Linear Eddy Viscosity Models Research Articles

Numerical analyses of water swirls in aquaculture tanks

Water swirls were observed in both model and prototype aquaculture tanks with bottom drains. This phenomenon was seldom replicated in studies of flow fields of water in aquaculture tanks with computational fluid dynamics. A commonality of these studies is that the linear eddy-viscosity turbulence models were employed. This study delves into the water swirls from the aspect of the generation of the turbulence kinetic energy; both the linear and nonlinear eddy-viscosity turbulence models are used, as well as a Reynolds stress turbulence model. A novel identification method is utilized to pinpoint vortex structures. Numerical results affirm the outstanding performance of the Reynolds stress model to estimate the flow fields of water in aquaculture tanks. Both the linear and nonlinear eddy-viscosity models overestimate the kinetic turbulence energy in the stagnation zone of swirls. The overestimation can be restrained by including the effect of the streamline curvature correction (SCC) to the turbulence production term. Even so, the SCC does not help to improve the estimating accuracy between the Reynolds stress tensor and the mean rate-of-strain tensor. While the nonlinear model offers some improvements, the Reynolds stress model provides a more effectvie solution. After that, we calculate the flow fields of water in aquaculture tanks with different inlet–outlet configurations with the Reynolds stress turbulence model. Both the vertical and horizontal scales of water swirls increase with inlet velocities and Froude numbers, where the Froude number is based on the bottom draining velocity and the water depth above the bottom drain. The vertical distributions of the water residence time in aquaculture tanks are shown to improve the design of tanks.

Data-driven turbulence modeling for fluid flow and heat transfer in peripheral subchannels of a rod bundle

This study presents a comparison of the performance of machine learning (ML) techniques, specifically multi-dimensional gene expression programming (MGEP), tensor basis neural network (TBNN), and also proposes a novel universally interpretable machine learning architecture to model the turbulent scalar flux (UIML-s) to enhance turbulence models for fluid flows at different Prandtl numbers in channels with complex shapes of walls in the channel cross section. In particular, peripheral subchannels of rod bundles are of primary interest. However, the accuracy of mean velocity and scalar distributions predicted by commonly used turbulence models still poses a challenge compared to data extracted from high-fidelity eddy-resolving numerical simulations, particularly for engineering applications involving complex geometry flows. In the present study, by utilizing an explicit algebraic expression for the nonlinear Reynolds-stress term obtained through both the evolutionary MGEP optimization and TBNN, the secondary flow structure has been adequately predicted in the cross-wise mean velocity distributions in the square duct and the rectangular channel with three longitudinal rods. This structure is also observed in the data from the concurrent runs performed by direct numerical simulation (DNS) but is completely absent in the results produced by a baseline Reynolds-averaged Navier–Stokes (RANS) closure, which employs the linear eddy viscosity model for the Reynolds stress tensor. Comparison of MGEP and TBNN has shown their nearly equal performance in a square duct flow; however, MGEP works better for the more complex geometry channel with three rods. Furthermore, based on the velocity field produced by the RANS-MGEP model, the ML modification of the gradient diffusion hypothesis, integrated into the aforementioned novel RANS-ML model called as UIML-s, significantly improves the mean scalar distributions in a flow with three bumps serving as a prototype for the peripheral subchannel of rod bundle. The normalized root mean squared error decreases from 13.5% to 7.6%, bringing the predicted distributions closer to the DNS data, particularly in the near-wall region. Another approach, MGEP-s, also yields the acceptable results, which are nearly identical to those from UIML-s. These findings highlight the potential of using data-driven calibration of turbulence models with nonlinear closures to enhance the predictability for RANS simulations of fluid flows, heat, and mass transfer in channels with complex geometry.

Improved prediction of turbomachinery flows using Reynolds stress model with [formula omitted] transition model

