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Non-Boussinesq experiments on natural ventilation in a 2D semi-confined enclosure
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International Journal of Heat and Fluid Flow 0 0 0 (2016) 1–5
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Non-Boussinesq experiments on natural ventilation in a 2D semi-confined enclosure O. Vauquelin a,∗, R. Mehaddi b, E. Casalé a, E. Valério a a b
Aix Marseille Université, CNRS, IUSTI UMR 7343, 13 453 Marseille, France Université de Lorraine, CNRS, LEMTA UMR 7563, 54 518 Vandoeuvre-lès-Nancy, France
a r t i c l e
i n f o
Article history: Received 21 June 2016 Revised 4 November 2016 Accepted 20 December 2016 Available online xxx Keywords: Natural ventilation Displacement and blocked regimes
a b s t r a c t This paper presents an experimental study which aimed to quantify the transition between displacement and blocked regimes in a box naturally ventilated through a lower and an upper opening. It complements and extends the results obtained by Paranthoën & Gonzalez (IJHFF, 2010) with heated air in a similar box. The ventilation was here driven by a buoyant air/helium mixture which was continuously released through a slot at the bottom of the box. The slot and the openings were the same width as the box, so that the experiments can be considered as quasi-2D. The Richardson number Ri of the buoyant source (determined by the ratio between buoyancy and momentum) varied over two orders of magnitude (between 0.1 and 10) in our experiments. An intermediate regime defined by the occurrence of bidirectional flow at the base opening, with incoming fresh air and outgoing buoyant fluid, was observed. This intermediate regime can be predicted by a simple correlation based on the Richardson number of the source and the dimensions of opening areas. The full blocked regime (with no more incoming fresh air) was only obtained for large flow rates and/or weakly buoyant releases corresponding to Richardson numbers lower than about 2 − 3. We found a good agreement with the classical relation published by Woods, Caulfield and Phillips (JFM, 2003) provided that the density differences involved were not too high. © 2016 Elsevier Inc. All rights reserved.
1. Introduction Natural ventilation generally refers to exchange flows generated by a difference in temperatures between an interior space (enclosure, building, ... ) and its external environment. The buoyant forces responsible for these flows result in a source of mass and buoyancy such as a heating or air conditioning system, a fire, or a gas leak located in the interior space. Two different regimes may be considered as described by Linden (1999): the displacement ventilation regime in which there is a marked stratification, and the mixed ventilation regime where the temperature (or density) can be considered as uniform in the interior space. If the enclosure has both lower and upper openings to the ambient environment, the displacement regime is characterized by an inflow through the lower opening and an outflow through the upper opening, assuming a source which releases a fluid lighter than the ambient fluid. As the source volume flux increases, a mixedlike regime can appear, in suitable conditions in which the inner buoyant fluid flows out through both openings. This situation
∗
Corresponding author. E-mail address: [email protected] (O. Vauquelin).
was analysed by Woods et al. (2003) and referred to as a blocked regime. Paranthoën and Gonzalez (2010) (hereafter referred to as PG10) studied displacement and blocked regimes in a rectangular box naturally ventilated by a continuous injection of heated air through a 2D slot located at the bottom of the box. They defined the onset of the blocked regime as the moment at which inner buoyant fluid began to flow out of both lower and upper openings, blocking any inward flow of external fresh air. PG10 then found that the transition between these two regimes (displacement and blocked) has to be associated with a so-called intermediate regime as long as the flow remains bidirectional at the base opening (see Fig. 1). In PG10, the required condition for the appearance of these regimes was based on the value of a Froude number derived from the mean density deficit in the box. They then estimated this density deficit by measuring the excess temperature at half box height. Since experiments in PG10 were performed for temperature differences lower than 80 ◦ C, one of our objectives has been to extend their work to large density contrasts, i.e. for non-Boussinesq flows, using air/helium mixtures. The experimental set-up and procedures are described in Section 2. Section 3 presents the experimental results for the transition between the displacement and intermediate regimes. The
http://dx.doi.org/10.1016/j.ijheatfluidflow.2016.12.006 0142-727X/© 2016 Elsevier Inc. All rights reserved.
