model
developed by Denison and Webb and the Discrete Ordinate Method (S8 quadrature).
Comparisons are carried out for four cases (without radiation, wall radiation only, gas
radiation only and combined gas and wall radiation) and compared in terms of velocity and
temperature fields as well as turbulent quantities (kinetic energy, turbulence intensity). Global
heat transfer is also investigated and compared for those different cases.
Keywords: turbulent natural convection, wall radiation, gas radiation, coupled heat transfer.
1 Correspondig author: [email protected] tel:+33 5 49 49 81 16 fax: +33 5 49 49 81 01
A. Ibrahim, D. Saury, D. Lemonnier, “Coupling of turbulent natural convection with radiation heat transfer in a square differentially-heated cavity for Ra = 1.5×109”, Computers and Fluids, Volume 88, December 2013, pp. 115–125, 2013. doi:10.1016/j.compfluid.2013.09.006
1. Introduction
Natural convection in differentially-heated cavities at Rayleigh numbers up to about 109 has
been widely investigated experimentally [1-4]. These studies provide a large and useful
database for validation of numerical simulations. During the 90’s, a paramount difference was
observed between experimental results [1] and 2D Direct Numerical Simulations (DNS) [5,
6]. It was interpreted as the potential existence of 3D effects or attributed to boundary
conditions along the passive walls, which might be different (adiabatic for calculations, more
likely conductive in experiments). In this context, many numerical studies have been carried
out using DNS [7-9] or LES [10] for 3D and turbulent flows. They show that only considering
the coupling of conduction and convection in passive walls is not enough to obtain accurate
results. Then the radiative effects and their coupling with other phenomena (conduction and
convection) appear to be essential to obtain reliable simulations (in particular regarding the
thermal stratification in the core of the cavity). In these numerical studies, the fluid within the
cavity is quite always considered as a transparent medium: emitting-absorbing effects of gas
radiation are neglected. Nevertheless at ambient temperature, the influence of H20 within air
in usual conditions of hygrometry has to be quantified. This is the goal of this paper.
2. Model and boundary conditions
A 1m2 air-filled square cavity is investigated. Its two vertical walls are kept isothermal at two
different temperatures, Tc (=300.5 K) and Tf (=285.5K), whereas the upper and lower
horizontal walls are adiabatic (figure 1). All these surfaces are considered as grey and
diffusively reflecting. They have constant emissivities, with values chosen according to the
experimental conditions provided by Salat et al. [10, 11] for Ra = 1.5×109 (ε1 = 0.1 on active
walls and ε2 = 0.2 on other walls).
A. Ibrahim, D. Saury, D. Lemonnier, “Coupling of turbulent natural convection with radiation heat transfer in a square differentially-heated cavity for Ra = 1.5×109”, Computers and Fluids, Volume 88, December 2013, pp. 115–125, 2013. doi:10.1016/j.compfluid.2013.09.006
The dynamic and thermal turbulent fields are computed using a 2D LES approach for a
Newtonian, incompressible fluid, assuming the Boussinesq approximation. Thus, the LES
filtered governing equations can be written as follow:
(1.a)
[
]
[
] (1.b)
[
]
[
] (1.c)
[
]
[
] (1.d)
where and are respectively the turbulent subgrid viscosity and the thermal subgrid
diffusivity. The achieve a better accuracy in natural convection, is calculated using the
model proposed by Sergent et al. [12, 13] instead of resorting to the Reynolds analogy (and its
associated turbulent Prandtl number) :
| |
| |
(2)
where is the mesh size, | |is a scalar representing the interaction between the thermal
gradient and the deformation tensor of the resolved scales , and is the heat flux at the cut-
off. They are respectively defined as:
| | √
( )( ) (3)
(
)
(4)
A. Ibrahim, D. Saury, D. Lemonnier, “Coupling of turbulent natural convection with radiation heat transfer in a square differentially-heated cavity for Ra = 1.5×109”, Computers and Fluids, Volume 88, December 2013, pp. 115–125, 2013. doi:10.1016/j.compfluid.2013.09.006
√( ) [( )
( )
]
(5)
Here, , and denote two time filtered quantities using a filter of size corresponding to a
cut-off length equals to the mesh size [12, 13]. In practice:
[
] (6)
The radiative source term appearing in the energy equation accounts
for gas radiation (absorption and emission) induced by the presence of water vapor in air.
