Beverley J. McKeon

Mean-flow-based linear analyses of turbulent flows, such as resolvent analysis, provide valuable insight about flow structures and their dynamics that has been widely leveraged to model, control and understand the underlying flow physics. However, these analyses are computationally expensive for flows over complex geometries and require the use of specialized codes that are typically only available in research environments. On the other hand, data-driven modal decompositions, such as the dynamic mode decomposition (DMD), identify turbulent flow structures that, although statistically relevant, do not provide insight into the physical mechanisms driving their dynamics. Here we introduce a novel data-driven method -- nonlinearity-subtracted DMD (NSDMD) -- that leverages knowledge of the structure of the Navier--Stokes equations to ensure that the learned operator is a low-rank approximation of the underlying mean-flow-linearized dynamics. Specifically, the method uses snapshots of the nonlinear terms in the perturbation equations to explicitly account for the contribution of the nonlinear forcing to the dynamics. We demonstrate the use of NSDMD to perform data-driven resolvent analysis on direct numerical simulation (DNS) and large-eddy simulation (LES) datasets, starting with a minimal channel flow and scaling up to the flow over a full aircraft model. As a result, NSDMD allows performing linear analyses of turbulent flows as a post-processing step on simulation data obtained with any available high-fidelity computational fluid dynamics (CFD) code.

Mean-flow-based linear analyses of turbulent flows, such as resolvent analysis, provide valuable insight about flow structures and their dynamics that has been widely leveraged to model, control and understand the underlying flow physics. However, these analyses are computationally expensive for flows over complex geometries and require the use of specialized codes that are typically only available in research environments. On the other hand, data-driven modal decompositions, such as the dynamic mode decomposition (DMD), identify turbulent flow structures that, although statistically relevant, do not provide insight into the physical mechanisms driving their dynamics. Here we introduce a novel data-driven method -- nonlinearity-subtracted DMD (NSDMD) -- that leverages knowledge of the structure of the Navier--Stokes equations to ensure that the learned operator is a low-rank approximation of the underlying mean-flow-linearized dynamics. Specifically, the method uses snapshots of the nonlinear terms in the perturbation equations to explicitly account for the contribution of the nonlinear forcing to the dynamics. We demonstrate the use of NSDMD to perform data-driven resolvent analysis on direct numerical simulation (DNS) and large-eddy simulation (LES) datasets, starting with a minimal channel flow and scaling up to the flow over a full aircraft model. As a result, NSDMD allows performing linear analyses of turbulent flows as a post-processing step on simulation data obtained with any available high-fidelity computational fluid dynamics (CFD) code.

The quadratic convection term in the incompressible Navier–Stokes equations is considered as a nonlinear forcing to the linear resolvent operator, and it is studied in the Fourier domain through the analysis of interactions between triadically compatible wavenumber–frequency triplets. A framework to quantify the triadic contributions to the forcing and response by each pair of triplets is developed and applied to data from direct numerical simulations of a turbulent channel at $ \textit{Re}_{\tau } \approx 550$ . The linear resolvent operator is incorporated to provide the missing link from energy transfer between modes to the effect on the spectral turbulent kinetic energy. The coefficients highlight the importance of interactions involving large-scale structures, providing a natural connection to the modelling assumptions in quasilinear (QL) and generalized QL (GQL) analyses. Specifically, it is revealed that the QL and GQL reductions efficiently capture important triadic interactions in the flow, especially when including of a small number of wavenumbers into the GQL large-scale base flow. Additionally, spatiotemporal analyses of the triadic contributions to a single mode representative of the near-wall cycle demonstrate the spatiotemporal nature of the triadic interactions and the effect of the resolvent operator, which selectively amplifies certain forcing profiles. The tools presented are expected to be useful for improving modelling of the nonlinearity, especially in QL, GQL and resolvent analyses, and understanding the amplitude modulation mechanism relating large-scale fluctuations to the modulation of near-wall structures.