Please cite this article as: O. Vauquelin et al., Non-Boussinesq experiments on natural ventilation in a 2D semi-confined enclosure, International Journal of Heat and Fluid Flow (2016), http://dx.doi.org/10.1016/j.ijheatfluidflow.2016.12.006
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Fig. 1. Scheme of the three regimes identified by Paranthoën and Gonzalez (2010): (a) is the displacement regime, (b) is the intermediate regime and (c) is the blocked regime.
helium) to 0.9 kg m−3 for the less buoyant mixtures. The ratio between the length jet (see Morton, 1959) and the box height is far from being small, which does not allow the ideal displacement flow theory (Linden et al., 1990; Kaye and Hunt, 2004; Vauquelin, 2015) to be considered as a suitable support for our experiments. For this reason, we have decided in this paper to simply use the experimental results by correlating, at transitions between regimes, the geometrical parameters Hb /H and Ht /H with a dimensionless parameter derived from the buoyant source characteristics (length e, velocity U and density ρ ). This dimensionless source parameter is the Richardson number Ri (also called the densimetric Froude number) which quantifies the ratio between buoyancy and momentum:
Ri =
Fig. 2. Schematic drawing of the experimental set-up.
transition between the intermediate and blocked regimes is addressed in Section 4. Conclusions are drawn in Section 5. 2. Experiments The experimental set-up is shown in Fig. 2. As in PG10, the enclosure was a cuboid box of height H = 200 mm, length L = 500 mm and width W = 250 mm. The buoyant fluid was released upwards in the box from a 2D slot (250 mm wide), flush with the bottom and located at the mid-point of the long wall of the box. The length of the slot (e = 30 mm) was kept constant for all experiments. The box was naturally ventilated through a lower 2D opening of height Hb and an upper 2D opening of height Ht (both 250 mm wide). The side walls could be moved vertically to allow the heights of the openings to be varied. All walls, including the top, were composed of 6 mm plexiglass which permitted flow visualization. To reach high density contrasts, a light air/helium mixture was used as the buoyant fluid. The mixture was first released into a plenum chamber located below the box. This plenum was connected to the box by a 2D slot. Satisfactorily uniform flow was obtained over its full width. The air and helium flow rates were controlled by two independent flow meters (Bronkhorst E-70 0 0 flow −1 meter) operating in the range 0 − 60 m3 h with an accuracy of ± 0.5% of the measurement. To visualize the flow, the mixture was seeded with ammonium salt (obtained by chemical reaction of ammonia vapour with hydrochloric acid) before entering the plenum. These particles (a few microns in size) act as a passive scalar in the flow since the mass flow rate added is negligible compared with the (air-helium) mixture mass flow rate and does not affect the buoyancy. A laser light sheet (argon 2 W) transected the longest plane of symmetry of the box for visualizations. In experiments, the mixture flow rate ranged from 20 to 700 l mn−1 , and the mixture’s density varied from 0.177 kg m−3 (pure
ρ ge , ρU 2
(1)
with g as the gravitational acceleration and ρ = ρ0 − ρ the density difference defined by reference to the ambient density ρ 0 . It should be noted that this form of the Richardson number uses ρ (and not ρ 0 ) in the denominator. This formulation has been found to be a suitable scaling parameter in many non-Boussinesq problems such as plumes (Crapper and Baines, 1978) and fountains (Mehaddi et al., 2015) to mention but a few. Two series of observations were performed for several flow rates and densities of the mixture, covering a range of source Richardson numbers between 0.1 and 10. In the first series, the lower opening height was fixed. The upper opening height was sufficiently high at the beginning of a test to ensure a displacement regime and then gradually decreased until the intermediate regime was reached (bidirectional flow at the lower opening). The upper opening area then continued to be decreased until the blocked regime appeared, when possible. The experimental procedure was similar in the second series of tests except that the upper opening height was fixed and the height of the lower opening height was varied. Transitions between natural, intermediate and blocked regimes were visually assessed leading to a statistical uncertainty of about ± 1 mm for Ht or Hb (based on a same test repeated several times). The three regimes are illustrated in Figs. 3 and 4 from photographs taken during experiments. 3. Intermediate regime This section presents the results obtained during the transition between the displacement and the intermediate regimes. The onset of the intermediate regime was easy to identify visually, corresponding to the appearance of a (weak) spill plume at the lower opening. Fig. 5 presents the results obtained for two different fixed values of the lower opening dimensionless height: Hb /H = 0.18 and Hb /H = 0.25. The upper opening dimensionless height Ht /H is then plotted as a function of the source Richardson number at transition. For a given value of Hb /H, the dispersion of the experimental points is very weak, suggesting that Ri is indeed the suitable scaling parameter. It may be seen that Ht /H monotonically decreases for increasing values of Ri. This indicates that the upper opening
Please cite this article as: O. Vauquelin et al., Non-Boussinesq experiments on natural ventilation in a 2D semi-confined enclosure, International Journal of Heat and Fluid Flow (2016), http://dx.doi.org/10.1016/j.ijheatfluidflow.2016.12.006
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Fig. 3. Photographs from experiments exhibiting (a) a displacement regime and (b) an intermediate regime.
