We propose a data-driven, closure model for Reynolds-averaged Navier-Stokes (RANS) simulations that incorporates aleatoric, model uncertainty. The proposed closure consists of two parts. A parametric one, which utilizes previously proposed, neural-network-based tensor basis functions dependent on the rate of strain and rotation tensor invariants. This is complemented by latent, random variables which account for aleatoric model errors. A fully Bayesian formulation is proposed, combined with a sparsity-inducing prior in order to identify regions in the problem domain where the parametric closure is insufficient and where stochastic corrections to the Reynolds stress tensor are needed. Training is performed using sparse, indirect data, such as mean velocities and pressures, in contrast to the majority of alternatives that require direct Reynolds stress data. For inference and learning, a Stochastic Variational Inference scheme is employed, which is based on Monte Carlo estimates of the pertinent objective in conjunction with the reparametrization trick. This necessitates derivatives of the output of the RANS solver, for which we developed an adjoint-based formulation. In this manner, the parametric sensitivities from the differentiable solver can be combined with the built-in, automatic differentiation capability of the neural network library in order to enable an end-to-end differentiable framework. We demonstrate the capability of the proposed model to produce accurate, probabilistic, predictive estimates for all flow quantities, even in regions where model errors are present, on a separated flow in the backward-facing step benchmark problem.
The volume of fluid (VOF) method is increasingly used in computational fluid dynamics (CFD) simulations of turbulent two-phase flows. The Reynolds-Averaged Navier-Stokes (RANS) approach is an economic and practical way for turbulent VOF simulations. Even though RANS-VOF simulations are widely conducted, the underlying physics is barely discussed. This study reveals the very basic closure requirement for RANS-VOF simulations.
In this short note we apply the recently proposed data-driven RANS closure modelling framework of Schmelzer et al. (2020) to fully three-dimensional, high Reynolds number flows: namely wall-mounted cubes and cuboids at Re=40,000, and a cylinder at Re=140,000. For each flow, a new RANS closure is generated using sparse symbolic regression based on LES or DES reference data. This new model is implemented in a CFD solver, and subsequently applied to prediction of the other flows. We see consistent improvements compared to the baseline $k-ω$ SST model in predictions of mean-velocity in the complet…
Multi-fidelity optimization methods promise a high-fidelity optimum at a cost only slightly greater than a low-fidelity optimization. This promise is seldom achieved in practice, due to the requirement that low- and high-fidelity models correlate well. In this article, we propose an efficient bi-fidelity shape optimization method for turbulent fluid-flow applications with Large-Eddy Simulation (LES) and Reynolds-averaged Navier-Stokes (RANS) as the high- and low-fidelity models within a hierarchical-Kriging surrogate modelling framework. Since the LES-RANS correlation is often poor, we use the…
The state-of-the-art in wind-farm flow-physics modeling is Large Eddy Simulation (LES) which makes accurate predictions of most relevant physics, but requires extensive computational resources. The next-fidelity model types are Reynolds-Averaged Navier-Stokes (RANS) which are two orders of magnitude cheaper, but resolve only mean quantities and model the effect of turbulence. They often fail to accurately predict key effects, such as the wake recovery rate. Custom RANS closures designed for wind-farm wakes exist, but so far do not generalize well: there is substantial room for improvement. In …
Data-driven Reynolds-Averaged Navier-Stokes (RANS) turbulence closures are increasing seen as a viable alternative to general-purpose RANS closures, when LES reference data is available -- also in wind-energy. Parsimonious closures with few, simple terms have advantages in terms of stability, interpret-ability, and execution speed. However experience suggests that closure model corrections need be made only in limited regions -- e.g.\ in the near-wake of wind turbines and not in the majority of the flow. A parsimonious model therefore must find a middle ground between precise corrections in th…
Data assimilation of flow MRI data into RANS models with algebraic closures
Department of Engineering, Trumpington Street, Cambridge CB2 1PZ, UK（英国剑桥大学工程系）；Department of Bioengineering, Stanford University, Stanford, CA, USA（美国斯坦福大学生物工程系）；Department of Radiology, Stanford University, Stanford, CA, USA（美国斯坦福大学放射系）；Stanford Cardiovascular Institute, Stanford University, Stanford, CA, USA（美国斯坦福大学心血管研究所）
For the purpose of Uncertainty Quantification (UQ) of Reynolds-Averaged Navier-Stokes closures, we introduce a framework in which perturbations in the eigenvalues of the anisotropy tensor are made in order to bound a Quantity-of-Interest based on limiting states of turbulence. To make the perturbations representative of local flow features, we introduce two additional transport equations for linear combinations of these aforementioned eigenvalues. The location, magnitude and direction of the eigenvalue perturbations are now governed by the model transport equations. The general behavior of our…
