We demonstrate how incorporating physics constraints into convolutional neural networks (CNNs) enables learning subgrid-scale (SGS) closures for stable and accurate large-eddy simulations (LES) in the small-data regime (i.e., when the availability of high-quality training data is limited). Using several setups of forced 2D turbulence as the testbeds, we examine the {\it a priori} and {\it a posteriori} performance of three methods for incorporating physics: 1) data augmentation (DA), 2) CNN with group convolutions (GCNN), and 3) loss functions that enforce a global enstrophy-transfer conservation (EnsCon). While the data-driven closures from physics-agnostic CNNs trained in the big-data regime are accurate and stable, and outperform dynamic Smagorinsky (DSMAG) closures, their performance substantially deteriorate when these CNNs are trained with 40x fewer samples (the small-data regime). We show that CNN with DA and GCNN address this issue and each produce accurate and stable data-driven closures in the small-data regime. Despite its simplicity, DA, which adds appropriately rotated samples to the training set, performs as well or in some cases even better than GCNN, which uses a sophisticated equivariance-preserving architecture. EnsCon, which combines structural modeling with aspect of functional modeling, also produces accurate and stable closures in the small-data regime. Overall, GCNN+EnCon, which combines these two physics constraints, shows the best {\it a posteriori} performance in this regime. These results illustrate the power of physics-constrained learning in the small-data regime for accurate and stable LES.
Structural subgrid stress models for large eddy simulation often allow for backscatter of energy from unresolved to resolved turbulent scales, but excessive model backscatter can eventually result in numerical instability. A commonly employed strategy to overcome this issue is to set predicted subgrid stresses to zero in regions of model backscatter. This clipping procedure improves the stability of structural models, however, at the cost of reduced correlation between the predicted subgrid stresses and the exact subgrid stresses. In this article, we propose an alternative strategy that remove…
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…
Two approaches for closing the turbulence subgrid-scale stress tensor in terms of matrix exponentials are introduced and compared. The first approach is based on a formal solution of the stress transport equation in which the production terms can be integrated exactly in terms of matrix exponentials. This formal solution of the subgrid-scale stress transport equation is shown to be useful to explore special cases, such as the response to constant velocity gradient, but neglecting pressure-strain correlations and diffusion effects. The second approach is based on an Eulerian-Lagrangian change o…
A class of spectral subgrid models based on a self-similar and reversible closure is studied with the aim to minimize the impact of subgrid scales on the inertial range of fully developed turbulence. In this manner, we improve the scale extension where anomalous exponents are measured by roughly one order of magnitude, when compared to direct numerical simulations or to other popular subgrid closures at the same resolution. We found a first indication that intermittency for high order moments is not captured by many of the popular phenomenological models developed so far.
We use approximate Bayesian computation (ABC) combined with an "improved" Markov chain Monte Carlo (IMCMC) method to estimate posterior distributions of model parameters in subgrid-scale (SGS) closures for large eddy simulations (LES) of turbulent flows. The ABC-IMCMC approach avoids the need to directly compute a likelihood function during the parameter estimation, enabling a substantial speed-up and greater flexibility as compared to full Bayesian approaches. The method also naturally provides uncertainties in parameter estimates, avoiding the artificial certainty implied by many optimizatio…
In the present study, we investigate different data-driven parameterizations for large eddy simulation of two-dimensional turbulence in the \emph{a priori} settings. These models utilize resolved flow field variables on the coarser grid to estimate the subgrid-scale stresses. We use data-driven closure models based on localized learning that employs multilayer feedforward artificial neural network (ANN) with point-to-point mapping and neighboring stencil data mapping, and convolutional neural network (CNN) fed by data snapshots of the whole domain. The performance of these data-driven closure …
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…
We discuss a theoretical framework to define an optimal sub-grid closure for shell models of turbulence. The closure is based on the ansatz that consecutive shell multipliers are short-range correlated, following the third hypothesis of Kolmogorov formulated for similar quantities for the original three-dimensional Navier-Stokes turbulence. We also propose a series of systematic approximations to the optimal model by assuming different degrees of correlations across scales among amplitudes and phases of consecutive multipliers. We show numerically that such low-order closures work well, reprod…
