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1. Physics‐based Pathline Clustering and Exploration

   https://doi.org/10.1111/cgf.14093  

   Nguyen, Duong B.; Zhang, Lei; Laramee, Robert S.; Thompson, David; Monico, Rodolfo Ostilla; Chen, Guoning (October 2020, Computer Graphics Forum)

   Abstract Most existing unsteady flow visualization techniques concentrate on the depiction of geometric patterns in flow, assuming the geometry information provides sufficient representation of the underlying physical characteristics, which is not always the case. To address this challenge, this work proposes to analyse the time‐dependent characteristics of the physical attributes measured along pathlines which can be represented as a series of time activity curves (TAC). We demonstrate that the temporal trends of these TACs can convey the relation between pathlines and certain well‐known flow features (e.g. vortices and shearing layers), which enables us to select pathlines that can effectively represent the physical characteristics of interest and their temporal behaviour in the unsteady flow. Inspired by this observation, a new TAC‐based unsteady flow visualization and analysis framework is proposed. The centre of this framework is a new similarity measure that compares the similarity of two TACs, from which a new spatio‐temporal, hierarchical clustering that classifies pathlines based on their physical attributes, and a TAC‐based pathline exploration and selection strategy are proposed. A visual analytic system incorporating the TAC‐based pathline clustering and exploration is developed, which also provides new visualizations to support the user exploration of unsteady flow using TACs. This visual analytic system is applied to a number of unsteady flow in 2D and 3D to demonstrate its utility. The new system successfully reveals the detailed structure of vortices, the relation between shear layer and vortex formation, and vortex breakdown, which are difficult to convey with conventional methods. 

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2. Compute and Visualize Discontinuity Among Neighboring Integral Curves of 2D Vector Fields

   https://doi.org/10.1007/978-3-319-44684-4_11  

   Lei Zhang, Robert S. (January 2017, Topological Methods in Data Analysis and Visualization)

   This paper studies the discontinuity in the behavior of neighboring integral curves. The discontinuity is measured by a number of selected attributes of integral curves. A variety of attribute fields are defined. The attribute value at any given spatio-temporal point in these fields is assigned by the attribute of the integral curve that passes through this point. This encodes the global behavior of integral curves into a number of scalar fields in an Eulerian fashion, which differs from the previous pathline attribute approach that focuses on the discrete representation of individual pathlines. With this representation, the discontinuity of the integral curve behavior now corresponds to locations in the derived fields where the attribute values have sharp gradients. We show that based on the selected attributes, the extracted discontinuity from the corresponding attribute fields may relate to a number of flow features, such as LCS, vortices, and cusp-like seeding curves. In addition, we study the correlations among different attributes via their pairwise scatter plots. We also study the behavior of the combined attribute fields to understand the spatial correlation that cannot be revealed by the scatter plots. Finally, we integrate our attribute field computation and their discontinuity detection into an interactive system to guide the exploration of various 2D flows. 

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3. Field inversion machine learning augmented turbulence modeling for time-accurate unsteady flow

   https://doi.org/10.1063/5.0207704  

   Fang, Lean; He, Ping (May 2024, Physics of Fluids)

   Field inversion machine learning (FIML) has the advantages of model consistency and low data dependency and has been used to augment imperfect turbulence models. However, the solver-intrusive field inversion has a high entry bar, and existing FIML studies focused on improving only steady-state or time-averaged periodic flow predictions. To break this limit, this paper develops an open-source FIML framework for time-accurate unsteady flow, where both spatial and temporal variations of flow are of interest. We augment a Reynolds-Averaged Navier–Stokes (RANS) turbulence model's production term with a scalar field. We then integrate a neural network (NN) model into the flow solver to compute the above augmentation scalar field based on local flow features at each time step. Finally, we optimize the weights and biases of the built-in NN model to minimize the regulated spatial-temporal prediction error between the augmented flow solver and reference data. We consider the spatial-temporal evolution of unsteady flow over a 45° ramp and use only the surface pressure as the training data. The unsteady-FIML-trained model accurately predicts the spatial-temporal variations of unsteady flow fields. In addition, the trained model exhibits reasonably good prediction accuracy for various ramp angles, Reynolds numbers, and flow variables (e.g., velocity fields) that are not used in training, highlighting its generalizability. The FIML capability has been integrated into our open-source framework DAFoam. It has the potential to train more accurate RANS turbulence models for other unsteady flow phenomena, such as wind gust response, bubbly flow, and particle dispersion in the atmosphere. 

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4. Bayesian conditional diffusion models for versatile spatiotemporal turbulence generation

   https://doi.org/10.1016/j.cma.2024.117023  

   Gao, Han; Han, Xu; Fan, Xiantao; Sun, Luning; Liu, Li-Ping; Duan, Lian; Wang, Jian-Xun (July 2024, Computer Methods in Applied Mechanics and Engineering)

   Turbulent flows, characterized by their chaotic and stochastic nature, have historically presented formidable challenges to predictive computational modeling. Traditional eddy-resolved numerical simulations often require vast computational resources, making them impractical or infeasible for numerous engineering applications. As an alternative, deep learning-based surrogate models have emerged, offering data-drive solutions. However, these are typically constructed within deterministic settings, leading to shortfall in capturing the innate chaotic and stochastic behaviors of turbulent dynamics. In this study, we introduce a novel generative framework grounded in probabilistic diffusion models for versatile generation of spatiotemporal turbulence under various conditions. Our method unifies both unconditional and conditional sampling strategies within a Bayesian framework, which can accommodate diverse conditioning scenarios, including those with a direct differentiable link between specified conditions and generated unsteady flow outcomes, as well as scenarios lacking such explicit correlations. A notable feature of our approach is the method proposed for long-span flow sequence generation, which is based on autoregressive gradient-based conditional sampling, eliminating the need for cumbersome retraining processes. We evaluate and showcase the versatile turbulence generation capability of our framework through a suite of numerical experiments, including: (1) the synthesis of Large Eddy Simulations (LES) simulated instantaneous flow sequences from unsteady Reynolds-Averaged Navier–Stokes (URANS) inputs; (2) holistic generation of inhomogeneous, anisotropic wall-bounded turbulence, whether from given initial conditions, prescribed turbulence statistics, or entirely from scratch; (3) super-resolved generation of high-speed turbulent boundary layer flows from low-resolution data across a range of input resolutions. Collectively, our numerical experiments highlight the merit and transformative potential of the proposed methods, making a significant advance in the field of turbulence generation. 

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5. Unsteady Friction in Mixed-Flow Models Based on the Saint-Venant Equations

   Pinto, Stephanie I_G; Vasconcelos, Jose G; Soares, Alexandre K (September 2025, Journal of hydraulic engineering)

   Transient flow models are expected to represent rapid flow changes, whereby acceleration and deceleration significantly influence energy dissipation. Such effects on energy dissipation can be expressed in terms of unsteady friction (UF) losses, which is a well-established process for closed-pipe flow models, but not in the context of mixed-flow models. Mixed flow refers to flow conditions where pressurized and free-surface flow regimes coexist or transition between each other within the same system. Many water systems experience significant flow acceleration and deceleration while in mixed-flow conditions, but current models have only the ability to account for these effects through steady roughness terms. This work builds from existing modeling approaches to adapt mixed-flow models based on the Saint-Venant equations that incorporate unsteady friction losses. The approach used to incorporate unsteady friction losses is a modification of a well-established formula based on local acceleration and spatial velocity gradient. The proposed numerical model, referred to as SVUF, is compared with three experimental data sets, and the results were improved, particularly for longer-duration flow simulations. 

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