Transition is a prominent flow phenomenon in turbomachinery, especially at low and moderate Reynolds numbers. The turbomachinery internal flow often encounters complex vortex structures, typically accompanied by strong turbulence anisotropy and non–equilibrium. Conventional Reynolds–averaged Navier–Stokes (RANS) modeling with linear eddy–viscosity models (LEVMs) may fail short in capturing certain complex flow features. The Reynolds stress model (RSM) based on a higher turbulence closure level has more potential to reasonably consider turbulence anisotropy and non–equilibrium for such complex flows within turbomachinery than the LEVMs. However, the current RSM lacks an effective representation of transition phenomena. This paper presents an improved prediction of turbomachinery flows using RSM with transition modeling. The transition effects are incorporated into the RSM (SSG/LRR–ω model) via Menter's γ transport equation. The blending of the SSG/LRR–ω model with the γ transition model (SSG/LRR–ω–γ model) is firstly validated based on the T3 series flat plate cases with zero pressure gradient (T3A, T3B, T3A–) and variable pressure gradients (T3C1–C5). Comparisons with experiments show that the SSG/LRR–ω–γ model predicts corner separation flows with transition more accurately, although there is still room for further improvement. Then, the performance of the SSG/LRR–ω–γ model is assessed in a linear highly loaded prescribed velocity distribution (PVD) cascade and a NACA65 K48 compressor cascade. Comparisons with experiments indicate that, in both compressor cascades, the SSG/LRR–ω–γ model can predict more accurately corner flows with boundary layer transition on suction surfaces than the SSG/LRR–ω and SST model. Nevertheless, it still exhibits a tendency to overestimate the size of corner separation, especially in cases characterized by strong adverse pressure gradients. Comparisons of skin friction and velocity profiles predicted by the SSG/LRR–ω model with and without transition modeling illustrate the rationale behind the improved prediction. Further improvement of the SSG/LRR–ω–γ model for predicting turbomachinery corner flows should be conducted in future.

Progressive augmentation of Reynolds stress tensor models for secondary flow prediction by computational fluid dynamics driven surrogate optimisation

Generalisability and the consistency of the a posteriori results are the most critical points of view regarding data-driven turbulence models. This study presents a progressive improvement of turbulence models using simulation-driven Bayesian optimisation with Kriging surrogates where the optimisation of the models is achieved by a multi-objective approach based on duct flow quantities. We aim for the augmentation of secondary-flow prediction capability in the linear eddy-viscosity model k−ω SST without violating its original performance on canonical cases e.g. channel flow. Progressively data-augmented explicit algebraic Reynolds stress models (PDA-EARSMs) for k−ω SST are obtained enabling the prediction of secondary flows that the standard model fails to predict. The new models are tested on channel flow cases guaranteeing that they preserve the successful performance of the original k−ω SST model. Subsequently, numerical verification is performed for various test cases. Regarding the generalisability of the new models, results of unseen test cases demonstrate a significant improvement in the prediction of secondary flows and streamwise velocity. These results highlight the potential of the progressive approach to enhance the performance of data-driven turbulence models for fluid flow simulation while preserving the robustness and stability of the solver.

Large Eddy Simulation of a Low-Pressure Turbine Cascade with Turbulent End Wall Boundary Layers

We present results of implicit large eddy simulation (LES) and different Reynolds-averaged Navier–Stokes (RANS) models of the MTU 161 low pressure turbine at an exit Reynolds number of 90000\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$90\\,000$$\\end{document} and exit Mach number of 0.6. The LES results are based on a high-order discontinuous Galerkin method and the RANS is computed using a classical finite-volume approach. The paper discusses the steps taken to create realistic inflow boundary conditions in terms of end wall boundary layer thickness and freestream turbulence intensity. This is achieved by tailoring the input distribution of total pressure and temperature, Reynolds stresses and turbulence length scale to a Fourier series based synthetic turbulence generator. With this procedure, excellent agreement with the experiment can be achieved in terms of blade loading at midspan and wake total pressure losses at midspan and over the channel height. Based on the validated setup, we focus on the discussion of secondary flow structures emerging due to the interaction of the incoming boundary layer and the turbine blade and compare the LES to two commonly used RANS models. Since we are able to create consistent setups for both LES and RANS, all discrepancies can be directly attributed to physical modelling problems. We show that both a linear eddy viscosity model and a differential Reynolds stress model coupled with a state-of-the-art correlation-based transition model fail, in this case, to predict the separation induced transition process around midspan. Moreover, their prediction of secondary flow losses leaves room for improvement as shown by a detailed discussion of turbulence kinetic energy and anisotropy fields.

