Boussinesq and non-Boussinesq turbulent plumes in a corner with applications to natural ventilation

A previous formulation of plume merger [Rooney, J. Fluid Mech. 796, 712 (2016)] is generalized to model both Boussinesq and non-Boussinesq plume rise in a corner of arbitrary angle $2\pi /n$ where $n\ge 1$. The Boussinesq plume theory predicts the correct near- and far-field similarity solutions when $n$ is noninteger. Moreover, an alternate entrainment assumption is proposed whereby the rate of entrainment per unit height correlates directly to the plume perimeter. Model predictions made using this alternative entrainment assumption agree well with a previous prediction for the plume volume flux when $n=2$. For non-Boussinesq plumes, the theory also approaches the correct near- and far-field similarity limits. When the source area is compact, and regardless of the corner angle, the non-Boussinesq height, i.e., height over which non-Boussinesq effects are important, is small compared to the contact height between the plume and the corner. When the source area is relatively large, the non-Boussinesq height can be comparable to the contact height; enhanced non-Boussinesq effects are observed for smaller corner angles. Our Boussinesq theory is adapted to the natural ventilation model developed by Linden et al. [J. Fluid Mech. 212, 309 (1990)] and agrees well with previous experimental and theoretical predictions for the steady-state depth of the layer of discharged plume fluid that accumulates along the ceiling of the (ventilated) interior space. For non-Boussinesq plumes, the counterpart theory compares satisfactorily with previously measured results of fire plume mass flux.

Figure 1

(a) Top view of a plume in a corner of angle $\beta =2\pi /n$. Here $n<2$, but our analysis applies equally to the case $n\ge 2$. (b) Velocity potential contours for $n=4/3$ ($\beta =3\pi /2$). Contours are labeled according to their $k$ value, $k$ defined in (6). For $k<1$, the solid and dashed curves denote the positive and negative square roots of (7), respectively. The thick solid curve corresponding to $k=1$ represents the level of first contact.

Figure 2

Log-log plot showing the vertical variation of $\widehat{w}$ (a) and $\widehat{Q}$ (b) for $n=7/3$ ($\beta =6\pi /7$). The dotted lines denote the respective near- and far-field similarity solutions, as given in Sec. 4b but with $\eta =1$ where $\eta$ is the ratio of plume density to ambient density. The lower and upper dashed horizontal lines denote ${\widehat{z}}_{c,1}=0.60$ and ${\widehat{z}}_{c,2}=1.28$, respectively. ${\widehat{z}}_{c,1}$ (${\widehat{z}}_{c,2}$) is determined as the height where $k$ is closest to 1 ($n$).

Figure 3

${\widehat{z}}_{c,1}, {\widehat{z}}_{c,2}$, and ${\widehat{z}}_{vf}$ vs $n$ ($\beta =2\pi /n$). ${\widehat{z}}_{c,1}$ and ${\widehat{z}}_{c,2}$ are two contact heights corresponding to $k=1$ and $k=n$, respectively. ${\widehat{z}}_{vf}$ is the far-field virtual origin. In all cases, ${\mathrm{\Gamma }}_0=1$. Virtual origin corrections are included in the manner of (24).

Figure 4

Comparison of $f_e, f_l$, and $2{\left(\pi \widehat{A}\right)}^{1/2}$ as functions of $k$ for $n=2$ (a) and $n=4$ (b). $f_e$ is defined by (21), and $f_l=l/a$ where $l$ denotes the contour length.

Figure 5

Effective entrainment vs height for four different theoretical models. (a) $n=2$. The vertical dashed line represents the far-field limit of $2^{-1/2}$. (b) $n=4$. The vertical dashed line represents the far-field limit of $4^{-1/2}=1/2$. The Cenedese and Linden [13] solution is here omitted because their model is restricted to $n=2$.

Figure 6

Representative results of the nondimensional vertical velocity (a), volume flux (b), density (c), and modified flux-balance parameter (d) for $n=2, {\eta }_0=0.4$ and ${\mathrm{\Gamma }}_{nb,0}=1$. The dotted lines in (a) and (b) denote the near- and far-field similarity limits, respectively. The vertical dashed line in (c) denotes $\eta =0.9$. In all panels, the lower and upper horizontal dashed lines denote the contact heights defined in Sec. 2d.

Figure 7

${\widehat{z}}_{c,1}, {\widehat{z}}_{c,2}$, and ${\widehat{z}}_{nb}$ vs $n$ ($\beta =2\pi /n$). ${\widehat{z}}_{c,1}$ and ${\widehat{z}}_{c,2}$ are two contact heights corresponding to $k=1$ and $k=n$, respectively. ${\widehat{z}}_{nb}$ is defined as the point at which $\eta =0.9$. In all cases, ${\mathrm{\Gamma }}_0=1$ and ${\eta }_0=0.4$. Virtual origin corrections are included in the manner of (43).

Figure 8

(a) ${\widehat{z}}_{nb}$ and ${\widehat{Q}}_0^{3/5}$ vs ${\mathrm{\Gamma }}_{nb,0}$ for $n=2$ and ${\eta }_0=0.2$. (b) ${\widehat{z}}_{nb}$ vs ${\mathrm{\Gamma }}_{nb,0}$ for $n=2$ and different ${\eta }_0$. ${\widehat{z}}_{nb}$ is defined as the point at which $\eta =0.9$. ${\widehat{Q}}_0^{3/5}$ (50) is a theoretically derived scaling height over which non-Boussinesq effects are important.

Figure 9

${\widehat{z}}_{nb}, {\widehat{Q}}_0^{3/5}$ and ${\widehat{z}}_{c,1}$ vs ${\mathrm{\Gamma }}_{nb,0}$ for $n=2$ (a) and $n=4$ (b). In both panels, ${\eta }_0=0.4$. ${\widehat{z}}_{nb}$ is defined as the point at which $\eta =0.9$. ${\widehat{Q}}_0^{3/5}$ (50) is a theoretically derived scaling height over which non-Boussinesq effects are important.

Figure 10

Schematics of centered (left) and wall-bounded (right) plumes in naturally ventilated enclosures. The interface height is $h$, and the total room height is $H$.

Figure 11

Comparison of different model predictions with the experimental data of Linden and Kaye [12] for two merging plumes. (a) Interface height as a function of the distance from the plume source to the adjacent wall with $A^{*}/H^2=0.02$ (data reproduced from Fig. 8 of Linden and Kaye [12] with permission). (b) Interface height as a function of the effective vent area with $a/H=0.05$ (data reproduced from Fig. 11 of Linden and Kaye [12] with permission).

Figure 12

Comparison of different model predictions for the ambient interface height $\zeta$ in the case of a plume in a right-angle corner. (a) Interface height as a function of $a/H$ for $A^{*}/H^2=0.04$. (b) Interface height as a function of $A^{*}/H^2$ for $a/H=0.1$. The red curve represents the solution of (57).

Figure 13

Illustration of plume rise in a right-angle corner of a room (shaded gray). The plume boundaries are shaded according to the nondimensional height, and the contours are projected onto the bottom plane.

Figure 14

Top view of a square ($D\times D$) fire source near a corner. The distance between the burner (i.e., fire source) edge and the wall surface is given by $S$, whereas $a$ denotes the distance between the burner center and the corner vertex [27].

Figure 15

Comparison of theory vs experiment in terms of mass flux at different heights for $S/D=0.5$ (a) and 2 (b). The configuration is a fire in a right-angle corner as shown in Fig. 14.