Bifurcation analysis of two-dimensional Rayleigh-Bénard convection using deflation

We perform a bifurcation analysis of the steady states of Rayleigh-Bénard convection with no-slip boundary conditions in two dimensions using a numerical method called deflated continuation. By combining this method with an initialization strategy based on the eigenmodes of the conducting state, we are able to discover multiple solutions to this nonlinear problem, including disconnected branches of the bifurcation diagram, without the need for any prior knowledge of the solutions. One of the disconnected branches we find contains an $\mathsf{S}$-shaped curve with hysteresis, which is the origin of a flow pattern that may be related to the dynamics of flow reversals in the turbulent regime. Linear stability analysis is also performed to analyze the steady and unsteady regimes of the solutions in the parameter space and to characterise the type of instabilities.

Figure 1

(a) Critical Rayleigh number with respect to the aspect ratio $\mathrm{\Gamma }$ (blue dots). The red dot indicates the aspect ratio $\mathrm{\Gamma }=1$, which is the focus of our study. (b) Growth rates of the eigenmodes that emanate from the conducting steady state.

Figure 2

Eigenmodes of the primary bifurcations emanating from the conducting steady state (4) in the Rayleigh number range $0\le \text{Ra}\le {10}^5$. The top rows show the magnitude of the velocity $\mathbf{v}$ (red color indicates a high magnitude and blue a zero-velocity magnitude), while the bottom rows show the temperature $\theta$ (red and blue display positive and negative temperature values, respectively).

Figure 3

(a) Bifurcation diagram using the kinetic energy ${\parallel \mathbf{u}\parallel }_2^2$ of the branches studied in the paper. Also represented are (b) the potential energy ${\parallel T\parallel }_2^2$ and (c) the Nusselt number $\text{Nu}$ of the branches as a function of $\text{Ra}$. The branches that originate from the linear conducting state are plotted with dashed lines, while branches emanating from secondary instabilities and disconnected branches are depicted with solid lines. The horizontal black line in (b) indicates the potential energy of the conducting steady state.

Figure 4

Evolution of the steady states in branch (1) arising from the first eigenmode, illustrated via (a) the kinetic energy and (b) the potential energy. The color scheme for the temperature ranges from 0 (blue) to 1 (red). Also shown are (c) the largest growth rates and (d) the corresponding frequencies.

Figure 5

Evolution of the steady states in branch (2) arising from the second eigenmode, illustrated via (a) the kinetic energy and (b) the potential energy. Also shown are (c) the largest growth rates and (d) the corresponding frequencies. The two largest growth rates are shown in red and green, while the subsequent ones are depicted in blue.

Figure 6

Evolution of the steady states in branch (10), bifurcating from branch (2) at $\text{Ra}\approx 13550$, illustrated via (a) the kinetic energy and (b) the potential energy of branch (10). Also shown are (c) the largest growth rates and (d) the corresponding frequencies.

Figure 7

Evolution of the steady states in branch (3) arising from the third eigenmode, illustrated via (a) the kinetic energy and (b) the potential energy. The upper and lower parts of the branch are colored in blue and red, respectively. Also shown are (c) the largest growth rates and (d) the corresponding frequencies of the upper part of the branch. (e) and (f) Same as (c) and (d) but for the lower part of the branch.

Figure 8

Evolution of the steady states in branch (12), bifurcating from branch (3) around $\text{Ra}\approx 23800$ and illustrated via (a) the kinetic energy and (b) the potential energy. Also shown are (c) the largest growth rates and (d) the corresponding frequencies.

Figure 9

Evolution of the steady states in branch (4) arising from the fourth eigenmode, illustrated via (a) the kinetic energy and (b) the potential energy. Also shown are (c) the largest growth rates and (d) the corresponding frequencies.

Figure 10

Evolution of the steady states in branch (5) illustrated via (a) the kinetic energy and (b) the potential energy. Similarly to branch (4), this branch bifurcates from the fourth eigenmode. Also shown are (c) the largest growth rates and (d) the corresponding frequencies.

Figure 11

Velocity magnitude of the steady state in branches (4) and (5) at $\text{Ra}={10}^5$.

Figure 12

Evolution of the steady states in branch (6), bifurcating from branch (4) at $\text{Ra}\approx 38150$, illustrated via (a) the kinetic energy and (b) the potential energy. Also shown are (c) the largest growth rates and (d) the corresponding frequencies.

Figure 13

Evolution of the steady states in branch (7), bifurcating from branch (5) at $\text{Ra}\approx 46300$, illustrated via (a) the kinetic energy and (b) the potential energy. Also shown are (c) the largest growth rates and (d) the corresponding frequencies.

Figure 14

Evolution of the steady states in the disconnected branch (11), illustrated via (a) the kinetic energy and (b) the potential energy. The upper and lower parts of the branch are colored in blue and red, respectively. Also shown are (c) the largest growth rates and (d) the corresponding frequencies of the upper part of the branch. (e) and (f) Same as (c) and (d) but for the lower part of the branch.

Figure 15

Evolution of the steady states in branch (8), bifurcating from the lower part of the disconnected branch (11) at $\text{Ra}\approx 37000$, illustrated via (a) the kinetic energy and (b) the potential energy. Also shown are (c) the largest growth rates and (d) the corresponding frequencies.

Figure 16

Evolution of the steady states in the disconnected branch (9), illustrated via (a) the kinetic energy and (b) the potential energy. The upper and lower parts of the branch are colored in blue and red, respectively. Also shown are (c) the largest growth rates and (d) the corresponding frequencies of the upper part of the branch. (e) and (f) Same as (c) and (d) but for the lower part of the branch.