Deep Learning Closure Models for Large-Eddy Simulation of Flows around Bluff Bodies

A deep learning (DL) closure model for large-eddy simulation (LES) is developed and evaluated for incompressible flows around a rectangular cylinder at moderate Reynolds numbers. Near-wall flow simulation remains a central challenge in aerodynamic modeling: RANS predictions of separated flows are often inaccurate, while LES can require prohibitively small near-wall mesh sizes. The DL-LES model is trained using adjoint PDE optimization methods to match, as closely as possible, direct numerical simulation (DNS) data. It is then evaluated out-of-sample (i.e., for new aspect ratios and Reynolds numbers not included in the training data) and compared against a standard LES model (the dynamic Smagorinsky model). The DL-LES model outperforms dynamic Smagorinsky and is able to achieve accurate LES predictions on a relatively coarse mesh (downsampled from the DNS grid by a factor of four in each Cartesian direction). We study the accuracy of the DL-LES model for predicting the drag coefficient, mean flow, and Reynolds stress. A crucial challenge is that the LES quantities of interest are the steady-state flow statistics; for example, the time-averaged mean velocity $\bar{u}(x) = \displaystyle \lim_{t \rightarrow \infty} \frac{1}{t} \int_0^t u(s,x) ds$. Calculating the steady-state flow statistics therefore requires simulating the DL-LES equations over a large number of flow times through the domain; it is a non-trivial question whether an unsteady partial differential equation model whose functional form is defined by a deep neural network can remain stable and accurate on $t \in [0, \infty)$. Our results demonstrate that the DL-LES model is accurate and stable over large physical time spans, enabling the estimation of the steady-state statistics for the velocity, fluctuations, and drag coefficient of turbulent flows around bluff bodies relevant to aerodynamic applications

Large eddy simulation of airfoil flows using adjoint-trained deep learning closure models

There is a wide interest in developing methods that reduce the cost of resolving the near-wall and small-scale turbulence in order to enable the use of turbulence-resolving methods (such as large eddy simulation) at higher, more industrially relevant, Reynolds numbers. Existing techniques for this can struggle to accurately predict scenarios with complex flow physics such as transition, large-scale unsteadiness, and smooth-body separation/reattachment. Deep-learning based subgrid-scale (DL-SGS) models may help address this. However, the standard a priori approach for training these is unable to account for interactions of the DL-SGS closure with the numerics and resolved physics. Optimizing the DL-SGS closure over the governing equations using the adjoint equation has previously been found to improve predictions and stability in incompressible flows. In this paper, an adjoint training method is developed for compressible LES, and is applied the flow around a NACA 0012 airfoil at Re = 50,000 and Ma = 0.4. The DL-SGS closure is trained on a single angle of attack, and improves predictions compared to classical SGS models when extrapolating to out-of-sample angles of attack, even when it has not been trained over fully representative flow physics

Deep Learning Closure of the Navier-Stokes Equations for Transition-Continuum Flows

The predictive accuracy of the Navier-Stokes equations is known to degrade at the limits of the continuum assumption, thereby necessitating expensive and often highly approximate solutions to the Boltzmann equation. While tractable in one spatial dimension, their high dimensionality makes multi-dimensional Boltzmann calculations impractical for all but canonical configurations. It is therefore desirable to augment the Navier-Stokes equations in these regimes. We present an application of a deep learning method to extend the validity of the Navier-Stokes equations to the transition-continuum flows. The technique encodes the missing physics via a neural network, which is trained directly from Boltzmann solutions. While standard DL methods can be considered ad-hoc due to the absence of underlying physical laws, at least in the sense that the systems are not governed by known partial differential equations, the DL framework leverages the a-priori known Boltzmann physics while ensuring that the trained model is consistent with the Navier-Stokes equations. The online training procedure solves adjoint equations, constructed using algorithmic differentiation, which efficiently provide the gradient of the loss function with respect to the learnable parameters. The model is trained and applied to predict stationary, one-dimensional shock thickness in low-pressure argon

Understanding Latent Timescales in Neural Ordinary Differential Equation Models for Advection-Dominated Dynamical Systems

