Transition to time-dependent flow in highly viscous horizontal convection

Horizontal convection is studied numerically in two dimensions for the case of an infinite Prandtl number and shear-free boundary conditions on all sides. Three different temperature distributions are applied at the top boundary. Steady states become unstable, giving way to unsteady flow with increasing Rayleigh number Ra. The transition to time-varying flow depends on the heating configuration. For $\text{Ra}\ge 5\times {10}^9$ simulations show a time-dependent flow behavior for a linearly varying temperature at the top, whereas other temperature distributions show this transition at even lower Rayleigh numbers. The scaling laws of heat flow and circulation strength are investigated and the results indicate that both are changed by the upcoming time-dependent flow. While in the steady boundary layer flow regime the heat flux scales as ${\text{Ra}}^{1/5}$, the exponent increases to a value of approximately $0.25$ in the unsteady regime. A similar transition is observed for the circulation strength. The observed scaling laws are in good agreement with established theories in both the steady and unsteady regimes. After the transition, the flow shows a periodic oscillation of the boundary layer circulation, which turns into a more irregular flow behavior as the Rayleigh number is increased further. We conclude that, even for an infinite Prandtl number, horizontal convection exhibits unsteady flow at sufficiently high Rayleigh numbers and that this flow resembles the turbulent circulation at low Prandtl numbers.

Figure 1

Model setup for the numerical simulations showing the different boundary conditions and a simplified grid including 21 elements in each direction and an exponentially decreasing grid spacing from $z=0$ to $z=1$. Actual grids have at least $101\times 101$ and up to $251\times 251$ elements.

Figure 2

Streamlines of the linear and the step profile for (a) and (d) $\text{Ra}={10}^4$, (b) and (e) $\text{Ra}={10}^8$, and (c) and (f) $\text{Ra}={10}^{10}$ after a thermal equilibrium is reached. Graphics (a), (b), and (d) evolved to a steady state, while (c), (e), and (f) remained unsteady, so these figures represent only snapshots of a temporal behavior. The intervals between streamlines are (a) and (d) 0.5, (b) and (e) 5, (c) 15, and (f) 25.

Figure 3

Temporal evolution of (a) Nu and (b) Pe in the step profile. Labels in the plots indicate Ra. In (b) $\text{Ra}={10}^9$ is not shown because it overlaps with the adjacent values.

Figure 4

Scaling laws of (a), (c), and (e) heat flux and (b), (d), and (f) circulation strength for the three heating configurations applied at the top boundary. Crosses indicate runs reaching a steady state and squares stand for runs that remain unsteady. The lines represent different scaling laws, indicated by the labels in the plot.

Figure 5

Phase plots of the Péclet number for (a) $\text{Ra}={10}^8$, (b) $\text{Ra}=2\times {10}^8$, and (c) $\text{Ra}={10}^{10}$ with the step profile. In (d) the linear profile is used for the same Rayleigh number as in (c): $\text{Ra}={10}^{10}$. Time derivatives are calculated with a centered-difference approximation using the number of the time step instead of the time in the denominator. This neglects the variable step size, but accounts for the loss of significance due to the very small time steps. The qualitative appearance of the phase diagrams is not affected.

Figure 6

(a)–(d) Temporal evolution of the Nusselt number for $\text{Ra}=2\times {10}^9$ and different initial temperatures in the step profile. (e) Depth dependence of the temperature in the final states of convection at $x=0.5$. The labels in the plot (e.g., $T=0.0$) mark the initial temperature of the run. The lines for $T=0.2$ and $T=0.076$ lie exactly on top of each other. The ones of $T=0.0$ and $T=0.07$ are also very similar to each other.