Reynolds-averaged Navier--Stokes (RANS) simulations with turbulence closure models continue to play important roles in industrial flow simulations. However, the commonly used linear eddy viscosity models are intrinsically unable to handle flows with non-equilibrium turbulence. Reynolds stress models, on the other hand, are plagued by their lack of robustness. Recent studies in plane channel flows found that even substituting Reynolds stresses with errors below 0.5% from direct numerical simulation (DNS) databases into RANS equations leads to velocities with large errors (up to 35%). While such…
In the present paper a new data-driven model is proposed to close and increase accuracy of RANS equations. The divergence of the Reynolds Stress Tensor (RST) is obtained through a Neural Network (NN) whose architecture and input choice guarantee both Galilean and coordinates-frame rotation. The former derives from the input choice of the NN while the latter from the expansion of the divergence of the RST into a vector basis. This approach has been widely used for data-driven models for the anisotropic RST or the RST discrepancies and it is here proposed for the divergence of the RST. Hence, a …
In this investigation, we outline an enveloping models methodology for estimating structural uncertainty bounds on RANS closures. This methodology incorporates both eigenvalue and eigenvector perturbations in the spectral representation of the Reynolds stress tensor. The underlying rationale of this enveloping models methodology are explicated in detail and the mathematical nuances are outlined. This methodology is validated via its application to a canonical case of separated turbulent flow, while contrasted against numerical data. It is exhibited that this procedure is able to provide pruden…
Despite well-known limitations of Reynolds-averaged Navier-Stokes (RANS) simulations, this methodology remains the most widely used tool for predicting many turbulent flows, due to computational efficiency. Machine learning is a promising approach to improve the accuracy of RANS simulations. One major area of improvement is using machine learning models to represent the complex relationship between the mean flow field gradients and the Reynolds stress tensor. In the present work, modifications to improve the stability of previous optimal eddy viscosity approaches for RANS simulations are prese…
In this work, model closures of the multiphase Reynolds-Average Navier-Stokes (RANS) equations are developed for homogeneous, fully-developed gas--particle flows. To date, the majority of RANS closures are based on extensions of single-phase turbulence models, which fail to capture complex two-phase flow dynamics across dilute and dense regimes, especially when two-way coupling between the phases is important. In the present study, particles settle under gravity in an unbounded viscous fluid. At sufficient mass loadings, interphase momentum exchange between the phases results in the spontaneou…
A data-driven framework for formulation of closures of the Reynolds-Average Navier--Stokes (RANS) equations is presented. In recent years, the scientific community has turned to machine learning techniques to distill a wealth of highly resolved data into improved RANS closures. While the body of work in this area has primarily leveraged Neural Networks (NNs), we alternately leverage a sparse regression framework. This methodology has two important properties: (1) The resultant model is in a closed, algebraic form, allowing for direct physical inferences to be drawn and naive integration into e…
Despite their well-known limitations, RANS models remain the most commonly employed tool for modeling turbulent flows in engineering practice. RANS models are predicated on the solution of the RANS equations, but these equations involve an unclosed term, the Reynolds stress tensor, which must be modeled. The Reynolds stress tensor is often modeled as an algebraic function of mean flow field variables and turbulence variables. This introduces a discrepancy between the Reynolds stress tensor predicted by the model and the exact Reynolds stress tensor. This discrepancy can result in inaccurate me…
Reynolds-averaged Navier-Stokes (RANS) is one of the most cost-efficient approaches to simulate wind-farm-atmosphere interactions. However, the applicability of RANS-based methods is always limited by the accuracy of turbulence closure models, which introduce various uncertainties into the models. In this study, we estimate model-form uncertainties in RANS simulations of wind farms. For this purpose, we compare different RANS models to a large-eddy simulation (LES). We find that the realizable k-epsilon model is a representative RANS model for predicting the mean velocity, the turbulence inten…
Variable density flows occur in a variety of different systems with a wide range of scales, from astrophysics to atmospheric flows to inertial confinement fusion or reacting flows. Given the inherent limitations of RANS simulations, it is important to find ways to quantify the uncertainty in the predictions. The aleatoric uncertainty of two different variable density flows to inlet and initial conditions, respectively, is studied. The two cases are the turbulent mixing of a jet in a co-flow at small Atwood number $A$, and the Rayleigh-Taylor mixing in a tilted rocket rig at medium $A$. Uncerta…
Turbulent problems in industrial applications are predominantly solved using Reynolds Averaged Navier Stokes (RANS) turbulence models. The accuracy of the RANS models is limited due to closure assumptions that induce uncertainty into the RANS modeling. We propose the use of deep learning algorithms via convolution neural networks along with data from direct numerical simulations to extract the optimal set of features that explain the evolution of turbulent flow statistics. Statistical tests are used to determine the correlation of these features with the variation in the quantities of interest…
