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 mean flow field predictions. In this paper, we introduce a data-informed approach for arriving at Reynolds stress models with improved predictive performance. Our approach relies on learning the components of the Reynolds stress discrepancy tensor associated with a given Reynolds stress model in the mean strain-rate tensor eigenframe. These components are typically smooth and hence simple to learn using state-of-the-art machine learning strategies and regression techniques. Our approach automatically yields Reynolds stress models that are symmetric, and it yields Reynolds stress models that are both Galilean and frame invariant provided the inputs are themselves Galilean and frame invariant. To arrive at computable models of the discrepancy tensor, we employ feed-forward neural networks and an input space spanning the integrity basis of the mean strain-rate tensor, the mean rotation-rate tensor, the mean pressure gradient, and the turbulent kinetic energy gradient, and we introduce a framework for dimensional reduction of the input space to further reduce computational cost. Numerical results illustrate the effectiveness of the proposed approach for data-informed Reynolds stress closure for a suite of turbulent flow problems of increasing complexity.
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, …
A Lagrangian particle cloud model is proposed that accounts for the effects of Reynolds-averaged particle and turbulent stresses and the averaged carrier-phase velocity of the sub-particle-cloud scale on the averaged motion and velocity of the cloud. The SPARSE (Subgrid Particle Average Reynolds Stress Equivalent) model is based on a combination of a truncated Taylor expansion of a drag correction function and Reynolds averaging. It reduces the required number of computational parcels to trace a cloud of particles in Eulerian-Lagrangian methods for the simulation of particle-laden flow. Closur…
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…
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 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…
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.
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…
Here we introduce new coarse-grained variables for a turbulent flow in the form of moments of its Reynolds-averaged Boltzmann equation. With the exception of the collision moments, the transport equations for the new variables are identical to the usual moment equations, and thus naturally lend themselves to the variety of already existing closure methods. Under the anelastic turbulence approximation, we derive equations for the Reynolds-averaged turbulent fluctuations around the coarse-grained state. We show that the global relative entropy of the coarse-grained state is bounded from above by…
Fluid turbulence is an important problem for physics and engineering. Turbulence modeling deals with the development of simplified models that can act as surrogates for representing the effects of turbulence on flow evolution. Such models correspond to a range of different fidelities, from simple eddy-viscosity-based closures to Reynolds Stress Models. Till now the focus of the data-driven turbulence modeling efforts has focused on Machine Learning augmented eddy-viscosity models. In this communication, we illustrate the manner in which the eddy-viscosity framework delimits the efficacy and pe…
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…
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…
Although an increased availability of computational resources has enabled high-fidelity simulations of turbulent flows, the RANS models are still the dominant tools for industrial applications. However, the predictive capabilities of RANS models are limited by potential inaccuracy driven by hypotheses in the Reynolds stress closure. Recently, a Physics-Informed Machine Learning (PIML) approach has been proposed to learn the functional form of Reynolds stress discrepancy in RANS simulations based on available data. It has been demonstrated that the learned discrepancy function can be used to im…
There exists continuous demand of improved turbulence models for the closure of Reynolds Averaged Navier-Stokes (RANS) simulations. Machine Learning (ML) offers effective tools for establishing advanced empirical Reynolds stress closures on the basis of high fidelity simulation data. This paper presents a turbulence model based on the Deep Neural Network(DNN) which takes into account the non-linear relationship between the Reynolds stress anisotropy tensor and the local mean velocity gradient as well as the near-wall effect. The construction and the tuning of the DNN-turbulence model are detai…
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 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…
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 paper we investigate whether the features of the non-equilibrium cascade, which have been identified in recent studies using high-fidelity tools, can be captured in the case of the classical dissipation scaling by turbulence closures based on the statistical description of freely decaying isotropic turbulence. Numerical results obtained using the EDQNM model over a very large range of Reynolds numbers (from $Re_λ=50$ up to $Re_λ=10^6$) are analyzed to perform an extensive investigation of the scaling region identified as inertial range in Kolmogorov's theory. It is observed that EDQNM …
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…
The paper examines the impact of different modelling choices in second-moment closures by assessing model performance in predicting 3-D duct flows. The test-cases (developing flow in a square duct [Gessner F.B., Emery A.F.: {\em ASME J. Fluids Eng.} {\bf 103} (1981) 445--455], circular-to-rectangular transition-duct [Davis D.O., Gessner F.B.: {\em AIAA J.} {\bf 30} (1992) 367--375], and \tsn{S}-duct with large separation [Wellborn S.R., Reichert B.A., Okiishi T.H.: {\em J. Prop. Power} {\bf 10} (1994) 668--675]) include progressively more complex strains. Comparison of experimental data with s…