Prediction of urban airflow fields around isolated high-rise buildings using data-driven non-linear correction models

When it comes to predicting urban airflow, steady Reynolds-averaged Navier-Stokes (SRANS) models that rely on Reynolds stress often face a challenge called the closure problem. This problem involves unresolved structural flaws and uncertainties in the closure coefficients used in the models. Previous attempts to recalibrate coefficients for specific urban flows without breaking the linear constitutive relation have resulted in simulation results constrained by the baseline turbulence model. Therefore, this study aims to enhance the performance of SRANS models by addressing these structural flaws. To achieve this, a novel data-driven framework is proposed. It leverages the deterministic symbolic regression algorithm to discover explicit algebraic expressions for a non-linear Reynolds stress correction model. The robustness of the correction model is ensured by maintaining the linear eddy viscosity model for iterative calculations while keeping the non-linear component frozen. The proposed framework is evaluated using three isolated building cases with varying geometric configurations and inflow boundary conditions. Findings demonstrate that computational fluid dynamics (CFD) predictions incorporating the data-driven non-linear correction model consistently align closer to wind tunnel experimental results compared to both standard and non-linear versions of the k-ε turbulence model. This improvement is reflected in reduced reattachment lengths and more accurate mean velocity distributions in the wake of buildings. However, it should be noted that there is a possibility of overpredicting wind velocity in the windward area. This study introduces valuable insights and additional strategies to enhance the prediction accuracy of SRANS models in urban airflow simulations.

Structural uncertainty quantification of Reynolds-Averaged Navier–Stokes closures for various shock-wave/boundary layer interaction flows

Accurate prediction of Shock-Wave/Boundary Layer Interaction (SWBLI) flows has been a persistent challenge for linear eddy viscosity models. A major limitation lies in the isotropic representation of the Reynolds stress, as assumed under the Boussinesq approximation. Recent studies have shown promise in improving the prediction capability for incompressible separation flows by perturbing the Reynolds-stress anisotropy tensor. However, it remains uncertain whether this approach is effective for SWBLI flows, which involve compressibility and discontinuity. To address this issue, this study systematically quantifies the structural uncertainty of the anisotropy for oblique SWBLI flows. The eigenspace perturbation method is applied to perturb the anisotropy tensor predicted by the Menter Shear–Stress Transport (SST) model and reveal the impacts of anisotropy on the prediction of quantities of interest, such as separation and reattachment positions, wall static pressure, skin friction, and heat flux. The results demonstrate the potential and reveal the challenges of eigenspace perturbation in improving the SST model. Furthermore, a detailed analysis of turbulent characteristics is performed to identify the source of uncertainty. The findings indicate that eigenspace perturbation primarily affects turbulent shear stress, while the prediction error of the SST model is more related to turbulent kinetic energy.

Realizability-preserving time-stepping for the differential Reynolds stress turbulence models

In the context of incompressible turbulence modelling, differential Reynolds stress models allow a better representation of turbulence anisotropy than classical linear eddy viscosity models and are less expensive than large eddy simulations in terms of computational cost. They rely on a modelled transport equation for the Reynolds stress tensor, which is by definition a covariance tensor and must remain symmetric positive semi-definite. However, the solution of the modelled and discretized transport equation may not preserve these mathematical properties. In the present article, a methodology to time-split Reynolds stress R__ terms in the transport equation is proposed using a decomposition theorem which proves the realizability of R__ provided realizable initial conditions. Realizability-preserving time-steppings are designed for several differential Reynolds stress models. The numerical schemes are presented in a finite volume context but may be used with others space discretization schemes. The realizability, precision and robustness of the numerical scheme is verified through the anisotropy time evolution in homogeneous shear-flow simulations with various initial conditions.

Physical interpretation of neural network-based nonlinear eddy viscosity models

Neural network-based turbulence modeling has gained significant success in improving turbulence predictions by incorporating high–fidelity data. However, the interpretability of the learned model is often not fully analyzed, which has been one of the main criticisms of neural network-based turbulence modeling. Therefore, it is increasingly demanding to provide physical interpretation of the trained model, which is of significant interest for guiding the development of interpretable and unified turbulence models. The present work aims to interpret the predictive improvement of turbulence flows based on the behavior of the learned model, represented with tensor basis neural networks. The ensemble Kalman method is used for model learning from sparse observation data due to its ease of implementation and high training efficiency. Two cases, i.e., flow over the S809 airfoil and flow in a square duct, are used to demonstrate the physical interpretation of the ensemble-based turbulence modeling. For the flow over the S809 airfoil, our results show that the ensemble Kalman method learns an optimal linear eddy viscosity model, which improves the prediction of the aerodynamic lift by reducing the eddy viscosity in the upstream boundary layer and promoting the early onset of flow separation. For the square duct case, the method provides a nonlinear eddy viscosity model, which predicts well secondary flows by capturing the imbalance of the Reynolds normal stresses. The flexibility of the ensemble-based method is highlighted to capture characteristics of the flow separation and secondary flow by adjusting the nonlinearity of the turbulence model.

