An Implicit Scheme for Least-Square Gradient in Coupled Algorithm

Abstract

In this paper, an implicit scheme that uses the least-square method to compute the pressure gradient term in the momentum equation, mainly for coupled algorithm was proposed. Accurate computation of the pressure gradient is crucial in computational fluid dynamics, directly influencing the precision of calculation results. The least-square gradient can reach unconditional second-order accuracy in the finite volume method. Currently, the least-square gradient method is predominantly employed in segregated algorithms, primarily utilizing explicit schemes that are not applicable to coupled algorithms. The scarcity of high-accuracy schemes for computing pressure gradients in coupled algorithms underscores a significant research gap. It contributes by presenting a derivation of an implicit scheme for the least-square gradient, complemented by an extensive discussion on boundary treatment methods. The efficacy of proposed least-square method through comparative analysis involving the Green-Gauss method, as well as benchmarking against existing literature or analytical solutions across distinct cases. The findings demonstrate that, in the majority of cases, the least-square method offers superior accuracy and convergence rates compared with the Green-Gauss method.