An improvement of collocated finite difference scheme with regard to kinetic energy conservation

Abstract

Finite difference method in a collocated grid system has the merits of both regular and staggered grid systems, and has mainly been used for steady flow simulations. Recently, some unsteady flows have been simulated by using the collocated grid system. The author pointed out a violation of kinetic energy conservation in the collocated grid system, which is important to unsteady turbulent flow simulations. In this paper, a modified algorithm for improving the conservation property of the collcated grid system is proposed. Van Kan type pressure correction is introduced to minimize the defect. The modified algorithm with fourth order accuracy in space is also proposed. Conservation properties of the numerical algorithms in the collocated grid system are demonstrated on the example of two-dimensional periodic inviscid flow simulations. Plane turbulent channel flow simulations are also performed to confirm the reliavility of the modified algorithm.