This paper proposes a simple new closure principle for turbulent shear flows. The turbulent flow field is divided into an outer and an inner region. The inner region is made up of a log-law region and a wall layer. The wall layer is viewed in terms of the well known inrush-sweep-burst sequence observed since 1967. It is modelled as a transient laminar sub-boundary layer, which obeys the Stokes solution for an impulsively started flat plate. The wall layer may also be modelled with a steady state solution by adding a damping function to the log-law. Closure is achieved by matching the unsteady …
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 str…
It has been demonstrated that the Euler equations of inviscid fluid are incomplete: according to the principle of release of constraints, absence of shear stresses must be compensated by additional degrees of freedom, and leads to Reynolds-type multivalued velocity field. however unlike the Reynolds equations, the enlarged Euler's (EE) model provides additional equations for fluctuations, and that eliminates the closure problem. Therefore the (EE) equations are applicable to fully developed turbulent motions where the physical viscosity is vanishingly small compare to the turbulent viscosity, …
Developing deterministic neighborhood-informed point-particle closure models using machine learning has garnered interest in recent times from dispersed multiphase flow community. The robustness of neural models for this complex multi-body problem is hindered by the availability of particle-resolved data. The present work addresses this unavoidable limitation of data paucity by implementing two strategies: (i) by using a rotation and reflection equivariant neural network and (ii) by pursuing a physics-based hierarchical machine learning approach. The resulting machine learned models are observ…
Numerical simulations based on Reynolds-Averaged Navier--Stokes (RANS) equations are widely used in engineering design and analysis involving turbulent flows. However, RANS simulations are known to be unreliable in many flows of engineering relevance, which is largely caused by model-form uncertainties associated with the Reynolds stresses. Recently, a machine-learning approach has been proposed to assist RANS modeling by building a functional mapping from mean flow features to discrepancies in RANS modeled Reynolds stresses as compared to high-fidelity data. However, it remains a challenge to…
In large eddy simulations, the Reynolds averages of nonlinear terms are not directly computable in terms of the resolved variables and require a closure hypothesis or model, known as a subgrid scale term. Inspired by the renormalization group (RNG),we introduce an expansion for the unclosed terms, carried out explicitly to all orders. In leading order, this expansion defines subgrid scale unclosed terms, which we relate to the dynamic subgrid scale closure models. The expansion, which generalizes the Leonard stress for closure analysis, suggests a systematic higher order determination of the m…
The statistics of the velocity gradient tensor in turbulent flows are of both theoretical and practical importance. The Lagrangian view provides a privileged perspective for studying the dynamics of turbulence in general, and of the velocity gradient tensor in particular. Stochastic models for the Lagrangian evolution of velocity gradients in isotropic turbulence, with closure models for the pressure Hesssian and viscous Laplacian, have been shown to reproduce important features such as non-Gaussian probability distributions, skewness and vorticity strain-rate alignments. The Recent Fluid Defo…
The Reynolds-averaged Navier-Stokes (RANS) equations are widely used in turbulence applications. They require accurately modeling the anisotropic Reynolds stress tensor, for which traditional Reynolds stress closure models only yield reliable results in some flow configurations. In the last few years, there has been a surge of work aiming at using data-driven approaches to tackle this problem. The majority of previous work has focused on the development of fully-connected networks for modeling the anisotropic Reynolds stress tensor. In this paper, we expand upon recent work for turbulent chann…
We demonstrate the coupling between the Algebraic Structure-Based Model (ASBM) and the one-equation Spalart-Allmaras (SA) model, which provides an easy route to bringing structure information in engineering turbulence closures. The estimation ability of the hybrid model was tested for a flow over a hump model with no-flow control and steady suction. ASBM-SA model produced satisfactory predictions for the streamwise Reynolds stress component, while a qualitative agreement with the experiments was achieved for the transverse component. Regarding the shear stress component, ASBM-SA closure provid…
The partially-averaged Navier-Stokes (PANS) equations are used to predict the variable-density Rayleigh-Taylor (RT) flow at Atwood number 0.5 and maximum Reynolds number $500$. This is a prototypical problem of material mixing featuring laminar, transitional, and turbulent flow, instabilities and coherent structures, density fluctuations, and production of turbulence kinetic energy by both shear and buoyancy mechanisms. These features pose numerous challenges to modeling and simulation, making the RT flow ideal to develop the validation space of the recently proposed PANS BHR-LEVM closure. The…
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）
We propose a new data-driven reduced order model (ROM) framework that centers around the hierarchical structure of the variational multiscale (VMS) methodology and utilizes data to increase the ROM accuracy at a modest computational cost. The VMS methodology is a natural fit for the hierarchical structure of the ROM basis: In the first step, we use the ROM projection to separate the scales into three categories: (i) resolved large scales, (ii) resolved small scales, and (iii) unresolved scales. In the second step, we explicitly identify the VMS-ROM closure terms, i.e., the terms representing t…
We model the dynamics of Jupiter's jets by averaging the dynamics of eddies, in a barotropic beta-plane model, and explicitly predicting the balance between Reynolds' stresses and dissipation, thus predicting the average velocity profile explicitly.In order to obtain this result, we adopt a non-equilibrium statistical mechanics approach. We consider a relevant limit for Jupiter troposphere, of a time scale separation between inertial dynamics on one hand, and stochastic forcing and dissipation on the other hand. We assume that the forcing acts on scales much smaller than the jet scale, and we …
