Convective flow in the presence of a small obstacle: Symmetry breaking, attractors, hysteresis, and information

This work explores the stability and hysteresis effects that occur when a small sink of momentum is introduced into a heat-driven, two-dimensional convective flow. As per standard fluid mechanical intuition, the system minimizes work generation and dissipation when one component of momentum is extracted. However, when the sink absorbs all incoming momentum, the system configures itself such that one of the convection plumes aligns directly with the sink. This state is the most hydrodynamically stable, but it maximizes, rather than minimizes extracted mechanical work. Furthermore, in the case of only vertical momentum extraction, there are two attractors, with different stabilities. Numerical experiments involving slow variations of the horizontal momentum extraction show a clear history dependence. This hysteresis preserves information about the system's past states, and hence represents a primitive memory. The momentum sink can also be used to manipulate the horizontal position of the flow field, with potential applications in microfluidics and laminar convection systems. This simple system exhibits the phenomena of autocatalysis (during the initial growth of the convection plumes), negative feedback (the attractors are either fully or quasistable), memory, and elementary computation.

Figure 1

Flow structure of a convective fluid driven at a Rayleigh number of $\text{Ra}={10}^5$. (a) Flow and temperature fields (dimensionless temperature scale is shown on the right). (b) Components and magnitude of the fluid momentum at the horizontal midplane of the system. Red dot-dashed line corresponds to horizontal momentum, solid green to vertical momentum, and dotted blue to total momentum magnitude. Values are normalized by the maximum momentum magnitude. $\rho$ is fluid density.

Figure 2

Graphical definition of the phase parameter $\varphi$, which measures the minimum distance between a convection cell center and the momentum sink, that is shown with white crosshairs.

Figure 3

Typical flow structure of a convective fluid driven at a Rayleigh number of $\text{Ra}={10}^5$ in the presence of a small sink of horizontal momentum placed at $[x,y]=[W/2,H/2]$ (the white crosshairs show its location, its size is only $\sim 0.7\%$ of the system height). (a) Flow and temperature fields (dimensionless temperature scale is shown on the right). (b) Components and magnitude of the fluid momentum at the horizontal midplane of the system. Red dot-dashed line corresponds to horizontal momentum, solid green to vertical momentum, and dotted blue to total momentum magnitude. Values are normalized by the maximum momentum magnitude, which was essentially invariant to the type of momentum sink present.

Figure 4

Typical flow structure of a convective fluid driven at a Rayleigh number of $\text{Ra}={10}^5$ in the presence of a small sink of vertical momentum placed at $[x,y]=[W/2,H/2]$ (the white crosshairs show its location, its size is only $\sim 0.7\%$ of the system height). (a) Flow and temperature fields (dimensionless temperature scale is shown on the right). (b) Components and magnitude of the fluid momentum at the horizontal midplane of the system. Red dot-dashed line corresponds to horizontal momentum, solid green to vertical momentum, and dotted blue to total momentum magnitude. Values are normalized by the maximum momentum magnitude, which was essentially invariant to the type of momentum sink present.

Figure 5

Typical flow structure of a convective fluid driven at a Rayleigh number of $\text{Ra}={10}^5$ in the presence of a small sink of total momentum placed at $[x,y]=[W/2,H/2]$ (the white crosshairs show its location; its size is only $\sim 0.7\%$ of the system height). (a) Flow and temperature fields (dimensionless temperature scale is shown on the right). Inset figure shows the local flow around the obstacle, with the red box indicating the area from which all momentum is extracted. (b) Components and magnitude of the fluid momentum at the horizontal midplane of the system. Red dot-dashed line corresponds to horizontal momentum, solid green to vertical momentum, and dotted blue to total momentum magnitude. Values are normalized by the maximum momentum magnitude, which was essentially invariant to the type of momentum sink present.

Figure 6

Work extracted from the total momentum sink as a function of time during a simulation with $\text{Ra}=2\times {10}^5$. Time is normalized by the reference time $\tau$, and work is normalized by the characteristic kinetic energy $W_0$. The maxima in the oscillatory part of the curve correspond to the plume pointing directly at the obstacle, and the minima correspond to the plume pointing to either side of the obstacle.

Figure 7

Kinetic energy extracted from the total momentum sink as a function of Rayleigh number. Curves were derived from simulations in which the Rayleigh number was slowly varied, as shown in the inset figure that displays Rayleigh number as a function of time (blue line). Time units are normalized by the characteristic timescale of the initial Rayleigh number, i.e., that corresponding to $\text{Ra}=2\times {10}^5$. The red line in the inset figure corresponds to the critical Rayleigh number of ${\text{Ra}}_c\approx 1760$. In the main figure, the four curves illustrate the influence of obstacle diameter $d$, given as a fraction of the system height $H$, upon the extracted work. As the obstacle radius becomes a larger fraction of the system height, the extracted work decreases. Work values are normalized by the reference kinetic energy $W_{0_{\text{Ra}=2\times {10}^5}}$, corresponding to the initial Rayleigh number of $2\times {10}^5$, and adjusted for each sink size (see beginning of Sec. 6).

Figure 8

Phase-space behavior for a convective obstacle simulation with time-varying horizontal momentum extraction. Vertical momentum was extracted fully at all times. (a) Time variation of the phase parameter $\varphi$ and fraction of extracted horizontal momentum $m_u$ (time is normalized by the reference time $\tau$). (b) Phase parameter $\varphi$ as a function of extracted horizontal momentum $m_u$ showing irreversibility. Note that the hysteresis path is not a loop, since the system cannot leave the plume attractor once it has arrived there (see text).

Figure 9

The two stable attractors of a convective fluid driven at a Rayleigh number of $\text{Ra}={10}^5$ in the presence of a small sink of horizontal momentum placed at $[x,y]=[W/2,H/2]$ (the black crosshairs show its location; its size is only $\sim 0.7\%$ of the system height). (a) Upwelling plume attractor. (b) Downwelling plume attractor.

Figure 10

The two most commonly occurring attractors of a convective fluid driven at a Rayleigh number of $\text{Ra}={10}^5$ in the presence of a small sink of vertical momentum placed at $[x,y]=[W/2,H/2]$ (the black crosshairs show its location; its size is only $\sim 0.7\%$ of the system height). (a) Clockwise vortex attractor. (b) Anticlockwise vortex attractor.

Figure 11

The two less frequently occurring attractors of a convective fluid driven at a Rayleigh number of $\text{Ra}={10}^5$ in the presence of a small sink of vertical momentum placed at $[x,y]=[W/2,H/2]$ (the black crosshairs show its location; its size is only $\sim 0.7\%$ of the system height). (a) Upwelling plume attractor. (b) Downwelling plume attractor. Note that these two attractors exhibit strong oscillations.

Figure 12

The two stable attractors of a convective fluid driven at a Rayleigh number of $\text{Ra}={10}^5$ in the presence of a small sink of total momentum placed at $[x,y]=[W/2,H/2]$ (the black crosshairs show its location; its size is only $\sim 0.7\%$ of the system height). Inset figures show the local flow around the obstacle, with the red box indicating the area from which all momentum is extracted. (a) Upwelling plume attractor. (b) Downwelling plume attractor.