Fig. 4. Photograph from experiments exhibiting a (fully) blocked regime.
Fig. 6. Evolution of the coefficient of proportionality k in relation (2) as a function of the lower opening dimensionless height Hb /H for Ht /H = 0.15. White crosses correspond to the values of k found from the best fits in previous experiments carried out for the variable Ht /H.
a real value for the slope is a choice that we made). This enables us to establish the following correlation for the transition between displacement and intermediate regimes:
Ri
Fig. 5. Upper opening dimensionless height as a function of the source Richardson number at transition between displacement and intermediate regimes. Squares represent Hb /H = 0.18, black circles Hb /H = 0.25. The straight lines correspond to a best fit with a slope of about −2/7.
area required to reach the intermediate regime can be reduced as the ratio between buoyancy and momentum increases in the released fluid. Although this is out of the range investigated, it is expected that Ht /H should tend towards 0 as Ri tends towards infinity. In a log-log plot, the experimental points for both Hb /H = 0.18 and Hb /H = 0.25 in Fig. 5 are good fit for a straight line of slope close to −2/7 (note that using a fraction of integers rather than
−2/7
≈k
Ht , H
(2)
with k ≈ 7.8 for Hb /H = 0.18 and k ≈ 6.3 for Hb /H = 0.25. These correlations are also plotted in Fig. 5 together with the experimental data. The coefficient k might obviously depend on the ratio Hb /H. Experiments were then carried out for a fixed value of the upper opening dimensionless height: Ht /H = 0.15. Given the relation (2), −2/7 the coefficient of proportionality k = 0.667 Ri was plotted as a function of the lower opening dimensionless height. These results are presented in Fig. 6 and show that k may be considered as a linearly decreasing function of Hb /H, at least in the range investigated. In particular, it should be noted that the coefficients of proportionality (7.8 and 6.3 indicated by white crosses in Fig. 6) found from previous experiments carried out for variable Ht /H fit very well with the data points obtained for variable Hb /H. On the basis of our experimental results only, a general correlation for the transition between displacement and intermediate regimes can therefore be proposed as follows:
1−2
Hb Ht −2/7 ≈ 0.08 Ri . H H
(3)
Please cite this article as: O. Vauquelin et al., Non-Boussinesq experiments on natural ventilation in a 2D semi-confined enclosure, International Journal of Heat and Fluid Flow (2016), http://dx.doi.org/10.1016/j.ijheatfluidflow.2016.12.006
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Fig. 7. Upper opening dimensionless height as a function of the source Richardson number at transition between intermediate and blocked regimes. Squares represent Hb /H = 0.18, black circles Hb /H = 0.25. The straight line of the best fit exhibits a slope of about −2/3 for Ri < 1. The dashed line shows the trend for greater values of Ri.
Fig. 8. Upper opening dimensionless height as a function of the source Richardson number at the transition between intermediate and blocked regimes. Black circles are experimental data (obtained for Hb /H = 0.18 and Hb /H = 0.25) and the straight line corresponds to the theoretical prediction by Woods et al. (2003).
4. Blocked regime The transition between the intermediate and blocked regimes was investigated for two fixed values of the lower opening dimensionless height (Hb /H = 0.18 and Hb /H = 0.25) by decreasing the size of the upper opening from an intermediate position. The blocked regime was relatively easy to reach for sources of low or moderate buoyancy. For sources of high buoyancy (in particular for the tests performed with pure helium), a strong stratification remained whatever the source momentum (at least in the range investigated). A thin layer of incoming fresh air was continually present and a truly blocked regime could not be obtained. Fig. 7 presents the results obtained. The upper opening dimensionless height Ht /H is plotted as a function of the source Richardson number at transition between the intermediate and the blocked regimes. It can be seen that Ht /H monotonically decreases as Ri increases. In practice, the blocked regime was never observed for Ri greater than 3, more or less. Since the experimental points for Hb /H = 0.18 and Hb /H = 0.25 are very close, we can suppose that the lower opening area weakly influences the appearance of the blocked regime which is mainly due to the blockage effect caused by the reduction of the area for the outflow. In a log-log plot, the experimental points for Hb /H = 0.18 and Hb /H = 0.25 in Fig. 7 align rather well along a straight line whose slope is about −2/3 as long as Ri < 1. For greater values of Ri, the slope is much steeper, indicating the difficulty of reaching a blocked regime as long as the buoyancy of the mixture allows stratification to exist in the box. Finally, it is interesting to compare these experimental data with the condition described by Woods et al. (2003) (and also retrieved theoretically in PG10) for the appearance of the blocked regime. For these authors, the natural exchange flow became fully blocked when:
Q0 = (CD Ht )2/3 (2B0 H )1/3 ,
(4)
where Q0 and B0 are the source volume flux and the source buoyancy flux, respectively, and CD is the discharge coefficient. As Q0 = √ Ue and B0 = g ρρ Q0 , and considering that CD ≈ 1/ 2, we have:
Ht = H
e 3 / 2 H
Ri
−1/2
.
