Turbulent convection and large scale circulation in a cube with rough horizontal surfaces

Large-eddy simulations of thermal convection are presented and discussed for a cube with rough horizontal surfaces. Two types of roughness are considered: uniformly placed pyramids, and grooves aligned parallel to one set of sidewalls. The Rayleigh number is ${10}^8$, the Prandtl number 0.7, and the aspect ratio 1, as in a previous study [N. Foroozani, J. J. Niemela, V. Armenio, and K. R. Sreenivasan, Phys. Rev. E 95, 033107 (2017)], except that the meshes here are finer. When the thermal boundary layers are sufficiently large relative to the characteristic roughness height, i.e., for hydrodynamically smooth conditions, the mean properties of the large scale circulation (LSC) are qualitatively similar to the case of smooth surfaces. In particular, the LSC is always aligned along one of the diagonals of the cube. When the boundaries are hydrodynamically rough, the same result holds true only for the case of pyramidal structures; for grooved surfaces, the LSC is forced to be parallel to the sidewalls on average, alternating rapidly between the two diagonals of the cube with a mean period of the order 10 turnover times. Our analysis suggests that the difference from the pyramidal case is due to the breaking of the horizontal $x-z$ symmetry under conditions of hydrodynamical roughness, and the corresponding directional concentration of plume emission along the grooves, from which the LSC is generated, providing a strong restoring force. Furthermore, in this study we observed a small reduction in heat transport for both roughness configurations which is in good agreement with past studies.

Figure 1

(a) Pyramidal rough surfaces with $k_1=0.025H$. A 3D structure of a zoomed area is also shown $0\le x,z\le 0.25$ and $0\le y\le 0.025$; (b) 3D plot of grooved surface with height $k_1$. We studied three different heights with $k_1=0.025H,k_2=0.0125H$, and $k_3=0.00625H$; the geometry of roughness is constant in all cases with $2a=0.125H$ [see the sketch in (b)]. Examples of stretched mesh along the $y$ axis are shown to the right of (b).

Figure 2

Schematic of the horizontal midplane showing the azimuthal positions for probes, placed at azimuthal angles ${\varphi }_i=(i\pi /8),i=0,...,15$. The distance from the vertical walls are $0.1H$ for all cases. In the case of grooves, the alignment is along the 0-$\pi$ direction.

Figure 3

Time series of the vertical velocity and its phase for different “probe” positions for grooved plate with $k_1=0.025$ (see Fig. 2 also). (a) Probes P7 and P15 at the midheight corresponding to the same diagonal. (b) Probes P3 and P11 at midheight in the opposite diagonal. (c) Time series of phase $\mathrm{\Phi }\left(t\right)$ of the first Fourier mode of the vertical velocity used as an approximate measure for the time evolution of the orientation of the LSC, for $\text{Ra}={10}^8,\text{Pr}=0.7$, and $\mathrm{\Gamma }=1$.

Figure 4

Time-averaged velocity streamlines and velocity vectors in the cube with (a), (d) grooved plate with height $k_3=0.00625H$, (b), (e) grooved plate with height $k_1=0.025H$, and (c), (f) pyramidal roughness with $k_1=0.025H$ for $\text{Ra}={10}^8,\text{Pr}=0.7$, and $\mathrm{\Gamma }=1$. The color coding depicts the magnitude of the vertical velocity normalized by the free fall velocity $U_f$. Note that the time averaging for (a) and (d) and (c) and (e) is for a period of time in which there are no major reorientations of the mean wind, for the purpose of reducing the effects of small turbulent fluctuations. In the case of (b) and (e), on the other hand, long-time averaging gives a representation of the mean orientation of the LSC which otherwise has large and rapid (compared to the averaging time) fluctuations in direction between adjacent diagonals. It serves to graphically illustrate the qualitative difference occurring when the mean wind interacts with roughness elements lacking $x-z$ symmetry for boundaries that are furthermore hydrodynamically rough and, in particular, illustrate the time-averaged orientation of flow relative to the groove channels.

Figure 5

Instantaneous normalized density isosurfaces over the bottom plate of the grooved surface with (a) height $k_3=0.00625H$, (b) height $k_1=0.025H$ for $\text{Ra}={10}^8$ and $\text{Pr}=0.7$.

Figure 6

Velocity vector on a vertical section over grooves with height $k_1$. Color contour denotes time-averaged streamwise velocity normalized by the free fall velocity $\langle w\rangle /U_f$ at $\text{Ra}={10}^8,\text{Pr}=0.7$, and $\mathrm{\Gamma }=1$.