Using the SLW model of Denison and Webb [14, 15] to take into account the spectral
properties of the air-H2O mixture, and within the frame of the discrete ordinate approximation
[16], this term can be written as:
∑ [∑
]
(7)
where Ng is the number of gray gases involved in the SLW model and aj the weighting factor
associated to the jth
gaz. Similarly, M is the number of discrete directions being used in the
discrete ordinate approximation, and wm the weight attributed to the mth
direction. This
expression of Sr neglects subgrid correlations between the absorption coefficient ), the
weights (aj) and the temperature fluctuations. It amounts to assuming that:
and where (8)
A. Ibrahim, D. Saury, D. Lemonnier, “Coupling of turbulent natural convection with radiation heat transfer in a square differentially-heated cavity for Ra = 1.5×109”, Computers and Fluids, Volume 88, December 2013, pp. 115–125, 2013. doi:10.1016/j.compfluid.2013.09.006
The first hypothesis is commonly used, since small structures (namely, at the subgrid scale)
are optically thin and, consequently, the fluctuations of κ and L are statistically uncoupled.
The second one is valid when small temperature fluctuations around the mean values are
encountered: such an assumption can reasonably be accepted in the context of this study
(Boussinesq approximation).
The thermal boundary conditions for the studied configuration can be detailed as:
- Vertical active walls:
(9)
(10)
- Horizontal walls:
|
inc | (11)
|
inc | (12)
where:
inc | ∑ ∑
(13)
inc | ∑ ∑
(14)
A. Ibrahim, D. Saury, D. Lemonnier, “Coupling of turbulent natural convection with radiation heat transfer in a square differentially-heated cavity for Ra = 1.5×109”, Computers and Fluids, Volume 88, December 2013, pp. 115–125, 2013. doi:10.1016/j.compfluid.2013.09.006
All the CFD computations have been carried out using the AQUILON software (now, named
THETIS [17]), which has been modified in order to take into account our specific needs
(subgrid thermal diffusivity, radiative calculations using discrete ordinates method and the
SLW model). A centered and conservative second order scheme in space is applied to the
energy equation, whereas a QUICK scheme is used for the momentum equation. This last
choice is guided by the fact that the scheme is dissipative enough to avoid a subgrid viscous
model, letting numerical diffusion play this role [13, 18]. Finally, the temporal discretization
is based on a second order Gear scheme. The pressure velocity coupling is treated by using an
augmented Lagrangian projection method.
The grid size for computations is 66(y)×130(x). Points are evenly distributed in the y-direction
and a hyperbolic tangent law is used in the x-direction, with a refinement coefficient equal to
6.81. The Discrete Ordinate Method [16] is used with a S8 quadrature (80 directions), using
11 grey gas in the SLW model. The dimensionless time step, relative to the convective time
scale H²/(αRa1/2
) ,is about 0.0084. The mean quantities are computed once the quasi-steady-
state regime is achieved by integrating over more than 500 dimensionless time units.
Validation of the CFD code was carried out against data provided by Salat et al. [10].
3. Results and discussion
In this numerical study, four specific cases have been investigated:
1 no radiation (NR),
2 only the wall radiation (WR) is considered (no gas radiation and ε1 = 0.1,
ε2 = 0.2),
3 only the gas radiation (GR) is considered (no wall radiation, i.e. ε1 = 0 and ε2 = 0),
A. Ibrahim, D. Saury, D. Lemonnier, “Coupling of turbulent natural convection with radiation heat transfer in a square differentially-heated cavity for Ra = 1.5×109”, Computers and Fluids, Volume 88, December 2013, pp. 115–125, 2013. doi:10.1016/j.compfluid.2013.09.006
4 gas and wall radiation (GR+WR) are both considered (air is assumed to have a
relative humidity of 50 %, which, at room pressure and temperature, corresponds
to a molar fraction of H2O of 1,15%; ε1 = 0.1, ε2 = 0.2).
