The Persistent Clock of Turbulent Thermal Convection

Abstract

The large-scale circulation (LSC) of turbulent convection is a prominent feature of its dynamics and forms the basis for descriptive theories. We show, using experimental and numerical results from thermal convection in a cylindrical cell, that the LSC possesses a persistent internal `clock': its pulsating velocity as a function of time is described by a constant value of the parameter U/(lf) where U is the mean velocity, f the pulsation frequency, and l the characteristic length scale. By introducing a narrow sidewall barrier, we can trip the LSC, forming a pair of interconnected rolls stacked above and below the barrier. They independently exhibit the same value for the ratio U/(lf), even for vertically asymmetric pairs, indicating signs of synchrony. Thus, this parameter establishes a direct connection between plume-shedding dynamics and the flow topology.

Figures (5)

• Figure 1: (a) Sketch of the $\Gamma = 1$ cylindrical cell, where a narrow ring-shaped barrier is mounted at mid-height along the sidewall; it can be removed or repositioned. Oblique views of streamlines from two perpendicular cross-sections reveal the nearly axisymmetric flow structures inside. (b), (c) Measurements of the mean vertical velocity $U_z$ overlaid with 2D streamlines. The field of view covers about $80\%$ of the full height $L$. (b) For a smooth sidewall, the LSC dominates. (c) With the barrier, the LSC is replaced by a four-roll structure within the measurement field; the barrier appears as a white band where it obstructs a small portion of the field. (a) Data from numerical simulations; (b,c) experiments, all at $\mathrm{Ra}=1.03\times10^9$.
• Figure 2: Flow topology at different driving strengths ($\mathrm{Ra}$), shown through mean vertical velocity ($U_z$) fields overlaid with 2D streamlines for the barrier case. (a,b) At lower thermal driving, the LSC persists as the dominant flow component. (c) As driving increases the flow breaks into four rolls in the vertical plane. (d) The HCC remains robust even at higher levels of turbulence.
• Figure 3: Normalized BL thickness as a function of $\mathrm{Ra}$ and $\mathrm{Re}$. In the blue-shaded region we observe a single LSC; in the pink-shaded region we find two stacked HCCs. Black triangles mark direct measurements from the velocity profiles, using the maximum-velocity position relative to the wall. Circular markers indicate the Reynolds-number scaling laws using the measured $U_{max}$; blue points use the classical definition $\mathrm{Re}_L$, red circles use the rescaled $\mathrm{Re}_{L/2}$. Dashed lines represent the scaling laws based on $\mathrm{Ra}_L$ and $\mathrm{Ra}_{L/2}$.
• Figure 4: (a) $\mathrm{Re}$ as function of $\mathrm{Ra}$ for the barrier case. Squares denote experimental measurements (blue for LSC, pink for HCC), purple crosses indicate DNS results, the dashed line is a fit. (b) $\mathrm{Re}_f$ as function of $\mathrm{Ra}$ for the barrier case (dashed blue and pink lines represent the fits corresponding to the one-roll and two-pairs states). (c) Dependence of $U/lf$ on $\mathrm{Ra}$. The two points per $\mathrm{Ra}$ correspond to measurements taken at the left/right sides when the LSC is dominant and at the top/bottom rolls after the transition (see Supplemental Material Fig. S1). The horizontal discontinuous line represents the fitted value $c=2.1$. Open symbols for $\mathrm{Ra}=1.03\times10^9$ correspond to cases where the barrier is not at $L/2$: LSC (without the barrier), $L/3$, and $L/4$. (d) Time series of near-wall velocity at $\mathrm{Ra}=1.03\times10^9$ in non-barrier and barrier cases (dark-shaded curves). After the transition, the flow pulsates at a higher frequency. The light-shaded lines correspond to the dominant frequency component of the signal’s Fourier decomposition.
• Figure 5: Temporal evolution of the near-wall vertical velocity signal phase $\phi(t)$ for different barrier positions at $\mathrm{Ra}=1.03 \times 10^{9}$. (a) Using the LSC as a reference, the signal phases on opposite walls evolve synchronously at the same frequency $2\pi f=\partial\phi/\partial t$, with $f_l \approx f_r = 10\;\mathrm{mHz}$ (evaluated at left $l$ and right $r$ sides). (b) When the barrier is placed at $z=L/2$, the frequency doubles ($f_t \approx f_b = 20\;\mathrm{mHz}$), while the phases remain locked. (c) With barrier at $z=L/3$, the top and bottom rolls become asymmetric ($f_t = 10\;\mathrm{mHz}, f_b = 26\;\mathrm{mHz}$) and the phases no longer evolve together. (d) With barrier at $z=L/4$, the disparity increases ($f_t = 4\;\mathrm{mHz}, f_b = 28\;\mathrm{mHz}$).