Confined turbulent convection driven by a combination of line and distributed sources of buoyancy

We study the flow and thermal stratification of a closed domain subjected to different combinations of line and distributed surface heating and cooling. Our observations are drawn from a set of direct numerical simulations in which the ratio of the strength of the distributed sources to the localized sources $\mathrm{\Gamma }$ is varied and shown to play a decisive role in determining the system's statistically steady state. Domains of sufficient horizontal extent that are heated from below and cooled from above in equal amounts by two line sources ($\mathrm{\Gamma }=0$) produce a stable two-layer stratification. The planar plumes generated by each line source are connected by a large-scale circulation over the full depth of the domain and induce secondary circulations within each layer. As the distributed component of the heating, and therefore $\mathrm{\Gamma }$, increases, the buoyancy difference between the layers decreases, before being destroyed when $\mathrm{\Gamma }>1$. For increasing $\mathrm{\Gamma }\in [0,1]$, we observe an increasing tilt of the interface between the layers and the eventual disappearance of the secondary circulation cells. The mean buoyancy transport between the two layers of the stable stratification is dominated by the plumes for all $\mathrm{\Gamma }<1$ because the buoyancy flux associated with interfacial mixing is negligible. Building on existing approaches that typically assume uniform buoyancy within each layer, we develop a model that admits a lateral buoyancy gradient. The model predictions of the buoyancy difference between the layers, the tilt of the interface, and the large-scale circulation strength exhibit a reasonably good agreement with the direct numerical simulation data.

Figure 1

Combined localized (${\widehat{F}}_{\ell }$) and distributed (${\widehat{F}}_{\text{d}}$) buoyancy sources investigated in the literature (${\widehat{F}}_{\ell },{\widehat{F}}_{\text{d}}>0$). Depicted are the configurations for confined volume heating from Wells, Griffiths, and Turner [10], using sources on (a) the same and (b) opposite boundaries. Also sketched are the configurations for the heating of a ventilated domain as considered by (c) Chenvidyakarn and Woods [11] (based on textual description) and (d) Hunt, Holford, and Linden [12] and Partridge and Linden [13].

Figure 2

Schematic of the investigated configurations. The horizontal boundaries are set to a constant local buoyancy flux ${\widehat{f}}_{\ell }$ within a strip of width $r$; the remainder of each boundary is set to ${\widehat{f}}_{\text{d}}$ (top and bottom graphs). The structure of buoyancy $\widehat{b}$ sketched in the middle depicts two plumes in a stable two-layer stratification.

Figure 3

Isopycnals of two oppositely oriented plumes, arising from two localized line sources of equal strength ($\mathrm{\Gamma }=0$). Red marks an isosurface at buoyancy $b\approx 5$, blue corresponds to $b\approx -5$; the color saturation indicates the kinetic energy. The fluid in the bulk of the domain is stably stratified in two layers, which are separated by an interface of zero buoyancy (violet). The displayed isosurface of zero buoyancy is cropped at the boundaries so as not to obscure the plumes.

Figure 4

Instances of the buoyancy field: the individual figures correspond to (a) $\mathrm{\Gamma }=0$, (b) $\mathrm{\Gamma }=0.10$, (c) $\mathrm{\Gamma }=0.23$, (d) $\mathrm{\Gamma }=0.39$, (e) $\mathrm{\Gamma }=0.60$, and (f) $\mathrm{\Gamma }=1.29$.

Figure 5

Time-averaged probability density function of the buoyancy field.

Figure 6

Mean buoyancy $\overline{b}$ along a vertical slice in the middle of the domain ($x=0$). At the chosen position $x=0$, the interface elevation is $z=0$ for all configurations.