These four cases are compared thereafter in terms of velocity and temperature fields as well as
global heat transfer for Ra = 1.5×109. In the present study, this Ra-value is achieved by setting
H = 1 m, ΔT = 15 °C, T0 = 293 K.
3.1 Airflow structure
The mean velocity vectors are plotted in figure 2 for each case under consideration, with a
focus set to the upper and the lower part of the cavity (Y ∈ [0;0.25] and Y ∈ [0.75;1.00]). In
case 1 (no radiation), the velocity field shows a typical pattern, with hydrodynamic
instabilities in the corners at the top hot side and the bottom cold side of the cavity (eddy
structure in figure 2). This instability remains when gas radiation is only taken into account
(case 3), but with a stretching in the x-direction. Nevertheless, the flow structure changes as
soon as wall radiation is introduced (cases 2 and 4): it then becomes similar to what was
observed by Salat et al. [19] with conductive (rather than adiabatic) passive walls (see
figure 3). Secondary recirculation airflow occurs, which is more intense when wall and gas
radiation are considered together (case 4). The corner instabilities disappear, probably due to
the redistribution of energy by radiation towards the other walls, entailing a local temperature
modification. It is to notice that, globally, gas radiation has a weaker effect on the flow
structure than wall radiation.
A. Ibrahim, D. Saury, D. Lemonnier, “Coupling of turbulent natural convection with radiation heat transfer in a square differentially-heated cavity for Ra = 1.5×109”, Computers and Fluids, Volume 88, December 2013, pp. 115–125, 2013. doi:10.1016/j.compfluid.2013.09.006
3.2 Mean velocity profiles
Mean profiles of vertical and horizontal velocity are plotted in figures 4 and 5 for all the NR,
WR, GR and GR+WR cases described before, and still at a Rayleigh number of 1.5×109.
Once again, the corner hydrodynamic instability shows up in the upper hot corner (y/H = 0.8,
0.9) and the lower cold one (y/H = 0.1, 0.2), except when wall radiation is considered.
Whatever the case, figures 4 and 5 clearly indicate that gas or wall radiation intensifies the
magnitude of the mean velocity, with an overshoot on the horizontal and vertical components.
This phenomenon is all the more important when gas radiation and wall radiation are
simultaneously considered. This acceleration of the fluid motion, combined to the changes in
the local energy balance (due to radiation), could explain the modification of the airflow
pattern when radiation is taken into account: eddy stretching (case 3) or eddy disappearance
(case 2 and 4). In addition, due to the fluid acceleration (induced by radiation, and especially
wall radiation), the top horizontal wall jet impinges vertical wall in the corner at higher speed.
Air then flows downwards and towards the core of the cavity giving birth to the secondary
airflow recirculation. The same scenario applies to the bottom horizontal wall jet.
3.3 Mean temperature profiles
Iso-values of the mean temperature are plotted in figure 6 for each investigated case. Due to
the airflow modifications, the temperature field is changed when radiation is introduced,
especially in the upper hot corner and the lower cold corner. The flow structures previously
encountered (eddy, secondary airflow) can be well identified on the isotherm field. In
particular, the air temperature in the upper or lower part of the cavity is all the more hot that
radiative effects are weak.
A. Ibrahim, D. Saury, D. Lemonnier, “Coupling of turbulent natural convection with radiation heat transfer in a square differentially-heated cavity for Ra = 1.5×109”, Computers and Fluids, Volume 88, December 2013, pp. 115–125, 2013. doi:10.1016/j.compfluid.2013.09.006
In addition, gas radiation causes a decrease in the stratification parameter as shown in table 1.
