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Several Tensor Basis Neural Network (TBNN) frameworks aimed at enhancing turbulence RANS modeling have recently been proposed in the literature as data-driven constitutive models for systems with known invariance properties. However, persistent ambiguities remain regarding the physical adequacy of applying the General Eddy Viscosity Model (GEVM). This work aims at investigating this aspect in an a priori stage for better predictions of the Reynolds stress anisotropy tensor, while preserving the Galilean and rotational invariances. In particular, we propose a general framework providing optimal tensor basis models for two types of canonical flows: Plane Channel Flow (PCF) and Square Duct Flow (SDF). Subsequently, deep neural networks based on these optimal models are trained using state-of-the-art strategies to achieve a balanced and physically sound prediction of the full anisotropy tensor. A priori results obtained by the proposed framework are in very good agreement with the reference DNS data. Notably, our shallow network with three layers provides accurate predictions of the anisotropy tensor for PCF at unobserved friction Reynolds numbers, both in interpolation and extrapolation scenarios. Learning the SDF case is more challenging because of its physical nature and a lack of training data at various regimes. We propose to alleviate this problem based on Transfer Learning (TL). To more efficiently generalize to an unseen intermediate $\mathrm{Re}_\tau$ regime, we take advantage of our prior knowledge acquired from a training with a larger and wider dataset. Our results indicate the potential of the developed network model, and demonstrate the feasibility and efficiency of the TL process in terms of training data size and training time. Based on these results, we believe there is a promising future by integrating these neural networks into an adapted in-house RANS solver.

Recently immersed boundary method-inspired physics-informed neural networks (PINNs) including the moving boundary-enabled PINNs (MB-PINNs) have shown the ability to accurately reconstruct velocity and recover pressure as a hidden variable for unsteady flow past moving bodies. Considering flow past a plunging foil, MB-PINNs were trained with global physics loss relaxation and also in conjunction with a physics-based undersampling method, obtaining good accuracy. The purpose of this study was to investigate which input spatial subdomain contributes to the training under the effect of physics loss relaxation and physics-based undersampling. In the context of MB-PINNs training, three spatial zones: the moving body, wake, and outer zones were defined. To quantify which spatial zone drives the training, two novel metrics are computed from the zonal loss component gradient statistics and the proportion of sample points in each zone. Results confirm that the learning indeed depends on the combined effect of the zonal loss component gradients and the proportion of points in each zone. Moreover, the dominant input zones are also the ones that have the strongest solution gradients in some sense.

Long-term operation of nuclear steam generators can result in the occurrence of clogging, a deposition phenomenon that may increase the risk of mechanical and vibration loadings on tube bundles and internal structures as well as potentially affecting their response to hypothetical accidental transients. To manage and prevent this issue, a robust maintenance program that requires a fine understanding of the underlying physics is essential. This study focuses on the utilization of a clogging simulation code developed by EDF R\&D. This numerical tool employs specific physical models to simulate the kinetics of clogging and generates time dependent clogging rate profiles for particular steam generators. However, certain parameters in this code are subject to uncertainties. To address these uncertainties, Monte Carlo simulations are conducted to assess the distribution of the clogging rate. Subsequently, polynomial chaos expansions are used in order to build a metamodel while time-dependent Sobol' indices are computed to understand the impact of the random input parameters throughout the whole operating time. Comparisons are made with a previous published study and additional Hilbert-Schmidt independence criterion sensitivity indices are computed. Key input-output dependencies are exhibited in the different chemical conditionings and new behavior patterns in high-pH regimes are uncovered by the sensitivity analysis. These findings contribute to a better understanding of the clogging phenomenon while opening future lines of modeling research and helping in robustifying maintenance planning.

Recently, physics informed neural networks (PINNs) have been explored extensively for solving various forward and inverse problems and facilitating querying applications in fluid mechanics applications. However, work on PINNs for unsteady flows past moving bodies, such as flapping wings is scarce. Earlier studies mostly relied on transferring to a body attached frame of reference which is restrictive towards handling multiple moving bodies or deforming structures. Hence, in the present work, an immersed boundary aware framework has been explored for developing surrogate models for unsteady flows past moving bodies. Specifically, simultaneous pressure recovery and velocity reconstruction from Immersed boundary method (IBM) simulation data has been investigated. While, efficacy of velocity reconstruction has been tested against the fine resolution IBM data, as a step further, the pressure recovered was compared with that of an arbitrary Lagrange Eulerian (ALE) based solver. Under this framework, two PINN variants, (i) a moving-boundary-enabled standard Navier-Stokes based PINN (MB-PINN), and, (ii) a moving-boundary-enabled IBM based PINN (MB-IBM-PINN) have been formulated. A fluid-solid partitioning of the physics losses in MB-IBM-PINN has been allowed, in order to investigate the effects of solid body points while training. This enables MB-IBM-PINN to match with the performance of MB-PINN under certain loss weighting conditions. MB-PINN is found to be superior to MB-IBM-PINN when {\it a priori} knowledge of the solid body position and velocity are available. To improve the data efficiency of MB-PINN, a physics based data sampling technique has also been investigated. It is observed that a suitable combination of physics constraint relaxation and physics based sampling can achieve a model performance comparable to the case of using all the data points, under a fixed training budget.