(5)
Fig. 8 presents a comparison between Eq. (5) and our experimental data. Order of magnitudes are satisfactorily recovered for the lowest Richardson numbers investigated. Nevertheless, for higher values of the Richardson number (i.e. for low density releases when buoyancy dominates momentum), it seems that the
Fig. 9. Evolution of the ratio between Ht /H given by relation (5) and Ht /H obtained experimentally as a function of the source density deficit ρ /ρ .
theoretical relation (5) tends to overestimate the value of the upper opening which would allow the blocked regime to appear. In order to be sure that density effects are responsible for these differences, the ratio between Ht /H given by relation (5) and Ht /H obtained experimentally has been plotted as a function of the density deficit ρ /ρ . Fig. 9 depicts the divergence between theory and experiment with higher values of ρ /ρ . This may be interpreted as a non-Boussinesq effect. 5. Conclusions Experiments were performed for a quasi-2D flow produced by a source of mass and buoyancy in a box ventilated to the exterior by two openings (a lower and an upper opening). The size of the openings was variable as were the flow rate and density of the fluid released. The box and the slot for injection had fixed dimensions. Following Paranthoën and Gonzalez (2010), the transition between the different flow regimes (displacement, intermediate and blocked) was studied for several values of the source buoyant flux and several dimensions of the openings. In contrast with previous studies our experiments were carried out for large density contrasts, i.e. outside the Boussinesq approximation, using air and helium mixtures. The buoyant release was associated with a non-Boussinesq Richardson number Ri that varied between 0.1 and 10 in our experiments. For a given dimension of the lower opening, correlations derived from our experiments indicated that the height of the upper opening Ht varies with Ri−2/7 at the transition
Please cite this article as: O. Vauquelin et al., Non-Boussinesq experiments on natural ventilation in a 2D semi-confined enclosure, International Journal of Heat and Fluid Flow (2016), http://dx.doi.org/10.1016/j.ijheatfluidflow.2016.12.006
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between displacement and intermediate regimes and with Ri−2/3 to reach a fully blocked regime from an intermediate regime. Our experimental data suggest that the theoretical relations published in Woods et al. (2003); Paranthoën and Gonzalez (2010) seem to overestimate the height of the upper opening required to reach a blocked regime as soon as density contrasts become significant. References Crapper, P.F., Baines, W.D., 1978. Some remarks on non-Boussinesq forced plumes. Atmos. Environ. 12, 1939–1941. Kaye, N.B., Hunt, G.R., 2004. Time-dependent flows in an emptying filling box. J. Fluid Mech. 520, 135–156.
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Linden, P.F., 1999. The fluid mechanics of natural ventilation. Annu. Rev. Fluid Mech. 31, 201–238. Linden, P.F., Lane-Serff, G.F., Smeed, D.A., 1990. Emptying filling boxes, the fluid mechanics of natural ventilation. J. Fluid Mech. 212, 309–335. Mehaddi, R., Vauquelin, O., Candelier, F., 2015. Experimental non-Boussinesq fountains. J. Fluid Mech. 784, R6. Morton, B.R., 1959. Forced plumes. J. Fluid Mech. 5, 151–163. Paranthoën, P., Gonzalez, M., 2010. Mixed convection in a ventilated enclosure. Int. J. Heat Fluid Flow 31, 172–178. Vauquelin, O., 2015. Oscillatory behaviour in an emptying-filling box. J. Fluid Mech. 781, 712–726. Woods, A.W., Caulfield, C.P., Phillips, J.C., 2003. Blocked natural ventilation: the effect of a source mass flux. J. Fluid Mech. 495, 119–133.