Data-driven Reynolds-averaged turbulence modeling with generalizable non-linear correction and uncertainty quantification using Bayesian deep learning

The past few years have witnessed a renewed blossoming of data-driven turbulence models. Quantification of the concomitant modeling uncertainty, however, has mostly been omitted, and the generalization performance of the data-driven models is still facing great challenges when predicting complex flows with different flow physics not seen during training. A robust data-driven Reynolds-averaged turbulence model with uncertainty quantification and non-linear correction is proposed in this work with the Bayesian deep neural network. In this model, the Reynolds stress tensor is decomposed into linear and non-linear parts. The linear part is taken as the usual linear eddy viscosity model while the non-linear counterpart is learned by a Bayesian deep neural network. Independent tensor bases of invariants and tensors constituted by mean strain rate tensor and rotation rate tensor are embedded into the neural network to effectively consider key turbulence features in different flows. The proposed model is well validated through numerical simulations of four canonical flows that significantly deviate in geometrical configurations and/or Reynolds numbers from those in the training data. With the non-linear corrections of embedded invariants and tensors representing key features of turbulence, the proposed model not only improves the predictive capabilities of Reynolds-averaged turbulence models on the same mesh but also has better generalization performance when simulating complex turbulent flows with large scale separation. In addition, this model allows us to quantitatively demonstrate the confidence interval of the predicted flow quantities that are originated from the model itself.

Data-Driven RANS Turbulence Closures for Forced Convection Flow in Reactor Downcomer Geometry

Recent progress in data-driven turbulence modeling has shown its potential to enhance or replace traditional equation-based Reynolds-averaged Navier-Stokes (RANS) turbulence models. This work utilizes invariant neural network (NN) architectures to model Reynolds stresses and turbulent heat fluxes in forced convection flows (when the models can be decoupled). As the considered flow is statistically one dimensional, the invariant NN architecture for the Reynolds stress model reduces to the linear eddy viscosity model. To develop the data-driven models, direct numerical and RANS simulations in vertical planar channel geometry mimicking a part of the reactor downcomer are performed. Different conditions and fluids relevant to advanced reactors (sodium, lead, unitary-Prandtl-number fluid, and molten salt) constitute the training database. The models enabled accurate predictions of velocity and temperature, and compared to the baseline k − τ turbulence model with the simple gradient diffusion hypothesis, do not require tuning of the turbulent Prandtl number. The data-driven framework is implemented in the open-source graphics processing unit–accelerated spectral element solver nekRS and has shown the potential for future developments and consideration of more complex mixed convection flows.

Discovering explicit Reynolds-averaged turbulence closures for turbulent separated flows through deep learning-based symbolic regression with non-linear corrections

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. Specifically, the Reynolds stress tensor is decomposed into linear and non-linear parts. While the linear part is taken as the regular linear eddy viscosity model, a long short-term memory neural network is employed to generate symbolic terms on which tractable mathematical expressions for the non-linear counterpart are built. A novel reinforcement learning algorithm is employed to train the neural network to produce best-fitted symbolic expressions. Within the proposed framework, the Reynolds stress closure is explicitly expressed in algebraic forms, thus allowing for direct functional inference. On the other hand, the Galilean and rotational invariance are craftily respected by constructing the training feature space with independent invariants and tensor basis functions. The performance of the present methodology is validated through numerical simulations of three different canonical flows that deviate in geometrical configurations. The results demonstrate promising accuracy improvements over traditional RANS models, showing the generalization ability of the proposed method. Moreover, with the given explicit model equations, it can be easier to interpret the influence of input features on generated models.

Development of explicit algebraic models of Reynolds stresses for flows in channels using gene expression programming

To build a data-driven approximation for the Reynolds-stress anisotropy (RSA), the symbolic regression method of gene expression programming (GEP) is applied. Two tensor-basis terms from the algebraic expansion for the RSA tensor are used in the GEP algorithm. A new RANS-GEP model is tested in several flows in channels without/with bumps at different physical and geometrical parameters, where DNS data of high fidelity involved as a target for RSA are available. The results of RANS-DNS runs are also obtained where the RSA values in the mean momentum equation are taken directly from DNS to show ability to improve the model performance versus the conventional linear eddy viscosity model (LEVM). Next, the training with carefully selected input features is performed to get an explicit non-linear algebraic model for RSA. The results of RANS-GEP study show potentials of the new tool to improve predictions.

