Chaotic dynamics of a convection roll in a highly confined, vertical, differentially heated fluid layer

Direct numerical simulation is used to study the air flow between two vertical plates maintained at different temperatures. The periodic dimensions of the plates are small so as to accommodate only one flow structure, which consists of a convection roll with oblique vorticity braids. At lower Rayleigh numbers, the roll and the braids grow and shrink alternatively following a cyclical process. As the Rayleigh number is increased, the flow becomes temporally chaotic through a period-doubling cascade. Windows corresponding to multiperiodic regimes and interior crises are observed. As the Rayleigh number is further increased, the structure intermittently switches between two vertical positions, which is seen to correspond to an “attractor-merging” crisis. The chaotic flow dynamics are characterized and the corresponding physical mechanisms are identified. We show that some of the flow key features, such as the chaotic oscillation and intermittency, can be captured by a low-order model.

Figure 1

The simulation domain is constituted by two vertical plates, separated by a distance $D$ and maintained at different temperatures. Periodic boundary conditions for the plates are enforced in both transverse and vertical directions $(y$ and $z)$. The aspect ratios of the periodic dimensions are $A_y=L_y/D=1$ and $A_z=L_z/D=2.5$. The temperature of the back plate at $x=0$ (in red/black in grayscale) is $\frac{\Delta T}{2}$, while that of the front plate at $x=1$ (in blue/gray in grayscale) is $-\frac{\Delta T}{2}$. The distance between the plates is $D$.

Figure 2

Q-criterion visualization of flow structures colored by the vertical vorticity ${\Omega }_z$. (a) Biperiodic domain at $\mathrm{Ra}=11500,Q=0.25$ in the present numerical configuration described in the legend of Fig. 1, i.e., with periodic boundary conditions in both $y$ and $z$ directions $\left(A_y=A_z=1\right)$; (b) Tall cavity of vertical aspect ratio $A_z=20$ at $\mathrm{Ra}=15000,Q=0.2$. The transverse direction is periodic $\left(A_y=1\right)$. Adiabatic boundary conditions are used for the horizontal walls.

Figure 3

Temporal evolution of the flow structures at $\mathrm{Ra}=11500$: 2D sections of streamlines and temperature isocontours of the instantaneous flow on two orthogonal planes at five different instants spanning one oscillation period $T_{\text{osc}}^{\text{DNS}}=28$. Top row: streamline plots on the vertical plane $y=0.5$, using the velocity components $u$ and $w$; middle row: streamline plots on the vertical plane $x=0.5$, using velocity components $v$ and $w$; bottom row: temperature isocontour on the vertical plane $x=0.5$.

Figure 4

Temporal evolutions of Fourier modes ${\widehat{w}}_{lk}$ obtained by the Fourier transform of the vertical velocity $w$ distribution on an arbitrarily chosen vertical plane at $x=0.0381,\mathrm{Ra}=11500$: (a) modulus, (b) phase.

Figure 5

Bifurcation diagram obtained by using the local maxima ${\theta }_n$ of the temperature time series at the point (0.038, 0.097, 0.983). Vertical line in the figure correspond to the $\text{Ra}=12350$, at which a “period-6 window” is observed.

Figure 6

First return maps $|{\widehat{w}}_{01}{|}_{n+1}$ as a function of $|{\widehat{w}}_{01}{|}_n$. $|{\widehat{w}}_{01}{|}_n$ is the local maxima of Fourier time series $|{\widehat{w}}_{01}|$ calculated from the vertical velocity $w$ distribution on the vertical plane $x=0.0381$.

Figure 7

(a) The largest Lyapunov exponent ${\lambda }_1$ for different Rayleigh numbers; Error bars are $1.96$ times the standard error assuming a Gaussian distribution and a 95% confidence interval. (b) Fractal dimension obtained by application of the Kaplan-Yorke formula as a function of the Rayleigh number. The position of the solid line spanning each figure represents the value of the Rayleigh number at the onset of the crisis. Error bars are calculated from the errors estimated in (a).

Figure 8

Temporal evolution of flow streamlines at three instants separated by a basic oscillation period $T_{\text{osc}}^{\text{DNS}}=28$ at $\mathrm{Ra}=12380$ on the vertical plane $y=0.5$, using velocity components $u$ and $w$.

