Three-dimensional instabilities of natural convection between two differentially heated vertical plates: Linear and nonlinear complementary approaches

The transition to the chaos of the air flow between two vertical plates maintained at different temperatures is studied in the Boussinesq approximation. After the first bifurcation at critical Rayleigh number ${\mathrm{Ra}}_c$, the flow consists of two-dimensional (2D) corotating rolls. The stability of the 2D rolls is examined, confronting linear predictions with nonlinear integration. In all cases the 2D rolls are destabilized in the spanwise direction. Efficient linear stability analysis based on an Arnoldi method shows competition between two eigenmodes, corresponding to different spanwise wavelengths and different types of roll distortion. Nonlinear integration shows that the lower-wave-number mode is always dominant. A partial route to chaos is established through the nonlinear simulations. The flow becomes temporally chaotic for $\mathrm{Ra}=1.05{\mathrm{Ra}}_c$, but remains characterized by the spatial patterns identified by linear stability analysis. This highlights the complementary role of linear stability analysis and nonlinear simulation.

Figure 1

Left: Configuration. Air flow ($\mathrm{Pr}=0.71$) between two vertical plates separated by a distance $D$ and maintained at different temperatures: the hot plate (left, in red) is at temperature $\frac{\mathrm{\Delta }T}{2}$ while the cold plate (right, in blue) is at temperature $-\frac{\mathrm{\Delta }T}{2}$. Periodic boundary conditions are imposed in the vertical and in the spanwise direction. The periodic height and width of the plates are, respectively, $L_z$ and $L_y$, and the corresponding aspect ratios are $A_y=L_y/D$ and $A_z=L_z/D$. The gravity $g$ is opposite to the vertical direction $z$. Right: Steady laminar conduction solution profiles for the vertical velocity $W\left(x\right)$ and temperature $\mathrm{\Theta }\left(x\right)$.

Figure 2

Rolls visualized with the $Q$ criterion using an isovalue of $Q=0.1$ at $\mathrm{Ra}=6050, t=2000$, and $A_z=9$.

Figure 3

Neutral curve, $\sigma \left({\text{Ra}}_{c2i}\right)=0$, as a function of the wave number $k_y, A_z=9$

Figure 4

Arnoldi eigenmode at $\mathrm{Ra}=6100$ with $A_z=9$ on the central plane $x=0.5$. Left, from top to bottom: (a–d) Velocity components $u,v,\text{and}w$ and temperature $\theta$. Right, from top to bottom: Fourier modulus $|\widehat{u}(i_y,i_z)|, |\widehat{v}(i_y,i_z)|, |\widehat{w}(i_y,i_z)|$, and $|\widehat{\theta }(i_y,i_z)|$.

Figure 5

Neutral curve, $\sigma \left({\text{Ra}}_{c2i}\right)=0$, as a function of the wave number $k_y, A_z=10$.

Figure 6

Arnoldi eigenmode at $\mathrm{Ra}=6050$ with $A_z=10$ on the central plane $x=0.5$. Left, from top to bottom: (a–d) Velocity components $u,v,\text{and}w$ and temperature $\theta$. Right, from top to bottom: Fourier modulus $|\widehat{u}(i_y,i_z)|, |\widehat{v}(i_y,i_z)|, |\widehat{w}(i_y,i_z)|$, and $|\widehat{\theta }(i_y,i_z)|$.

Figure 7

(a) Time series of temperature at point (0.0381, 0.122, 4.96) in the boundary layer near the hot wall at $\mathrm{Ra}=6100, A_z=9$. (b) An enlargement of (a) for $4000<t<6000$ on a logarithmic scale.

Figure 8

Isosurface of $Q$ criterion $Q=0.1$ and streamlines, at $t=6000, \mathrm{Ra}=6100, A_z=9$.

Figure 9

(a) Temperature contour on the vertical plane $x=0.0381$. (b) Corresponding 2D spatial spectrum $|\widehat{\theta }(i_y,i_z)|$ ($i_y,i_z=-5,5$) for the fluctuations, $t=6000, \mathrm{Ra}=6100, A_z=9$.

Figure 10

Isosurface of $Q$ criterion $Q=0.1$ and streamlines for the flow pattern at $t=9000, \mathrm{Ra}=6100, A_z=9$.

Figure 11

Left: Flow pattern on plane $x=0.0381$ at $t=10000$ and $\mathrm{Ra}=6100$ with $A_z=9$. (a–c) Velocity components $u,v,\text{and}w$. (d) Temperature $\theta$ isocontours. Right, from top to bottom: Fourier coefficients (e) $|\widehat{u}(i_y,i_z)|$, (f) $|\widehat{v}(i_y,i_z)|$, (g) $|\widehat{w}(i_y,i_z)|$, and (h) $|\widehat{\theta }(i_y,i_z)|$.

