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Natural convection in an air-filled cavity: Experimental results at large Rayleigh numbers
International Communications in Heat and Mass Transfer 38 (2011) 679–687
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International Communications in Heat and Mass Transfer j o u r n a l h o m e p a g e : w w w. e l s ev i e r. c o m / l o c a t e / i c h m t
Natural convection in an air-filled cavity: Experimental results at large Rayleigh numbers☆ Didier Saury ⁎, Nicolas Rouger, Francis Djanna, François Penot Institut Pprime, UPR CNRS 3346, CNRS-ENSMA-Université de Poitiers, Département Fluides Thermique Combustion, ENSMA, Téléport 2, 1 avenue Clément Ader - BP 40109, F-86961 Futuroscope CEDEX, France
a r t i c l e
i n f o
Available online 22 March 2011 Keywords: Turbulent natural convection Buoyancy-driven flows Experimental characterization
a b s t r a c t A large-scale experimental setup is built and instrumented. It consists in a 4 m-high cavity with a horizontal cross-section equal to 0.86 × 1.00 m². Two opposite vertical walls are heated and cooled down; other walls (lateral walls, ceiling and floor) are made of insulating medium covered with a thin and low-emissivity film designed to minimize radiative effects into the cavity. The temperatures of active walls are imposed, constant and equally distributed around the ambient temperature in order to reduce heat losses. The temperature difference between the hot and cold walls is chosen to respect the Boussinesq approximation. Under these assumptions, Rayleigh number values up to 1.2 × 10 11 (ΔT = 20 °C) can be obtained. The centre-symmetry is verified on the thermal stratification. Influence of the temperature difference and of wall emissivities on the stratification parameter (dimensionless vertical temperature gradient) is discussed. Velocity measurements allow the velocity field to be obtained and provide information on flows encountered in the cavity. Temperature measurements are also carried out in the whole cavity. In the paper, a complete experimental characterization is provided: airflow inside the cavity is analyzed and the Nusselt number along the hot and the cold wall is presented. © 2011 Elsevier Ltd. All rights reserved.
1. Introduction For many decades, studies on phenomena occurring in differentially-heated cavities have been of great interest [1–37]. This is particularly the case today, since public figures have come to realize that economic development must be accompanied by a reduction of greenhouse gas and by policies aimed at reducing energy consumption. In many areas the stakes are high and it is of paramount interest to understand the phenomena occurring in order to control them. Natural convection occurs “naturally” almost everywhere as soon as a density gradient exists and it entails heat exchanges. We need to understand and control the attendant phenomena and to thereby promote or reduce the associated heat transfers. The differentially-heated cavity is an interesting academic configuration allowing for a worthwhile study of the natural convection phenomenon and its chaotic behavior [34–36]. Moreover, applications to concrete cases (building, aeronautics…) often require precise understanding of phe-
☆ Communicated by J. Taine and A. Soufiani. ⁎ Corresponding author. E-mail address: [email protected] (D. Saury). 0735-1933/$ – see front matter © 2011 Elsevier Ltd. All rights reserved. doi:10.1016/j.icheatmasstransfer.2011.03.019
nomena occurring at high Rayleigh numbers (N1011). Numerous numerical or experimental studies have already been carried out at low Rayleigh numbers [1–5,8–33] where airflow remains laminar or weakly turbulent (Ra H b 10 9 ). Recently, a few numerical investigations at higher Rayleigh numbers have appeared, but only experimental validation will confirm their credibility [6,7,37].
2. Experimental setup description This study was carried out in a differentially-heated cavity shown in Fig. 1. The cavity is 3.84 m high, 1 m wide and 0.86 m deep. Two opposite vertical walls are differentially-heated (one is heated while the other is cooled). Their temperature is imposed so that the core of the cavity is at ambient temperature. The two walls, made of aluminum plate (k = 134 W m− 1 K− 1 and ε = 0.15), are kept at a constant temperature by a glycol-water circuit flowing in 2 separated cryothermostats imposing the temperature. The other walls (back, front walls, floor and ceiling) are made of 8 cm thick polyurethane foam panels (k = 0.035 W m− 1 K− 1) placed between the heated and cooled walls. The floor is also made with extruded polyurethane foam panels. Finally, 3 cm-thick panels are added around the cavity on the outer side. On the inner side of the cavity, a thin (40 μm) low-emissivity film (ε = 0.1 with the film and ε = 0.6 without) is added in order to reduce wall radiation.
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Nomenclature AH AV Cf D g H L k Q NuH 〈Nu〉 RaH Ray S T Tm Tamb U V VREF x y z X⁎ Y⁎ Z⁎
Horizontal aspect ratio: AH = D/L [−] Vertical aspect ratio: AV = H/L [−] Friction coefficient [−] Depth of the cavity: D = 0.86 m [m] Gravitational acceleration [m.s− 2] Height of the cavity: H = 3.84 m [m] Width of the cavity: L = 1 m [m] Thermal conductivity [W.m− 1.K− 1] Flow rate per unit depth [g.s− 1.m− 1] Nusselt number [−] Overall Nusselt number: Nu = H1 ∫0H NuH dY =∫01 NuH dY α [−] 3 gβΔTH [−] Rayleigh number: RaH = αv 3 gβΔTy [−] Rayleigh number: Ray = αv Stratification parameter [−] Air temperature [°C] Mean temperature, Tm = ½ (Tc + Th) [°C] Ambient temperature (≈ 20 °C) [°C] Horizontal component of the velocity (in the x-direction) [m.s− 1] Vertical component of the velocity (in the y-direction) [m.s− 1] pffiffiffiffiffiffiffiffiffi −1 Reference velocity: VREF = α ] H RaH [m.s Horizontal Cartesian coordinate [m] Vertical Cartesian coordinate [m] Transverse Cartesian coordinate [m] Dimensionless horizontal Cartesian coordinate X⁎ = x/H [−] Dimensionless vertical Cartesian coordinate Y⁎ = y/H [−] Dimensionless transverse Cartesian coordinate Z⁎ = z/H [−]
Greek characters α Thermal diffusivity [m2.s− 1] β Coefficient of thermal expansion [K− 1] δ Thickness of the flow [m] ΔT Temperature difference, ΔT = Th − Tc [°C] ε Wall emissivity [−] μ Dynamic viscosity [Pa.s] ν Kinematic viscosity [m2.s− 1] m θ Dimensionless temperature : θ = TT ΔT [−] −3 ρ0 Air density at T = Tm [kg.m ] τW Friction stress [Pa]
Subscripts and superscripts c Relative to the cold wall h Relative to the hot wall max Relative to maximum quantity R Relative to the recirculation flow under the ceiling jet
The horizontal (X direction) and the vertical (Y direction) components of the velocity are measured by 2D Laser Doppler Velocimetry. These components were measured in the upper part of the cavity at Y* = 0.7 (Y = 2.688 m from the floor) and Y* = 0.85 (Y = 3.264 m) and near the ceiling at mid-width X* = 0.130 m. Three depths (Z* = 0.112, 0.075 and 0.038) were investigated but only
