This paper presents a novel CFD-driven machine learning framework to develop Reynolds-averaged Navier-Stokes (RANS) models. For the CFD-driven training, the gene expression programming (GEP) method (Weatheritt & Sandberg, J. Comput. Phys., 325, 22-37 (2016)) uses RANS calculations in an integrated way to evaluate the fitness of candidate models. The resulting model, which is the one providing the most accurate CFD results at the end of the training process, is thus expected to show good performance in RANS calculations. To demonstrate the potential of this new approach, the CFD-driven machine learning is applied to develop a model for improved prediction of wake mixing in turbomachines. A new model is trained based on a high-pressure turbine training case with particular physical features. The developed model is shown to have a more compact functional form than models trained without CFD assistance. Furthermore, the trained model has been evaluated a posteriori for the training case and three additional test cases with different physical flow features, and the predicted wake mixing profiles are significantly improved in all cases. With the present framework, the model equation is explicitly given and available for analysis, thus it could be deduced that the enhanced wake prediction is due to the extra diffusion introduced by the CFD-driven model.

We report a Lagrangian study on the evolution of material surfaces in the Klebanofftype temporal transitional channel flow. Based on the Eulerian velocity field from the direct numerical simulation, a backward-particle-tracking method is applied to solve the transport equation of the Lagrangian scalar field, and then the isosurfaces of the Lagrangian field can be extracted as material surfaces in the evolution. Three critical issues for Lagrangian investigations on the evolution of coherent structures using material surfaces are addressed. First, the initial scalar field is uniquely determined based on the proposed criteria, so that the initial material surfaces can be approximated as vortex surfaces, and remain invariant in the initial laminar state. Second, the evolution of typical material surfaces initially from different wall distances is presented, and then the influential material surface with the maximum deformation is identified. Large vorticity variations with the maximum curvature growth of vortex lines are also observed on this surface. Moreover, crucial events in the transition can be characterized in a Lagrangian approach by conditional statistics on the material surfaces. Finally, the influential material surface, which is initially a vortex surface, is demonstrated as a surrogate of the vortex surface before significant topological changes of vortical structures. Therefore, this material surface can be used to elucidate the continuous temporal evolution of vortical structures in transitional wall-bounded flows in a Lagrangian perspective. The evolution of the influential material surface is divided into three stages: the formation of a triangular bulge from an initially disturbed streamwise-spanwise sheet, rolling up of the vortex sheet near the bulge ridges with the vorticity intensification and the generation and evolution of signature hairpin-like structures with self-induced dynamics of vortex filaments.

Vortex reconnection, as the topological change of vortex lines or surfaces, is a critical process in transitional flows, but is challenging to accurately characterize, particularly in shear flows. We apply the vortex-surface field (VSF), whose isosurface is the vortex surface consisting of vortex lines, to study vortex reconnection in the Klebanoff-type temporal transition in channel flow. The VSF evolution can capture the reconnection of the hairpin-like vortical structures evolving from the initial vortex sheets in opposite halves of the channel. The incipient vortex reconnection is characterized by the vanishing minimum distance between a pair of vortex surfaces and the reduction of vorticity flux through the region enclosed by the wall and the VSF isoline of the channel half-height on the spanwise symmetric plane. We find that the surge of the wall-friction coefficient begins at the identified reconnection time. From the Biot–Savart law, the rapid reconnection of vortex lines can induce a velocity opposed to the mean flow, which partially blocks the flow near the central region and generally accelerates the near-wall fluid motion in the flow with constant mass flux. Therefore, the vortex reconnection appears to play an important role in the sudden increase of wall friction in transitional channel flows.

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In low-pressure-turbines (LPT) around 60–70% of losses are generated away from end-walls, while the remaining 30–40% is controlled by the interaction of the blade profile with the end-wall boundary layer. Experimental and numerical studies have shown how the strength and penetration of the secondary flow depends on the characteristics of the incoming end-wall boundary layer. Experimental techniques did shed light on the mechanism that controls the growth of the secondary vortices, and scale-resolving CFD allowed to dive deep into the details of the vorticity generation. Along these lines, this paper discusses the end-wall flow characteristics of the T106 LPT profile at Re = 120K and M = 0.59 by benchmarking with experiments and investigating the impact of the incoming boundary layer state. The simulations are carried out with proven Reynolds-averaged Navier–Stokes (RANS) and large-eddy simulation (LES) solvers to determine if Reynolds Averaged models can capture the relevant flow details with enough accuracy to drive the design of this flow region. Part I of the paper focuses on the critical grid needs to ensure accurate LES, and on the analysis of the overall time averaged flow field and comparison between RANS, LES and measurements when available. In particular, the growth of secondary flow features, the trace and strength of the secondary vortex system, its impact on the blade load variation along the span and end-wall flow visualizations are analysed. The ability of LES and RANS to accurately predict the secondary flows is discussed together with the implications this has on design.