Instability of natural convection of air in a laterally heated cube with perfectly insulated horizontal boundaries and perfectly conducting spanwise boundaries

Steady and oscillatory instabilities of buoyancy convection in a laterally heated cube with perfectly thermally insulated horizontal boundaries and perfectly thermally conducting spanwise boundaries is studied. The problem is treated by Newton and Arnoldi methods based on Krylov subspace iteration. The finite volume grid is gradually refined from ${100}^3$ to ${256}^3$ finite volumes. It is revealed that en route to unsteadiness the flow undergoes a steady symmetry-breaking pitchfork bifurcation, which breaks its reflection and two-dimensional rotational symmetries. With a further increase of the Grashof number, the nonsymmetric flow bifurcates into an oscillatory state via a Hopf bifurcation. It is argued that the steady symmetry-breaking bifurcation takes place owing to the instability of counter-rotating vortical motions. The oscillatory instability develops as a result of the interaction between the unstable boundary layer flow and the stabilizing effect of the stable temperature stratification.

Figure 1

Visualization of a subcritical steady 3D flow at $\mathrm{Pr}=0.71$, $\mathrm{Gr}={10}^7$ by isotherms (a) and divergence-free projections of the velocity field on the coordinate planes (b)–(e). The projected velocity fields are depicted by vectors. Isosurfaces of the velocity potentials, to which the projected velocities fields are tangent, are shown by colors. The minimal and maximal values of the potentials are $\pm 0.0101,\left(-0.0460,0.000587\right)$, and $\pm 0.0104$ for ${\mathrm{\Psi }}^{\left(x\right)}$, ${\mathrm{\Psi }}^{\left(y\right)}$, and ${\mathrm{\Psi }}^{\left(z\right)}$, respectively. The isosurfaces are plotted for the levels: $-0.0092$ (b) and $-0.0023$, $-0.0138$, and $-0.0276$ (c) for ${\mathrm{\Psi }}^{\left(y\right)}$; $\pm 0.005$ for ${\mathrm{\Psi }}^{\left(x\right)}$ (d); and $\pm 0.005$ for ${\mathrm{\Psi }}^{\left(z\right)}$ (e).

Figure 2

Visualization of a subcritical steady 3D flow at $\mathrm{Pr}=0.71$, $\mathrm{Gr}=6\times {10}^7$ by isotherms (a) and divergence-free projections of the velocity field on the coordinate planes (b)–(e). The projected velocities fields are depicted by vectors. Isosurfaces of the velocity potentials, to which the projected velocities fields are tangent, are shown by colors. The minimal and maximal values of the potentials are $\pm 0.00736,\left(-0.0434,0.000662\right)$, and $\pm 0.00901$ for ${\mathrm{\Psi }}^{\left(x\right)}$, ${\mathrm{\Psi }}^{\left(y\right)}$, and ${\mathrm{\Psi }}^{\left(z\right)}$, respectively. The isosurfaces are plotted for the levels: $-0.000866$ (b) and $-0.00217$, $-0.013$, and $-0.026$ (c) for ${\mathrm{\Psi }}^{\left(y\right)};\pm 0.00365$ for ${\mathrm{\Psi }}^{\left(x\right)}$ (d), $\pm 0.00455$ for ${\mathrm{\Psi }}^{\left(z\right)}$ (e).

Figure 3

Visualization of a subcritical steady 3D flow at $\mathrm{Pr}=0.71$, $\mathrm{Gr}=1.1\times {10}^8$ by isotherms (a) and divergence-free projections of velocity field on the coordinate planes (b)–(e). The projected velocities fields are depicted by vectors. Isosurfaces of the velocity potentials, to which the projected velocities fields are tangent, are shown by colors. The minimal and maximal values of the potentials are $(-0.00665,0.00720),(-0.0422,0.000951)$ and $\left(-0.00823,0.00880\right)$ for ${\mathrm{\Psi }}^{\left(x\right)}$, ${\mathrm{\Psi }}^{\left(y\right)}$, and ${\mathrm{\Psi }}^{\left(z\right)}$, respectively. The isosurfaces are plotted for the levels: $-0.00844$ (b) and $-0.00211$, $-0.0127$, and $-0.0253$ (c) for ${\mathrm{\Psi }}^{\left(y\right)}$; $\pm 0.0035$ for ${\mathrm{\Psi }}^{\left(x\right)}$ (d); and $\pm 0.00425$ for ${\mathrm{\Psi }}^{\left(z\right)}$ (e).

Figure 4

Profiles of the velocities and the temperatures along the midline $x=z=0.5$.

Figure 5

Profiles of the velocities and the temperatures along the line $x=z=0.1$.

Figure 6

Perturbation of the temperature and the three velocity components that causes the symmetry-breaking steady bifurcation. The plotted levels are shown in the lower right corner of each frame. All functions are real.

Figure 7

Velocity potentials of the perturbation that causes the symmetry-breaking steady bifurcation. The plotted levels are shown in the lower right corner of each frame. All functions are real.

Figure 8

Perturbation of the three velocity components that causes the symmetry-breaking steady bifurcation with the zeroed temperature perturbation. The plotted levels are shown in the lower right corner of each frame. All functions are real.

Figure 9

(a)–(c) Flow at $\mathrm{Gr}=2.32\times {10}^6$ visualized by divergence-free projections of velocity field on the coordinate planes. The levels shown are (a) $\pm 0.008$, (b) $-0.015,-0.02$, −0.03, and (c) $\pm 0.005$. (d)–(f) Perturbation of the three potentials that causes the symmetry breaking steady bifurcation with the zeroed temperature perturbation. The levels shown are (a) $\pm 6\times {10}^{-5}$, (b) $\pm {10}^{-4}$, and (c) $\pm 1.5\times {10}^{-4}$. All functions are real.

Figure 10

Isotherms at $\mathrm{Gr}=2.32\times {10}^6$ plotted on the coordinate planes for (a) $x=0.05,0.95;$ (b) $y=0.2,0.8;$ and (c) $z=0.07,0.7$. White arrows show schematically the direction of the perturbed motion as depicted in Figs. 8–8.

Figure 11

(a) Isosurfaces of the amplitude of the most unstable temperature perturbation. The maximal value and levels plotted are 0.308, 0.04, 0.07, and 0.1. (b)–(d) Amplitude isolines in characteristic $y$, $x$, and $z$ cross sections respectively. $\mathrm{Gr}=1.13\times {10}^8$.

Figure 12

Isosurfaces of the absolute values of the most unstable perturbations. The plotted levels are 0.1 and 0.2 in all frames. $\mathrm{Gr}=1.13\times {10}^8$.

Figure 13

Snapshots of the most unstable perturbation of the temperature and velocity. The plotted levels are reported near the corresponding frames. The amplitudes are the same as those in Fig. 6. The snapshots at the 1/2 and 3/4 of the time period can be obtained by reversing the colors. $\mathrm{Gr}=1.13\times {10}^8$. See Supplemental Material Animation 1 [33] for change of the perturbation over the time period.

Figure 14

Snapshots of the velocity potentials of the most unstable perturbation. The minimal and maximal values and the plotted levels are shown in the upper frames. The snapshots at the 1/2 and 3/4 of the time period can be obtained by reversing the colors. The color maps show the temperature field in the characteristic cross sections. $\mathrm{Gr}=1.13\times {10}^8$. See Supplemental Material Animation 2 [33] for change of the perturbation potentials over the time period.

Figure 15

Snapshots of the temperature perturbations superimposed with snapshots of isolines of vector potentials of the velocity disturbances. $\mathrm{Gr}=1.13\times {10}^8$.