results in the Z* = 0.112 plane are being presented, since the results obtained on mean and RMS flows are identical. 3. Velocity and temperature fields 3.1. Thermal stratification The stratification parameter S is the dimensionless temperature gradient along a verticalline at the center of the cavity: S=
∂θ ∂Y
X = 0:13;
Z = 0:112:
ð1Þ
In order to characterize the influence of the Rayleigh number on the stratification parameter in this cavity, stratification is studied for 4 temperature differences and 2 passive wall configurations (ε = 0.1 and ε = 0.6). The temperature differences are set between 10 and 20 K entailing Rayleigh numbers ranging from 5.8 × 1010 to 1.2 × 1011. Fig. 2 displays the dimensionless temperature profiles in the vertical mid-axis for the 4 temperature differences (10, 15, 17.4 and 20 K). The inner passive face emissivity is = 0.6. This figure shows that the stratification parameter is not modified significantly when changing the Rayleigh number. S remains almost constant at a value S = 0.44 ± 0.03. The dimensionless temperature profiles along the vertical midaxis for the same 4 temperature differences are plotted in Fig. 3 for a passive wall emissivity = 0.1. As regards the case = 0.6, S remains almost constant and equal to S = 0.57 ± 0.03. In order to know which parameter has a real influence on the stratification parameter, sets of experimental results have been assembled in Table 1, which shows that while the Rayleigh number has a small influence on the stratification parameter, the aspect ratio (vertical Av and horizontal AH) and the wall emissivity () both play an important role. Influence of wall radiation is explained by the modification of the wall temperature on passive walls entailed by radiative exchanges (especially on the floor and the ceiling). It results in a reduction of the vertical temperature gradient: the stratification parameter is consequently lower when wall radiation is significant. Influence of the horizontal aspect ratio on thermal stratification (see Table 1) is also connected with front and rear passive wall radiation. When the aspect ratio decreases, the front and the rear walls come closer. Wall radiation consequently increases and entails, as previously, a larger stratification parameter. The last point on thermal stratification is that while the stratification parameter S is strongly connected with wall radiation, gas radiation and/or turbulence are not preponderant factors; this is clearly shown in experimental settings. 3.2. Velocity profiles ( = 0.1) 3.2.1. Y* = 0.7 The dimensionless vertical component profiles for the hot and the cold boundary layer at Y* = 0.7 are plotted in Figs. 4 and 5 and the main characteristics of the boundary layers at this height are assembled in Tables 2 (for the hot side) and 3 (for the cold side). In the vertical boundary layers, the airflow rate per unit depth is calculated as follows (δ is the distance to the wall where vertical velocity is equal to 0): δ
Q = ρ0 ∫0 VdX:
ð2Þ
Whatever the Rayleigh number, measurements show the existence of a reverse flow on the outer edge of the hot boundary layer at Y* = 0.7. Furthermore, the maximum velocity is quite low when compared to the reference natural convection velocity (less than 14%).
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Fig. 1. Scheme of the inner cavity and photography of the upper part of the cavity (about 1/3 of the cavity).
whereas this ratio was about 23% in the differentially-heated cavity studied in [33] (RaH = 1.48 × 109 and AH = 1). At the same time, the boundary layer is thicker in this study than in [33]. For the cold boundary layer at Y* = 0.7, there is no backflow on its outer edge. It is important to note that the maximum velocity for Ra H = 4.0 × 10 10 or Ra H = 8.1 × 10 10 is higher than this for RaH = 1.2 × 1011. As expected, the air flow rate increases with the Rayleigh number. At this height, dimensionless vertical profiles in the cold boundary layer indicate a flow modification when RaH increases. Indeed, for RaH = 1.2 × 1011, the local Rayleigh number (calculated using the real height, i.e. y = H × (1-Y⁎) for the cold boundary layer) indicates the beginning of the transition to the turbulent state (Ray N 3.0 × 109). For other RaH this state is not reached at this height (i.e. Ray b 3.0 × 109). This finding explains the two airflow trends observed in Fig. 5: the airflow for RaH = 1.2 × 1011 is turbulent or in transition towards turbulence, whereas for the other two Rayleigh numbers (4.0 × 1010 and 8.1 × 1010) are still laminar.
3.2.2. Y* = 0.85 Figs. 6 and 7 plot respectively the dimensionless profiles of the vertical velocity component for the hot and the cold boundary layers at Y* = 0.85 and at mid-depth. The main characteristics of the boundary layers at this level are also assembled in Table 2 (for the hot side) and Table 3 (for the cold side). As expected, the airflow rate in the hot boundary layer at Y* = 0.85, increases with the Rayleigh number. In addition, at this height, the flow rate (and the velocity) is lower than those measured at Y* = 0.7, whatever the Rayleigh number considered. The horizontal component of the velocity indicates that the outer part of the ascending airflow is ejected towards the outer region of the boundary layer before flowing downwards. This could explain why the upward airflow rate decreases along with the height. In the cold boundary layer, regardless of the Rayleigh number the dimensionless velocity profiles are similar. The maximum velocity for RaH = 1.2 × 1011 is higher at Y* = 0.85 than at Y*= 0.7, whereas the flow
Fig. 2. Thermal stratification in the experimental cavity with ε = 0.6.
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Fig. 3. Thermal stratification in the experimental cavity with ε = 0.1.
Table 1 Stratification parameter comparisons in various configurations in experimental differentially-heated cavity.
Present study
Mergui [31] Salat and Penot [33]
Ndame [32]
AV
AH
RaH
S
3.84 3.84 3.84 3.84 3.84 3.84 3.84 3.84 1 1 1 1 1 1 4
0.86 0.86 0.86 0.86 0.86 0.86 0.86 0.86 0.29 0.32 0.32 1 1 0.80 1.33
5.8 × 1010 8.6 × 1010 1.0 × 1011 1.2 × 1011 5.8 × 1010 8.6 × 1010 1.0 × 1011 1.2 × 1011 1.69 × 109 1.48 × 109 1.48 × 109 1.48 × 109 1.48 × 109 1.48 × 109 1.01 × 106
0.1 0.1 0.1 0.1 0.6 0.6 0.6 0.6 0.1 0.95* 0.1* 0.95* 0.1* 0.2* 0.04
0.59 0.57 0.56 0.54 0.47 0.44 0.44 0.41 0.37 0.375 0.44 0.54 0.72 0.65 0.71
*only for the front and rear wall emissivities.
0,14 1.2×10^11 0,12 8.1×10^10 0,1 4.0×10^10 0,08
V*
0,06 0,04 0,02 0 -0,02 -0,04
0
0,02
0,04
0,06
0,08
X* Fig. 4. Vertical component of the velocity in the hot boundary layer at Y* = 0.7 and mid-depth.
0,1
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683
0,05
0
V*
-0,05
-0,1
-0,15
1.2×10^11 8.1×10^10
-0,2
4.0×10^10 -0,25 0,19
0,2
0,21
0,22
0,23
0,24
0,25
0,26
X* Fig. 5. Vertical component of the velocity in the cold boundary layer at Y* = 0.7 and mid-depth.
