Posteriori Tests Research Articles

A transformer-based convolutional method to model inverse cascade in forced two-dimensional turbulence

The present work proposes a novel transformer-based convolutional neural network (TransCNN) method to effectively model the inverse energy cascade in two dimensional (2D) turbulence. The TransCNN structure combines large-scale features extracted by transformer with small-scale features from convolutional layers, thus is considered suitable for multi-scale modeling. The novel TransCNN method has been applied to model sub-grid scale (SGS) stress for large-eddy simulation (LES) of 2D turbulence, under the extremely challenging situation that the LES grid is too coarse to resolve the external forcing scale. The data-driven model trained by the novel TransCNN structure is compared to two deep CNN models with varying complexities. All models exhibit proficiency during a priori tests. Notably, TransCNN surpasses its counterparts in predictive accuracy and generalizability in a posteriori tests. An investigation into the receptive fields reveals that the TransCNN model can efficiently leverage global information with the transformer structure, which is key to its superior performance in representing the inverse energy cascade in the 2D turbulent simulations.

Viscous-Layer Compressibility Correction for Two-Equation Reynolds-Averaged Navier–Stokes Models

The baseline two-equation Reynolds-averaged Navier–Stokes (RANS) models include fluid density but lack calibration for compressible flows, making them inadequate for high Mach numbers. Various compressibility corrections have been proposed to integrate the van Driest transformation or the semilocal transformation, i.e., a given compressible law of the wall, into the RANS formulation. Prior work has focused mainly on the logarithmic layer, but upon evaluating these log-layer compressibility corrections, we find that they do not significantly improve skin friction estimates. To overcome this challenge, we develop viscous-layer compressibility corrections. We do that by altering the dissipation terms. These corrections conform the RANS model to the semilocal scaling, resulting in more accurate predictions of mean velocity and temperature in a posteriori tests. Although not the primary focus of this study, we also find that the baseline one-equation Spalart–Allmaras model, which was proposed before the concept of semilocal scaling, produces results consistent with the semilocal scaling in a posteriori tests without requiring any compressibility correction.

Artificial neural-network-based subgrid-scale model for large-eddy simulation of isotropic turbulence

This study is concerned with accurately predicting the subgrid-scale (SGS) stress using an artificial neural network (ANN) with a linear eddy-viscosity term and a nonlinear term as the input variables. A priori and a posteriori tests are conducted to examine the prediction performance of the ANN-based SGS stress model in decaying homogeneous isotropic turbulence. In a priori test, the present ANN-based SGS model shows high correlation coefficients between the true and predicted SGS stresses, and excellent predictions of the SGS stress and dissipation. In a posteriori test, it is found that the ANN-based SGS model can predict the turbulence statistics more accurately than the traditional dynamic SGS models. The generalization capabilities of the model to untrained flow conditions and unstrained types of turbulent flow have been evaluated. It is found that the proposed ANN-based model can provide an accurate prediction of the SGS stress under different Reynolds numbers and flow types. A comparison among several existing ANN-based models with different input variables is presented, demonstrating a significant advantage of the present model.

Fourier neural operator for large eddy simulation of compressible Rayleigh–Taylor turbulence

The Fourier neural operator (FNO) framework is applied to the large eddy simulation (LES) of three-dimensional compressible Rayleigh–Taylor turbulence with miscible fluids at Atwood number At=0.5, stratification parameter Sr = 1.0, and Reynolds numbers Re = 10 000 and 30 000. The FNO model is first used for predicting three-dimensional compressible turbulence. The different magnitudes of physical fields are normalized using root mean square values for an easier training of FNO models. In the a posteriori tests, the FNO model outperforms the velocity gradient model, the dynamic Smagorinsky model, and implicit large eddy simulation in predicting various statistical quantities and instantaneous structures, and is particularly superior to traditional LES methods in predicting temperature fields and velocity divergence. Moreover, the computational efficiency of the FNO model is much higher than that of traditional LES methods. FNO models trained with short-time, low Reynolds number data exhibit a good generalization performance on longer-time predictions and higher Reynolds numbers in the a posteriori tests.

