A time-dependent nodal-integral method for the investigation of bifurcation and nonlinear phenomena in fluid flow and natural convection

Two new low- and high-order time-dependent nodal-integral methods were developed and applied to both incompressible fluid flow and natural convection. These new methods have a high level of accuracy on a coarse mesh, high efficiency, and an ability to reproduce results using various time-step sizes independent of a Courant condition. These new methods are applied to various bench-mark problems, such as double-glazing,to verify their accuracy in space and time. Other applications to bifurcation searches and stability of flow fields verify their accuracy and their ability to duplicate natural phenomena without exhibiting problems with spurious solutions, turning points, and bifurcation points. The new methods are also used to verify the existence of critical values of the aspect ratio. The means by which alternative stable solutions can be obtained form a no-flow initial condition for a critical aspect ratio are also examined.