Please cite this article as: O. Vauquelin et al., Non-Boussinesq experiments on natural ventilation in a 2D semi-confined enclosure, International Journal of Heat and Fluid Flow (2016), http://dx.doi.org/10.1016/j.ijheatfluidflow.2016.12.006
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International Journal of Heat and Fluid Flow 0 0 0 (2016) 1–5
Contents lists available at ScienceDirect
International Journal of Heat and Fluid Flow journal homepage: www.elsevier.com/locate/ijhff
Non-Boussinesq experiments on natural ventilation in a 2D semi-confined enclosure O. Vauquelin a,∗, R. Mehaddi b, E. Casalé a, E. Valério a a b
Aix Marseille Université, CNRS, IUSTI UMR 7343, 13 453 Marseille, France Université de Lorraine, CNRS, LEMTA UMR 7563, 54 518 Vandoeuvre-lès-Nancy, France
a r t i c l e
i n f o
Article history: Received 21 June 2016 Revised 4 November 2016 Accepted 20 December 2016 Available online xxx Keywords: Natural ventilation Displacement and blocked regimes
a b s t r a c t This paper presents an experimental study which aimed to quantify the transition between displacement and blocked regimes in a box naturally ventilated through a lower and an upper opening. It complements and extends the results obtained by Paranthoën & Gonzalez (IJHFF, 2010) with heated air in a similar box. The ventilation was here driven by a buoyant air/helium mixture which was continuously released through a slot at the bottom of the box. The slot and the openings were the same width as the box, so that the experiments can be considered as quasi-2D. The Richardson number Ri of the buoyant source (determined by the ratio between buoyancy and momentum) varied over two orders of magnitude (between 0.1 and 10) in our experiments. An intermediate regime defined by the occurrence of bidirectional flow at the base opening, with incoming fresh air and outgoing buoyant fluid, was observed. This intermediate regime can be predicted by a simple correlation based on the Richardson number of the source and the dimensions of opening areas. The full blocked regime (with no more incoming fresh air) was only obtained for large flow rates and/or weakly buoyant releases corresponding to Richardson numbers lower than about 2 − 3. We found a good agreement with the classical relation published by Woods, Caulfield and Phillips (JFM, 2003) provided that the density differences involved were not too high. © 2016 Elsevier Inc. All rights reserved.
1. Introduction Natural ventilation generally refers to exchange flows generated by a difference in temperatures between an interior space (enclosure, building, ... ) and its external environment. The buoyant forces responsible for these flows result in a source of mass and buoyancy such as a heating or air conditioning system, a fire, or a gas leak located in the interior space. Two different regimes may be considered as described by Linden (1999): the displacement ventilation regime in which there is a marked stratification, and the mixed ventilation regime where the temperature (or density) can be considered as uniform in the interior space. If the enclosure has both lower and upper openings to the ambient environment, the displacement regime is characterized by an inflow through the lower opening and an outflow through the upper opening, assuming a source which releases a fluid lighter than the ambient fluid. As the source volume flux increases, a mixedlike regime can appear, in suitable conditions in which the inner buoyant fluid flows out through both openings. This situation
∗
Corresponding author. E-mail address: [email protected] (O. Vauquelin).
was analysed by Woods et al. (2003) and referred to as a blocked regime. Paranthoën and Gonzalez (2010) (hereafter referred to as PG10) studied displacement and blocked regimes in a rectangular box naturally ventilated by a continuous injection of heated air through a 2D slot located at the bottom of the box. They defined the onset of the blocked regime as the moment at which inner buoyant fluid began to flow out of both lower and upper openings, blocking any inward flow of external fresh air. PG10 then found that the transition between these two regimes (displacement and blocked) has to be associated with a so-called intermediate regime as long as the flow remains bidirectional at the base opening (see Fig. 1). In PG10, the required condition for the appearance of these regimes was based on the value of a Froude number derived from the mean density deficit in the box. They then estimated this density deficit by measuring the excess temperature at half box height. Since experiments in PG10 were performed for temperature differences lower than 80 ◦ C, one of our objectives has been to extend their work to large density contrasts, i.e. for non-Boussinesq flows, using air/helium mixtures. The experimental set-up and procedures are described in Section 2. Section 3 presents the experimental results for the transition between the displacement and intermediate regimes. The
http://dx.doi.org/10.1016/j.ijheatfluidflow.2016.12.006 0142-727X/© 2016 Elsevier Inc. All rights reserved.
