RollCells2D

• Problem description
• Visualizations
• References

Problem description

The results and descriptions presented on this page are based on the work of Weatherhill et al. [1].

When denser, more saline ground water overlies less dense, fresher ground water, the denser water has a tendency to sink, and the less dense water to rise. These tendencies are mitigated by the dissipation of salinity gradients by solute dispersion and the dissipation of convective transport by viscous forces. Whether the system remains stable or whether it undergoes unstable convection is determined by the so-called "solute Rayleigh number," which is the ratio of buoyancy forces (which drive free convective transport of solute) to dispersive/viscous forces (which disperse solute and dissipate free convective transport). When the solute Rayleigh number exceeds a critical value, unstable convection occurs. For solute Rayleigh numbers below the critical value, the density stratification remains stable.

The results shown on this page are for unstable ("natural") convection in a finite, two-dimensional (2D) box filled with a saturated porous medium, whose length is an integer multiple of its height, and whose top and bottom boundaries are held at different solute concentrations -- the maximum concentration at the top, and the minimum concentration at the bottom. For this configuration, the critical solute Rayleigh number has been computed theoretically to be 4p² (approximately 39.48). The system is initially at an unstable steady state, with a mildly nonlinear vertical concentration profile that corresponds to diffusion between the top and bottom plates, and a hydrostatic pressure profile. To facilitate the onset of convection, the system is seeded with a small concentration anomaly in the center of the box. The physical properties of the system are such that the Rayleigh number exceeds the critical value, and a system of convective roll cells develops.

Visualizations

The results shown below were generated using SutraGUI 2D3D.1, SUTRA 2.0 (2D3D.1), and Model Viewer 1.1.

	Concentration. Plot of the near-steady concentration distribution. Concentrations range from a minimum value (blue) at the bottom boundary to a maximum value (red) at the top boundary. Animation shows evolution toward steady state over four years. Images generated using Model Viewer. Larger image (GIF, 8 KB) Animation (GIF, 648 KB)
	Flow vectors. Plot of the near-steady velocity distribution. Note that roll cell width is nearly equal to spacing between top and bottom boundaries, in agreement with theory [1]. Animation shows evolution toward steady state over four years. Images generated using Model Viewer. Larger image (GIF, 11 KB) Animation (GIF, 889 MB)

References

[1]  Weatherill, D., Simmons, C. T., Voss, C. I., and Robinson, N. I., 2004, Testing density-dependent groundwater models: Two-dimensional steady state unstable convection in infinite, finite and inclined porous layers: Advances in Water Resources, v. 11, no. 5, p. 547-562.