This parameter is evaluated for 0.4 < y/H < 0.6 (around the central part of the cavity). Table 1
shows that wall radiation has a weaker effect on the thermal stratification than gas radiation,
but both induce the same trend (decrease of ST). More specifically, wall radiation changes the
temperature distribution along the passive walls. Conversely, the influence of gas radiation is
more observable in the core of the cavity where the fluid is at rest. In this area, heat transfer is
dominated by diffusion and by radiation (no air motion). Gas radiation between the upper hot
fluid layers and the lower cold ones tends to decrease the vertical temperature gradient in the
central part of the cavity and thus the stratification parameter ST. This may be observed when
plotting the vertical distribution of temperature at mid-width for all the cases and
Ra = 1.5×109 (see figure 7). The stratification parameter calculated in case 4 compares well
with measurements provided by Salat et al. [19] (ST = 0.72), at least when radiative coupling
with the “front” and “back” walls of their 3D mock-up is low enough (plates covered by a low
emissivity sheet, ε = 0.1). Otherwise, if these bounding surfaces in the third dimension are
almost black, the measured value is much lower (ST = 0.38), due a strong additional wall
radiation effect that certainly breaks up the assumption of a purely bidimensional solution.
3.4 Turbulent quantities
3.4.1 Mean turbulent kinetic energy
The bidimensional mean turbulent kinetic energy, plotted in figure 8, is evaluated as:
[
√ ]
(15)
A. Ibrahim, D. Saury, D. Lemonnier, “Coupling of turbulent natural convection with radiation heat transfer in a square differentially-heated cavity for Ra = 1.5×109”, Computers and Fluids, Volume 88, December 2013, pp. 115–125, 2013. doi:10.1016/j.compfluid.2013.09.006
In all the investigated cases, this quantity remains low in the core of the cavity, which is
almost at rest. In addition, gas radiation, when considered alone (case 3, GR), tends to further
“laminarize” the airflow by lowering the turbulent kinetic energy with respect to the no-
radiation case (case 1, NR).
Wall radiation has the opposite effect. Comparison of cases 1 (NR) and 2 (WR) clearly
indicates an increase in the turbulence kinetic energy near the walls, especially in the top cold
corner or the bottom hot corner. In these places, the horizontal wall jet impinges the vertical
wall with an enhanced intensity, entailing the appearance of a secondary airflow recirculation
associated to a higher level of turbulence.
It worth noting, however, that gas radiation loses its stabilizing effect when combined to wall
radiation (case 4: WR+GR). It even slightly increases the turbulence level compared to case 2
(WR), by further accelerating the horizontal flow layers strengthened by wall radiation.
3.4.2 Temperature turbulent intensity
The thermal turbulent intensity is defined as:
√
(16)
As shown in figure 9, temperature fluctuations are weak (It lower than 3% whatever the case)
and remains below 0.5% in the core of the cavity. As for turbulent kinetic energy, wall
radiation tends to increase the thermal turbulent intensity whereas gas radiation seems to play
a stabilizing role. This is in agreement with the previous observation (see Part 3.4.1).
A. Ibrahim, D. Saury, D. Lemonnier, “Coupling of turbulent natural convection with radiation heat transfer in a square differentially-heated cavity for Ra = 1.5×109”, Computers and Fluids, Volume 88, December 2013, pp. 115–125, 2013. doi:10.1016/j.compfluid.2013.09.006
3.5 Heat transfer
Heat transfer is investigated by using global convective and radiative Nusselt numbers
respectively denoted by Nuc and Nur. These dimensionless numbers are evaluated on the
vertical (active) and horizontal (passive) walls. They are classically defined as the ratio of
conductive or net radiative heat flux relative to the reference kΔT/H (k being the gas
conductivity, ΔT the temperature difference between the cold and hot walls, and H the cavity
width).
Along horizontal walls, which are adiabatic, the net surface radiation must exactly balance
heat conduction from the fluid. This means that Nuc =-Nur for passive walls and, in case 1 (no
radiation) or case 3 (gas radiation only), this amounts to Nuc = Nur = 0. In the other two cases,
the corresponding values of Nuc are listed in table 2. It is to notice that gas radiation tends to
slightly reduce the convective heat transfer to horizontal walls.