Sparse Bayesian Learning of Explicit Algebraic Reynolds-Stress models for turbulent separated flows

A novel Sparse Bayesian Learning (SBL) framework is introduced for generating stochastic Explicit Algebraic Reynolds Stress (EARSM) closures for the Reynolds-Averaged Navier–Stokes (RANS) equations from high-fidelity data. Building on the recently proposed SpaRTA (Sparse Regression of Turbulent Stress Anisotropy) algorithm of Schmelzer et al. (2020), corrections to an underlying Linear-Eddy-Viscosity Models (LEVM) (namely, the k−ω SST model) are formulated as physically-interpretable, frame-invariant tensor polynomials and built from a redundant dictionary of candidate functions. The SBL-SpaRTA algorithm yields a sparse model structure (which favors interpretability and reduces overfitting), while endowing model coefficients with a measure of uncertainty, namely, posterior probability distributions. The framework is used to learn customized stochastic closure models for three separated flow configurations, characterized by different geometries but similar Reynolds number. The resulting stochastic models are then propagated through a CFD solver for all three configurations by means of probabilistic chaos collocation (Loeven et al., 2007). SBL-SpaRTA predictions of velocity profiles and friction coefficient distributions outperform those of the baseline LEVM, for training as well as for test cases. Furthermore, the prediction uncertainty intervals encompass reasonably well the reference data and tend to become large in regions of large discrepancy between RANS and high-fidelity predictions, thus warning the user about model reliability. Finally, a global sensitivity analysis of the stochastic models is carried out, illustrating the role of the dominant corrective terms and providing insights for future improvement of data-driven turbulence models.

Efficient assimilation of sparse data into RANS-based turbulent flow simulations using a discrete adjoint method

Turbulent flow simulations based on the Reynolds-averaged Navier–Stokes (RANS) equations continue to be the workhorse approach for industrial flow problems. However, due to the inherent averaging and the closure models required for the Reynolds stresses, their accuracy is limited. Experimental data, on the other hand, represents the true flow features but is potentially only available in limited locations or at low resolution. Data assimilation (DA) can be used to combine closure models with such experimental data to obtain accurate simulation results. In this scenario, the result of the DA process can be interpreted as a physics-based interpolation and serve as a basis for further studies of the flow. Moreover, such tuned models may be used in machine learning approaches aiming at a priori closure model enhancement. The main objective of this work is to recover a spatially varying eddy viscosity correction factor from sparsely distributed reference data.We use an efficient in-house data assimilation implementation based on the discrete adjoint method for RANS simulations in OpenFOAM to recover an optimal eddy viscosity field. A gradient optimization procedure is used to minimize a velocity-based cost function with a regularization term. All the partial derivatives in the gradient computation are evaluated with an efficient semi-analytical approach at the cost of approximately one forward solution step.The effectiveness of the proposed approach is demonstrated for a periodic hill test case at Re=10595 with different spatial distributions of the reference data. The results show improvements in the agreement between simulation and reference for a range of reference data configurations and highlight the performance of linear eddy viscosity models.

CFD-driven symbolic identification of algebraic Reynolds-stress models

A CFD-driven deterministic symbolic identification algorithm for learning explicit algebraic Reynolds-stress models (EARSM) from high-fidelity data is developed building on the frozen-training SpaRTA algorithm of [1]. Corrections for the Reynolds stress tensor and the production of transported turbulent quantities of a baseline linear eddy viscosity model (LEVM) are expressed as functions of tensor polynomials selected from a library of candidate functions. The CFD-driven training consists in solving a blackbox optimization problem in which the fitness of candidate EARSM models is evaluated by running RANS simulations. The procedure enables training models against any target quantity of interest, computable as an output of the CFD model. Unlike the frozen-training approach, the proposed methodology is not restricted to data sets for which full fields of high-fidelity data, including second flow order statistics, are available. However, the solution of a high-dimensional expensive blackbox function optimization problem is required. Several steps are then undertaken to reduce the associated computational burden. First, a sensitivity analysis is used to identify the most influential terms and to reduce the dimensionality of the search space. Afterwards, the Constrained Optimization using Response Surface (CORS) algorithm, which approximates the black-box cost function using a response surface constructed from a limited number of CFD solves, is used to find the optimal model parameters. Model discovery and cross-validation is performed for three configurations of 2D turbulent separated flows in channels of variable section using different sets of training data to show the flexibility of the method. The discovered models are then applied to the prediction of an unseen 2D separated flow with higher Reynolds number and different geometry. The predictions of the discovered models for the new case are shown to be not only more accurate than the baseline LEVM, but also of a multi-purpose EARSM model derived from purely physical arguments. The proposed deterministic symbolic identification approach constitutes a promising candidate for building accurate and robust RANS models customized for a given class of flows at moderate computational cost.