An Investigation of LES Wall Modeling for Rayleigh–Bénard Convection via Interpretable and Physics-Aware Feedforward Neural Networks with DNS

Abstract The traditional approach of using the Monin–Obukhov similarity theory (MOST) to model near-surface processes in large-eddy simulations (LESs) can lead to significant errors in natural convection. In this study, we propose an alternative approach based on feedforward neural networks (FNNs) trained on output from direct numerical simulation (DNS). To evaluate the performance, we conduct both a priori and a posteriori tests. In the a priori (offline) tests, we compare the statistics of the surface shear stress and heat flux, computed from filtered DNS input variables, to the stress and flux obtained from the filtered DNS. Additionally, we investigate the importance of various input features using the Shapley additive explanations value and the conditional average of the filter grid cells. In the a posteriori (online) tests, we implement the trained models in the System for Atmospheric Modeling (SAM) LES and compare the LES-generated surface shear stress and heat flux with those in the DNS. Our findings reveal that vertical velocity, a traditionally overlooked flow quantity, is one of the most important input features for determining the wall fluxes. Increasing the number of input features improves the a priori test results but does not always improve the model performance in the a posteriori tests because of the differences in input variables between the LES and DNS. Last, we show that physics-aware FNN models trained with logarithmic and scaled parameters can well extrapolate to more intense convection scenarios than in the training dataset, whereas those trained with primitive flow quantities cannot. Significance Statement The traditional near-surface turbulence model, based on a shear-dominated boundary layer flow, does not represent near-surface turbulence in natural convection. Using a feedforward neural network (FNN), we can construct a more accurate model that better represents the near-surface turbulence in various flows and reveals previously overlooked controlling factors and process interactions. Our study shows that the FNN-generated models outperform the traditional model and highlight the importance of the near-surface vertical velocity. Furthermore, the physics-aware FNN models exhibit the potential to extrapolate to convective flows of various intensities beyond the range of the training dataset, suggesting their broader applicability for more accurate modeling of near-surface turbulence.

A POD-mode-augmented wall model and its applications to flows at non-equilibrium conditions

Insights gained from modal analysis are invoked for predictive large-eddy simulation (LES) wall modelling. Specifically, we augment the law of the wall (LoW) by an additional mode based on a one-dimensional proper orthogonal decomposition (POD) applied to a two-dimensional turbulent channel. The constructed wall model contains two modes, i.e. the LoW-based mode and the POD-based mode, and the model matches with the LES at two, instead of one, off-wall locations. To show that the proposed model captures non-equilibrium effects, we perform a priori and a posteriori tests in the context of both equilibrium and non-equilibrium flows. The a priori tests show that the proposed wall model captures extreme wall-shear stress events better than the equilibrium wall model. The model also captures non-equilibrium effects due to adverse pressure gradients. The a posteriori tests show that the wall model captures the rapid decrease and the initial decrease of the streamwise wall-shear stress in channels subjected to suddenly imposed adverse and transverse pressure gradients, respectively, both of which are missed by currently available wall models. These results show promise in applying modal analysis for turbulence wall modelling. In particular, the results show that employing multiple modes helps in the modelling of non-equilibrium flows.

Large-eddy simulations of compressible Rayleigh–Taylor turbulence with miscible fluids using spatial gradient model

Large-eddy simulations (LES) and implicit LES (ILES) of three-dimensional compressible Rayleigh–Taylor turbulence with miscible fluids are performed and compared with direct numerical simulation (DNS) at the Atwood number At=0.5 and stratification parameters Sr = 1.0 and 4.0. Three sub-grid-scale (SGS) models including constant-coefficient spatial gradient model (CSGM), dynamic Smagorinsky model (DSM), and dynamic mixed model (DMM) are considered. The CSGM model achieves a high accuracy by using the velocity gradients at the neighboring LES grids. The a priori tests show that the CSGM model has significantly higher correlation coefficients and lower relative errors than traditional SGS models. In the a posteriori tests, the probability density functions of SGS terms predicted by the CSGM model are consistent with the filtered DNS results. The CSGM model can accurately predict the small bubble and spike structures, resulting in good predictions of mixing heights and concentration fields. The instantaneous structures, spectra, and statistics of velocity and vorticity fields are also examined, showing the excellent performance of the CSGM model compared to the ILES, DSM, and DMM models. Moreover, the predictions of the temperature and pressure fields by the CSGM model are significantly better than the traditional SGS models and ILES.

