Re-scaling of a fractional step method for low Reynolds number flows and fluid-structure-interaction

Abstract

Fractional step methods, even with implicit time integration, suffer from severe time step restrictions at low Reynolds numbers ($Re$). Our analysis shows the splitting error of the implicit fractional step method is inversely proportional to the Reynolds number, which results in the time step restriction. Our solution involves introducing a new Reynolds number scaling in the fractional step method, which eliminates the restriction on time step size without sacrificing the accuracy of the results. We demonstrate the power of this simple scaling in two cases at low Reynolds numbers: (1) flow through a channel commonly encountered in microfluidic applications, and (2) fluid-particle interaction commonly arises in suspensions of micro- or nano-particles. Our results show that, with the new scaling, simulations for Reynolds numbers as low as $Re$ = 0.0001 can be conducted using the same time step as for $Re=1$, which increases the size of time steps by 10000 folds, thereby reducing the computational cost/time. This simple re-scaling can easily be implemented in any fractional-step method.