Data Assimilation (DA) and Uncertainty quantification (UQ) are extensively used in analysing and reducing error propagation in high-dimensional spatial-temporal dynamics. Typical applications span from computational fluid dynamics (CFD) to geoscience and climate systems. Recently, much effort has been given in combining DA, UQ and machine learning (ML) techniques. These research efforts seek to address some critical challenges in high-dimensional dynamical systems, including but not limited to dynamical system identification, reduced order surrogate modelling, error covariance specification and model error correction. A large number of developed techniques and methodologies exhibit a broad applicability across numerous domains, resulting in the necessity for a comprehensive guide. This paper provides the first overview of the state-of-the-art researches in this interdisciplinary field, covering a wide range of applications. This review aims at ML scientists who attempt to apply DA and UQ techniques to improve the accuracy and the interpretability of their models, but also at DA and UQ experts who intend to integrate cutting-edge ML approaches to their systems. Therefore, this article has a special focus on how ML methods can overcome the existing limits of DA and UQ, and vice versa. Some exciting perspectives of this rapidly developing research field are also discussed.

The Reynolds-Averaged Navier-Stokes (RANS) approach remains a backbone for turbulence modeling due to its high cost-effectiveness. Its accuracy is largely based on a reliable Reynolds stress anisotropy tensor closure model. There has been an amount of work aiming at improving traditional closure models, while they are still not satisfactory to some complex flow configurations. In recent years, advances in computing power have opened up a new way to address this problem: the machine-learning-assisted turbulence modeling. In this paper, we employ neural networks to fully predict the Reynolds stress anisotropy tensor of turbulent channel flows at different friction Reynolds numbers, for both interpolation and extrapolation scenarios. Several generic neural networks of Multi-Layer Perceptron (MLP) type are trained with different input feature combinations to acquire a complete grasp of the role of each parameter. The best performance is yielded by the model with the dimensionless mean streamwise velocity gradient $\alpha$, the dimensionless wall distance $y^+$ and the friction Reynolds number $\mathrm{Re}_\tau$ as inputs. A deeper theoretical insight into the Tensor Basis Neural Network (TBNN) clarifies some remaining ambiguities found in the literature concerning its application of Pope's general eddy viscosity model. We emphasize the sensitivity of the TBNN on the constant tensor $\textbf{T}^{*(0)}$ upon the turbulent channel flow data set, and newly propose a generalized $\textbf{T}^{*(0)}$, which considerably enhances its performance. Through comparison between the MLP and the augmented TBNN model with both $\{\alpha, y^+, \mathrm{Re}_\tau\}$ as input set, it is concluded that the former outperforms the latter and provides excellent interpolation and extrapolation predictions of the Reynolds stress anisotropy tensor in the specific case of turbulent channel flow., Comment: 35 pages, 10 figures

Mapping near-field pollutant concentration is essential to track accidental toxic plume dispersion in urban areas. By solving a large part of the turbulence spectrum, large-eddy simulations (LES) have the potential to accurately represent pollutant concentration spatial variability. Finding a way to synthesize this large amount of information to improve the accuracy of lower-fidelity operational models (e.g. providing better turbulence closure terms) is particularly appealing. This is a challenge in multi-query contexts, where LES become prohibitively costly to deploy to understand how plume flow and tracer dispersion change with various atmospheric and source parameters. To overcome this issue, we propose a non-intrusive reduced-order model combining proper orthogonal decomposition (POD) and Gaussian process regression (GPR) to predict LES field statistics of interest associated with tracer concentrations. GPR hyperpararameters are optimized component-by-component through a maximum a posteriori (MAP) procedure informed by POD. We provide a detailed analysis of the reducedorder model performance on a two-dimensional case study corresponding to a turbulent atmospheric boundary-layer flow over a surface-mounted obstacle. We show that near-source concentration heterogeneities upstream of the obstacle require a large number of POD modes to be well captured. We also show that the component-by-component optimization allows to capture the range of spatial scales in the POD modes, especially the shorter concentration patterns in the high-order modes. The reduced-order model predictions remain acceptable if the learning database is made of at least fifty to hundred LES snapshot providing a first estimation of the required budget to move towards more realistic atmospheric dispersion applications.