Table 2 Hot boundary layer characteristics (at mid-depth) for different heights and different values of the Rayleigh number at mid-depth. ΔT [°C] 7 14 20 7 14 20
RaH 10
4.0 × 10 8.1 × 1010 1.2 × 1011 4.0 × 1010 8.1 × 1010 1.2 × 1011
Y*
VREF [m/s]
Q [g.s− 1.m− 1]
Vmax [m.s− 1]
Vmax/VREF
0.70 0.70 0.70 0.85 0.85 0.85
1.12 1.59 1.90 1.12 1.59 1.90
12.3 15.3 18.0 9.4 9.8 13.4
0.136 0.196 0.214 0.117 0.151 0.180
0.12 0.12 0.11 0.10 0.095 0.095
istics of the ceiling jet are specified in Table 4. In the jet under the ceiling, the flow rates per unit depth are calculated as follows:
δ
Q = ρ0 ∫0 ∫UdY:
ð3Þ
Measurements show that the flow rate is the same for the three Rayleigh numbers even though the maximum velocity increases with RaH. In the jet, as in the boundary layers, maximum dimensionless velocity decreases when the Rayleigh number increases.
rate is lower, which shows that the boundary layer becomes thicker. The horizontal component of the velocity confirms this trend and shows that the flow sucks air from the core and drives it downwards. 3.2.3. X* = 0.5 (at the ceiling in the mid-plane at mid-depth) Fig. 8 represents the dimensionless horizontal component of the velocity at mid-width and mid-depth near the ceiling. The character-
3.2.4. General flow behavior in the upper part of the cavity The velocity vectors are obtained using the vertical and horizontal components of the velocity. They are plotted in Fig. 9 for RaH = 4.0 × 1010 and RaH = 1.2 × 1011. When RaH increases, a slight modification of the upper recirculation area and the intensification of the downward airflow on the outer edge of the hot boundary layer is worth noting. Based on the velocity vector fields
0,12 1.2x10^11 0,1 8.1x10^10 0,08 4.0x10^10
V*
0,06 0,04 0,02 0 -0,02 -0,04
0
0,01
0,02
0,03
0,04
0,05
0,06
X* Fig. 6. Vertical component of the velocity in the hot boundary layer at Y* = 0.85 and mid-depth.
0,07
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0,02 0 -0,02 -0,04
V*
-0,06 -0,08 -0,1
1.2x10^11
-0,12
8.1x10^10
-0,14 -0,16
4.0x10^10
0,2
0,21
0,22
0,23
0,24
0,25
0,26
X* Fig. 7. Vertical component of the velocity in the cold boundary layer at Y* = 0.85 and mid-depth.
Table 3 Cold boundary layer characteristics (at mid-depth) for different heights and different values of the Rayleigh number. ΔT [°C] 7 14 20 7 14 20
RaH 10
4.0 × 10 8.1 × 1010 1.2 × 1011 4.0 × 1010 8.1 × 1010 1.2 × 1011
Y*
VREF [m/s]
Q [g.s− 1.m− 1]
Vmax [m.s− 1]
Vmax/VREF
0.70 0.70 0.70 0.85 0.85 0.85
1.12 1.59 1.90 1.12 1.59 1.90
− 6.6 − 9.2 − 14.8 − 5.7 − 5.8 − 6.1
− 0.221 − 0.286 − 0.244 − 0.166 − 0.233 − 0.273
− 0.20 − 0.18 − 0.13 − 0.15 − 0.15 − 0.14
considered: case A occurs at heights exceeding about Y* = 0.8 and whereas case B occurs below. 3.3. Friction stress (ε = 0.1) The friction stress calculated as follows: !! ∂V ! τw = −μ ∂x
ð4Þ
wall
(Fig. 9), two kinds of airflow behavior (denoted A and B in Fig. 10) can be extrapolated: • In case A, the downward flow directly feeds the cold boundary layer. • In case B, the downward airflow feeds the hot boundary layer at a lower height. 3D LES computations, obtained by A. Sergent [38], show that, for RaH = 1.2 × 1011, each scenario occurs depending on the heights
yields the friction coefficient, corresponding to the dimensionless friction stress
Cf =
! 2 τw 2 ρ0 VREF
:
ð5Þ
1
0,98
0,96
Y*
0,94
0,92 4.0x10^10 0,9 8.1x10^10 0,88
0,86 -0,04
1.2x10^11
-0,02
0
0,02
0,04
0,06
U* Fig. 8. Horizontal component of the velocity near the ceiling at mid-width and mid-depth.
0,08
D. Saury et al. / International Communications in Heat and Mass Transfer 38 (2011) 679–687 Table 4 Ceiling jet characteristics at mid-width and mid-depth (X⁎ = 0.130; Z⁎ = 0.112). ΔT [°C]
RaH
Q [g.s− 1.m− 1]
Umax [m.s− 1]
Umax/VREF
7 14 20
4.0 × 1010 8.1 × 1010 1.2 × 1011
9.0 8.9 9.2
0.068 0.075 0.085
0.061 0.047 0.045
685
The friction stresses and the friction coefficients are shown in Tables 5 and 6 respectively for several values of the Rayleigh number. Along the boundary layers, friction stress increases with the rise of RaH. This trend is due to the fact that velocity increases with the Rayleigh number (see for instance Vmax in Table 2) whereas the thickness of these boundary layers does not significantly change (see
Fig. 9. Dimensionless airflow map in the upper part of the cavity.
Fig. 10. Scenario of flow behavior in the upper part of the cavity.
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Table 5 Friction stress, τW [Pa]. RaH
Table 7 Nusselt number: comparisons for various RaH.
