# A Parallelizable Mathematical Framework for Linearized Analysis of Flows in the Presence of $n$-Periodic Structures
In this manuscript we outline a practical application of the framework presented by Schmid et al. (Phys. Rev. Fluids, 2 (11), 2017) for the linear analysis of fluid systems consisting of periodic arrays of $n$ identical units (i.e. $n$-periodic systems). We generalize and apply the methodology in the case of large-scale fluid flow solvers, implementing an operator-free approach. Using the roots-of-unity and the corresponding complex transformation, we show that we can decouple the original linear system and massively parallelize its solution. We exploit an operator-free approach to extract information about any arbitrary number of units from our reduced simulations. In this work, we apply this methodology to a simple flow configuration: a boundary layer over a flat plate with $n$-periodic synthetic wakes, approximating the flow downstream of a roughness array. The proposed mathematical framework enables the study of wake synchronization downstream of such arrays with a reduced-cost system of independent simulations that can run in parallel. From a limited number of simulations, it is possible to extract the linear response of the full geometry for arbitrary number of units $n$, thus relaxing the assumption of single-unit periodicity that is commonly imposed in such simulations. We use an in-house high-fidelity direct numerical simulation code to solve the nonlinear and the linearized systems of the Navier-Stokes equations. Potential coupling with the corresponding adjoint solver would enable efficient sensitivity analysis and optimization of the full geometry, with reduced simulation cost compared to the full simulations. This manuscript presents the necessary background and some preliminary results for the proposed methodology. Development of supplementary analysis tools and extension of the applications are part of ongoing work.