Cardiac disease and clinical intervention may both lead to an increased risk for thrombosis events due to modified blood flow in the heart, and thereby a change in the mechanical stimuli of blood cells passing through the chambers of the heart. Specifically, the degree of platelet activation is influenced by the level and type of mechanical stresses in the blood flow. Here we analyze the blood flow in the left ventricle of the heart through a computational model constructed from patient-specific data. The blood flow in the ventricle is modeled by the Navier-Stokes equations, and the flow through the mitral valve by a parameterized model which represents the projected opening of the valve. A finite element method is used to solve the equations, from which a simulation of the velocity and pressure of the blood flow is constructed. A triple decomposition of the velocity gradient tensor is then used to distinguish between rigid body rotational flow, irrotational straining flow, and shear flow. The triple decomposition enables the separation of three fundamentally different flow structures, each generating a distinct type of mechanical stimulus on the blood cells in the flow. We compare the results to simulations where a mitral valve clip intervention is modelled, which leads to a significant modification of the ventricular flow. It was found that the shear in the simulation cases treated with clips increased more compared to the untreated case than the rotation and strain did. A decrease in valve opening area of 64 % in one of the cases led to a 90 % increase in rotation and strain, but a 150 % increase in shear. The computational analysis suggests a process for patient-specific simulation of clinical interventions in the heart with a detailed analysis of the resulting blood flow, which could support clinical risk assessment with respect to platelet activation and thrombosis events., Comment: The following article has been submitted to Frontiers in Physiology

Mathematical models of thrombosis are currently used to study clinical scenarios of pathological thrombus formation. Most of these models involve inherent uncertainties that must be assessed to increase the confidence in model predictions and identify avenues of improvement for both thrombosis modeling and anti-platelet therapies. In this work, an uncertainty quantification analysis of a multi-constituent thrombosis model is performed considering a common assay for platelet function (PFA-100). The analysis is performed using a polynomial chaos expansion as a parametric surrogate for the thrombosis model. The polynomial approximation is validated and used to perform a global sensitivity analysis via computation of Sobol' coefficients. Six out of fifteen parameters were found to be influential in the simulation variability considering only individual effects. Nonetheless, parameter interactions are highlighted when considering the total Sobol' indices. In addition to the sensitivity analysis, the surrogate model was used to compute the PFA-100 closure times of 300,000 virtual cases that align well with clinical data. The current methodology could be used including common anti-platelet therapies to identify scenarios that preserve the hematological balance., Comment: 17 pages, 10 figures, 3 tables, original research article

Accurate estimation of error covariances (both background and observation) is crucial for efficient observation compression approaches in data assimilation of large-scale dynamical problems. We propose a new combination of a covariance tuning algorithm with existing PCA-type data compression approaches, either observation- or information-based, with the aim of reducing the computational cost of real-time updating at each assimilation step. Relying on a local assumption of flow-independent error covariances, dynamical assimilation residuals are used to adjust the covariance in each assimilation window. The estimated covariances then contribute to better specify the principal components of either the observation dynamics or the state-observation sensitivity. The proposed approaches are first validated on a shallow water twin experiment with correlated and non-homogeneous observation error. Proper selection of flow-independent assimilation windows, together with sampling density for background error estimation, and sensitivity of the approaches to the observations error covariance knowledge, are also discussed and illustrated with various numerical tests and results. The method is then applied to a more challenging industrial hydrological model with real-world data and a non-linear transformation operator provided by an operational precipitation-flow simulation software.

Recent works have explored the potential of machine learning as data-driven turbulence closures for RANS and LES techniques. Beyond these advances, the high expressivity and agility of physics-informed neural networks (PINNs) make them promising candidates for full fluid flow PDE modeling. An important question is whether this new paradigm, exempt from the traditional notion of discretization of the underlying operators very much connected to the flow scales resolution, is capable of sustaining high levels of turbulence characterized by multi-scale features? We investigate the use of PINNs surrogate modeling for turbulent Rayleigh-B{\'e}nard (RB) convection flows in rough and smooth rectangular cavities, mainly relying on DNS temperature data from the fluid bulk. We carefully quantify the computational requirements under which the formulation is capable of accurately recovering the flow hidden quantities. We then propose a new padding technique to distribute some of the scattered coordinates-at which PDE residuals are minimized-around the region of labeled data acquisition. We show how it comes to play as a regularization close to the training boundaries which are zones of poor accuracy for standard PINNs and results in a noticeable global accuracy improvement at iso-budget. Finally, we propose for the first time to relax the incompressibility condition in such a way that it drastically benefits the optimization search and results in a much improved convergence of the composite loss function. The RB results obtained at high Rayleigh number Ra = 2 $\bullet$ 10 9 are particularly impressive: the predictive accuracy of the surrogate over the entire half a billion DNS coordinates yields errors for all flow variables ranging between [0.3% -- 4%] in the relative L 2 norm, with a training relying only on 1.6% of the DNS data points.

