Multiple States in Turbulent Large-Aspect-Ratio Thermal Convection: What Determines the Number of Convection Rolls?

Wall-bounded turbulent flows can take different statistically stationary turbulent states, with different transport properties, even for the very same values of the control parameters. What state the system takes depends on the initial conditions. Here we analyze the multiple states in large-aspect ratio ($\mathrm{\Gamma }$) two-dimensional turbulent Rayleigh-Bénard flow with no-slip plates and horizontally periodic boundary conditions as model system. We determine the number $n$ of convection rolls, their mean aspect ratios ${\mathrm{\Gamma }}_r=\mathrm{\Gamma }/n$, and the corresponding transport properties of the flow (i.e., the Nusselt number Nu), as function of the control parameters Rayleigh (Ra) and Prandtl number. The effective scaling exponent $\beta$ in $\mathrm{Nu}\sim {\mathrm{Ra}}^{\beta }$ is found to depend on the realized state and thus ${\mathrm{\Gamma }}_r$, with a larger value for the smaller ${\mathrm{\Gamma }}_r$. By making use of a generalized Friedrichs inequality, we show that the elliptical shape of the rolls and viscous damping determine the ${\mathrm{\Gamma }}_r$ window for the realizable turbulent states. The theoretical results are in excellent agreement with our numerical finding $2/3\le {\mathrm{\Gamma }}_r\le 4/3$, where the lower threshold is approached for the larger Ra. Finally, we show that the theoretical approach to frame ${\mathrm{\Gamma }}_r$ also works for free-slip boundary conditions.

Figure 1

(a), (b) Time evolution of (a) ${\mathrm{Re}}_z/{\mathrm{Re}}_x$ and (b) Nu for different initial roll states, $\mathrm{Ra}={10}^{10}$, $\mathrm{Pr}=10$, $\mathrm{\Gamma }=8$. ${\mathrm{\Gamma }}_r^{(i)}=\mathrm{\Gamma }/n^{(i)}$ is the initial and ${\mathrm{\Gamma }}_r=\mathrm{\Gamma }/n$ the final mean aspect ratio of the rolls; $n^{(i)}$ is the initial and $n$ the final number of the rolls. Note the logarithmic scales of the time axes. (c) Snapshots of the temperature fields for the three statistically stable turbulent states for ${\mathrm{\Gamma }}_r=1$ ($n=8$), ${\mathrm{\Gamma }}_r=4/5$ ($n=10$), and ${\mathrm{\Gamma }}_r=2/3$ ($n=12$) and the corresponding Nusselt numbers, for $\mathrm{Ra}={10}^{10}$, $\mathrm{Pr}=10$, and $\mathrm{\Gamma }=8$. All subfigures are for no-slip BCs.

Figure 2

(a) Nu and (b) final aspect ratio ${\mathrm{\Gamma }}_r=\mathrm{\Gamma }/n$ of individual rolls, as function of $\mathrm{\Gamma }$, for different turbulent states. The numbers and colors in the legend denote the number $n$ of convection rolls of that state. (c) Nu and (d) Re as functions of ${\mathrm{\Gamma }}_r=\mathrm{\Gamma }/n$ for three different values of $\mathrm{\Gamma }$. In this figure $\mathrm{Ra}={10}^9$ and $\mathrm{Pr}=10$ and the BCs are no slip.

Figure 3

(a) Compensated Nusselt number $\mathrm{Nu}/{\mathrm{Ra}}^{1/3}$ and (b) compensated Reynolds number $\mathrm{Re}/{\mathrm{Ra}}^{2/3}$, as functions of Ra, for four different turbulent states as characterized by ${\mathrm{\Gamma }}_r$, for $\mathrm{Pr}=10$, $\mathrm{\Gamma }=8$, and no-slip BCs. The effective scaling exponents $\beta$ and $\gamma$ in the scaling relations $\mathrm{Nu}\sim {\mathrm{Ra}}^{\beta }$ and $\mathrm{Re}\sim {\mathrm{Ra}}^{\gamma }$ are shown next to the curves in the respective color of the state and curve.

Figure 4

Phase diagram in the (a),(c) $\mathrm{Ra}-{\mathrm{\Gamma }}_r$ and (b),(d) $\mathrm{Pr}-{\mathrm{\Gamma }}_r$ parameter space for (a),(b) no-slip BCs and (c),(d) free-slip BCs: (a) $\mathrm{Pr}=10$, $\mathrm{\Gamma }=8$; (b) $\mathrm{Ra}={10}^9$, $\mathrm{\Gamma }=8$; (c) $\mathrm{Pr}=10$, $\mathrm{\Gamma }=16$; (d) $\mathrm{Ra}={10}^8$, $\mathrm{\Gamma }=16$. Black circles denote that the corresponding roll state is stable, whereas red crosses mean that it is not stable. The theoretical estimates for the transitions between the regimes are shown as solid lines.

Figure 5

$B=\{[{\mathrm{Re}}^2{\mathrm{Pr}}^2]/[\mathrm{Ra}(\mathrm{Nu}-1)]\}$, as functions of (a),(c) Ra and (b),(d) Pr, for (a),(b) no-slip BCs and (c),(d) free-slip BCs. (a) $\mathrm{Pr}=10$, $\mathrm{\Gamma }=8$; (b) $\mathrm{Ra}={10}^9$, $\mathrm{\Gamma }=8$; (c) $\mathrm{Pr}=10$, $\mathrm{\Gamma }=16$; (d) $\mathrm{Ra}={10}^8$, $\mathrm{\Gamma }=16$. The solid lines are fits to the data (see the Supplemental Material [40]). Note that the values of $B$ in the no-slip case (a),(b) are much smaller than those in the free-slip case (c),(d).