Transition from steady to chaotic flow of natural convection on a section-triangular roof

Natural convection over a roof-shaped triangular surface is investigated using direct numerical simulations. The Rayleigh number (Ra) was varied from 1 to $5\times {10}^6$ with air as working fluid (Prandtl number of 0.71) at a fixed geometrical aspect ratio of 0.1, defined as the ratio of roof height to half-width. The transition route from a steady flow to a chaotic flow on the surface is characterized by the topological method with the increase of Ra. A weak flow, dominated by conduction, occurs when Ra was relatively small. As Ra increases, the convective flow becomes stronger and a sequence of bifurcations is found. Between $\mathrm{Ra}={10}^2$ and ${10}^3$, a primary pitchfork bifurcation occurs. Secondary and tertiary pitchfork bifurcations are observed in the range $\mathrm{Ra}=\left[{10}^3,{10}^4\right]$ and $\left[{10}^4,{10}^5\right]$, respectively. After another pitchfork bifurcation at $\mathrm{Ra}=\left[1.4,1.5\right]\times {10}^6$, which makes the plume tilt to either side of the roof top edge, a Hopf bifurcation is observed in $\mathrm{Ra}=\left[1.9,2\right]\times {10}^6$, after which both the slope flow and plume become periodic. This is followed by further bifurcations including a period doubling bifurcation at $\mathrm{Ra}\approx 3\times {10}^6$ and a quasiperiodic bifurcation firstly arising at $\mathrm{Ra}\approx 3.4\times {10}^6$. Finally, the flow becomes chaotic for $\mathrm{Ra}>3.7\times {10}^6$. The state space, the maximum Lyapunov exponent, the fractal dimension, and the power spectral density are presented to analyze the flows in the transition to chaos. This work is a comprehensive description of the flow transition from steady state to chaos on surface of a section-triangular roof that is pertinent to various settings where fluid flow develops.

Figure 1

Schematic of the computational domain and boundary conditions of natural convection around a section-triangular roof model. The surface on the bottom in red is isothermal, whereas the shadow surface is adiabatic.

Figure 2

Validation of the present numerical model based on the local heat transfer coefficient of a plate. The experimental result was from Kimura et al.. [41] for a slightly inclined, heated plate.

Figure 3

Conduction-dominant weak flow and temperature for (a) $\mathrm{Ra}=1$, (b) $\mathrm{Ra}=10$, and (c) $\mathrm{Ra}={10}^2$, plotted for (1) X-velocity contour on $\theta =0.2$ isothermal surface, and temperature contours on the planes (2) $Z=2$, (3) $X=0$, and (4) $Y=1$.

Figure 4

Flow and temperature after the transition to a convection-dominant regime. Contours for(a) $\mathrm{Ra}={10}^3$, (b) ${10}^4$, (c) ${10}^5$, and (d) ${10}^6$ are plotted for (1) X velocity on $\theta =0.2$ isothermal surface, (2) temperature on $Z=2$ plane, (3) temperature on $X=0$ plane with streamlines.

Figure 5

Streamlines on quarter surface plane $(X^{\prime }-Z^{\prime })$ situated 0.01 from the heated plate for (a) $\mathrm{Ra}={10}^3$, (b) Ra = $\mathrm{Ra}={10}^4$, (c) $\mathrm{Ra}={10}^5$, and (d) $\mathrm{Ra}={10}^6$.

Figure 6

Velocity in the X direction at point (0, 0.2, 1.5) on plane $X=0$. The shaded regions indicate the Ra ranges where pitchfork bifurcations occurred, based on the topological method.

Figure 7

Temperature history at point (0, 0.2, 1.5) for (a) $\mathrm{Ra}=1.9\times {10}^6$ and (b) $2\times {10}^6$. Z velocity at point (0, 0.2, 1.5) for (c) $\mathrm{Ra}=2\times {10}^6$. The corresponding power spectral density for (b) is plotted in (d), where $f_f=0.209$.

Figure 8

Flow and temperature contours at $\mathrm{Ra}=2\times {10}^6$ after the onset of a Hopf bifurcation. Contours at (a) ${\tau }^{\prime }$, (b) ${\tau }^{\prime }+3$, (c) ${\tau }^{\prime }+6$ are plotted for (1) X velocity on $\theta =0.2$ isothermal surface, and temperature contours on the planes (2) $Z=2$, (3) $X=0$, and (4) $Y=1$.

Figure 9

Z velocity at point (0, 0.2, 1.5) for (a) $\mathrm{Ra}=2.9\times {10}^6$, and (b) $\mathrm{Ra}=3\times {10}^6$, with corresponding power spectral densities in (c) and (d).

Figure 10

Flow and temperature for $\mathrm{Ra}=3\times {10}^6$ after onset of period-doubling bifurcation. Contours at (a) ${\tau }^{\prime }$, (b) ${\tau }^{\prime }+6$, (c) ${\tau }^{\prime }+12$ are plotted for (1) X velocity on $\theta =0.2$ isothermal surface, and temperature on the orthogonal planes of (2) $Z=2$, (3) $X=0$, and (4) $Y=1$.

Figure 11

Z velocity at point (0, 0.2, 1.5) for (a) $\mathrm{Ra}=3.4\times {10}^6$ and (b) $\mathrm{Ra}=3.5\times {10}^6$ with respective power spectral densities in (c) and (d).

Figure 12

Temperature evolution at point (0, 0.2, 1.5) for (a) $\mathrm{Ra}=3.7\times {10}^6$ and (b) $\mathrm{Ra}=5\times {10}^6$ with corresponding power spectral densities in (c) and (d).

Figure 13

The maximum Lyapunov exponent ${\lambda }_L$ for different Rayleigh numbers.

Figure 14

Flow and temperature profiles for $\mathrm{Ra}=4\times {10}^6$. Contours at (a) ${\tau }^{\prime }$, (b) ${\tau }^{\prime }+75$, (c) ${\tau }^{\prime }+150$, (d) ${\tau }^{\prime }+225$ are plotted for (1) X velocity on $\theta =0.2$ isothermal surface, temperatures on the planes (2) $Z=2$, (3) $X=0$.

Figure 15

Portrait of the X and Z velocities and temperature at point (0, 0.2, 1.5) as Ra is increased, showing the transitions of flow regime from laminar to chaos.

Figure 16

Nusselt number of the flow over the inclined plate as a function of Ra.