Hot Wall Y* = 0.7 10
4.0 × 10 8.1 × 1010 1.2 × 1011
−4
5.8 × 10 9.4 × 10− 4 1.2 × 10− 3
Cold wall Y* = 0.85 −4
3.9 × 10 7.4 × 10− 4 8.6 × 10− 4
Ceiling
Y* = 0.7
Y* = 0.85
−4
−4
9.5 × 10 1.7 × 10− 3 2.0 × 10− 3
8.5 × 10 1.4 × 10− 3 1.8 × 10− 3
X = 0.5 m −6
3.2 × 10 2.5 × 10− 6 2.2 × 10− 6
Table 6 Friction coefficient, Cf [−]. RaH
4.0 × 1010 8.1 × 1010 1.2 × 1011
Hot wall
Cold wall
Ceiling
Y* = 0.7
Y* = 0.85
Y* = 0.7
Y* = 0.85
X = 0.5 m
7.7 × 10− 4 6.3 × 10− 4 5.5 × 10− 3
5.2 × 10− 4 4.9 × 10− 4 4.0 × 10− 4
1.3 × 10− 3 1.1 × 10− 3 9.2 × 10− 4
1.1 × 10− 3 9.7 × 10− 4 8.3 × 10− 4
4.3 × 10− 6 1.6 × 10− 6 1.0 × 10− 6
for example the vertical velocity profiles Figs. 4 and 6). It entails an in ! and consequently of increase of the wall velocity gradient ∂∂xV wall friction stress (τw). On the contrary, the friction coefficient decreases when the Rayleigh number increases. Indeed, the dimensionless velocity decreases slightly when the Rayleigh number increases whereas the pffiffiffiffiffiffiffiffiffi reference velocity increases as RaH . Furthermore, as the velocity in the cold boundary layer is higher than in the hot one, the friction is greater at the same height in the cold boundary layer (about twice as great). 3.4. Wall heat transfer–Nusselt number (ε = 0.1)
AV
AH
RaH
bNuN
3.84 3.84
0.86 0.86
1.2 × 1011 1.2 × 1011
231 254
The difficulty in carrying out these measurements is due to the thickness of the thermal boundary layer in the large cavity (4 m3). Indeed, temperature profile is actually linear over a few millimeters. In this area, measurement must be carried out carefully in order to determine the slope of the profile in the near wall area. The same problem appears for numerical simulations where a suitable mesh must be used to obtain an accurate Nusselt number. The experimental values obtained using Eq. (6) are plotted in Fig. 11 for RaH = 1.2 × 1011. The comparison between these experimental values and a 3D LES simulation performed by A. Sergent [38] provided in Fig. 11 shows good agreement. More specifically, similar trends can be found in spatial variation of the Nusselt number, i.e. a first part where NuH decreases ([0; 0.2]), followed by an increase of the local Nusselt number ([0.2 ; 0.3]) probably due to transition state, and finally an almost constant value for NuH is achieved ([0.3 ; 1]).The reason for the sudden modification of the Nusselt profile is still under investigation. Two possible explanations have been investigated but they await confirmation. Integrating the mean local Nusselt number over the Y direction, the overall Nusselt number 〈 is determined. A value of bNuN = 231 ± 30 is then calculated for RaH = 1.2 × 1011. This value can be compared to the one obtained by A. Sergent for RaH = 1.2 × 1011, i.e. bNuN = 254 (Table 7). 4. Conclusion
Using dimensionless temperature profile in the vicinity of the wall determined with a micro-thermocouple (12.7 μm), the mean wall heat flux can be calculated:
H ∂T ∂θ NuH Y = = : ΔT ∂x Y ∂X Y
Present study Sergent [38]
ð6Þ
This paper provides an initial experimental characterization of a differentially-heated cavity at large Rayleigh numbers (on the order of 1011). Influence of wall radiation on the stratification parameter has been highlighted. Velocity profiles in the upper part of the cavity for the three Rayleigh numbers (4.0 × 1010, 8.1 × 1010 and 1.2 × 1011) are given as well as the air flow rate or the friction coefficient. The mean local Nusselt number is indicated along with recommendations for its
Fig. 11. Local Nusselt along the hot wall for RaH = 1.2 × 1011.
D. Saury et al. / International Communications in Heat and Mass Transfer 38 (2011) 679–687
determination (experimentations or calculations). Using these values of the local Nusselt number, the overall Nusselt number has been calculated for this Rayleigh number. Other Rayleigh numbers are currently under investigation and should be fully specified before too long. Acknowledgments This work was conducted under research project COCORACOPHA of the French CNRS “programme énergie”. References [1] G. De Vahl Davis, Laminar natural convection in an enclosed rectangular cavity, International Journal of Heat and Mass Transfer 11 (1968) 1675–1693. [2] A.I. Catton, Natural convection in enclosures, National Res. Council of Canada Canadian Society for Chemical Engineering, and Canadian Society for Mechanical Engineering, Proceedings of The 6th International Heat Transfer Conference, vol. 6, Hemisphere Publishing Corp, Toronto, Canada, 1978, pp. 13–31. [3] G.D. Mallinson, G. De Wahl Davis, Three-dimensional natural convection in box: a numerical study, J. Fluid Mech. 33 (part. 1) (1977) 1–31. [4] D.G. Briggs, D.N. Jones, Two dimensional periodic natural convection in a rectangular enclosure of aspect ratio one, Journal of Heat Transfer (107) (1985) 850–854. [5] D.N. Jones, D.G. Briggs, Periodic two-dimensional cavity flow: effect of linear horizontal thermal boundary conditions, Journal of Heat Transfer (111) (1988) 850–854. [6] F.X. Trias, A. Gorobets, M. Soria, A. Oliva, Direct numerical simulation of a differentially heated cavity of aspect ratio 4 with Rayleigh numbers up to 1011 — Part I: numerical methods and time-averaged flow, International Journal of Heat and Mass Transfer 53 (4) (2010) 665–673. [7] F.X. Trias, A. Gorobets, M. Soria, A. Oliva, Direct numerical simulation of a differentially heated cavity of aspect ratio 4 with Rayleigh numbers up to 1011 — Part II: heat transfer and flow dynamics, International Journal of Heat and Mass Transfer 53 (4) (2010) 674–683. [8] G. Colomer, M. Costa, R. Cònsul, A. Oliva, Three-dimensional numerical simulation of convection and radiation in a differentially heated cavity using the discrete ordinates method, International Journal of Heat and Mass Transfer 47 (2) (2004) 257–269. [9] A. Baïri, N. Laraqi, J.M. Garcia de Maria, Numerical and experimental study of natural convection in tilted parallelepipedic cavities for large Rayleigh numbers, Experimental Thermal and Fluid Science 31 (2007) 309–324. [10] O.G. Martynenko, P.P. Khramtsov, Free convective heat transfer, Natural Convection in Enclosures, Springer-Verlag, 2005, pp. 291–310. [11] M.W. Nansteel, R. Grief, Natural convection in enclosures with two- and threedimensional partitions, Int. J. Heat Mass Transfer 27 (1984) 561–571. [12] R.L. Frederick, A. Valencia, Heat transfer in a square cavity with a conducting partition on its hot wall, Int. Commun. Heat Mass Transfer 16 (1989) 347–354. [13] E.K. Lakhal, M. Hasnaoui, E. Bilgen, P. Vesseur, Natural convection heat transfer in rectangular enclosures with perfectly conducting fins attached on the heated wall, in: G.F. Hewitt (Ed.), Proc. of the Tenth Int. Heat Transfer Conf, Taylor and Francis, Brighton, UK, 1994, pp. 97–102. [14] M. Hasnaoui, P. Vesseur, E. Bilgen, Natural convection in rectangular enclosures with adiabatic fins attached on the heated wall, WarmeStoffubertrag 27 (1992) 357–368. [15] P.H. Oosthuizen, J.T. Paul, Free convective heat transfer in a cavity fitted with a horizontal plate on the cold wall, Adv. Enhance Heat Transfer ASME HTD 43 (1985) 101–107. [16] A. Bowles, R. Cheesewright, Direct measurements of turbulence heat flux in a large rectangular air cavity, Exp. Heat Transfer 2 (1989) 59–69.