The neural ordinary differential equation (ODE) framework has shown promise in developing accelerated surrogate models for complex systems described by partial differential equations (PDEs). In PDE-based systems, neural ODE strategies use a two-step approach for acceleration: a nonlinear dimensionality reduction via an autoencoder, and a time integration step through a neural-network based model (neural ODE). This study explores the effectiveness of autoencoder-based neural ODE strategies for advection-dominated PDEs. It includes predictive demonstrations and delves into the sources of model acceleration, focusing on how neural ODEs achieve this. The research quantifies the impact of autoencoder and neural ODE components on system time-scales through eigenvalue analysis of dynamical system Jacobians. It examines how various training parameters, like training methods, latent space dimensionality, and training trajectory length, affect model accuracy and latent time-scales. Particularly, it highlights the influence of training trajectory length on neural ODE time-scales, noting that longer trajectories enhance limiting time-scales, with optimal neural ODEs capturing the largest time-scales of the actual system. The study conducts demonstrations on two distinct unsteady fluid dynamics settings influenced by advection: the Kuramoto-Sivashinsky equations and Hydrogen-Air channel detonations, using the compressible reacting Navier--Stokes equations

Influence of adversarial training on super-resolution turbulence reconstruction

Supervised super-resolution deep convolutional neural networks (CNNs) have gained significant attention for their potential in reconstructing velocity and scalar fields in turbulent flows. Despite their popularity, CNNs currently lack the ability to accurately produce high-frequency and small-scale features, and tests of their generalizability to out-of-sample flows are not widespread. Generative adversarial networks (GANs), which consist of two distinct neural networks (NNs), a generator and discriminator, are a promising alternative, allowing for both semi-supervised and unsupervised training. The difference in the flow fields produced by these two NN architectures has not been thoroughly investigated, and a comprehensive understanding of the discriminator's role has yet to be developed. This study assesses the effectiveness of the unsupervised adversarial training in GANs for turbulence reconstruction in forced homogeneous isotropic turbulence. GAN-based architectures are found to outperform supervised CNNs for turbulent flow reconstruction for in-sample cases. The reconstruction accuracy of both architectures diminishes for out-of-sample cases, though the GAN's discriminator network significantly improves the generator's out-of-sample robustness using either an additional unsupervised training step with large eddy simulation input fields and a dynamic selection of the most suitable upsampling factor. These enhance the generator's ability to reconstruct small-scale gradients, turbulence intermittency, and velocity-gradient probability density functions. The extrapolation capability of the GAN-based model is demonstrated for out-of-sample flows at higher Reynolds numbers. Based on these findings, incorporating discriminator-based training is recommended to enhance the reconstruction capability of super-resolution CNNs

Effects of combustion heat release on velocity and scalar statistics in turbulent premixed jet flames at low and high Karlovitz numbers

Theoretical scaling arguments for turbulent premixed combustion indicate that the pressure-dilatation source of turbulent kinetic energy becomes significant at low Karlovitz numbers, leading to potential invalidation of commonly-used turbulence models developed for non-reacting flow. Based on these arguments, a critical Karlovitz number is defined, below which dilatation effects are expected to become significant. Velocity and scalar statistics are obtained from Direct Numerical Simulation (DNS) calculations of low Mach number spatially-evolving turbulent premixed planar jet flames. At fixed bulk Reynolds number and stoichiometric equivalence ratio, two simulations are performed at Karlovitz numbers above and below the critical Karlovitz number. Hydrogen combustion with detailed transport is modeled using a detailed nine-species chemical kinetic mechanism, and coflows of combustion products are used to ensure flame stability at uniform equivalence ratio. The analysis of these statistics focuses on three key areas. First, the influence of the velocity-pressure gradient source of turbulent kinetic energy is confirmed at a low Karlovitz number, and the unimportance of these effects is confirmed at a high Karlovitz number. Similar effects are observed for the chemical source term in the scalar variance budgets. Second, the degree of alignment between the Reynolds stress tensor (scalar flux) and the strain-rate tensor (scalar gradient), the foundation of a majority of the turbulence models used in reacting flows, is assessed with the DNS databases. Additionally, consistency of anisotropic Reynolds stress and strain-rate tensor invariants is assessed using invariant maps. While good alignment and consistency are obtained for statistics and invariants at a high Karlovitz number, both alignment and consistency degrade at a low Karlovitz number. Third, turbulence models formulated for non-reacting flow are modified algebraically in the Bray–Moss–Libby (BML) formalism for turbulent premixed combustion. A variable efficiency function is defined to capture the regime dependence of heat release effects in these models. Model performance is evaluated at Karlovitz numbers above and below the critical Karlovitz number using the DNS databases, and satisfactory prediction of counter-gradient transport in the flame-normal direction is obtained. However, heat release effects are also observed in the flame-parallel directions in the low-Karlovitz number simulation, and the models developed in the formalism for statistically planar flames fail to capture these effects. Furthermore, in the low-Karlovitz number case, redistributive effects are active on the shear components of the Reynolds stress, which are not considered in the BML formalism. More advanced turbulence models are therefore necessary for turbulent premixed jet flames below the critical Karlovitz number.</p

