Adaptive gradient-enhanced successive over-relaxation for efficient solution of the Reynolds equation in spherical coordinates

AĞÖREN G., KARAÇAY T.

Proceedings of the Institution of Mechanical Engineers, Part C: Journal of Mechanical Engineering Science, 2026 (SCI-Expanded, Scopus)

Özet

In this study, the Reynolds equation derived in spherical coordinates for compressible fluids has been solved using a hybrid approach combining the differential transformation method (DTM) and the finite difference method (FDM). To accelerate convergence, the classical successive over-relaxation (SOR) method has been enhanced by adding a slope parameter (β). The effects of the relaxation parameter (α) and the slope parameter (β) on pressure distribution, load-carrying capacity, iteration count, and total solution time have been investigated for different eccentricity ratios. The results show that increasing (Formula presented) significantly reduces the iteration count, while the addition of (Formula presented) provides a further reduction in computation time, especially when (Formula presented) is already optimized. For fixed (Formula presented) and (Formula presented) values, the accelerated scheme introduces a practical speed deviation trade-off in the load-carrying-capacity results under the adopted stopping criterion. The shortest solution time is obtained with (Formula presented) and (Formula presented), approximately 50 s, compared with 145 s for the baseline case (Formula presented) ) = (1, 0). To recover the baseline solution while retaining the computational benefit of acceleration, an adaptive strategy is introduced in which (Formula presented) and (Formula presented) are gradually reduced to 1 and 0, respectively, as convergence is approached. The findings demonstrate that the developed SOR approach can significantly accelerate the numerical solution of the Reynolds equation in spherical coordinates, thereby enhancing computational efficiency in engineering applications. Experimental validation is performed by measuring the shaft lift at different supply pressures, and the corresponding eccentricity ratios showed reasonable agreement with the simulation results.