Computational Fluid Dynamics (CFD) simulations using turbulence models are commonly used in engineering design. Of the different turbulence modeling approaches that are available, eddy viscosity based models are the most common for their computational economy. Eddy viscosity based models utilize many simplifications for this economy such as the gradient diffusion and the isotropic eddy viscosity hypotheses. These simplifications limit the degree to which eddy viscosity models can replicate turbulence physics and lead to model form uncertainty. The Eigenspace Perturbation Method (EPM) has been developed for purely physics based estimates of this model form uncertainty in turbulence model predictions. Due to its physics based nature, the EPM weighs all physically possible outcomes equally leading to overly conservative uncertainty estimates in many cases. In this investigation we use data driven Machine Learning (ML) approaches to address this limitation. Using ML models, we can weigh the physically possible outcomes by their likelihood leading to better calibration of the uncertainty estimates. Specifically, we use ML models to predict the degree of perturbations in the EPM over the flow domain. This work focuses on a Convolutional Neural Network (CNN) based model to learn the discrepancy between Reynolds Averaged Navier Stokes (RANS) and Direct Numerical Simulation (DNS) predictions. This model acts as a marker function, modulating the degree of perturbations in the EPM. We show that this physics constrained machine learning framework performs better than the purely physics or purely ML alternatives, and leads to less conservative uncertainty bounds with improved calibration.
Calibrating a Reynolds-averaged Navier-Stokes (RANS) model against data leads to an improvement. Determining {\it a priori} if such an improvement generalizes to flows outside the calibration data is an outstanding challenge. This work attempts to address this challenge via global epistemic Uncertainty Quantification (UQ). Unlike the available epistemic UQ methods that are local and tell us a model's uncertainty at one specific flow condition, the global epistemic UQ method presented in this work tells us also whether a perturbation of the original model would generalize. Specifically, the glo…
White paper: The aim of this work is to apply and analyze machine learning methods for uncertainty quantification of turbulence models. In this work we investigate the classical and data-driven variants of the eigenspace perturbation method. This methodology is designed to estimate the uncertainties related to the shape of the modeled Reynolds stress tensor in the Navier-Stokes equations for Computational Fluid Dynamics (CFD). The underlying methodology is extended by adding a data-driven, physics-constrained machine learning approach in order to predict local perturbations of the Reynolds str…
Reynolds-averaged Navier-Stokes (RANS)-based transition modeling is widely used in aerospace applications but suffers inaccuracies due to the Boussinesq turbulent viscosity hypothesis. The eigenspace perturbation method can estimate the accuracy of a RANS model by injecting perturbations to its predicted Reynolds stresses. However, there lacks a reliable method for choosing the strength of the injected perturbation, while existing machine learning models are often complex and data craving. We examined two light-weighted machine learning models to help select the strength of the injected pertur…
In order to achieve a more virtual design and certification process of jet engines in aviation industry, the uncertainty bounds for computational fluid dynamics have to be known. This work shows the application of a machine learning methodology to quantify the epistemic uncertainties of turbulence models. The underlying method in order to estimate the uncertainty bounds is based on an eigenspace perturbation of the Reynolds stress tensor in combination with random forests.
For the purpose of Uncertainty Quantification (UQ) of Reynolds-Averaged Navier-Stokes closures, we introduce a framework in which perturbations in the eigenvalues of the anisotropy tensor are made in order to bound a Quantity-of-Interest based on limiting states of turbulence. To make the perturbations representative of local flow features, we introduce two additional transport equations for linear combinations of these aforementioned eigenvalues. The location, magnitude and direction of the eigenvalue perturbations are now governed by the model transport equations. The general behavior of our…
This document serves as the manual for using the EQUiPS (Enabling Quantification of Uncertainty in Physics Simulations) module in SU2. The EQUiPS module uses the Eigenspace Perturbation methodology to provide interval bounds on Quantities of Interest (QoIs) that capture epistemic uncertainties arising from assumptions made in RANS turbulence models. This has been implemented and tested in SU2 for a variety of benchmark turbulence cases as well as flows of aerodynamic interest.