Figure 12

(a): Time series of spanwise velocity $v$ at point (0.0381, 0.585, 6.985) at $\mathrm{Ra}=6050, A_z=10$. (b) Enlargement of (a) for $8000<t<12000$. The red dots correspond to the instantaneous fields at times $t=10000$ and $25000$ shown, respectively, in Figs. 13 and 14.

Figure 13

Flow perturbation pattern on the central plane $x=0.5$ at $t=10000$ and $\mathrm{Ra}=6050$ with $A_z=10$. Left: Velocity components ($\mathrm{\Delta }u, v, \mathrm{\Delta }w$) (a–c) and temperature (d) $\mathrm{\Delta }\theta$ contours. Right, from top to bottom: Fourier coefficients (e) $|\mathrm{\Delta }\widehat{u}(i_y,i_z)|$, (f) $|\mathrm{\Delta }\widehat{v}(i_y,i_z)|$, (g) $|\mathrm{\Delta }\widehat{w}(i_y,i_z)|$, and (h) $|\mathrm{\Delta }\widehat{\theta }(i_y,i_z)|$.

Figure 14

Flow field on the central plane $x=0.5$ at $t=25000$ and $\mathrm{Ra}=6050$ with $A_z=10$. Left: Velocity components $u,v,w$ (a–c) and temperature (d) $\theta$ contours. Right, from top to bottom: Fourier modulus $|\widehat{u}(i_y,i_z)|, |\widehat{v}(i_y,i_z)|, |\widehat{w}(i_y,i_z)|$, and $|\widehat{\theta }(i_y,i_z)|$.

Figure 15

(a) Time series of temperature at the point (0.0381, 0.122, 4.96) in the boundary layer near the hot wall, $\mathrm{Ra}=6150$. (b) Temporal spectrum of the periodic portion $t\in [8000,10000]$ of the signal (a).

Figure 16

Nonlinear simulation of flow structures at two different instants: isosurface of $Q$ criterion $Q=0.1$ and streamlines, $\mathrm{Ra}=6150, A_z=9$.

Figure 17

Left column: The temperature contours on the vertical plane $x=0.0381$. Right column: Corresponding spatial 2D Fourier modes $|\widehat{\theta }(i_y,i_z)|$ ($i_y,i_z=-5,5$) for the fluctuations, $\mathrm{Ra}=6150$.

Figure 18

Temporal evolution at $\mathrm{Ra}=6150$ of the spectral coefficients $|\widehat{\theta }(i_y,i_z)|$ on the vertical plane $x=0.0381$ (close to the hot plate) for the two most energetic modes.

Figure 19

(a) Time series at $\mathrm{Ra}=6250$ of the temperature perturbation at the point (0.0381, 0.122, 4.96) in the hot boundary layer. (b) Temporal spectrum of the periodic portion of the signal represented in (a) for $t\in [5000,8000]$.

Figure 20

Flow structures $\mathrm{Ra}=6250$ at different instants spanning half of one temporal oscillation: isosurface of $Q=0.1$ and streamlines. The four chosen times correspond to the dash-point lines in Fig. 22.

Figure 21

Evolution at $\mathrm{Ra}=6250$. Left: Temperature contours at $\mathrm{Ra}=6250$ on a vertical plane $x=0.0381$ for four different times (indicated in Fig. 20). Right: Corresponding spatial spectrum of the temperature perturbation $|\mathrm{\Delta }\widehat{\theta }(i_y,i_z)|$ ($i_y,i_z=-5,5$).

Figure 22

Temporal evolution at $\mathrm{Ra}=6250$ of the four most energetic spectral coefficients $|\widehat{\theta }(i_y,i_z)|$ on the plane $x=0.0381$. The vertical dash-point lines correspond to the four instants shown in Fig. 21.

Figure 23

(a) Time series of temperature perturbation at the point (0.0381, 0.122, 4.96) in the boundary layer near the hot wall, $\mathrm{Ra}=6300$. (b) Temporal spectrum of the chaotic portion of the signal in (a) limited to $t\in [5000,8000]$.

Figure 24

Flow structures at $\mathrm{Ra}=6300$ for six different instants [indicated by vertical dash-point lines in Fig. 25 (right)]: isosurface of $Q=0.1$ and streamlines.

Figure 25

Temperature field at $\mathrm{Ra}=6300$ (the spatial average has been removed). (a) Time-averaged 2D Fourier coefficient of the temperature $|\widehat{\theta }(i_y,i_z)|$ ($i_y,i_z=-5,5$) on the vertical plane $x=0.0381$; the average is performed over times $t\in [7000,8000]$. (b) Temporal evolution of the four most energetic spectral coefficients $|\widehat{\theta }(i_y,i_z)|$ on the plane $x=0.0381$. The vertical dash-point lines correspond to the six instances of the flow pattern shown in Fig. 24.