A data-driven approach to model enstrophy transfers in large eddy simulation of forced two-dimensional turbulence

Contemporary research on the application of data-driven techniques to modeling subgrid closure in two-dimensional turbulence has been limited. Furthermore, the incorporation of the enstrophy cascade and other two-dimensional turbulence-specific physics has received insufficient attention. To address these gaps, a novel physics-based shallow feed-forward neural network framework was designed in this study to model subgrid closure in three selected cases of forced two-dimensional turbulence with a forcing that provides energy and enstrophy at a particular wavenumber. As a novel approach, we trained our framework to learn the subgrid vorticity transport vector from a set of appropriate resolved flow variables. Another framework used in recent works which directly learned the subgrid forcing field was also investigated. Both frameworks were assessed using a priori and a posteriori tests for two selected filter widths. Both frameworks performed accurately for the lower filter width but less accurately for the higher filter width. However, we demonstrate that our new framework has wider usefulness for model diagnosis. Ad hoc clipping procedures were used to make the models more generalizable to higher filter widths, and stable and consistent a posteriori tests were observed for all test cases and filter widths when the subgrid forcing field was modified to enhance the model's subgrid dissipative characteristics. In contrast, modifying the enstrophy fluxes did not perform as consistently. These findings demonstrate the potential of the novel physics-based framework for improving subgrid modeling in two-dimensional turbulence.

Neural-network-based mixed subgrid-scale model for turbulent flow

An artificial neural-network-based subgrid-scale (SGS) model, which is capable of predicting turbulent flows at untrained Reynolds numbers and on untrained grid resolution is developed. Providing the grid-scale strain-rate tensor alone as an input leads the model to predict a SGS stress tensor that aligns with the strain-rate tensor, and the model performs similarly to the dynamic Smagorinsky model. On the other hand, providing the resolved stress tensor as an input in addition to the strain-rate tensor is found to significantly improve the prediction of the SGS stress and dissipation, and thereby the accuracy and stability of the solution. In an attempt to apply the neural-network-based model trained for turbulent flows with a limited range of the Reynolds number and grid resolution to turbulent flows at untrained conditions on untrained grid resolution, special attention is given to the normalisation of the input and output tensors. It is found that the successful generalization of the model to turbulence for various untrained conditions and resolution is possible if distributions of the normalised inputs and outputs of the neural network remain unchanged as the Reynolds number and grid resolution vary. In a posteriori tests of the forced and the decaying homogeneous isotropic turbulence and turbulent channel flows, the developed neural-network model is found to predict turbulence statistics more accurately, maintain the numerical stability without ad hoc stabilisation such as clipping of the excessive backscatter, and to be computationally more efficient than the algebraic dynamic SGS models.

A unified understanding of scale-resolving simulations and near-wall modelling of turbulent flows using optimal finite-element projections

The main objective of this work is to develop a unified framework that can be used as a lens to quantitatively assess and augment a wide range of coarse-grained models of turbulence, namely large eddy simulations (LES), hybrid Reynolds-averaged/LES methods and wall-modelled (WM)LES. Taking a turbulent channel flow as an example, optimality is assessed in the wall-resolved limit, the hybrid RANS–LES limit and the WMLES limit, via projections at different resolutions suitable for these approaches. These optimal a priori estimates are shown to have similar characteristics to existing a posteriori solutions reported in the literature. Consistent accuracy metrics are developed for scale-resolving methods using the optimal solution as a reference, and evaluations are performed. We further characterise the slip velocity in WMLES in terms of the near-wall under-resolution and develop a universal scaling relationship. Insights from the a priori tests are used to augment existing slip-based wall models. Various a posteriori tests reveal superior performance over the dynamic slip wall model. Guidance for the development of improved slip-wall models is provided, including a target for the dynamic procedure.

Applying physics-informed enhanced super-resolution generative adversarial networks to turbulent premixed combustion and engine-like flame kernel direct numerical simulation data

Models for finite-rate-chemistry in underresolved flows still pose one of the main challenges for predictive simulations of complex configurations. The problem gets even more challenging if turbulence is involved. This work advances the recently developed PIESRGAN modeling approach to turbulent premixed combustion. For that, the physical information processed by the network and considered in the loss function are adjusted, the training process is smoothed, and especially effects from density changes are considered. The resulting model provides good results for a priori and a posteriori tests on direct numerical simulation data of a fully turbulent premixed flame kernel. The limits of the modeling approach are discussed. Finally, the model is employed to compute further realizations of the premixed flame kernel, which are analyzed with a scale-sensitive framework regarding their cycle-to-cycle variations. The work shows that the data-driven PIESRGAN subfilter model can very accurately reproduce direct numerical simulation data on much coarser meshes, which is hardly possible with classical subfilter models, and enables studying statistical processes more efficiently due to the smaller computing cost.