Structure tensor Reynolds-averaged Navier-Stokes turbulence models with equivariant neural networks
Computational Science and Engineering Laboratory, Harvard University（哈佛大学计算科学与工程实验室）；Oden Institute for Computational Engineering and Sciences, The University of Texas at Austin（德克萨斯大学奥斯汀分校计算工程与科学研究所）；Walker Department of Mechanical Engineering, The University of Texas at Austin（德克萨斯大学奥斯汀分校沃克机械工程系）
Deep learning (DL)-based Reynolds stress with its capability to leverage values of large data can be used to close Reynolds-averaged Navier-Stoke (RANS) equations. Type I and Type II machine learning (ML) frameworks are studied to investigate data and flow feature requirements while training DL-based Reynolds stress. The paper presents a method, flow features coverage mapping (FFCM), to quantify the physics coverage of DL-based closures that can be used to examine the sufficiency of training data points as well as input flow features for data-driven turbulence models. Three case studies are fo…
Reynolds-averaged Navier-Stokes (RANS) equations are presently one of the most popular models for simulating turbulence. Performing RANS simulation requires additional modeling for the anisotropic Reynolds stress tensor, but traditional Reynolds stress closure models lead to only partially reliable predictions. Recently, data-driven turbulence models for the Reynolds anisotropy tensor involving novel machine learning techniques have garnered considerable attention and have been rapidly developed. Focusing on modeling the Reynolds stress closure for the specific case of turbulent channel flow, …
This paper presents numerical simulations of a bichromatic wave group propagating and breaking over a fixed breaker bar. The simulations are performed using a newly stabilized Reynolds-averaged Navier Stokes (RANS) two-equation turbulence closure, which solves the longstanding problem of over-production of turbulence beneath surface waves in the nearly potential flow region prior to breaking. This model has previously been tested on small-scale spilling breaking regular waves, whereas in this work focus is on full (rather than model) scale application, wave groups (rather than regular waves) a…
This work introduces a novel data-driven framework to formulate explicit algebraic Reynolds-averaged Navier-Stokes (RANS) turbulence closures. Recent years have witnessed a blossom in applying machine learning (ML) methods to revolutionize the paradigm of turbulence modeling. However, due to the black-box essence of most ML methods, it is currently hard to extract interpretable information and knowledge from data-driven models. To address this critical limitation, this work leverages deep learning with symbolic regression methods to discover hidden governing equations of Reynolds stress models…
A new slow growth formulation for DNS of wall-bounded turbulent flow is developed and demonstrated to enable extension of slow growth modeling concepts to complex boundary layer flows. As in previous slow growth approaches, the formulation assumes scale separation between the fast scales of turbulence and the slow evolution of statistics such as the mean flow. This separation enables the development of approaches where the fast scales of turbulence are directly simulated while the forcing provided by the slow evolution is modeled. The resulting model admits periodic boundary conditions in the …
Buoyant shear layers are encountered in many engineering and environmental applications and have been studied by researchers in the context of experiments and modeling for decades. Often, these flows have high Reynolds and Richardson numbers, and this leads to significant/intractable space-time resolution requirements for DNS or LES modeling. On the other hand, many of the important physical mechanisms in these systems, such as stress anisotropy, wake stabilization, and regime transition, inherently render eddy viscosity-based RANS modeling inappropriate. Accordingly, we pursue second-moment c…
Machine-learning (ML) techniques provide a new and encouraging perspective for constructing turbulence models for Reynolds-averaged Navier--Stokes (RANS) simulations. In this study, an iterative ML-RANS computational framework is proposed that combines an ML algorithm with transport equations of a conventional turbulence model. This framework maintains a consistent procedure for obtaining the input features of an ML model in both the training and predicting stages, ensuring a built-in reproducibility. The effective form of the closure term is discussed to determine suitable target variables fo…
In this paper, we propose normalizing flows (NF) as a novel probability density function (PDF) turbulence model (NF-PDF model) for the Reynolds-averaged Navier-Stokes (RANS) equations. We propose to use normalizing flows in two different ways: firstly, as a direct model for the Reynolds stress tensor, and secondly as a second-moment closure model, i.e. for modeling the pressure-strain and dissipation tensor in the Reynolds stress transport equation. In classical PDF closure models, a stochastic differential equation has to be modeled and solved to obtain samples of turbulent quantities. The NF…