An original graph clustering approach to efficient localization of error covariances is proposed within an ensemble-variational data assimilation framework. Here the localization term is very generic and refers to the idea of breaking up a global assimilation into subproblems. This unsupervised localization technique based on a linearizedstate-observation measure is general and does not rely on any prior information such as relevant spatial scales, empirical cut-off radius or homogeneity assumptions. It automatically segregates the state and observation variables in an optimal number of clusters (otherwise named as subspaces or communities), more amenable to scalable data assimilation.The application of this method does not require underlying block-diagonal structures of prior covariance matrices. In order to deal with inter-cluster connectivity, two alternative data adaptations are proposed. Once the localization is completed, an adaptive covariance diagnosis and tuning is performed within each cluster. Numerical tests show that this approach is less costly and more flexible than a global covariance tuning, and most often results in more accurate background and observations error covariances.

In order to leverage the information embedded in the background state and observations, covariance matrices modelling is a pivotal point in data assimilation algorithms. These matrices are often estimated from an ensemble of observations or forecast differences. Nevertheless, for many industrial applications the modelling still remains empirical based on some form of expertise and physical constraints enforcement in the absence of historical observations or predictions. We have developed two novel robust adaptive assimilation methods named CUTE (Covariance Updating iTerativE) and PUB (Partially Updating BLUE). These two non-parametric methods are based on different optimization objectives, both capable of sequentially adapting background error covariance matrices in order to improve assimilation results under the assumption of a good knowledge of the observation error covariances. We have compared these two methods with the standard approach using a misspecified background matrix in a shallow water twin experiments framework with a linear observation operator. Numerical experiments have shown that the proposed methods bear a real advantage both in terms of posterior error correlation identification and assimilation accuracy.

We propose a hemodynamic reduced-order model bridging macroscopic and meso-scopic blood flow circulation scales from arteries to capillaries. In silico tree like vascular geometries, mathematically described by graphs, are synthetically generated by means of stochastic growth algorithms constrained by statistical morphological and topological principles. Scale-specific pruning gradation of the tree is then proposed in order to fit computational budget requirement. Different compliant structural models with respect to pressure loads are used depending on vessel walls thicknesses and structures, which vary considerably from macroscopic to mesoscopic circulation scales. Nonlinear rheological properties of blood are also included and microcirculation network responses are computed for different rheologies. Numerical results are in very good agreement with available experimental measurements. The computational model captures the dynamic transition between large-to small-scale flow pulsatility speeds and magnitudes and wall shear stresses, which have wide-ranging physiological influences.

The simulation of complex nonlinear engineering systems such as compressible fluid flows may be targeted to make more efficient and accurate the approximation of a specific (scalar) quantity of interest of the system. Putting aside modeling error and parametric uncertainty, this may be achieved by combining goal-oriented error estimates and adaptive anisotropic spatial mesh refinements. To this end, an elegant and efficient framework is the one of (Riemannian) metric-based adaptation where a goal-based a priori error estimation is used as indicator for adaptivity. This work proposes a novel extension of this approach to the case of aforementioned system approximations bearing a stochastic component. In this case, an optimisation problem leading to the best control of the distinct sources of errors is formulated in the continuous framework of the Riemannian metric space. Algorithmic developments are also presented in order to quantify and adaptively adjust the error components in the deterministic and stochastic approximation spaces. The capability of the proposed method is tested on various problems including a supersonic scramjet inlet subject to geometrical and operational parametric uncertainties. It is demonstrated to accurately capture discontinuous features of stochastic compressible flows impacting pressure-related quantities of interest, while balancing computational budget and refinements in both spaces.

A stochastic EDQNM approach is used to investigate self-similar decaying isotropic turbulence at high Reynolds number ($400 \leq Re_\lambda \leq 10^4$). The realistic energy spectrum functional form recently proposed by Meyers & Meneveau is generalised by considering some of the model constants as random parameters, since they escape measure in most experimental set-ups. The induced uncertainty on the solution is investigated building response surfaces for decay power-law exponents of usual physical quantities. Large-scale uncertainties are considered, the emphasis being put on Saffman and Batchelor turbulence. The sensitivity of the solution to initial spectrum uncertainties is quantified through probability density functions of the decay exponents. It is observed that initial spectrum shape at very large scales governs the long-time evolution, even at high Reynolds number, a parameter which is not explicitly taken into account in many theoretical works. Therefore, a universal asymptotic behavior in which kinetic energy decays as $t^{-1}$ is not detected. But this decay law is observed at finite Reynolds number with low probability for some initial conditions.