Effects of small-scale heat release on turbulence scaling in premixed and nonpremixed flames

The effect of combustion heat release on small scales of turbulence is studied using static statistical analysis of low Mach number Direct Numerical Simulations (DNS) of hydrogen turbulent planar jet flames, both premixed and nonpremixed. For the premixed flames, the central fuel/air jet is surrounded by turbulent co-flows of equilibrium combustion products to ensure a constant mixture fraction and high turbulence intensity. For the nonpremixed flames, the central fuel jet is surrounded by turbulent co-flows of air in order to ensure high turbulence intensity. The calculations are performed with detailed chemistry and detailed transport, and grid convergence tests verify that the calculations are fully resolved. Previous scaling studies have suggested that the dilatation from small-scale heat release has a much more significant effect on the turbulence dynamics in premixed flames than in nonpremixed flames, with the turbulence in nonpremixed flames being scaled with Kolmogorov/Batchelor scales but in premixed flames being scaled with flame scales. The validity and applicability of this hypothesis is verified against the DNS data using density-weighted energy spectra in both reacting and nonreacting portions of the flame domains.</p

Multi-modal counterflow flames under autoignitive conditions

In practical systems, combustion occurs in multi-model regimes rather than in asymptotic limits of nonpremixed flames, premixed flames, or autoignition as commonly assumed in turbulent combustion models. In canonical configurations, multi-modal combustion is critical in the stabilization of lifted jet flames. At low temperatures, stabilization occurs kinetmatically through a "triple" flame with regions of both premixed and nonpremixed combustion. At high temperatures, autoignition is activated, and the role of autoignition versus premixed flame propagation for flame stabilization depends on the residence times and flow speed. While detailed simulations of laminar lifted jet flames are computationally tractable, for turbulent lifted jet flames, the lift-off heights require very large computational domains, and simulations of such flames with DNS becomes extremely expensive. Therefore, in work, the counterflow configuration is investigate as a more compact alternative. A series of detailed simulations of DME/air counter flames at elevated pressure are performed spanning a range of boundary conditions in both streams. The counterflow configuration is shown to exhibit multi-reaction zone structures, that is double and triple "flame" structures, analogous to lifted jet flames but only if the fuel and oxidizer streams are changes from pure components to partially premixed components. In other words, fuel/air boundary conditions gives only one reaction zone, but, under certain conditions, rich/lean boundary conditions can give up to three reaction zones.</p

Dynamic mode decomposition of a direct numerical simulation of a turbulent premixed planar jet flame: convergence of the modes

Dynamic Mode Decomposition (DMD) is a technique that enables investigation of unsteady and dynamic phenomena by decomposing data into coherent modes with corresponding growth rates and oscillatory frequencies. Because the method identifies structures unbiased by energy, it is particularly well suited to exploring dynamic processes having phenomena that span disparate temporal and spatial scales. In turbulent combustion, DMD has been previously applied to the analysis of narrowband phenomena such as combustion instabilities utilising both experimental and computational data. In this work, DMD is used as a tool to analyse broadband turbulent combustion phenomena from a three-dimensional direct numerical simulation of a low Mach number spatially-evolving turbulent planar premixed hydrogen/air jet flame. The focus of this investigation is on defining the metric of convergence of the DMD modes for broadband phenomena when both the temporal resolution and number of data snapshots can be varied independently. The residual is identified as an effective, even if imperfect, metric for judging convergence of the DMD modes. Other metrics–specifically, the convergence of the mode eigenvalues and the decay of the amplitudes of the modes–fail to capture convergence of the modes independently but do complete the information needed to evaluate the quality of the DMD analysis.</p