Calibrating a Reynolds-averaged Navier-Stokes (RANS) model against data leads to an improvement. Determining {\it a priori} if such an improvement generalizes to flows outside the calibration data is an outstanding challenge. This work attempts to address this challenge via global epistemic Uncertainty Quantification (UQ). Unlike the available epistemic UQ methods that are local and tell us a model's uncertainty at one specific flow condition, the global epistemic UQ method presented in this work tells us also whether a perturbation of the original model would generalize. Specifically, the glo…
A Priori Uncertainty Quantification of Reacting Turbulence Closure Models using Bayesian Neural Networks
Oden Institute for Computational Engineering and Sciences, The University of Texas at Austin（德克萨斯大学奥斯汀分校奥登计算工程与科学学院）；Computational Science Center, National Renewable Energy Laboratory（美国国家可再生能源实验室计算科学中心）
The recent surge in machine learning augmented turbulence modelling is a promising approach for addressing the limitations of Reynolds-averaged Navier-Stokes (RANS) models. This work presents the development of the first open-source dataset, curated and structured for immediate use in machine learning augmented turbulence closure modelling. The dataset features a variety of RANS simulations with matching direct numerical simulation (DNS) and large-eddy simulation (LES) data. Four turbulence models are selected to form the initial dataset: $k$-$\varepsilon$, $k$-$\varepsilon$-$ϕ_t$-$f$, $k$-$ω$…
Harvard University（哈佛大学）；Imperial College London（帝国理工学院）；Dassault Systemes, Simulia Corp.（达索系统Simulia公司）
The Reynolds-averaged Navier-Stokes (RANS) equations require accurate modeling of the anisotropic Reynolds stress tensor. Traditional closure models, while sophisticated, often only apply to restricted flow configurations. Researchers have started using machine learning approaches to tackle this problem by developing more general closure models informed by data. In this work we build upon recent convolutional neural network architectures used for turbulence modeling and propose a multi-task learning-based fully convolutional neural network that is able to accurately predict the normalized anis…
The goal of this dissertation is to investigate the PANS model capabilities in providing significant improvement over RANS predictions at slightly higher computational expense and producing LES quality results at significantly lower computational cost. The objectives of this study are: (i) investigate the model fidelity at a fixed level of scale resolution (Generation1-PANS/G1-PANS) for smooth surface separation, (ii) Derive the PANS closure model in regions of resolution variation (Generation2-PANS/G2-PANS), and (iii) Validate G2-PANS model for attached and separated flows. The separated flow…
Reynolds-averaged Navier-Stokes (RANS) equations are widely used in engineering turbulent flow simulations. However, RANS predictions may have large discrepancies due to the uncertainties in modeled Reynolds stresses. Recently, Wang et al. demonstrated that machine learning can be used to improve the RANS modeled Reynolds stresses by leveraging data from high fidelity simulations (Physics informed machine learning approach for reconstructing Reynolds stress modeling discrepancies based on DNS data. Physical Review Fluids. 2, 034603, 2017). However, solving for mean flows from the improved Reyn…
In this work, we demonstrate a framework for developing closure models in turbulent combustion using experimental multi-scalar measurements. The framework is based on the construction of conditional means and joint scalar PDFs from experimental data based on the parameterization of composition space using principal component analysis (PCA). The resulting principal components (PCs) act as both conditioning variables and transport variables. Their chemical source terms are constructed starting from instantaneous temperature and species measurements using a variant of the pairwise mixing stirred …
Generalizability of machine-learning (ML) based turbulence closures to accurately predict unseen practical flows remains an important challenge. At the Reynolds-averaged Navier-Stokes (RANS) level, NN-based turbulence closure modeling is rendered difficult due to two important reasons: inherent complexity of the constitutive relation arising from flow-dependent non-linearity and bifurcations; and, inordinate difficulty in obtaining high-fidelity data covering the entire parameter space of interest. In this context, the objective of the work is to investigate the approximation capabilities of s…
Direct numerical simulation (DNS) is very accurate however, the computational cost increases significantly with the increase in Reynolds number. On the other hand, we have the Reynolds Averaged Navier Stokes (RANS) method for simulating turbulent flows, which needs less computational power. Turbulence models based on linear eddy viscosity models (LEVM) in the RANS method, which use a linear stress-strain rate relationship for modelling the Reynolds stress tensor do not perform well for complex flows \cite{shih1995new} . In this work, we intend to study the performance of non linear eddy viscos…
Reynolds Averaged Navier Stokes (RANS) models represent the workhorse for studying turbulent flows in industrial applications. Such single-point turbulence models have limitations in accounting for the influence of the non-local physics and flow history on the evolution of the turbulent flow. In this context, we investigate the sensitivity inherent to single-point models due to their characterization of the internal structure of homogeneous turbulent flows solely by the means of the Reynolds stresses. For a variety of mean flows, we study the prediction intervals engendered due to this coarse-…
