A locally implicit improvement of the equilibrium Eulerian method

The equilibrium Eulerian method is a simple way to determine the velocity field of a disperse system of particles. It avoids solving a partial differential equation for particle velocity, which makes it more efficient than the standard Eulerian–Eulerian method. It captures such essential disperse-phase physics as preferential concentration and turbophoretic migration––effects which are ignored by methods that set the particle velocity equal to the fluid velocity. Although the equilibrium Eulerian method works well for small particles, it fails for particles that are too large. This paper presents a straightforward improvement to the method which minimizes the error for larger particles, thereby extending the method’s range of applicability. In particular, it is demonstrated that the modified method captures a physical mechanism neglected by unmodified method: the memory a particle retains when it migrates to an adjacent layer of fluid. The improvement is demonstrated in a direct numerical simulation (DNS) of turbulent channel flow, where particles migrate toward the wall through a strong shear. It is also demonstrated that the modified method performs well in a case where no mean shear is present: a DNS of isotropic turbulence.