Development of the static-dynamic flow decomposition method

The static-dynamic flow decomposition (SD-FD) is rooted in existing literature mathematical decompositions, yet a specific name is coined to emphasize its novelty of studying flows which is best demonstrated through its application. The SD-FD splits a flowfield variable into the sum of a spatial-dependent static component and a space–time-dependent dynamic component of infinitesimal or finite amplitude. Once the decomposition is inserted into the governing equations of fluid mechanics, it yields a static flow with all associated convective, diffusive, and reactive space-dependent effects as well as a dynamic flow encompassing explicitly all time-dependent processes: turbulence, several flow patterns, and waves propagation. Similarities and differences with established decompositions and their assumptions are discussed. Furthermore, additional observations are made supporting the definition and justifying the introduction of this decomposition. The discussed observations are based on spatially developing transitional pipe flow direct numerical simulation data analysis, quantitative analytical signals analysis of an interface submitted to modulation, transient synthetic field analysis with dynamic mode decomposition, and modulated premixed flames results documented in the literature and discussed here. The observations made demonstrate the advantages of the decomposition to perfectly separate spatial vs time-dependent processes within the flow. Building on the state of the art, the advances provided by the SD-FD method relies on (i) bridging and unifying existing established definitions from the literature in a rigorous manner, (ii) removing any ambiguity with its introduction, id est being well-defined, and (iii) opening new perspectives in flow understanding. The relevance of the method is demonstrated in a companion paper.