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[17] F. Ampofo, T.G. Karayiannis, Experimental benchmark data for turbulent natural convection in an air filled square cavity, Int. J. Heat Mass Transfer 46 (2003) 3551–3572. [18] Y.S. Tian, T.G. Karayiannis, Low turbulence natural convection in an air filled square cavity. Part I. Thermal and fluid flow fields, Int. J. Heat Mass Transfer 43 (2000) 849–866. [19] Y.S. Tian, T.G. Karayiannis, Low turbulence natural convection in an air filled square cavity. Part II. The turbulence quantities, Int. J. Heat Mass Transfer 43 (2000) 867–884. [20] A.M. Lankhorst, Laminar and Turbulent Natural Convection in Cavities — Numerical Modelling and Experimental Validation, PhD Thesis, Technology University of Delft, 1991. [21] C. Beghein, F. Allard, A. Draoui, Numerical modelling of turbulent convection in a thermally-driven square cavity, in: R.A.W.M. Henkes, C.J. Hoogendoorn (Eds.), Turbulent Natural Convection in Enclosures — A Computational and Experimental Benchmark Study, Proceedings of the Eurotherm Seminar Delft, The Netherlands, No. 22, 1992, pp. 31–42. [22] S. Ziai, Turbulent Natural Convection in a Large Rectangular Air Cavity, PhD Thesis, Queen Mary College, University of London,1983. [23] K.J. King, Turbulent Natural Convection in Rectangular Air Cavities, PhD Thesis, QueenMary College, University of London, 1989. [24] S. Paolucci, Direct numerical simulation of two-dimensional turbulent natural convection in an enclosed cavity, J. Fluid Mech. 215 (1990) 229–262. [25] F. Ampofo, Turbulent natural convection of air in a non-partitioned or partitioned cavity with differentially heated vertical and conducting horizontal walls, Experimental Thermal and Fluid Science 29 (2) (2005) 137–157. [26] P.L. Betts, I.H. Bokhari, Experiments on turbulent natural convection in an enclosed tall cavity, International Journal of Heat and Fluid Flow 21 (6) (2000) 675–683. [27] S. Xin, P. Le Quéré, Direct numerical simulation of two dimensional chaotic natural convection in a differentially heated cavity of aspect ratio 4, J. Fluid Mech. (304) (1995) 87–118. [28] S. Xin, P. Le Quéré, An extended Chebyshev pseudo-spectral bench-mark for the 8: 1 differentially heated cavity, Int. J. Num. Methods Fluids 8 (2002) 981–998. [29] M. Soria, F.X. Trias, C.D. Pérez-Segarra, A. Oliva, Direct simulation of a threedimensional natural-convection flow in a differentially heated cavity of aspect ratio 4, Numerical Heat Transfer Part A (45) (2004) 649–673. [30] A. Bejan, Convection Heat Transfer, second ed., Wiley, New-York, 1995, pp. 189–190. [31] S. Mergui, F. Penot, Convection naturelle en cavité carrée différentiellement chauffée investigation expérimentale à Ra = 1.69 × 109, Int. J. Heat Mass Transfer 39 (3) (1996) 563–574. [32] F. Penot, A. N'Dame, Successive bifurcations of natural convection in a vertical enclosure heated from the side, in: Institute of Chemical Engineers Staff (Ed.), Heat Transfer: Third UK National Conference Incorporating First European Conference on Thermal Sciences volume 1, Institution of Chemical Engineers, Birmingham, UK, ISBN: 0852952821, 1992, pp. 507–513. [33] J. Salat, S. Xin, P. Joubert, A. Sergent, F. Penot, P. Le Quéré, Experimental and numerical investigation of turbulent natural convection in a large air-filled cavity, Int. J. Heat Fluid Flow (2004) 824–832. [34] P. Le Quéré, F. Penot, Numerical an experimental investigation of the transition to unsteady natural convection of air in a vertical differentially heated cavity, in: H.H. Bau, S.A. Korpela (Eds.), Bifurcation phenomena in thermal processes and convection, ASME HTD, 94, 1989, pp. 75–82. [35] S. Xin, P. Le Quéré, Linear stability analyses of natural convection flows in a differentially heated square cavity with conducting horizontal walls, Physics of Fluids 13 (9) (2001) 2529–2542. [36] J.P. Gollub, S.V. Benson, Many routes to turbulent convection, J. Fluid Mech. 100 (part. 3) (1980) 449–470. [37] F.X. Trias, M. Soria, A. Oliva, C.D. Pérez-Segarra, Direct numerical simulations of two- and three-dimensional turbulent natural convection flows in a differentially heated cavity of aspect ratio 4, Journal of Fluid Mechanics 586 (2007) 259–293. [38] A. Sergent, Private communication on wall heat transfer in a differentially-heated cavity at RaH = 1011, 2009.
Contents lists available at ScienceDirect
International Communications in Heat and Mass Transfer j o u r n a l h o m e p a g e : w w w. e l s ev i e r. c o m / l o c a t e / i c h m t
Natural convection in an air-filled cavity: Experimental results at large Rayleigh numbers☆ Didier Saury ⁎, Nicolas Rouger, Francis Djanna, François Penot Institut Pprime, UPR CNRS 3346, CNRS-ENSMA-Université de Poitiers, Département Fluides Thermique Combustion, ENSMA, Téléport 2, 1 avenue Clément Ader - BP 40109, F-86961 Futuroscope CEDEX, France
a r t i c l e
i n f o
Available online 22 March 2011 Keywords: Turbulent natural convection Buoyancy-driven flows Experimental characterization
a b s t r a c t A large-scale experimental setup is built and instrumented. It consists in a 4 m-high cavity with a horizontal cross-section equal to 0.86 × 1.00 m². Two opposite vertical walls are heated and cooled down; other walls (lateral walls, ceiling and floor) are made of insulating medium covered with a thin and low-emissivity film designed to minimize radiative effects into the cavity. The temperatures of active walls are imposed, constant and equally distributed around the ambient temperature in order to reduce heat losses. The temperature difference between the hot and cold walls is chosen to respect the Boussinesq approximation. Under these assumptions, Rayleigh number values up to 1.2 × 10 11 (ΔT = 20 °C) can be obtained. The centre-symmetry is verified on the thermal stratification. Influence of the temperature difference and of wall emissivities on the stratification parameter (dimensionless vertical temperature gradient) is discussed. Velocity measurements allow the velocity field to be obtained and provide information on flows encountered in the cavity. Temperature measurements are also carried out in the whole cavity. In the paper, a complete experimental characterization is provided: airflow inside the cavity is analyzed and the Nusselt number along the hot and the cold wall is presented. © 2011 Elsevier Ltd. All rights reserved.
1. Introduction For many decades, studies on phenomena occurring in differentially-heated cavities have been of great interest [1–37]. This is particularly the case today, since public figures have come to realize that economic development must be accompanied by a reduction of greenhouse gas and by policies aimed at reducing energy consumption. In many areas the stakes are high and it is of paramount interest to understand the phenomena occurring in order to control them. Natural convection occurs “naturally” almost everywhere as soon as a density gradient exists and it entails heat exchanges. We need to understand and control the attendant phenomena and to thereby promote or reduce the associated heat transfers. The differentially-heated cavity is an interesting academic configuration allowing for a worthwhile study of the natural convection phenomenon and its chaotic behavior [34–36]. Moreover, applications to concrete cases (building, aeronautics…) often require precise understanding of phe-
☆ Communicated by J. Taine and A. Soufiani. ⁎ Corresponding author. E-mail address: [email protected] (D. Saury). 0735-1933/$ – see front matter © 2011 Elsevier Ltd. All rights reserved. doi:10.1016/j.icheatmasstransfer.2011.03.019
nomena occurring at high Rayleigh numbers (N1011). Numerous numerical or experimental studies have already been carried out at low Rayleigh numbers [1–5,8–33] where airflow remains laminar or weakly turbulent (Ra H b 10 9 ). Recently, a few numerical investigations at higher Rayleigh numbers have appeared, but only experimental validation will confirm their credibility [6,7,37].