Investigation of reaction-induced subgrid scalar mixing in LES/FDF simulations of turbulent premixed flames

Modeling molecular mixing remains a key issue for the filtered density function (FDF) method. We propose a new closure for the mixing timescale of each individual chemical species. A posteriori tests demonstrate the advantages of the proposed model, highlighting its potential to be employed in FDF simulations of turbulent reacting flows.

Dynamic nonlocal passive scalar subgrid-scale turbulence modeling

Extensive experimental evidence highlights that scalar turbulence exhibits anomalous diffusion and stronger intermittency levels at small scales compared to that in fluid turbulence. This renders the corresponding subgrid-scale dynamics modeling for scalar turbulence a greater challenge to date. We develop a new large eddy simulation (LES) paradigm for efficiently and dynamically nonlocal LES modeling of the scalar turbulence. To this end, we formulate the underlying nonlocal model starting from the filtered Boltzmann kinetic transport equation, where the divergence of subgrid-scale scalar fluxes emerges as a fractional-order Laplacian term in the filtered advection–diffusion model, coding the corresponding superdiffusive nature of scalar turbulence. Subsequently, we develop a robust data-driven algorithm for estimation of the fractional (noninteger) Laplacian exponent, where we, on the fly, calculate the corresponding model coefficient employing a new dynamic procedure. Our a priori tests show that our new dynamically nonlocal LES paradigm provides better agreement with the ground-truth filtered direct numerical simulation data in comparison to the conventional static and dynamic Prandtl–Smagorinsky models. Moreover, in order to analyze the numerical stability and assessing the model's performance, we carry out comprehensive a posteriori tests. They unanimously illustrate that our new model considerably outperforms other existing functional models, correctly predicting the backscattering phenomena and, at the same time, providing higher correlations at small-to-large filter sizes. We conclude that our proposed nonlocal subgrid-scale model for scalar turbulence is amenable for coarse LES and very large eddy simulation frameworks even with strong anisotropies, applicable to environmental applications.

Artificial neural network-based one-equation model for simulation of laminar-turbulent transitional flow

A mapping function between the Reynolds-averaged Navier-Stokes mean flow variables and transition intermittency factor is constructed by fully connected artificial neural network (ANN), which replaces the governing equation of the intermittency factor in transition-predictive Spalart-Allmaras (SA)-γ model. By taking SA-γ model as the benchmark, the present ANN model is trained at two airfoils with various angles of attack, Mach numbers and Reynolds numbers, and tested with unseen airfoils in different flow states. The a posteriori tests manifest that the mean pressure coefficient, skin friction coefficient, size of laminar separation bubble, mean streamwise velocity, Reynolds shear stress and lift/drag/moment coefficient from the present two-way coupling ANN model almost coincide with those from the benchmark SA-γ model. Furthermore, the ANN model proves to exhibit a higher calculation efficiency and better convergence quality than traditional SA-γ model.

A New Random Positioning Machine Modification Applied for Microgravity Simulation in Laboratory Experiments with Rats

The study presents a newly constructed modification of a random positioning machine (RPM) used in 3D-clinostat and in random mode. The main purpose is to provide an RPM animal model that uses up to four experimental animals simultaneously. In order to validate our RPM, the gravity dispersion and its magnitude are compared with the ones of a traditional machine. The results showed no crucial deviations in gravity dispersion and its time-averaged value in all sets of parameters. Furthermore, a posteriori stress tests are conducted on three Wistar male rats groups in order to estimate the level of stress from the setup. The social trait results suggest that the group exposed to our device has no increase in anxiety.