Compressible magnetohydrodynamic (MHD) turbulence is ubiquitous in astrophysical phenomena ranging from the intergalactic to the stellar scales. In studying them, numerical simulations are nearly inescapable, due to the large degree of nonlinearity involved. However the dynamical ranges of these phenomena are much larger than what is computationally accessible. In large eddy simulations (LES), the resulting limited resolution effects are addressed explicitly by introducing to the equations of motion additional terms associated with the unresolved, subgrid-scale (SGS) dynamics. This renders the…
Even though compressible plasma turbulence is encountered in many astrophysical phenomena, its effect is often not well understood. Furthermore, direct numerical simulations are typically not able to reach the extreme parameters of these processes. For this reason, large-eddy simulations (LES), which only simulate large and intermediate scales directly, are employed. The smallest, unresolved scales and the interactions between small and large scales are introduced by means of a subgrid-scale (SGS) model. We propose and verify a new set of nonlinear SGS closures for future application as an SGS…
Stanford University（斯坦福大学）；SLAC National Accelerator Laboratory（SLAC国家加速器实验室）
Many practical combustion systems such as those in rockets, gas turbines, and internal combustion engines operate under high pressures that surpass the thermodynamic critical limit of fuel-oxidizer mixtures. These conditions require the consideration of complex fluid behaviors that pose challenges for numerical simulations, casting doubts on the validity of existing subgrid-scale (SGS) models in large-eddy simulations of these systems. While data-driven methods have shown high accuracy as closure models in simulations of turbulent flames, these models are often criticized for lack of physical …
The present work is concerned with a study of large eddy simulations (LES) of unsteady turbulent jet flows. In particular, the present analysis is focused on the effects of the subgrid-scale modeling used when a second-order spatial discretization methodology is employed for the numerical simulations. The present effort addresses perfectly expanded supersonic jets, because the authors want to emphasize the effects of the jet mixing phenomena. The LES formulation is discretized using the finite difference approach, after the equations are rewritten in a generalized coordinate system. Both space…
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…
In this paper, we introduce a relaxation filtering closure approach to account for subgrid scale effects in explicitly filtered large eddy simulations using the concept of anisotropic diffusion. We utilize the Perona-Malik diffusion model and demonstrate its shock capturing ability and spectral performance for solving the Burgers turbulence problem, which is a simplified prototype for more realistic turbulent flows showing the same quadratic nonlinearity. Our numerical assessments present the behavior of various diffusivity functions in conjunction with a detailed sensitivity analysis with res…
Current design constraints have encouraged the studies of aeroacoustics fields around compressible jet flows. The present work addresses the numerical study of subgrid scale modeling for unsteady turbulent jet flows as a preliminary step for future aeroacoustic analyses of main engine rocket plumes. An in-house large eddy simulation (LES) tool is developed in order to reproduce high fidelity results of compressible jet flows. In the present study, perfectly expanded jets are considered because the authors want to emphasize the effects of the jet mixing phenomena. The large eddy simulation form…
Filtering the passive scalar transport equation in the large-eddy simulation (LES) of turbulent transport gives rise to the closure term corresponding to the unresolved scalar flux. Understanding and respecting the statistical features of subgrid-scale (SGS) flux is a crucial point in robustness and predictability of the LES. In this work, we investigate the intrinsic nonlocal behavior of the SGS passive scalar flux through studying its two-point statistics obtained from the filtered direct numerical simulation (DNS) data for passive scalar transport in homogeneous isotropic turbulence (HIT). …
The issue of the parameterization of small scale (``subgrid'') turbulence is addressed in the context of passive scalar transport. We focus on the Kraichnan advection model which lends itself to the analytical investigation of the closure problem. We derive systematically the dynamical equations which rule the evolution of the coarse-grained scalar field. At the lowest-order approximation in $l/r$, $l$ being the characteristic scale of the filter defining the coarse-grained scalar field and $r$ the inertial range separation, we recover the classical eddy-diffusivity parameterization of small s…
Turbulence in compressible plasma plays a key role in many areas of astrophysics and engineering. The extreme plasma parameters in these environments, e.g. high Reynolds numbers, supersonic and super-Alfvenic flows, however, make direct numerical simulations computationally intractable even for the simplest treatment -- magnetohydrodynamics (MHD). To overcome this problem one can use subgrid-scale (SGS) closures -- models for the influence of unresolved, subgrid-scales on the resolved ones. In this work we propose and validate a set of constant coefficient closures for the resolved, compressib…