2. Experimental setup description This study was carried out in a differentially-heated cavity shown in Fig. 1. The cavity is 3.84 m high, 1 m wide and 0.86 m deep. Two opposite vertical walls are differentially-heated (one is heated while the other is cooled). Their temperature is imposed so that the core of the cavity is at ambient temperature. The two walls, made of aluminum plate (k = 134 W m− 1 K− 1 and ε = 0.15), are kept at a constant temperature by a glycol-water circuit flowing in 2 separated cryothermostats imposing the temperature. The other walls (back, front walls, floor and ceiling) are made of 8 cm thick polyurethane foam panels (k = 0.035 W m− 1 K− 1) placed between the heated and cooled walls. The floor is also made with extruded polyurethane foam panels. Finally, 3 cm-thick panels are added around the cavity on the outer side. On the inner side of the cavity, a thin (40 μm) low-emissivity film (ε = 0.1 with the film and ε = 0.6 without) is added in order to reduce wall radiation.
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Nomenclature AH AV Cf D g H L k Q NuH 〈Nu〉 RaH Ray S T Tm Tamb U V VREF x y z X⁎ Y⁎ Z⁎
Horizontal aspect ratio: AH = D/L [−] Vertical aspect ratio: AV = H/L [−] Friction coefficient [−] Depth of the cavity: D = 0.86 m [m] Gravitational acceleration [m.s− 2] Height of the cavity: H = 3.84 m [m] Width of the cavity: L = 1 m [m] Thermal conductivity [W.m− 1.K− 1] Flow rate per unit depth [g.s− 1.m− 1] Nusselt number [−] Overall Nusselt number: Nu = H1 ∫0H NuH dY =∫01 NuH dY α [−] 3 gβΔTH [−] Rayleigh number: RaH = αv 3 gβΔTy [−] Rayleigh number: Ray = αv Stratification parameter [−] Air temperature [°C] Mean temperature, Tm = ½ (Tc + Th) [°C] Ambient temperature (≈ 20 °C) [°C] Horizontal component of the velocity (in the x-direction) [m.s− 1] Vertical component of the velocity (in the y-direction) [m.s− 1] pffiffiffiffiffiffiffiffiffi −1 Reference velocity: VREF = α ] H RaH [m.s Horizontal Cartesian coordinate [m] Vertical Cartesian coordinate [m] Transverse Cartesian coordinate [m] Dimensionless horizontal Cartesian coordinate X⁎ = x/H [−] Dimensionless vertical Cartesian coordinate Y⁎ = y/H [−] Dimensionless transverse Cartesian coordinate Z⁎ = z/H [−]
Greek characters α Thermal diffusivity [m2.s− 1] β Coefficient of thermal expansion [K− 1] δ Thickness of the flow [m] ΔT Temperature difference, ΔT = Th − Tc [°C] ε Wall emissivity [−] μ Dynamic viscosity [Pa.s] ν Kinematic viscosity [m2.s− 1] m θ Dimensionless temperature : θ = TT ΔT [−] −3 ρ0 Air density at T = Tm [kg.m ] τW Friction stress [Pa]
Subscripts and superscripts c Relative to the cold wall h Relative to the hot wall max Relative to maximum quantity R Relative to the recirculation flow under the ceiling jet
The horizontal (X direction) and the vertical (Y direction) components of the velocity are measured by 2D Laser Doppler Velocimetry. These components were measured in the upper part of the cavity at Y* = 0.7 (Y = 2.688 m from the floor) and Y* = 0.85 (Y = 3.264 m) and near the ceiling at mid-width X* = 0.130 m. Three depths (Z* = 0.112, 0.075 and 0.038) were investigated but only
results in the Z* = 0.112 plane are being presented, since the results obtained on mean and RMS flows are identical. 3. Velocity and temperature fields 3.1. Thermal stratification The stratification parameter S is the dimensionless temperature gradient along a verticalline at the center of the cavity: S=
∂θ ∂Y
X = 0:13;
Z = 0:112:
ð1Þ
In order to characterize the influence of the Rayleigh number on the stratification parameter in this cavity, stratification is studied for 4 temperature differences and 2 passive wall configurations (ε = 0.1 and ε = 0.6). The temperature differences are set between 10 and 20 K entailing Rayleigh numbers ranging from 5.8 × 1010 to 1.2 × 1011. Fig. 2 displays the dimensionless temperature profiles in the vertical mid-axis for the 4 temperature differences (10, 15, 17.4 and 20 K). The inner passive face emissivity is = 0.6. This figure shows that the stratification parameter is not modified significantly when changing the Rayleigh number. S remains almost constant at a value S = 0.44 ± 0.03. The dimensionless temperature profiles along the vertical midaxis for the same 4 temperature differences are plotted in Fig. 3 for a passive wall emissivity = 0.1. As regards the case = 0.6, S remains almost constant and equal to S = 0.57 ± 0.03. In order to know which parameter has a real influence on the stratification parameter, sets of experimental results have been assembled in Table 1, which shows that while the Rayleigh number has a small influence on the stratification parameter, the aspect ratio (vertical Av and horizontal AH) and the wall emissivity () both play an important role. Influence of wall radiation is explained by the modification of the wall temperature on passive walls entailed by radiative exchanges (especially on the floor and the ceiling). It results in a reduction of the vertical temperature gradient: the stratification parameter is consequently lower when wall radiation is significant. Influence of the horizontal aspect ratio on thermal stratification (see Table 1) is also connected with front and rear passive wall radiation. When the aspect ratio decreases, the front and the rear walls come closer. Wall radiation consequently increases and entails, as previously, a larger stratification parameter. The last point on thermal stratification is that while the stratification parameter S is strongly connected with wall radiation, gas radiation and/or turbulence are not preponderant factors; this is clearly shown in experimental settings. 3.2. Velocity profiles ( = 0.1) 3.2.1. Y* = 0.7 The dimensionless vertical component profiles for the hot and the cold boundary layer at Y* = 0.7 are plotted in Figs. 4 and 5 and the main characteristics of the boundary layers at this height are assembled in Tables 2 (for the hot side) and 3 (for the cold side). In the vertical boundary layers, the airflow rate per unit depth is calculated as follows (δ is the distance to the wall where vertical velocity is equal to 0): δ
Q = ρ0 ∫0 VdX:
ð2Þ
Whatever the Rayleigh number, measurements show the existence of a reverse flow on the outer edge of the hot boundary layer at Y* = 0.7. Furthermore, the maximum velocity is quite low when compared to the reference natural convection velocity (less than 14%).
