Convection in a thermosyphon: Bifurcation and stability analysis.

摘要

When a closed vertical loop of fluid is heated from below, the more buoyant hot fluid at the bottom of the loop creates an unstable configuration. In flows such as this one, it is the interplay between the buoyancy, causing the fluid to tend to rise, and the viscosity, causing the fluid to resist flow, that produces the motion of the fluid.; Using the Navier-Stokes equations to model this flow, one arrives at a sequence of bifurcation problems. Historically, researchers have been interested in flow in a thermosyphon both as a bifurcation and an engineering problem. I have derived a model where, in the case of a circular loop, the first Fourier modes exactly decouple from all other Fourier modes, leaving a system of three coupled nonlinear PDEs that completely describe the flow in the thermosyphon. I have characterized the flow through two bifurcations and found a global stability limit, thereby narrowing the location of a third bifurcation. This model has identified periodic solutions for flows of Prandtl number greater than 19, a phenomenon that other, more simplified models do not.; Finding the trivial solution (pure conduction) and linearizing about this solution, one arrives at an eigenvalue problem from which the onset of convection can be found. Using continuation in Grashof number, one can follow this solution branch to the Hopf bifurcation, which changes from sub- to supercritical at Prandtl number ≈19.; I numerically analyze the equations using a spectral method with Chebyshev basis functions for the space dimension and use an implicit-explicit scheme to discretize the time dimension. The results obtained agree with those found analytically, where possible, and those obtained from the full three-dimensional equations with a FEM CFD code (MPSalsa), where available.; Because of the quadratic nonlinearity in this system of equations, it is possible to find the global stability limit, and I have proven it is identical to the first bifurcation point.