This paper introduces two novel concepts in data-driven turbulence modeling that enable the simultaneous development of multiple closure models and the training towards multiple objectives. The concepts extend the evolutionary framework by Weatheritt and Sandberg (2016), which derives interpretable and implementation-ready expressions from high-fidelity simulation data. By assigning a shared fitness value to the evolved closure models and utilizing the CFD-driven training approach by Zhao et al. (2020), the multi-expression training concept introduced here is able to account for the coupling b…
A novel machine learning algorithm is presented, serving as a data-driven turbulence modeling tool for Reynolds Averaged Navier-Stokes (RANS) simulations. This machine learning algorithm, called the Tensor Basis Random Forest (TBRF), is used to predict the Reynolds-stress anisotropy tensor, while guaranteeing Galilean invariance by making use of a tensor basis. By modifying a random forest algorithm to accept such a tensor basis, a robust, easy to implement, and easy to train algorithm is created. The algorithm is trained on several flow cases using DNS/LES data, and used to predict the Reynol…
Its is a well known fact that Turbulence exhibits non-locality, however, modeling has largely received local treatment following the work of Prandl over mixing-length model. Thus, in this article we report our findings by formulating a non-local closure model for Reynolds-averaged Navier-Stokes (RANS) equation using Fractional Calculus. Two model formulations are studied, namely one- and two-sided for Channel, Pipe and Couette flow, where the results shown have less 1% error. The motivation of two-sided model lies in recognising the fact that non-locality at a given spatial location is an aggr…
This work determines the inaccuracy of using Reynolds averaged Navier Stokes (RANS) turbulence models in transition to turbulent flow regimes by predicting the model-based discrepancies between RANS and large eddy simulation (LES) models and then incorporates the capabilities of machine learning algorithms to characterize the discrepancies which are defined as a function of mean flow properties of RANS simulations. First, three-dimensional CFD simulations using k-omega Shear Stress Transport (SST) and dynamic one-equation subgrid-scale models are conducted in a wall-bounded channel containing …
The application machine learning (ML) algorithms to turbulence modeling has shown promise over the last few years, but their application has been restricted to eddy viscosity based closure approaches. In this article we discuss rationale for the application of machine learning with high-fidelity turbulence data to develop models at the level of Reynolds stress transport modeling. Based on these rationale we compare different machine learning algorithms to determine their efficacy and robustness at modeling the different transport processes in the Reynolds Stress Transport Equations. Those data…
The Reynolds-averaged Navier-Stokes (RANS) equations for steady-state assessment of incompressible turbulent flows remain the workhorse for practical computational fluid dynamics (CFD) applications. Consequently, improvements in speed or accuracy have the potential to affect a diverse range of applications. We introduce a machine learning framework for the {surrogate modeling of steady-state turbulent eddy viscosities for RANS simulations, given the initial conditions. This modeling strategy} is assessed for parametric interpolation, while numerically solving for the pressure and velocity equa…
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…
University of Notre Dame（圣母大学）；Missouri University of Science and Technology（密苏里科技大学）；Virginia Tech（弗吉尼亚理工大学）
Modeled Reynolds stress is a major source of model-form uncertainties in Reynolds-averaged Navier-Stokes (RANS) simulations. Recently, a physics-informed machine-learning (PIML) approach has been proposed for reconstructing the discrepancies in RANS-modeled Reynolds stresses. The merits of the PIML framework has been demonstrated in several canonical incompressible flows. However, its performance on high-Mach-number flows is still not clear. In this work we use the PIML approach to predict the discrepancies in RANS modeled Reynolds stresses in high-Mach-number flat-plate turbulent boundary lay…
Numerical simulations of geophysical and atmospheric flows have to rely on parameterizations of subgrid scale processes due to their limited spatial resolution. Despite substantial progress in developing parameterization (or closure) models for subgrid scale (SGS) processes using physical insights and mathematical approximations, they remain imperfect and can lead to inaccurate predictions. In recent years, machine learning has been successful in extracting complex patterns from high-resolution spatio-temporal data, leading to improved parameterization models, and ultimately better coarse grid…
The purpose of the present note is to contribute in clarifying the relation between representation bases used in the closure for the redistribution (pressure-strain) tensor $ϕ_{ij}$, and to construct representation bases whose elements have clear physical significance. The representation of different models in the same basis is essential for comparison purposes, and the definition of the basis by physically meaningfull tensors adds insight to our understanding of closures. The rate-of-production tensor can be split into production by mean strain and production by mean rotation $P_{ij}=P_{\bar …
We present a macro-scale description of quasi-periodically developed flow in channels, which relies on double volume-averaging. We show that quasi-developed macro-scale flow is characterized by velocity modes which decay exponentially in the main flow direction. We prove that the closure force can be represented by an exact permeability tensor consisting of two parts. The first part, which is due to the developed macro-scale flow, is uniform everywhere, except in the side-wall region, where it is affected by the macro-scale velocity profile and its slip length. The second part expresses the re…