Please cite this article as: O. Vauquelin et al., Non-Boussinesq experiments on natural ventilation in a 2D semi-confined enclosure, International Journal of Heat and Fluid Flow (2016), http://dx.doi.org/10.1016/j.ijheatfluidflow.2016.12.006
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Fig. 1. Scheme of the three regimes identified by Paranthoën and Gonzalez (2010): (a) is the displacement regime, (b) is the intermediate regime and (c) is the blocked regime.
helium) to 0.9 kg m−3 for the less buoyant mixtures. The ratio between the length jet (see Morton, 1959) and the box height is far from being small, which does not allow the ideal displacement flow theory (Linden et al., 1990; Kaye and Hunt, 2004; Vauquelin, 2015) to be considered as a suitable support for our experiments. For this reason, we have decided in this paper to simply use the experimental results by correlating, at transitions between regimes, the geometrical parameters Hb /H and Ht /H with a dimensionless parameter derived from the buoyant source characteristics (length e, velocity U and density ρ ). This dimensionless source parameter is the Richardson number Ri (also called the densimetric Froude number) which quantifies the ratio between buoyancy and momentum:
Ri =
Fig. 2. Schematic drawing of the experimental set-up.
transition between the intermediate and blocked regimes is addressed in Section 4. Conclusions are drawn in Section 5. 2. Experiments The experimental set-up is shown in Fig. 2. As in PG10, the enclosure was a cuboid box of height H = 200 mm, length L = 500 mm and width W = 250 mm. The buoyant fluid was released upwards in the box from a 2D slot (250 mm wide), flush with the bottom and located at the mid-point of the long wall of the box. The length of the slot (e = 30 mm) was kept constant for all experiments. The box was naturally ventilated through a lower 2D opening of height Hb and an upper 2D opening of height Ht (both 250 mm wide). The side walls could be moved vertically to allow the heights of the openings to be varied. All walls, including the top, were composed of 6 mm plexiglass which permitted flow visualization. To reach high density contrasts, a light air/helium mixture was used as the buoyant fluid. The mixture was first released into a plenum chamber located below the box. This plenum was connected to the box by a 2D slot. Satisfactorily uniform flow was obtained over its full width. The air and helium flow rates were controlled by two independent flow meters (Bronkhorst E-70 0 0 flow −1 meter) operating in the range 0 − 60 m3 h with an accuracy of ± 0.5% of the measurement. To visualize the flow, the mixture was seeded with ammonium salt (obtained by chemical reaction of ammonia vapour with hydrochloric acid) before entering the plenum. These particles (a few microns in size) act as a passive scalar in the flow since the mass flow rate added is negligible compared with the (air-helium) mixture mass flow rate and does not affect the buoyancy. A laser light sheet (argon 2 W) transected the longest plane of symmetry of the box for visualizations. In experiments, the mixture flow rate ranged from 20 to 700 l mn−1 , and the mixture’s density varied from 0.177 kg m−3 (pure
ρ ge , ρU 2
(1)
with g as the gravitational acceleration and ρ = ρ0 − ρ the density difference defined by reference to the ambient density ρ 0 . It should be noted that this form of the Richardson number uses ρ (and not ρ 0 ) in the denominator. This formulation has been found to be a suitable scaling parameter in many non-Boussinesq problems such as plumes (Crapper and Baines, 1978) and fountains (Mehaddi et al., 2015) to mention but a few. Two series of observations were performed for several flow rates and densities of the mixture, covering a range of source Richardson numbers between 0.1 and 10. In the first series, the lower opening height was fixed. The upper opening height was sufficiently high at the beginning of a test to ensure a displacement regime and then gradually decreased until the intermediate regime was reached (bidirectional flow at the lower opening). The upper opening area then continued to be decreased until the blocked regime appeared, when possible. The experimental procedure was similar in the second series of tests except that the upper opening height was fixed and the height of the lower opening height was varied. Transitions between natural, intermediate and blocked regimes were visually assessed leading to a statistical uncertainty of about ± 1 mm for Ht or Hb (based on a same test repeated several times). The three regimes are illustrated in Figs. 3 and 4 from photographs taken during experiments. 3. Intermediate regime This section presents the results obtained during the transition between the displacement and the intermediate regimes. The onset of the intermediate regime was easy to identify visually, corresponding to the appearance of a (weak) spill plume at the lower opening. Fig. 5 presents the results obtained for two different fixed values of the lower opening dimensionless height: Hb /H = 0.18 and Hb /H = 0.25. The upper opening dimensionless height Ht /H is then plotted as a function of the source Richardson number at transition. For a given value of Hb /H, the dispersion of the experimental points is very weak, suggesting that Ri is indeed the suitable scaling parameter. It may be seen that Ht /H monotonically decreases for increasing values of Ri. This indicates that the upper opening
Please cite this article as: O. Vauquelin et al., Non-Boussinesq experiments on natural ventilation in a 2D semi-confined enclosure, International Journal of Heat and Fluid Flow (2016), http://dx.doi.org/10.1016/j.ijheatfluidflow.2016.12.006
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3
Fig. 3. Photographs from experiments exhibiting (a) a displacement regime and (b) an intermediate regime.
