Reorientations of the large-scale flow in turbulent convection in a cube

Large-eddy simulations of turbulent Rayleigh-Bénard convection were conducted for a fluid of Prandtl number $\text{Pr}=0.7$ confined in a cube, for Rayleigh numbers of ${10}^6$ and ${10}^8$. The model solves the unsteady Navier-Stokes equations under the Boussinesq approximation, using a dynamic Smagorinsky model with a Lagrangian averaging technique for the subgrid terms. Under fully developed conditions the flow topology is characterized by a large-scale circulation (LSC) developing in a plane containing one of the diagonals of the cell, while two counter-rotating vortices consequently develop in the other diagonal plane, resulting in a strong inflow at the horizontal midplane. This flow structure is not static, with the LSC undergoing nonperiodic reorientations, or switching, between the two diagonal planes; hence, we supplement the observations of the three-dimensional time-averaged flow structures with single point measurements (time series) to shed light on the dynamics of the reorientations. For all observations, this switching results from a lateral rotation of the LSC in which some finite time spent in a transient state where the large-scale circulation is parallel to one set of side walls; there are, importantly, no observations consistent with so-called cessations of the LSC, in which it decays and then reforms in another plane without such a rotation. The average switching rate for the LSC is in excellent agreement with the results of Bai et al. [K. Bai, D. Ji, and E. Brown, Phys. Rev. E 93, 023117 (2016)].

Figure 1

Schematic of an arbitrary horizontal plane showing the azimuthal positions for probes, placed at azimuthal angles ${\varphi }_i=(i\pi /4)$ ($i=0,...,7$). The distance from the vertical walls is $0.1H$ for all cases. Probe positions P6 and P4 are shown in blue (grey in print) to help illustrate the positions for corresponding data shown below in Fig. 8.

Figure 2

Instantaneous normalized density isosurfaces for (a) $\mathrm{Ra}={10}^6$, (b) $\mathrm{Ra}={10}^8$. $\mathrm{Pr}=0.7, \mathrm{\Gamma }=1$ with $64\times 96\times 64$ grids.

Figure 3

Time-averaged velocity streamlines at $\mathrm{Ra}={10}^8, \mathrm{Pr}=0.7$, and $\mathrm{\Gamma }=1$. The LSC is oriented along one of the diagonal directions as illustrated by the large arrow. The color coding (grey-scale in print) depicts the magnitude of the time-averaged vertical velocity normalized by the free-fall velocity $\langle v\rangle /U_f$.

Figure 4

Time-averaged velocity streamlines in a horizontal plane near the top boundary of the cube for $\mathrm{Ra}={10}^8, \text{Pr}=0.7$, and $\mathrm{\Gamma }=1$. The color coding (grey-scale in print) depicts the magnitude of the time-averaged vertical velocity normalized by the free-fall velocity $\langle v\rangle /U_f$. (a)–(d) Different instances in time, illustrating that the LSC does not remain fixed in its orientation.

Figure 5

Transient state during a reorientation of the LSC. Time-averaged velocity streamlines are shown for $\mathrm{Ra}={10}^8, \mathrm{Pr}=0.7$, and $\mathrm{\Gamma }=1$. The color coding (grey-scale in print) depicts the magnitude of the time-averaged vertical velocity normalized by the free-fall velocity $\langle v\rangle /U_f$. The large arrow shows the direction of the LSC.

Figure 6

Separated views for horizontal and vertical planes of Fig. 5 showing time-averaged velocity streamlines at $\mathrm{Ra}={10}^8, \mathrm{Pr}=0.7$, and $\mathrm{\Gamma }=1$ in the transition mode. (a) Top panel, (b) vertical wall $xy$, and (c) vertical wall $zy$. The color coding (grey-scale in print) depicts the magnitude of the time-averaged vertical velocity normalized by the free-fall velocity $\langle v\rangle /U_f$.

Figure 7

3D velocity vectors superimposed on the color-coded representation of the time-averaged, normalized time-averaged horizontal velocity $\langle u_h\rangle /U_f=\sqrt{{\langle u\rangle }^2+{\langle w\rangle }^2}$ at $\mathrm{Ra}={10}^8, \text{Pr}=0.7$, and $\mathrm{\Gamma }=1$: (a) horizontal plane near the bottom boundary at $Y\approx 0.02H$, (b) horizontal plane near the top boundary at $Y\approx 0.98H$, and (c) horizontal plane at midheight at $Y=0.5H$. In (a) we label the corners A–D for clarity of the discussion (see text). This same labeling is implied in the other two plots.

Figure 8

Time series of vertical velocity for different “probe” positions. Positions P4 and P6 are in blue (grey in print), corresponding to the placement shown in Fig. 1. (a) Probes P4(m) and P8(m) at midheight, (b) probes P6(m) and P2(m) at the midheight at the opposite diagonal, (c) probes P6(d) and P2(d) at a distance of one-quarter of the cell height from the bottom, and (d) probes P6(u) and P2(u) at a distance of one-quarter of the cell height from the top. The solid horizontal line shows the position of zero-velocity magnitude and the boxed regimes highlight a particular part of the simultaneous time series, illustrating the different flow regimes occurring on opposite diagonals (LSC or counter-rotating vortices). The dashed lines in (a) and (b) illustrate the mean values for each probe. $\mathrm{Ra}={10}^8, \text{Pr}=0.7$, and $\mathrm{\Gamma }=1$. See text for nomenclature and the interpretation of these signals.

Figure 9

Time-averaged vertical velocity normalized by free-fall velocity at the midheight as a function of diagonal distance at $\mathrm{Ra}={10}^6$ ($▴$) and $\mathrm{Ra}={10}^8$ ($\blacksquare$), $\text{Pr}=0.7$, and $\mathrm{\Gamma }=1$.

Figure 10

Vertical velocity profile measured by the sidewall probes at the midplane for $\mathrm{Ra}={10}^8, \text{Pr}=0.7$, and $\mathrm{\Gamma }=1$. The solid black line shows a cosine fit of Eq. (10) to our data. The fit yields the orientation $\varphi$ and an amplitude $\delta$ that describe the LSC.

Figure 11

Time series of (a) the amplitude $\delta$ and (b) phase $\varphi$ of the first Fourier mode of the vertical velocity used as an approximate measure for the reorientation of the LSC, for $\mathrm{Ra}={10}^8, \text{Pr}=0.7$, and $\mathrm{\Gamma }=1$. The boxed region highlights one of the LSC orientations, $\varphi \simeq {135}^{\circ }$ near $t\simeq 200$, corresponding to the same region boxed in Fig. 8. Nonperiodic switching of the LSC orientations between the corners is observed where $\mathrm{\Delta }\varphi <{180}^{\circ }$.

Figure 12

Zoom of a switching event occurring near $t\simeq 370$. (a) Throughout this event $\delta$ is seen to remain nonzero and (b) $\mathrm{\Delta }\varphi \simeq {90}^{\circ }$ for $\mathrm{Ra}={10}^8, \text{Pr}=0.7$, and $\mathrm{\Gamma }=1$.