Steady-state solutions of buoyancy-assisted internal flows using a fast false implicit transient scheme (FITS)

A fast false implicit transient scheme FITS is developed to predict the two-dimensional steady-state solutions of buoyancy-assisted laminar internal flows. This new scheme uses the control volume based on power law technique in conjugation with the alternating direction implicit (ADI) and the successive grid refinement (SGR) procedures to solve the transient vorticity and energy transport equations. The ADI procedure allows the power law, which gives an excellent approximation to the exact I-D solution, to be applied locally in one-dimensional sense for each sweep in the co-ordinates' directions. This in turn increased the solution accuracy and hence permits the use of a larger time increment. As a result a remarkable increase in the convergence rate to steady-state is achieved. The final solution is obtained by successively refining the grid as the solution advances in time. The efficiency of FITS is verified by comparing the present predictions with three steady-state benchmark solutions: natural convection of a heat generating fluid in a rectangular enclosure, natural convection inside a cavity with two isothermal walls, and a vertical buoyancy-assisted laminar backward-facing step flow.