Fig. 4. Photograph from experiments exhibiting a (fully) blocked regime.
Fig. 6. Evolution of the coefficient of proportionality k in relation (2) as a function of the lower opening dimensionless height Hb /H for Ht /H = 0.15. White crosses correspond to the values of k found from the best fits in previous experiments carried out for the variable Ht /H.
a real value for the slope is a choice that we made). This enables us to establish the following correlation for the transition between displacement and intermediate regimes:
Ri
Fig. 5. Upper opening dimensionless height as a function of the source Richardson number at transition between displacement and intermediate regimes. Squares represent Hb /H = 0.18, black circles Hb /H = 0.25. The straight lines correspond to a best fit with a slope of about −2/7.
area required to reach the intermediate regime can be reduced as the ratio between buoyancy and momentum increases in the released fluid. Although this is out of the range investigated, it is expected that Ht /H should tend towards 0 as Ri tends towards infinity. In a log-log plot, the experimental points for both Hb /H = 0.18 and Hb /H = 0.25 in Fig. 5 are good fit for a straight line of slope close to −2/7 (note that using a fraction of integers rather than
−2/7
≈k
Ht , H
(2)
with k ≈ 7.8 for Hb /H = 0.18 and k ≈ 6.3 for Hb /H = 0.25. These correlations are also plotted in Fig. 5 together with the experimental data. The coefficient k might obviously depend on the ratio Hb /H. Experiments were then carried out for a fixed value of the upper opening dimensionless height: Ht /H = 0.15. Given the relation (2), −2/7 the coefficient of proportionality k = 0.667 Ri was plotted as a function of the lower opening dimensionless height. These results are presented in Fig. 6 and show that k may be considered as a linearly decreasing function of Hb /H, at least in the range investigated. In particular, it should be noted that the coefficients of proportionality (7.8 and 6.3 indicated by white crosses in Fig. 6) found from previous experiments carried out for variable Ht /H fit very well with the data points obtained for variable Hb /H. On the basis of our experimental results only, a general correlation for the transition between displacement and intermediate regimes can therefore be proposed as follows:
1−2
Hb Ht −2/7 ≈ 0.08 Ri . H H
(3)
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Fig. 7. Upper opening dimensionless height as a function of the source Richardson number at transition between intermediate and blocked regimes. Squares represent Hb /H = 0.18, black circles Hb /H = 0.25. The straight line of the best fit exhibits a slope of about −2/3 for Ri < 1. The dashed line shows the trend for greater values of Ri.
Fig. 8. Upper opening dimensionless height as a function of the source Richardson number at the transition between intermediate and blocked regimes. Black circles are experimental data (obtained for Hb /H = 0.18 and Hb /H = 0.25) and the straight line corresponds to the theoretical prediction by Woods et al. (2003).
4. Blocked regime The transition between the intermediate and blocked regimes was investigated for two fixed values of the lower opening dimensionless height (Hb /H = 0.18 and Hb /H = 0.25) by decreasing the size of the upper opening from an intermediate position. The blocked regime was relatively easy to reach for sources of low or moderate buoyancy. For sources of high buoyancy (in particular for the tests performed with pure helium), a strong stratification remained whatever the source momentum (at least in the range investigated). A thin layer of incoming fresh air was continually present and a truly blocked regime could not be obtained. Fig. 7 presents the results obtained. The upper opening dimensionless height Ht /H is plotted as a function of the source Richardson number at transition between the intermediate and the blocked regimes. It can be seen that Ht /H monotonically decreases as Ri increases. In practice, the blocked regime was never observed for Ri greater than 3, more or less. Since the experimental points for Hb /H = 0.18 and Hb /H = 0.25 are very close, we can suppose that the lower opening area weakly influences the appearance of the blocked regime which is mainly due to the blockage effect caused by the reduction of the area for the outflow. In a log-log plot, the experimental points for Hb /H = 0.18 and Hb /H = 0.25 in Fig. 7 align rather well along a straight line whose slope is about −2/3 as long as Ri < 1. For greater values of Ri, the slope is much steeper, indicating the difficulty of reaching a blocked regime as long as the buoyancy of the mixture allows stratification to exist in the box. Finally, it is interesting to compare these experimental data with the condition described by Woods et al. (2003) (and also retrieved theoretically in PG10) for the appearance of the blocked regime. For these authors, the natural exchange flow became fully blocked when:
Q0 = (CD Ht )2/3 (2B0 H )1/3 ,
(4)
where Q0 and B0 are the source volume flux and the source buoyancy flux, respectively, and CD is the discharge coefficient. As Q0 = √ Ue and B0 = g ρρ Q0 , and considering that CD ≈ 1/ 2, we have:
Ht = H
e 3 / 2 H
Ri
−1/2
.