Regarding vertical active walls, radiation (from gas and/or wall) also modifies the amount of
heat brought by convection. Table 3 provides the values of the convective and radiative
Nusselt numbers for each case under consideration. For sake of comparison, a Nur value is
provided in the “no radiation” case. It corresponds to the radiative exchange between the two
active walls, owing to their own temperature and emissivity (ε2 = 0.1). This exchange does
not interact with the fluid, since there no sources within it (transparent medium) nor coupling
at the adiabatic boundaries (ε1 = 0). It can be evaluated from a simple radiosity calculation.
A. Ibrahim, D. Saury, D. Lemonnier, “Coupling of turbulent natural convection with radiation heat transfer in a square differentially-heated cavity for Ra = 1.5×109”, Computers and Fluids, Volume 88, December 2013, pp. 115–125, 2013. doi:10.1016/j.compfluid.2013.09.006
As shown in table 3, radiation from either wall or gas and wall radiation decrease of the
convective heat transfer (about 10% when combined) whereas radiative heat exchanges are
only slightly affected, and solely by wall radiation. It is worth pointing out that our prediction
in case 4 (the most likely to depict the real situation) are close to the experimental data
provided by Salat et al. [19], namely Nuc = 55 at the hot wall and 54 at the cold one.
The temporal evolution of the convective Nusselt fluctuations on the hot wall is plotted in
figure 10. Compared to the no radiation case (case 1, NR), fluctuations remain very weak
when considering gas radiation only (case 3, GR). This comes from the fact that absorption
and emission within the gas tends to homogenize the temperature field. Moreover, in this
case, the flow is quasi mono-periodic. On the contrary, wall radiation intensifies the
fluctuations of convective flux (due to an intensification of thermal turbulence in boundary
layers (case 2, WR). This trend is still verified (actually, slightly reduced) when gas radiation
is added (case 4, WR+GR).
A spectral analysis reveals that, for case 3 (gas radiation only), a mono-periodic mode
prevails at a dimensionless frequency of f*=0,675 (i.e. 0.56 Hz). This value was f*=0,552 (i.e.
0.46 Hz) in case 1 (no radiation). These “high” frequencies (about one hertz) are not observed
for other two cases (case 2 and 4) and could be caused by boundary layer instabilities
(Tollmien-Schlichting waves, for instance). In fact, due to the weakness of the heat flux
fluctuations (see figure 10), these frequencies are hidden by other more intense phenomena in
case 2 (WR) and 4 (WR+GR). In particular, an emerging low frequency (f* ≈ 0.06) is
observed, which has the same order of magnitude than the one observed by Le Quéré [20],
corresponding to the first non-stationary mode appearing at Ra=1.81×108 in a square cavity
(f*≈0.045). This mode is due to the boundary layer flow impingement on horizontal walls.
A. Ibrahim, D. Saury, D. Lemonnier, “Coupling of turbulent natural convection with radiation heat transfer in a square differentially-heated cavity for Ra = 1.5×109”, Computers and Fluids, Volume 88, December 2013, pp. 115–125, 2013. doi:10.1016/j.compfluid.2013.09.006
4. Conclusion
This paper provides trends caused by radiative effects on a 2D natural convection flow of
humid air in a 1m×1m differentially heated cavity. It was shown that wall radiation modifies
the airflow structure in particular at the top hot corner and the bottom cold corner. By
accelerating the flow motion along the horizontal wall, it increases the turbulence intensity,
especially in areas where the horizontal layers impinge the side walls. Comparatively, gas
radiation has little influence on the flow structure, at least when considered alone (without
wall radiation) and in the present configuration. It nevertheless tends to stabilize the flow and
homogenizes the temperature field in the core of the cavity. As a consequence, the central
thermal stratification is notably reduced; wall radiation has the same, but weaker effect. On
the other hand, when combined to wall radiation, gas absorption/emission (due to water
vapor) increases the turbulence level by further accelerating the flow motion along the
horizontal walls.
