Reversals in infinite-Prandtl-number Rayleigh-Bénard convection

Using direct numerical simulations, we study the statistical properties of reversals in two-dimensional Rayleigh-Bénard convection for infinite Prandtl number. We find that the large-scale circulation reverses irregularly, with the waiting time between two consecutive genuine reversals exhibiting a Poisson distribution on long timescales, while the interval between successive crossings on short timescales shows a power-law distribution. We observe that the vertical velocities near the sidewall and at the center show different statistical properties. The velocity near the sidewall shows a longer autocorrelation and $1/f^2$ power spectrum for a wide range of frequencies, compared to shorter autocorrelation and a narrower scaling range for the velocity at the center. The probability distribution of the velocity near the sidewall is bimodal, indicating a reversing velocity field. We also find that the dominant Fourier modes capture the dynamics at the sidewall and at the center very well. Moreover, we show a signature of weak intermittency in the fluctuations of velocity near the sidewall by computing temporal structure functions.

Figure 1

Two stable flow configurations of the convective flow in a 2D square box with stress-free walls: LSC in the clockwise direction (a) and in the counterclockwise direction (b). Temperature field is shown as density plots and velocity field is represented by vectors. Time evolutions of the temperature and velocity fields are tracked for the entire duration of simulation at the two probes located near the left wall (green circles) and at the center (green squares).

Figure 2

Evolution of the vertical velocity at the left probe (indicated in Fig. 1 by green circles). Velocity changes its sign irregularly, indicating occurrence of flow reversals.

Figure 3

(a) Magnification of Fig. 2 showing a few reversal and crossing events, which are indicated in panel (b).

Figure 4

(a) Probability distribution function of the waiting time between two consecutive crossings $\mathcal{P}\left(\mathrm{\Delta }{t}_1\right)$ on a double logarithmic scale, which exhibits a power-law distribution for the shorter waiting times. (b) Probability distribution of the waiting time between two consecutive reversals, however, follows an exponential distribution.

Figure 5

Moments of the generalized interswitch interval $\langle |\mathrm{\Delta }{t}_r{|}^q\rangle$ between reversals as function of $r$ for (a) $q=1$ to 6 and (b) $q\le 1$ ($q$ increases from bottom to top). We observe scaling regions for $r\le 6$ and for $r\ge 50$.

Figure 6

Scaling exponents ${\zeta }_q$ as function of the order $q$ of the moments of generalized interswitch interval $\mathrm{\Delta }{t}_r$ between reversals for (a) $q\ge 1$ and (b) $q\le 1$. Exponents obtained for $r\le 6$ (red squares) scale linearly for $q\ge 1$, whereas show a very weak nonlinear behavior for $q\le 1$. However for $r\ge 50$, the exponents scale as ${\zeta }_q\cong q$ (green circles), revealing a decorrelation between distant reversals.

Figure 7

(a) Vertical velocities at the left and the center probes as function of time, (b) and the evolution of the most dominant Fourier modes during the same time interval. A fraction of the full time series covering two reversals is shown here.

Figure 8

Velocity (arrows) and temperature fluctuation fields (greyscale) corresponding to Fourier modes with indices $(m,n)$ computed using Eqs. (5, 6, 7). Dark and bright regions represent the hottest and coldest fluids respectively.

Figure 9

Probability distribution function of the vertical velocities at the left and the center probes (a) and of the amplitudes of the dominant Fourier modes (b). The distribution is bimodal for $u_z\left(\mathrm{LP}\right)$ and the OO modes, consistent with the fact that they change sign after flow reversals. PDFs of $u_z\left(\mathrm{CP}\right)$ and ${\widehat{u}}_z(2,1)$ exhibit broad peaks centered at zero and are non-Gaussian. The exponential tails are observed for all the distributions, and the exponential regions are indicated by straight lines.

Figure 10

Power spectral densities (PSD) of the vertical velocities at the left and the center probes, and of the Fourier modes. For the OO modes and $u_z\left(\mathrm{LP}\right)$, we observe $\mathrm{PSD}\sim f^{-2}$ for a wide range of frequencies, whereas this scaling is observed only for a relatively narrower range of frequencies for $u_z\left(\mathrm{CP}\right)$ and ${\widehat{u}}_z(2,1)$.

Figure 11

Power spectral densities of the autocorrelation functions, $\mathrm{PSD}\left[C\right(\tau \left)\right]$, that also exhibit $f^{-2}$ scaling for a similar range of frequencies as those in Fig. 10.

Figure 12

Structure functions $S_q\left(\tau \right)$ for $q=1$ to 8 for (a) $u_z\left(\mathrm{LP}\right)$ and (b) ${\widehat{u}}_z(1,1)$. Two scaling regimes located between red solid and blue dashed lines can be identified.

Figure 13

Structure function exponents ${\mu }_q$ for $u_z\left(\mathrm{LP}\right)$ and ${\widehat{u}}_z(1,1)$. The ${\mu }_q$ increases nonlinearly with $q$ for the first scaling regime at small $\tau$ (open symbols). The blue dashed curve depicts the Kolmogorov scaling ${\mu }_q=q/3$ [28], whereas blue solid curve represents the scaling deduced by She and Leveque [35]. For the second scaling regime at larger $\tau$ (filled symbols), ${\mu }_q$ increases markedly only up to $q=3$, and saturates for $q\ge 4$.

Figure 14

Structure function exponents ${\mu }_q$ for $q\le 1$ for $u_z\left(\mathrm{LP}\right)$ and ${\widehat{u}}_z(1,1)$. The exponents deviate from both the Kolmogorov scaling [28] (dashed blue curve) as well as from the She and Leveque scaling [35] (solid blue curve).

Figure 15

Structure functions $S_q\left(\tau \right)$ plotted against $S_3\left(\tau \right)$ for (a) $u_z\left(\mathrm{LP}\right)$ and (b) ${\widehat{u}}_z(1,1)$. Enhanced scaling regimes compared to those in Fig. 12 can be observed here.

Figure 16

ESS exponents ${\nu }_q(={\mu }_q/{\mu }_3)$ for $u_z\left(\mathrm{LP}\right)$ and ${\widehat{u}}_z(1,1)$ for the first scaling regime at small $\tau$ (open symbols) and for the second scaling regime at larger $\tau$ (filled symbols). Blue crosses are the ESS exponents reported in Benzi et al. [52] for the homogeneous and isotropic turbulence.

Figure 17

The second-order (a) and the fourth-order (b) structure functions, and the flatness factor $\kappa \left(\tau \right)=S_4\left(\tau \right)/{\left[S_2\left(\tau \right)\right]}^2$ (c) for the vertical velocity at the left probe and for the OO modes as function of $\tau$. Dashed horizontal lines represent the estimated values of $S_2\left(\tau \right)$, $S_4\left(\tau \right)$, and $\kappa \left(\tau \right)$ in the decorrelation regime. The flatness factors are nearly constant in very large $\tau$ and small $\tau$ regimes, whereas they scale as $\kappa \left(\tau \right)\sim {\tau }^{-0.50}$ in the intermediate $\tau$ regime indicated between two black vertical lines.