Coupling of turbulent natural convection with radiation heat transfer in an air-filled square...

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


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).


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)


√( ) [( )

( )

]

(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)


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)


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),


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.


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.


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)


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).


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.


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.


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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http://thetis.enscbp.fr/documentation


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


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

Coupling of turbulent natural convection with radiation heat transfer in an air-filled square...

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