(5)
Fig. 8 presents a comparison between Eq. (5) and our experimental data. Order of magnitudes are satisfactorily recovered for the lowest Richardson numbers investigated. Nevertheless, for higher values of the Richardson number (i.e. for low density releases when buoyancy dominates momentum), it seems that the
Fig. 9. Evolution of the ratio between Ht /H given by relation (5) and Ht /H obtained experimentally as a function of the source density deficit ρ /ρ .
theoretical relation (5) tends to overestimate the value of the upper opening which would allow the blocked regime to appear. In order to be sure that density effects are responsible for these differences, the ratio between Ht /H given by relation (5) and Ht /H obtained experimentally has been plotted as a function of the density deficit ρ /ρ . Fig. 9 depicts the divergence between theory and experiment with higher values of ρ /ρ . This may be interpreted as a non-Boussinesq effect. 5. Conclusions Experiments were performed for a quasi-2D flow produced by a source of mass and buoyancy in a box ventilated to the exterior by two openings (a lower and an upper opening). The size of the openings was variable as were the flow rate and density of the fluid released. The box and the slot for injection had fixed dimensions. Following Paranthoën and Gonzalez (2010), the transition between the different flow regimes (displacement, intermediate and blocked) was studied for several values of the source buoyant flux and several dimensions of the openings. In contrast with previous studies our experiments were carried out for large density contrasts, i.e. outside the Boussinesq approximation, using air and helium mixtures. The buoyant release was associated with a non-Boussinesq Richardson number Ri that varied between 0.1 and 10 in our experiments. For a given dimension of the lower opening, correlations derived from our experiments indicated that the height of the upper opening Ht varies with Ri−2/7 at the transition
Please cite this article as: O. Vauquelin et al., Non-Boussinesq experiments on natural ventilation in a 2D semi-confined enclosure, International Journal of Heat and Fluid Flow (2016), http://dx.doi.org/10.1016/j.ijheatfluidflow.2016.12.006
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between displacement and intermediate regimes and with Ri−2/3 to reach a fully blocked regime from an intermediate regime. Our experimental data suggest that the theoretical relations published in Woods et al. (2003); Paranthoën and Gonzalez (2010) seem to overestimate the height of the upper opening required to reach a blocked regime as soon as density contrasts become significant. References Crapper, P.F., Baines, W.D., 1978. Some remarks on non-Boussinesq forced plumes. Atmos. Environ. 12, 1939–1941. Kaye, N.B., Hunt, G.R., 2004. Time-dependent flows in an emptying filling box. J. Fluid Mech. 520, 135–156.
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Linden, P.F., 1999. The fluid mechanics of natural ventilation. Annu. Rev. Fluid Mech. 31, 201–238. Linden, P.F., Lane-Serff, G.F., Smeed, D.A., 1990. Emptying filling boxes, the fluid mechanics of natural ventilation. J. Fluid Mech. 212, 309–335. Mehaddi, R., Vauquelin, O., Candelier, F., 2015. Experimental non-Boussinesq fountains. J. Fluid Mech. 784, R6. Morton, B.R., 1959. Forced plumes. J. Fluid Mech. 5, 151–163. Paranthoën, P., Gonzalez, M., 2010. Mixed convection in a ventilated enclosure. Int. J. Heat Fluid Flow 31, 172–178. Vauquelin, O., 2015. Oscillatory behaviour in an emptying-filling box. J. Fluid Mech. 781, 712–726. Woods, A.W., Caulfield, C.P., Phillips, J.C., 2003. Blocked natural ventilation: the effect of a source mass flux. J. Fluid Mech. 495, 119–133.
Please cite this article as: O. Vauquelin et al., Non-Boussinesq experiments on natural ventilation in a 2D semi-confined enclosure, International Journal of Heat and Fluid Flow (2016), http://dx.doi.org/10.1016/j.ijheatfluidflow.2016.12.006