Reference
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Int. Heat Transfer Conf. (San Francisco, CA)
vol. 4 p 1465.
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[3] Betts P L and Bokhtari I H 2000 Int. J. Heat Fluid Flow 21 675.
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[5] Paolucci S 1990 J. Fluid Mech. 215 229.
[6] Xin S and Le Quéré P 1995 J. Fluid Mech. 304 87.
[7] Xin S and Le Quéré P 2006 Num. Heat Transfer Part. A 50(5) 437.
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A. Ibrahim, D. Saury, D. Lemonnier, “Coupling of turbulent natural convection with radiation heat transfer in a square differentially-heated cavity for Ra = 1.5×109”, Computers and Fluids, Volume 88, December 2013, pp. 115–125, 2013. doi:10.1016/j.compfluid.2013.09.006
[9] Xin S, Salat J, Joubert P, Sergent A, Le Quéré P and Penot F 2006 3D numerical
simulations of turbulent natural convection in an air-filled differentially heated cavity
13th Int. Heat Transfer Conf. (Sydney, August 2006).
[10] Salat J, Xin S, Joubert P, Sergent A, Penot F and Le Quéré P 2004 Int. J. Heat Fluid
Flow 25(5) 824.
[11] Salat J and Penot F 2003 VIe Colloque Interuniversitaire Franco-Québécois (Thermique
des systèmes) (Québec, May 2003) p 75.
[12] Sergent A, Joubert P, Le Quéré P and Tenaud C 2000 C. R. Acad. Sci., 328(série II b)
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[17] THETIS website: http://thetis.enscbp.fr/documentation.
[18] Baraghi D G and Davidson L 2007 Phys. Fluids 19 1.
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A. Ibrahim, D. Saury, D. Lemonnier, “Coupling of turbulent natural convection with radiation heat transfer in a square differentially-heated cavity for Ra = 1.5×109”, Computers and Fluids, Volume 88, December 2013, pp. 115–125, 2013. doi:10.1016/j.compfluid.2013.09.006
Nomenclature
aj Weighting coefficient for the jth
grey gas in the SLW model -
Et Bidimensional turbulent energy m²/s²
f* Dimensionless frequency -
g Gravity m/s²
H Height of the cavity m
It
Bidimensional thermal turbulence intensity (dimensionless RMS
temperature fluctuation)
%
L Radiative intensity W/sr/m²/µm
Nuc Mean convective Nusselt number -
Nur Mean radiative Nusselt number -
Pr Prandtl number -
qinc Incident radiative flux W/m²
Ra Rayleigh number -
Sr Source term in the energy equation W/m3
ST Thermal stratification K/m
T0 Median temperature (=
) K
Tc Hot wall temperature K
Tf Cold wall temperature K
Tp Wall temperature K
u, v Horizontal and vertical components of the velocity vector m/s
U, V
Dimensionless horizontal and vertical components of the velocity
vector
-
x, y Dimensional coordinates m
A. Ibrahim, D. Saury, D. Lemonnier, “Coupling of turbulent natural convection with radiation heat transfer in a square differentially-heated cavity for Ra = 1.5×109”, Computers and Fluids, Volume 88, December 2013, pp. 115–125, 2013. doi:10.1016/j.compfluid.2013.09.006
X, Y Dimensionless coordinates (X=x/H and Y=y/H) -
Greek symbols
α Thermal diffusivity m²/s
β Coefficient of thermal expansion 1/K
Δ Mesh size m
ΔT Temperature difference (= Tc-Tf) K
ε1 Active wall emissivity -
ε2 Passive wall emissivity -
κ Absorption coefficient 1/m
η Direction cosine -
ν Kinematic viscosity m²/s
Upper scripts and lower scripts
X’ Fluctuating quantity
Xsm Subgrid quantity
Spatially filtered quantity
Double spatially filtered quantity
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