Tuning of Membership Functions in a Fuzzy Rule Set for Controlling Convergence of Laminar CFD Solutions

摘要

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. Four problems of laminar fluid flow and heat transfer are solved. They include buoyancy driven flow in a square cavity, lid driven flow in a square enclosure, mixed convection over a backward facing step and Dean flow. The consideration of buoyancy driven flow in a square cavity took into account a case with isothermal vertical walls, as well as a case with heat conduction in the solid wall on the right hand side of the cavity. 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, a set of triangular membership functions was 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 performance of triangular membership functions before optimization. The improvement in speed of convergence varied from 10% in the case of Dean flow to 70% in the case of buoyancy driven flow in a square enclosure with isothermal walls. For all the problems considered, the fuzzy rule set provided convergence comparable in speed to that obtained with the best choice of constant relaxation factors. In the case of buoyancy driven cavity flow with conjugate heat transfer, the fuzzy controller outperformed any choice of constant relaxation factors.