Tuning of a fuzzy rule set for controlling convergence of a CFD solver in turbulent flow

Under-relaxation in an iterative CFD solver is guided by fuzzy logic to achieve automatic convergence with minimum CPU time. The fuzzy rule set uses information from a Fourier transform of a set of characteristic values. The control algorithm adjusts the relaxation factors for the system variables on each iteration and restarts the solver if divergence occurs. Two turbulent problems based on a k–ε model are solved. They include buoyancy driven flow in a rectangular cavity and mixed convection over a backward facing step. The incompressible Newtonian conservation equations are solved by the SIMPLER algorithm with simple substitution. In order to achieve the best performance of the fuzzy controller, the membership functions were tuned by using a gradient method. The fuzzy control algorithm with the optimal membership functions significantly reduced the CPU time needed for solving the problem, compared to the highest set of constant relaxation factors which do not cause divergence. For turbulent flow over a backward facing step, the CPU time was more than five times shorter with the fuzzy controller than with the constant relaxation factors. For turbulent buoyancy driven flow in a rectangular cavity, the CPU time required for convergence with a fuzzy controller was reduced by a factor of two, compared to the constant relaxation factors.