A scalable spectral Stokes solver for simulation of time-periodic flows in complex geometries

Simulation of unsteady creeping flows in complex geometries has traditionally required the use of a time-stepping procedure, which is typically costly and unscalable. To reduce the cost and allow for computations at much larger scales, we propose an alternative approach that is formulated based on the unsteady Stokes equation expressed in the time-spectral domain. This transformation results in a boundary value problem with an imaginary source term proportional to the computed mode that is discretized and solved in a complex-valued finite element solver using Bubnov-Galerkin formulation. This transformed spatio-spectral formulation presents several advantages over the traditional spatio-temporal techniques. Firstly, for cases with boundary conditions varying smoothly in time, it provides a significant saving in computational cost as it can resolve time-variation of the solution using a few modes rather than thousands of time steps. Secondly, in contrast to the traditional time integration scheme with a finite order of accuracy, this method exhibits a super convergence behavior versus the number of computed modes. Thirdly, in contrast to the stabilized finite element methods for fluid, no stabilization term is employed in our formulation, producing a solution that is consistent and more accurate. Fourthly, the proposed approach is embarrassingly parallelizable owing to the independence of the solution modes, thus enabling scalable calculations at a much larger number of processors. The comparison of the proposed technique against a standard stabilized finite element solver is performed using two- and three-dimensional canonical and complex geometries. The results show that the proposed method can produce more accurate results at 1% to 11% of the cost of the standard technique for the studied cases.