Stability analysis of Rayleigh-Bénard convection in a cylinder with internal heat generation

The flow instabilities of Rayleigh-Bénard convection in a cylinder with effect of uniform internal heat source are investigated numerically. The instabilities of the static state and of axisymmetric flows are investigated by linear stability analysis. The convection threshold depends on the strength of internal heat source $q$ and the aspect ratio of the cylinder $\mathrm{\Gamma }$. The stability of axisymmetric flows is strongly affected by these two parameters, as well as the Prandtl number $\text{Pr}$. Depending on the value of $q$, three regimes are identified: weak internal heating, moderate internal heating, and strong internal heating regime. In a weak internal heating regime, the instability characteristics are similar to Rayleigh-Bénard convection. In a moderate internal heating regime, intense interaction of buoyancy instability and hydrodynamic instability result in complex instability curves. When $q$ is large enough, the internal heating effect overwhelms the boundary heating effect. Specifically, the influence of $\text{Pr}$ on instability is studied at a moderate internal heat strength $q=6.4$. An extremely multivalued stability curve is observed. At most five critical Rayleigh numbers can be determined for the axisymmetry-breaking instability at a certain Prandtl number. An axisymmetric unsteady instability mode is observed as well. By nonlinear simulation, the oscillatory flow patterns are obtained, and the axisymmetry-breaking bifurcation of the unsteady toroidal flow is studied.

Figure 1

Conductive temperature distribution for different heat strength.

Figure 2

Variation of critical Rayleigh number for onset of convection with heat strength at different aspect ratios.

Figure 3

Axisymmetric base flow at $\text{Pr}=0.02, \text{Ra}=4000$, and $q=3$ (a), $q=8$ (b), $q=16$ (c). Left: Stream function contours. Right: Isotherms.

Figure 4

Axisymmetric base flow at $\text{Pr}=0.4, \text{Ra}=10000$, and $q=3$ (a), $q=8$ (b), $q=16$ (c). Left: Stream function contours. Right: Isotherms.

Figure 5

Axisymmetric base flow at $\text{Pr}=1, \text{Ra}=80000$, and $q=3$ (a), $q=8$ (b), $q=16$ (c). Left: Stream function contours. Right: Isotherms.

Figure 6

Critical Rayleigh number, ${\text{Ra}}_{cr}$, as a function of the internal heat strength, $q$, for the transition from the basic state to different three-dimensional states for $\text{Pr}=0.02$ (a), $\text{Pr}=0.4$ (b), and $\text{Pr}=1$ (c). Note that the vertical coordinate for panel (c) is plotted in natural logarithm scale. In panel (d) the critical circular frequency, ${\omega }_{cr}$, as a function of the internal heat strength, $q$, for the unsteady transitions occurred in panels (b) and (c) are plotted. Solid curves with hollow symbols represent steady transitions and filled symbols oscillatory transitions.

Figure 7

(a) ${\text{Ra}}_{cr}$ as a function of $\text{Pr}$ for $q=6.4$. (b) Critical circular frequency ${\omega }_{cr}$ as a function of $\text{Pr}$ for $q=6.4$. The horizontal coordinates and vertical coordinate in panel (a) are plotted in natural logarithm scale. Solid curves with hollow symbols represent steady transitions and filled symbols oscillatory transitions.

Figure 8

Contours of azimuthal velocity on the $z=0.8$ plane in a period at $q=9, \text{Pr}=1, \text{Ra}=1.52\times {10}^5$, oscillation period $T=0.0294$, and patterns at $t=0,T$/6, $2T$/6, $3T$/6, $4T$/6, and $5T$/6 are plotted. Eight levels are drawn between the values of $-2$ and 2. Light area (solid lines) represents positive value, and dark area (dashed lines) represents negative value.

Figure 9

Contours of azimuthal velocity on the $z=0.8$ plane in a period at $q=10, \text{Pr}=1, \text{Ra}=1.4\times {10}^5$, oscillation period $T=0.0317$, and patterns at $t=0,T$/6, $2T$/6, $3T$/6, $4T$/6, and $5T$/6 are plotted. Eight levels are drawn between the values of $-2$ and 2. Light area (solid lines) represents positive value, and dark area (dashed lines) represents negative value.

Figure 10

Contours of azimuthal velocity on the $z=0.8$ plane in a period at $q=12, \text{Pr}=1, \text{Ra}=1.42\times {10}^5$, oscillation period $T=0.0315$, and patterns at $t=0,T$/6, $2T$/6, $3T$/6, $4T$/6, and $5T$/6 are plotted. Six levels are drawn between the values of $-1.4$ and 1.5. Light area (solid lines) represents positive value, and dark area (dashed lines) represents negative value.

Figure 11

Contours of azimuthal velocity on the $z=0.8$ plane in a period at $q=6.4, \text{Pr}=6.7, \text{Ra}=1.45\times {10}^5$, oscillation period $T=0.161$, and patterns at $t=0,T$/6, $2T$/6, $3T$/6, $4T$/6, and $5T$/6 are plotted. Eight levels are drawn between the values of $-1.2$ and 1.2. Light area (solid lines) represents positive value, and dark area (dashed lines) represents negative value.