Efficient matrix-free linear analysis for large-scale $n$-periodic systems: Application to spanwise arrays of flow irregularities*

This work presents a mathematical framework for the linear analysis of $n$-periodic systems, consisting of $n$ identical units. Such $n$-periodic configurations occur in a variety of applications in fluid dynamics, such as boundary layers over roughness arrays or arrays of injectors. The analysis of these systems as single periodic units is often insufficient as dynamics are likely to span over multiple units. In order to accommodate cross-unit interactions, the roots-of-unity framework is integrated into a large-scale compressible solver and tested on a case of spanwise roughness arrays. The benefit of this approach is the decoupling of the problem from the specific number of units, allowing the analysis of a large range of configurations at a reduced cost. The methodology and its applicability are demonstrated for a model problem of synthetic wakes downstream of roughness elements in subsonic and hypersonic boundary layers. The linear model dynamics are extracted using Dynamic Mode Decomposition.  Finally, capabilities for further non-modal analysis using resolvent modes, linear-adjoint sensitivity analysis and optimization or control are also introduced.