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Fig. 1. Scheme of the inner cavity and photography of the upper part of the cavity (about 1/3 of the cavity).
whereas this ratio was about 23% in the differentially-heated cavity studied in [33] (RaH = 1.48 × 109 and AH = 1). At the same time, the boundary layer is thicker in this study than in [33]. For the cold boundary layer at Y* = 0.7, there is no backflow on its outer edge. It is important to note that the maximum velocity for Ra H = 4.0 × 10 10 or Ra H = 8.1 × 10 10 is higher than this for RaH = 1.2 × 1011. As expected, the air flow rate increases with the Rayleigh number. At this height, dimensionless vertical profiles in the cold boundary layer indicate a flow modification when RaH increases. Indeed, for RaH = 1.2 × 1011, the local Rayleigh number (calculated using the real height, i.e. y = H × (1-Y⁎) for the cold boundary layer) indicates the beginning of the transition to the turbulent state (Ray N 3.0 × 109). For other RaH this state is not reached at this height (i.e. Ray b 3.0 × 109). This finding explains the two airflow trends observed in Fig. 5: the airflow for RaH = 1.2 × 1011 is turbulent or in transition towards turbulence, whereas for the other two Rayleigh numbers (4.0 × 1010 and 8.1 × 1010) are still laminar.
3.2.2. Y* = 0.85 Figs. 6 and 7 plot respectively the dimensionless profiles of the vertical velocity component for the hot and the cold boundary layers at Y* = 0.85 and at mid-depth. The main characteristics of the boundary layers at this level are also assembled in Table 2 (for the hot side) and Table 3 (for the cold side). As expected, the airflow rate in the hot boundary layer at Y* = 0.85, increases with the Rayleigh number. In addition, at this height, the flow rate (and the velocity) is lower than those measured at Y* = 0.7, whatever the Rayleigh number considered. The horizontal component of the velocity indicates that the outer part of the ascending airflow is ejected towards the outer region of the boundary layer before flowing downwards. This could explain why the upward airflow rate decreases along with the height. In the cold boundary layer, regardless of the Rayleigh number the dimensionless velocity profiles are similar. The maximum velocity for RaH = 1.2 × 1011 is higher at Y* = 0.85 than at Y*= 0.7, whereas the flow
Fig. 2. Thermal stratification in the experimental cavity with ε = 0.6.
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Fig. 3. Thermal stratification in the experimental cavity with ε = 0.1.
Table 1 Stratification parameter comparisons in various configurations in experimental differentially-heated cavity.
Present study
Mergui [31] Salat and Penot [33]
Ndame [32]
AV
AH
RaH
S
3.84 3.84 3.84 3.84 3.84 3.84 3.84 3.84 1 1 1 1 1 1 4
0.86 0.86 0.86 0.86 0.86 0.86 0.86 0.86 0.29 0.32 0.32 1 1 0.80 1.33
5.8 × 1010 8.6 × 1010 1.0 × 1011 1.2 × 1011 5.8 × 1010 8.6 × 1010 1.0 × 1011 1.2 × 1011 1.69 × 109 1.48 × 109 1.48 × 109 1.48 × 109 1.48 × 109 1.48 × 109 1.01 × 106
0.1 0.1 0.1 0.1 0.6 0.6 0.6 0.6 0.1 0.95* 0.1* 0.95* 0.1* 0.2* 0.04
0.59 0.57 0.56 0.54 0.47 0.44 0.44 0.41 0.37 0.375 0.44 0.54 0.72 0.65 0.71
*only for the front and rear wall emissivities.
0,14 1.2×10^11 0,12 8.1×10^10 0,1 4.0×10^10 0,08
V*
0,06 0,04 0,02 0 -0,02 -0,04
0
0,02
0,04
0,06
0,08
X* Fig. 4. Vertical component of the velocity in the hot boundary layer at Y* = 0.7 and mid-depth.
0,1
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683
0,05
0
V*
-0,05
-0,1
-0,15
1.2×10^11 8.1×10^10
-0,2
4.0×10^10 -0,25 0,19
0,2
0,21
0,22
0,23
0,24
0,25
0,26
X* Fig. 5. Vertical component of the velocity in the cold boundary layer at Y* = 0.7 and mid-depth.
Table 2 Hot boundary layer characteristics (at mid-depth) for different heights and different values of the Rayleigh number at mid-depth. ΔT [°C] 7 14 20 7 14 20
RaH 10
4.0 × 10 8.1 × 1010 1.2 × 1011 4.0 × 1010 8.1 × 1010 1.2 × 1011
Y*
VREF [m/s]
Q [g.s− 1.m− 1]
Vmax [m.s− 1]
Vmax/VREF
0.70 0.70 0.70 0.85 0.85 0.85
1.12 1.59 1.90 1.12 1.59 1.90
12.3 15.3 18.0 9.4 9.8 13.4
0.136 0.196 0.214 0.117 0.151 0.180
0.12 0.12 0.11 0.10 0.095 0.095
istics of the ceiling jet are specified in Table 4. In the jet under the ceiling, the flow rates per unit depth are calculated as follows:
δ
Q = ρ0 ∫0 ∫UdY:
ð3Þ
Measurements show that the flow rate is the same for the three Rayleigh numbers even though the maximum velocity increases with RaH. In the jet, as in the boundary layers, maximum dimensionless velocity decreases when the Rayleigh number increases.
rate is lower, which shows that the boundary layer becomes thicker. The horizontal component of the velocity confirms this trend and shows that the flow sucks air from the core and drives it downwards. 3.2.3. X* = 0.5 (at the ceiling in the mid-plane at mid-depth) Fig. 8 represents the dimensionless horizontal component of the velocity at mid-width and mid-depth near the ceiling. The character-
3.2.4. General flow behavior in the upper part of the cavity The velocity vectors are obtained using the vertical and horizontal components of the velocity. They are plotted in Fig. 9 for RaH = 4.0 × 1010 and RaH = 1.2 × 1011. When RaH increases, a slight modification of the upper recirculation area and the intensification of the downward airflow on the outer edge of the hot boundary layer is worth noting. Based on the velocity vector fields
0,12 1.2x10^11 0,1 8.1x10^10 0,08 4.0x10^10
V*
0,06 0,04 0,02 0 -0,02 -0,04
0
0,01
0,02
0,03
0,04
0,05
0,06
X* Fig. 6. Vertical component of the velocity in the hot boundary layer at Y* = 0.85 and mid-depth.
0,07
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0,02 0 -0,02 -0,04
V*
-0,06 -0,08 -0,1
1.2x10^11
-0,12
8.1x10^10
-0,14 -0,16
4.0x10^10
0,2
0,21
0,22
0,23
0,24
0,25
0,26
X* Fig. 7. Vertical component of the velocity in the cold boundary layer at Y* = 0.85 and mid-depth.