Figure 9

Temporal evolutions of Fourier modes ${\widehat{w}}_{lk}$ obtained by the Fourier transform of the vertical velocity $w$ distribution on an arbitrarily chosen vertical plane at $x=0.381,\mathrm{Ra}=12380$: (a) modulus, (b) phase.

Figure 10

Bifurcation diagram obtained by using the local peaks ${\theta }_n$ of the time series at the point (0.038, 0.097, 0.983). Note: the vertical line in the figure corresponds the largest Rayleigh number in Fig. 5.

Figure 11

Phase of the temporal evolution of the Fourier mode ${\widehat{w}}_{01}$ calculated on the vertical plane $x=0.0381,\mathrm{Ra}=12600$.

Figure 12

Phase portraits. Abscissa: real part of the temporal evolution of the Fourier mode ${\widehat{w}}_{01}$ calculated on vertical plane $x=0.0381$; ordinate: real part of the temporal evolution of the Fourier mode ${\widehat{w}}_{10}$. (a) $\mathrm{Ra}=12500$, (b) $\mathrm{Ra}=12600$.

Figure 13

Temporal evolution of flow structures at $\mathrm{Ra}=12600$. Upper row: flow streamlines at different instants on the plane $y=0.5$, using the velocity components $u$ and $w$; Middle row: flow streamlines at different instants on the plane $x=0.5$, using velocity components $v$ and $w$; Bottom row: temperature isocontours on the plane $x=0.5$. Figures (a) top row, (d) top row appeared in Ref. [7].

Figure 14

(a) Real part of the temporal evolution of the Fourier mode ${\widehat{w}}_{01}$ calculated on the vertical plane $x=0.0381$; (b) phase portraits. Abscissa: real part of the temporal evolution of the Fourier mode ${\widehat{w}}_{01}$; ordinate: real part of the temporal evolution of the Fourier mode ${\widehat{w}}_{10}$. $\mathrm{Ra}=13000$

Figure 15

(a) Real part of the temporal evolution of the Fourier mode ${\widehat{w}}_{01}$ calculated on the vertical plane $x=0.0381$. (b) Fourier spectrum of the temperature time series at the point (0.038, 0.097, 0.98) $\mathrm{Ra}=14000$

Figure 16

${log}_{10}\langle \tau \rangle$ vs. ${log}_{10}(\mathrm{Ra}-{\mathrm{Ra}}_c)$. The slope of the straight line gives $\gamma \approx 0.78$. Triangles: chaotic, nonintermittent cases; circles: chaotic, intermittent cases; squares: periodic cases.

Figure 17

Velocity and temperature 2D Fourier mode on the plane $x=0.5$: (a) (0,1), (b) (1,0).

Figure 18

Comparison between the model and the simulation at $\mathrm{Ra}=11500$: from top to bottom, $|{\widehat{w}}_{00}|,|{\widehat{w}}_{01}|,|{\widehat{w}}_{10}|$. DNS: dash line; model: continuous line.

Figure 19

Time-averaged power spectral density of $w$ over the vertical and spanwise wavenumber: (a) $\mathrm{Ra}=12500$, (b) $\mathrm{Ra}=12800$.

Figure 20

Velocity Fourier modes ${\widehat{w}}_{01}$ (dash line), ${\widehat{w}}_{10}$ (continuous line), ${\widehat{w}}_{00}$ (dash dot line) on the plane $x=0.5$ at $\mathrm{Ra}=12500$: (a) DNS, (b) model with $\beta =0.5,T^{\prime }=25$.

Figure 21

Phase portraits at $\mathrm{Ra}=12500$: (a) DNS, (b) model with $\beta =0.5,T^{\prime }=25$ at $\mathrm{Ra}=12500$.

Figure 22

Phase portraits at $\mathrm{Ra}=12500$: (a) DNS, (b) model with $\beta =1,f=0.4$.

Figure 23

Real part of the Fourier modes ${\widehat{w}}_{01}$ (dash line), ${\widehat{w}}_{10}$ (continuous line), ${\widehat{w}}_{00}$ (dash dot line) at $\mathrm{Ra}=12800$: (a) DNS, (b) model with Gaussian noise of 5% amplitude.

Figure 24

Phase portrait $\mathrm{Re}\left[{\widehat{w}}_{00}\right]$ as a function of $\text{Re}\left[{\widehat{w}}_{01}\right]$ at $\mathrm{Ra}=12800$: (a) DNS, (b) model with Gaussian noise of 5% amplitude.