Turbulence model assessment and heat transfer phenomena inside a rectangular channel under forced and mixed convection

A very high-temperature gas-cooled reactor (VHTR) is an attractive option for hydrogen production due to its high operating temperature. The reactor cavity cooling system (RCCS), a passive safety system of the VHTR, surrounds the reactor vessel with rectangular riser channels and removes decay heat from it. Under emergency operation conditions such as a decrease in the flow rate of the RCCS, the flow condition inside the RCCS riser can become a mixed convection condition accompanying heat transfer deterioration. To design and operate the VHTR with reliable safety, accurate predictions for various convective heat transfer conditions in RCCS riser channels are required. In this study, by comparing the results of an air flow visualization experiment, the predictability of the turbulence models was assessed for the convective heat transfer and complex flow characteristics near the corner region of the rectangular duct confirmed from the experimental study. In this paper, the heat removal rate through the test section of the visualization experiment was quantified under various convective heat transfer conditions, and local temperature measurement data from the experiment were presented to describe the heat transfer deterioration phenomena inside a rectangular riser in turbulent mixed convection. Linear and nonlinear eddy viscosity models were evaluated by comparing the heat transfer coefficient, local temperature, local flow structure including secondary flow, and turbulence quantities obtained from the experiment. Furthermore, improvements to turbulence models for enhancing the prediction of mixed convection heat transfer in rectangular riser ducts were proposed.

Particle subgrid scale modeling in hybrid RANS/LES of turbulent channel flow at low to moderate Reynolds number

Particle dispersion in a periodic channel is studied using the elliptic relaxation hybrid RANS/LES (ER-HRL) model. This approach employs a four-equation linear eddy viscosity (LEV) model while in Reynolds Averaged Navier-Stokes (RANS) mode near the wall, and switches to the LES Smagorinsky dynamic model in the outer flow region. We perform a systematic analysis of the dispersion of six sets of particles having Stokes numbers St = 0.2, 1, 5, 15, 25, 125 at shear Reynolds numbers of Reτ=150, 590. To account for the effect of the unresolved scales on particle dispersion, a novel subgrid-scale model (SGS) is proposed based on the wall-normal RMS of the velocity transport equation. The ER-HRL model is validated against DNS and LES databases, with a globally good agreement. For higher Reynolds number i.e. Reτ = 590, the model, with a much coarser grid, outperforms the LES subgrid stochastic acceleration (LES-SSAM) approach.

Analysis and extension of the quadratic constitutive relation for RANS methods

AbstractThe quadratic constitutive relation was proposed as an extension of minimal complexity to linear eddy-viscosity models in order to improve mean flow predictions by better estimating turbulent stress distributions. However, the successes of this modification have been relatively modest and are limited to improved calculations of flow along streamwise corners, which are influenced by weak secondary vortices. This paper revisits the quadratic constitutive relation in an attempt to explain its capabilities and deficiencies. The success in streamwise corner flows cannot be entirely explained by significant improvements in turbulent stress estimates in general, but is instead due to better prediction of the particular turbulent stress combinations which appear in the mean streamwise vorticity equation. As a consequence of this investigation, a new formulation of turbulent stress modification is proposed, which appears to better predict the turbulent stress distributions for a variety of flows: channel flow, equilibrium boundary layers, pipe flow, separated boundary layers and square duct flow.

End-to-end differentiable learning of turbulence models from indirect observations

The emerging push of the differentiable programming paradigm in scientific computing is conducive to training deep learning turbulence models using indirect observations. This paper demonstrates the viability of this approach and presents an end-to-end differentiable framework for training deep neural networks to learn eddy viscosity models from indirect observations derived from the velocity and pressure fields. The framework consists of a Reynolds-averaged Navier–Stokes (RANS) solver and a neural-network-represented turbulence model, each accompanied by its derivative computations. For computing the sensitivities of the indirect observations to the Reynolds stress field, we use the continuous adjoint equations for the RANS equations, while the gradient of the neural network is obtained via its built-in automatic differentiation capability. We demonstrate the ability of this approach to learn the true underlying turbulence closure when one exists by training models using synthetic velocity data from linear and nonlinear closures. We also train a linear eddy viscosity model using synthetic velocity measurements from direct numerical simulations of the Navier–Stokes equations for which no true underlying linear closure exists. The trained deep-neural-network turbulence model showed predictive capability on similar flows.