Multiple states and heat transfer in two-dimensional tilted convection with large aspect ratios

The coexistence of multiple turbulent states was reported in several recent studies in different flows. We present in this work that multiple turbulent states also exist for thermal convection in two-dimensional tilted cells with large aspect ratios $\left(\mathrm{\Gamma }=\mathrm{width}/\mathrm{height}\right)$ through direct numerical simulations for the Rayleigh number $\text{Ra}={10}^7$ and the Prandtl number $\text{Pr}=0.71$. The considered $\mathrm{\Gamma }$ ranges from 1 to 16. The tilt angle $\beta$ varies from $0^{\circ }$ to ${180}^{\circ }$. Multiple states are identified for small $\beta$ with $\mathrm{\Gamma }\ge 2$, where the different flow states are reflected in different numbers of convection rolls. The corresponding Nu is generally higher for the flow state with more convection rolls. Moreover, flow mode transitions between different roll states are observed for large $\mathrm{\Gamma }\ge 8$ when $\beta$ is larger than a critical value. The effect of cell tilting on Nu and Re is also investigated. It is found that for $\mathrm{\Gamma }\le 4$, Nu first increases with increasing $\beta$ and then declines after reaching its local maximum. However, Nu generally decreases monotonically with increasing $\beta$ for $\mathrm{\Gamma }=8$, 12, and 16. This indicates that the idea to enhance heat transfer by tilting the cell can be realized only for relatively small $\mathrm{\Gamma }$ for the present system. It is also found that the previous finding that Re decreases monotonically with increasing $\beta$ for large $\beta$ with $\mathrm{\Gamma }=1$ does not hold for large-$\mathrm{\Gamma }$ cases.

Figure 1

Sketch of the two-dimensional tilted cavity.

Figure 2

Instantaneous temperature fields superposed with velocity vectors for different $\beta$ and $\mathrm{\Gamma }=2$: (a) double-roll state for $\beta =0^{\circ }$, (b) single-roll state for $\beta =0^{\circ }$, (c) $\beta ={15}^{\circ }$, (d) $\beta ={30}^{\circ }$, (e) $\beta ={80}^{\circ }$, and (f) $\beta ={90}^{\circ }$. The black arrows indicate the direction of gravity.

Figure 3

Instantaneous temperature fields superposed with velocity vectors for different $\beta$ with $\mathrm{\Gamma }=4$: (a) Four-roll state for $\beta =0^{\circ }$, (b) three-roll state for $\beta =0^{\circ }$, (c) single-roll state for $\beta =0^{\circ }$, and (d) single-roll state for $\beta ={19}^{\circ }$. The black arrows indicate the direction of gravity. The color scale is the same as in Fig. 2.

Figure 4

Instantaneous temperature fields for $\text{Ra}={10}^7,\text{Pr}=0.71$, and $\mathrm{\Gamma }=8$ for different $\beta$. The color scale is the same as in Fig. 2.

Figure 5

Instantaneous temperature fields for different flow states obtained using different initial conditions for $\text{Ra}={10}^7,\mathrm{\Gamma }=8$, and $\beta =5^{\circ }$: (a) eight-roll state, (b) seven-roll state, (c) five-roll state, (d) three-roll state, and (e) time evolution of the Nu for the four different roll states in (a)–(d); for clarity the Nu for the seven-roll and eight-roll states are shifted upward by values of 3 and 7, respectively. The black arrow indicates the direction of gravity. The color scale is the same as in Fig. 2.

Figure 6

Time evolution of Nu for $\mathrm{\Gamma }=8$ and (a) $\beta ={23}^{\circ }$ and (b) $\beta ={25}^{\circ }$. (c) Instantaneous temperature fields at the different times in (b). The black arrow indicate the direction of gravity. The color scale is the same as in Fig. 2.

Figure 7

Phase diagram in the $\beta \text{−}\mathrm{\Gamma }$ plane. Red circles correspond to the coexistence of multiple states, blue diamonds denote the single-roll state, and green squares stand for the state with flow mode transitions.

Figure 8

Absolute and normalized Nusselt numbers Nu and Reynolds numbers Re as a function of $\beta$ for fixed $\text{Ra}={10}^7$ and $\text{Pr}=0.71$ with different aspect ratios: (a) absolute Nusselt numbers Nu, (b) normalized Nusselt numbers $\text{Nu}\left(\beta \right)/\text{Nu}\left(0\right)$, (c) absolute Reynolds numbers Re, and (d) normalized Reynolds numbers $\text{Re}\left(\beta \right)/\text{Re}\left(0\right)$. The normalization for $\mathrm{\Gamma }=2$ and 4 uses the corresponding values for single-roll state for RBC $\left(\beta =0^{\circ }\right)$.