Figure 7

Mean buoyancy field $\overline{b}$ and stream lines of the mean flow $\overline{\mathbf{u}}$: the individual figures correspond to (a) $\mathrm{\Gamma }=0$, (b) $\mathrm{\Gamma }=0.10$, (c) $\mathrm{\Gamma }=0.23$, (d) $\mathrm{\Gamma }=0.39$, (e) $\mathrm{\Gamma }=0.60$, and (f) $\mathrm{\Gamma }=1.29$. Dotted streamlines correspond to a finer scaling than solid ones, marking a fifth of the difference between the solid streamlines. The isosurface at buoyancy $b=0$ is marked by a red dashed line; the interface between the layers and its horizontal continuation to the boundaries is marked by a solid red line. The interface is separated from the continuation by a “$+$.” For definitions of the interface and the “$+$” markers, see Appendix pp2.

Figure 8

Profiles of mean velocities: (a) the horizontal velocity $\overline{u}$ along a vertical slice in the middle of the domain ($x=0$) and (b) the mean vertical velocity $\overline{w}$ in the left-hand plume at $z=0$. At the position $x=0$ depicted in (a), the interface elevation is $z=0$ for all configurations and the mean velocity is antisymmetric to $z=0$, $\overline{u}(0,z)=-\overline{u}(0,-z)$.

Figure 9

Mean flow and turbulent contribution to the specific kinetic energy in the domain.

Figure 10

Buoyancy exchange between the two layers of the stable stratification (i.e., across the red solid lines in Fig. 7). Depicted are the buoyancy fluxes $F_{\text{p}}$ and $F_{\text{I}}$, corresponding to the discharge of one plume and mixing across the interface, respectively. In case of $F_{\text{p}}$, the total, the mean flow, and the turbulent contribution are depicted.

Figure 11

Schematics of the buoyancy structure for the models of (a) Sec. 5a and (b) Sec. 5b. The highlighted section is the control volume used. The volume flux across all dashed boundaries is equal $Q_{\text{p}}$.

Figure 12

Difference $\mathrm{\Delta }b$ between the peaks of the buoyancy field's PDF: comparing simulation results and models. Depicted is the buoyancy difference from simulation data (Fig. 5), and the predictions of model 1 and 2 (19). The value $\mathrm{\Delta }b\left(Q_{\text{p}}\right)$ from (23) is a combination of modeling and observation (see Sec. 6c). The models are valid for stable stratification corresponding to $\mathrm{\Gamma }<1$.

Figure 13

Tilt of the interface between the layers of the stratification: comparing simulation results and model 2. The buoyancy field is separated into values above (red) and below (blue) zero; we consider the resulting separation line between the “$+$” markers as the interface (Sec. 4c). The red curve shows the interface $z_{\text{I}}\left(x\right)$ of model 2 (Sec. 5b). The individual figures correspond to (a) $\mathrm{\Gamma }=0$, (b) $\mathrm{\Gamma }=0.10$, (c) $\mathrm{\Gamma }=0.23$, (d) $\mathrm{\Gamma }=0.39$, (e) $\mathrm{\Gamma }=0.60$, and (f) $\mathrm{\Gamma }=1.29$.

Figure 14

Volume flux $Q_{\text{p}}$ of the plume where it penetrates the interface: comparing simulation results and models. Depicted are the predictions of models 1 and 2 (23) as well as the volume flux of the plumes from simulation data. The latter is defined as the volume flux through the horizontal continuation of the interface in Fig. 7 (see Appendix pp2 for a formal definition).

Figure 15

Gradient Richardson number ${\text{Ri}}_g$ along a vertical slice in the middle of the domain ($x=0$). The value ${\text{Ri}}_g=\frac{1}{4}$ is marked by a dotted line. At the chosen position $x=0$, the interface elevation is $z=0$ for all configurations.

Figure 16

Scaling of the plume's (a) volume flux $Q$ and (b) relative buoyancy flux $\tilde{F}$ at $\mathrm{\Gamma }=0$. Both the total buoyancy flux $\tilde{F}$ of the plume and its contribution from the mean flow are shown. $\tilde{F}$ and $Q$ are calculated from (A2) and (A3), respectively, for two integration intervals, ${\mathcal{I}}_1$ and ${\mathcal{I}}_2$ (see text).

Figure 17

Schematic for the definition of a separating surface between the two layers of the stratification, using the mean buoyancy field. The red dashed line marks the isosurface at zero buoyancy. The solid red line corresponds to the separating curve $s$ (see text).