Subgrid-scale (SGS) models are critical in large-eddy simulations (LES) of turbulent flows. In this paper we conduct a comparative study on different SGS models, including one-k-equation, wall-adapting local eddy-viscosity (WALE), Sigma and shear-constrained model. Wall-resolved LES simulations of channel flows are performed with a finite volume code at shear Reynolds number Re_τ= 395. In the simulations, the buffer sublayer turns out to be the most sensitive to the SGS model. Through the analysis of the mean velocity, the second-order moments, the SGS viscosity and the fluctuating vorticities…
Turbulence is still one of the main challenges for accurately predicting reactive flows. Therefore, the development of new turbulence closures which can be applied to combustion problems is essential. Data-driven modeling has become very popular in many fields over the last years as large, often extensively labeled, datasets became available and training of large neural networks became possible on GPUs speeding up the learning process tremendously. However, the successful application of deep neural networks in fluid dynamics, for example for subgrid modeling in the context of large-eddy simula…
Online learning of eddy-viscosity and backscattering closures for geophysical turbulence using ensemble Kalman inversion
Union College（联合学院）；The University of Chicago（芝加哥大学）；California Institute of Technology（加州理工学院）；Peking University（北京大学）；University of Wisconsin–Madison（威斯康星大学麦迪逊分校）；Google Research（谷歌研究）
Deep Learning of Solver-Aware Turbulence Closures from Nudged LES Dynamics
School of Mechanical Engineering（机械工程学院）；Purdue University（普渡大学）；College of Information Sciences and Technology（信息科学与技术学院）；Pennsylvania State University（宾夕法尼亚州立大学）
Generalizable data-driven turbulence closure modeling on unstructured grids with differentiable physics
School of Mechanical Engineering, Purdue University, West Lafayette, 47907, IN, USA（机械工程学院，普渡大学）；Mechanical Engineering Department, Carnegie Mellon University, Pittsburgh, 15213, PA, USA（机械工程系，卡内基梅隆大学）；Department of Aerospace Engineering, University of Michigan, Ann Arbor, 48109, MI, USA（航空航天工程系，密歇根大学）；College of Information Sciences and Technology, Pennsylvania State University, University Park, 16802, PA, USA（信息科学与技术学院，宾夕法尼亚州立大学）；Computer Science Division, Argonne National Laboratory, Lemont, 60439, IL, USA（计算机科学 division，阿贡国家实验室）
We formulate a data-driven, physics-constrained closure method for coarse-scale numerical simulations of turbulent fluid flows. Our approach involves a closure scheme that is non-local both in space and time, i.e. the closure terms are parametrized in terms of the spatial neighborhood of the resolved quantities but also their history. The data-driven scheme is complemented with a physical constrain expressing the energy conservation property of the nonlinear advection terms. We show that the adoption of this physical constrain not only increases the accuracy of the closure scheme but also impr…
Simulating complex physical systems often involves solving partial differential equations (PDEs) with some closures due to the presence of multi-scale physics that cannot be fully resolved. Therefore, reliable and accurate closure models for unresolved physics remains an important requirement for many computational physics problems, e.g., turbulence simulation. Recently, several researchers have adopted generative adversarial networks (GANs), a novel paradigm of training machine learning models, to generate solutions of PDEs-governed complex systems without having to numerically solve these PD…
In this work we introduce a computational framework for determining optimal closures of the eddy-viscosity type for Large-Eddy Simulations (LES) of a broad class of PDE models, such as the Navier-Stokes equation. This problem is cast in terms of PDE-constrained optimization where an error functional representing the misfit between the target and predicted observations is minimized with respect to the functional form of the eddy viscosity in the closure relation. Since this leads to a PDE optimization problem with a nonstandard structure, the solution is obtained computationally with a flexible…
Liquid metals play a central role in new generation liquid metal cooled nuclear reactors, for which numerical investigations require the use of appropriate thermal turbulence models for low Prandtl number fluids. Given the limitations of traditional modelling approaches and the increasing availability of high-fidelity data for this class of fluids, we propose a Machine Learning strategy for the modelling of the turbulent heat flux. A comprehensive algebraic mathematical structure is derived and physical constraints are imposed to ensure attractive properties promoting applicability, robustness…
Recent years have seen a surge in interest in building deep learning-based fully data-driven models for weather prediction. Such deep learning models if trained on observations can mitigate certain biases in current state-of-the-art weather models, some of which stem from inaccurate representation of subgrid-scale processes. However, these data-driven models, being over-parameterized, require a lot of training data which may not be available from reanalysis (observational data) products. Moreover, an accurate, noise-free, initial condition to start forecasting with a data-driven weather model …
