Numerical methods for Rayleigh -Benard convection inside a Hele -Shaw cell
Fluid dynamics inside a Hele-Shaw cell are investigated computationally. A Hele-Shaw cell is a rectangular chamber, filled with fluid, that consists of two closely placed parallel plates, i.e., it is very thin in one direction as compared with the other two. It effectively turns a three-dimensional situation into a “quasi” two-dimensional situation. Whenever a denser fluid is above a less dense fluid, a potentially unstable situation is created. For large enough density differences, convective motion occurs in the chamber. The fluid dynamics that result due to such an odd density arrangement is a specific example of the better known Rayleigh-Benard convection. Whereas, classic Rayleigh-Benard convection is temperature driven, the fluid dynamics in our chamber will be driven by a solute concentration gradient (an isometric problem). The incompressible Navier-Stoke's equations in the Boussinesq approximation are used to model and simulate the fluid motion. An additional energy equation is coupled to describe the solute concentration evolution. These equations are addressed numerically using a pseudo-spectral technique that utilizes a Fast Fourier Transform, and a projection method that allows for inflow/outflow boundary conditions. A stability analysis is performed to evaluate the strength of the simulation process.