Evolution of isolated turbulent trailing vortices

In this work, the temporal evolution of a low swirl-number turbulent Batchelor vortex is studied using pseudospectral direct numerical simulations. The solution of the governing equations in the vorticity-velocity form allows for accurate application of boundary conditions. The physics of the evolution is investigated with an emphasis on the mechanisms that influence the transport of axial and angular momentum. Excitation of normal mode instabilities gives rise to coherent large scale helical structures inside the vortical core. The radial growth of these helical structures and the action of axial shear and differential rotation results in the creation of a polarized vortex layer. This vortex layer evolves into a series of hairpin-shaped structures that subsequently breakdown into elongated fine scale vortices. Ultimately, the radially outward propagation of these structures results in the relaxation of the flow towards a stable high-swirl configuration. Two conserved quantities, based on the deviation from the laminar solution, are derived and these prove to be useful in characterizing the polarized vortex layer and enhancing the understanding of the transport process. The generation and evolution of the Reynolds stresses is also addressed

Data-driven Balanced Truncation for Predictive Model Order Reduction of Aeroacoustic Response

Rapid prediction of the aeroacoustic response is a key component in the design of aircraft and turbomachinery. While it is possible to achieve accurate predictions using direct solution of the compressible Navier-Stokes equations, applications of such solvers is not feasible in design optimization due to the high cost of resolving wave phenomena in an Eulerian setting. In this work, we propose a technique for highly accelerated predictions of aeroacoustic response using a data-driven model reduction approach based on the eigensystem realization algorithm (ERA), as a non-intrusive balanced truncation method. Specifically, we create and compare ERA ROMs based on the training data generated by solving the linearized and nonlinear Euler equations with Gaussian pulse inputs, and use them for prediction of the aeroacoustic response of an airfoil subject to different types of gust loading. The results show that both ROMs are in good agreement with the full-order model (FOM) solution in a purely predictive setting, while achieving orders of magnitude reduction in the online computation time. Using ERA for prediction of the acoustic response requires activating each input channel separately in the FOM for training ROMs, and operating on a large Hankel matrix, that can become computationally infeasible. We address this bottleneck in two steps: first, we propose a multi-fidelity gappy POD method to identify the most impactful input channels based on a coarser grid. Therefore, we reduce the computation cost on the FOM and ROM levels as we build the Markov sequence by querying the high-resolution FOM only for the input channels identified by gappy POD. Second, we use tangential interpolation at the ROM level to reduce the size of the Hankel matrix. The proposed methods enable application of ERA for highly accurate online acoustic response prediction and reduce the offline computation cost of ROMs

Extracting Koopman Operators for Prediction and Control of Non-linear Dynamics Using Two-stage Learning and Oblique Projections

The Koopman operator framework provides a perspective that non-linear dynamics can be described through the lens of linear operators acting on function spaces. As the framework naturally yields linear embedding models, there have been extensive efforts to utilize it for control, where linear controller designs in the embedded space can be applied to control possibly nonlinear dynamics. However, it is challenging to successfully deploy this modeling procedure in a wide range of applications. In this work, some of the fundamental limitations of linear embedding models are addressed. We show a necessary condition for a linear embedding model to achieve zero modeling error, highlighting a trade-off relation between the model expressivity and a restriction on the model structure to allow the use of linear systems theories for nonlinear dynamics. To address these limitations, neural network-based modeling is proposed based on linear embedding with oblique projection, which is derived from a weak formulation of projection-based linear operator learning. We train the proposed model using a two-stage learning procedure, wherein the features and operators are initialized with orthogonal projection, followed by the main training process in which test functions characterizing the oblique projection are learned from data. The first stage achieves an optimality ensured by the orthogonal projection and the second stage improves the generalizability to various tasks by optimizing the model with the oblique projection. We demonstrate the effectiveness of the proposed method over other data-driven modeling methods by providing comprehensive numerical evaluations where four tasks are considered targeting three different systems