Constant-coefficient spatial gradient models for the sub-grid scale closure in large-eddy simulation of turbulence

Constant-coefficient spatial gradient models (SGMs) are proposed for the sub-grid scale (SGS) closure in large-eddy simulation (LES) of turbulence. The model coefficients are determined either by expanding the neighboring first-order gradients using the local higher-order gradient or by directly discretizing the local higher-order gradients using first-order values among spatial stencil locations. The a priori tests show that the SGM model can have a correlation coefficient larger than 0.97, which is close to the machine-learning based model. In the a posteriori tests, the LESs with different SGS models are performed for the forced incompressible homogeneous isotropic turbulence (HIT) and weakly compressible turbulent mixing layer (TML). The performance of the SGM model is comprehensively examined through the prediction of the flow statistics including the velocity spectrum, the probability density functions of the strain rate, and velocity increments. The evolution of turbulent kinetic energy, the instantaneous structures of the vorticity field, and the Q-criterion are also examined to evaluate the spatial temporal performances of the LES. The predictions of the SGM model are consistently more satisfying compared to the traditional models, including the dynamic Smagorinsky model, the dynamic mixed model, and implicit-LES (ILES) while its computational cost is similar to traditional models. For the weakly compressible TML, most LESs perform better when the length scale of the initial perturbation field is larger than the filter width, providing a useful guidance for LES of turbulent mixing layers.

Optimal Clipping of Structural Subgrid Stress Closures for Large-Eddy Simulation

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 paper, we propose an alternative strategy that removes model backscatter from model predictions through the solution of a constrained minimization problem. This procedure, which we refer to as optimal clipping, results in a parameter-free mixed model, and it yields predicted subgrid stresses in higher correlation with the exact subgrid stresses as compared with those attained with the traditional clipping procedure. We perform a series of a priori and a posteriori tests to investigate the impact of applying the traditional and optimal clipping procedures to Clark’s gradient subgrid stress model, and we observe that optimal clipping leads to a significant improvement in model predictions as compared to the traditional clipping procedure.

Invariant data-driven subgrid stress modeling in the strain-rate eigenframe for large eddy simulation

We present a new approach for constructing data-driven subgrid stress models for large eddy simulation of turbulent flows. The key to our approach is representation of model input and output tensors in the filtered strain rate eigenframe. Provided inputs and outputs are selected and non-dimensionalized in a suitable manner, this yields a model form that is symmetric, Galilean invariant, rotationally invariant, reflectionally invariant, and unit invariant. We use this model form to train a simple and efficient neural network model using only one time step of filtered direct numerical simulation data from a forced homogeneous isotropic turbulence simulation. We demonstrate the accuracy of this model as well as the model’s ability to generalize to previously unseen filter widths, Reynolds numbers, and flow physics using a priori and a posteriori tests.

An approximate inertial manifold (AIM) based closure for turbulent flows

A closure model for turbulent flows is developed based on a dynamical system theory. An appropriately discretized formulation of the governing equations is considered for this process. The key ingredient is an approximation of the system’s attractor, where all the trajectories in phase space are confined. This approximate inertial manifold based approach provides a path to track trajectories of the system in a lower-dimensional subspace. Unlike conventional coarse-graining approaches, the turbulent field is decomposed into resolved and unresolved dynamics using the properties of the governing equations. The novelty of the approach relies on the reconstruction of the unresolved field constrained by the governing equations. A posteriori tests for homogeneous isotropic turbulence and the Kuramoto–Sivashinsky equation show promising results for considerable dimension reduction with strong convergence properties. The proposed model outperforms the dynamic Smagorinsky model, and the computational overhead is competitive with similar approaches.

A mixing timescale model for differential mixing in transported probability density function simulations of turbulent non-premixed flames

The modeling of scalar mixing timescale remains a primary challenge in the transported probability density function (TPDF) method. The variation of scalar mixing timescale among species, i.e., differential mixing, results from the difference in molecular diffusivity and reaction-induced scalar gradient. Nevertheless, the vast majority of TPDF studies on turbulent non-premixed flames simply apply a single mixing timescale determined by the mixture fraction. In this work, a reaction-induced differential mixing timescale (RIDM) model for the mixing timescale of individual species in turbulent non-premixed flames is proposed. The key idea of the RIDM model is to approximate the relative magnitude of the species dissipation rates by using their values in laminar flamelets. A direct numerical simulation dataset of a temporally evolving non-premixed ethylene flame is employed to thoroughly evaluate the model performance via a priori and a posteriori tests. Results show that specifying a single mixing timescale for all species results in a poor prediction of the species dissipation rate and thus the failure to predict the overall combustion process. By accounting for the difference in molecular diffusivity, a slightly better prediction can be obtained, but the improvement is very limited, illustrating that simply modeling the difference due to molecular diffusivities for differential mixing is not sufficient. In comparison, the RIDM model exhibits superior performance in both a priori and a posteriori tests. Moreover, all the components of the RIDM model are readily available in the TPDF method, making the RIDM model a promising candidate employed in practice.