Table 3 Cold boundary layer characteristics (at mid-depth) for different heights and different values of the Rayleigh number. ΔT [°C] 7 14 20 7 14 20
RaH 10
4.0 × 10 8.1 × 1010 1.2 × 1011 4.0 × 1010 8.1 × 1010 1.2 × 1011
Y*
VREF [m/s]
Q [g.s− 1.m− 1]
Vmax [m.s− 1]
Vmax/VREF
0.70 0.70 0.70 0.85 0.85 0.85
1.12 1.59 1.90 1.12 1.59 1.90
− 6.6 − 9.2 − 14.8 − 5.7 − 5.8 − 6.1
− 0.221 − 0.286 − 0.244 − 0.166 − 0.233 − 0.273
− 0.20 − 0.18 − 0.13 − 0.15 − 0.15 − 0.14
considered: case A occurs at heights exceeding about Y* = 0.8 and whereas case B occurs below. 3.3. Friction stress (ε = 0.1) The friction stress calculated as follows: !! ∂V ! τw = −μ ∂x
ð4Þ
wall
(Fig. 9), two kinds of airflow behavior (denoted A and B in Fig. 10) can be extrapolated: • In case A, the downward flow directly feeds the cold boundary layer. • In case B, the downward airflow feeds the hot boundary layer at a lower height. 3D LES computations, obtained by A. Sergent [38], show that, for RaH = 1.2 × 1011, each scenario occurs depending on the heights
yields the friction coefficient, corresponding to the dimensionless friction stress
Cf =
! 2 τw 2 ρ0 VREF
:
ð5Þ
1
0,98
0,96
Y*
0,94
0,92 4.0x10^10 0,9 8.1x10^10 0,88
0,86 -0,04
1.2x10^11
-0,02
0
0,02
0,04
0,06
U* Fig. 8. Horizontal component of the velocity near the ceiling at mid-width and mid-depth.
0,08
D. Saury et al. / International Communications in Heat and Mass Transfer 38 (2011) 679–687 Table 4 Ceiling jet characteristics at mid-width and mid-depth (X⁎ = 0.130; Z⁎ = 0.112). ΔT [°C]
RaH
Q [g.s− 1.m− 1]
Umax [m.s− 1]
Umax/VREF
7 14 20
4.0 × 1010 8.1 × 1010 1.2 × 1011
9.0 8.9 9.2
0.068 0.075 0.085
0.061 0.047 0.045
685
The friction stresses and the friction coefficients are shown in Tables 5 and 6 respectively for several values of the Rayleigh number. Along the boundary layers, friction stress increases with the rise of RaH. This trend is due to the fact that velocity increases with the Rayleigh number (see for instance Vmax in Table 2) whereas the thickness of these boundary layers does not significantly change (see
Fig. 9. Dimensionless airflow map in the upper part of the cavity.
Fig. 10. Scenario of flow behavior in the upper part of the cavity.
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Table 5 Friction stress, τW [Pa]. RaH
Table 7 Nusselt number: comparisons for various RaH.
Hot Wall Y* = 0.7 10
4.0 × 10 8.1 × 1010 1.2 × 1011
−4
5.8 × 10 9.4 × 10− 4 1.2 × 10− 3
Cold wall Y* = 0.85 −4
3.9 × 10 7.4 × 10− 4 8.6 × 10− 4
Ceiling
Y* = 0.7
Y* = 0.85
−4
−4
9.5 × 10 1.7 × 10− 3 2.0 × 10− 3
8.5 × 10 1.4 × 10− 3 1.8 × 10− 3
X = 0.5 m −6
3.2 × 10 2.5 × 10− 6 2.2 × 10− 6
Table 6 Friction coefficient, Cf [−]. RaH
4.0 × 1010 8.1 × 1010 1.2 × 1011
Hot wall
Cold wall
Ceiling
Y* = 0.7
Y* = 0.85
Y* = 0.7
Y* = 0.85
X = 0.5 m
7.7 × 10− 4 6.3 × 10− 4 5.5 × 10− 3
5.2 × 10− 4 4.9 × 10− 4 4.0 × 10− 4
1.3 × 10− 3 1.1 × 10− 3 9.2 × 10− 4
1.1 × 10− 3 9.7 × 10− 4 8.3 × 10− 4
4.3 × 10− 6 1.6 × 10− 6 1.0 × 10− 6
for example the vertical velocity profiles Figs. 4 and 6). It entails an in ! and consequently of increase of the wall velocity gradient ∂∂xV wall friction stress (τw). On the contrary, the friction coefficient decreases when the Rayleigh number increases. Indeed, the dimensionless velocity decreases slightly when the Rayleigh number increases whereas the pffiffiffiffiffiffiffiffiffi reference velocity increases as RaH . Furthermore, as the velocity in the cold boundary layer is higher than in the hot one, the friction is greater at the same height in the cold boundary layer (about twice as great). 3.4. Wall heat transfer–Nusselt number (ε = 0.1)
AV
AH
RaH
bNuN
3.84 3.84
0.86 0.86
1.2 × 1011 1.2 × 1011
231 254
The difficulty in carrying out these measurements is due to the thickness of the thermal boundary layer in the large cavity (4 m3). Indeed, temperature profile is actually linear over a few millimeters. In this area, measurement must be carried out carefully in order to determine the slope of the profile in the near wall area. The same problem appears for numerical simulations where a suitable mesh must be used to obtain an accurate Nusselt number. The experimental values obtained using Eq. (6) are plotted in Fig. 11 for RaH = 1.2 × 1011. The comparison between these experimental values and a 3D LES simulation performed by A. Sergent [38] provided in Fig. 11 shows good agreement. More specifically, similar trends can be found in spatial variation of the Nusselt number, i.e. a first part where NuH decreases ([0; 0.2]), followed by an increase of the local Nusselt number ([0.2 ; 0.3]) probably due to transition state, and finally an almost constant value for NuH is achieved ([0.3 ; 1]).The reason for the sudden modification of the Nusselt profile is still under investigation. Two possible explanations have been investigated but they await confirmation. Integrating the mean local Nusselt number over the Y direction, the overall Nusselt number 〈 is determined. A value of bNuN = 231 ± 30 is then calculated for RaH = 1.2 × 1011. This value can be compared to the one obtained by A. Sergent for RaH = 1.2 × 1011, i.e. bNuN = 254 (Table 7). 4. Conclusion
Using dimensionless temperature profile in the vicinity of the wall determined with a micro-thermocouple (12.7 μm), the mean wall heat flux can be calculated:
H ∂T ∂θ NuH Y = = : ΔT ∂x Y ∂X Y
Present study Sergent [38]
ð6Þ
This paper provides an initial experimental characterization of a differentially-heated cavity at large Rayleigh numbers (on the order of 1011). Influence of wall radiation on the stratification parameter has been highlighted. Velocity profiles in the upper part of the cavity for the three Rayleigh numbers (4.0 × 1010, 8.1 × 1010 and 1.2 × 1011) are given as well as the air flow rate or the friction coefficient. The mean local Nusselt number is indicated along with recommendations for its
Fig. 11. Local Nusselt along the hot wall for RaH = 1.2 × 1011.
D. Saury et al. / International Communications in Heat and Mass Transfer 38 (2011) 679–687
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