The Bayesian uncertainty quantification technique has become well established in turbulence modeling over the past few years. However, it is computationally expensive to construct a globally accurate surrogate model for Bayesian inference in a high-dimensional design space, which limits uncertainty quantification for complex flow configurations. Borrowing ideas from stratified sampling and inherited sampling, an adaptive model refinement approach is proposed in this work, which concentrates on asymptotically improving the local accuracy of the surrogate model in the high-posterior-density regi…
Reynolds-averaged Navier-Stokes (RANS) is one of the most cost-efficient approaches to simulate wind-farm-atmosphere interactions. However, the applicability of RANS-based methods is always limited by the accuracy of turbulence closure models, which introduce various uncertainties into the models. In this study, we estimate model-form uncertainties in RANS simulations of wind farms. For this purpose, we compare different RANS models to a large-eddy simulation (LES). We find that the realizable k-epsilon model is a representative RANS model for predicting the mean velocity, the turbulence inten…
Turbulence consists of interacting flow structures covering a wide range of length and time scales. A long-standing question looms over pair diffusion of particles in close proximity i.e. particle pair diffusion at small separations: what range of turbulence length scales governs pair diffusion? Here, we attempt to answer this question by addressing pair diffusion by both fine scales and larger scale coherent structures in which the fine scales are embedded - we unavoidably encounter a combination of both local and non-local interactions associated with the small and large length scales. The l…
Data-driven turbulence modelling approaches are gaining increasing interest from the CFD community. However, the introduction of a machine learning (ML) model introduces a new source of uncertainty, the ML model itself. Quantification of this uncertainty is essential since the predictive capability of a data-driven model diminishes when predicting physics not seen during training. In this work, we explore the suitability of Mondrian forests (MF's) for data-driven turbulence modelling. MF's are claimed to possess many of the advantages of the commonly used random forest (RF) machine learning al…
The present study revisits the amplitude modulation phenomenon, specifically for the robustness in its quantification. To achieve this, a well-resolved large-eddy simulation (LES) data set at Re_theta= 8200 is used. First, the fluctuating streamwise velocity signal is decomposed into its small- and large-scale components using both Fourier filters and empirical mode decomposition (EMD), allowing the comparison among different separation filters. The effects of these filters on various definitions for quantifying the amplitude modulation have been discussed. False positive identification of the…
With the advent of improved computational resources, aerospace design has testing-based process to a simulation-driven procedure, wherein uncertainties in design and operating conditions are explicitly accounted for in the design under uncertainty methodology. A key source of such uncertainties in design are the closure models used to account for fluid turbulence. In spite of their importance, no reliable and extensively tested modules are available to estimate this epistemic uncertainty. In this article, we outline the EQUiPS uncertainty estimation module developed for the SU2 CFD suite that …
The expected growth in wind energy capacity requires efficient and accurate models for wind farm layout optimization, control, and annual energy predictions. Although analytical wake models are widely used for these applications, several model components must be better understood to improve their accuracy. To this end, we propose a Bayesian uncertainty quantification framework for physics-guided data-driven model enhancement. The framework incorporates turbulence-related aleatoric uncertainty in historical wind farm data, epistemic uncertainty in the empirical parameters, and systematic uncert…
Computational fluid dynamics using the Reynolds-averaged Navier-Stokes (RANS) remains the most cost-effective approach to study wake flows and power losses in wind farms. The underlying assumptions associated with turbulence closures are one of the biggest sources of errors and uncertainties in the model predictions. This work aims to quantify model-form uncertainties in RANS simulations of wind farms at high Reynolds numbers under neutrally stratified conditions by perturbing the Reynolds stress tensor through a data-driven machine-learning technique. To this end, a two-step feature-selection…
A statistical survey of spectral anisotropy of space plasma turbulence is performed using five years measurements from MMS in the magnetosheath. By measuring the five-point second-order structure functions of the magnetic field, we have for the first time quantified the three-dimensional anisotropies and scalings at sub-ion-scales ($<$ 100 km). In the local reference frame $(\hat L_{\perp}, \hat l_{\perp}, \hat l_{\parallel})$ defined with respect to local mean magnetic field $\boldsymbol {B}_0$ (Chen et al. 2012), the "statistical eddies" are found to be mostly elongated along $\boldsymbol {B…
A Priori Uncertainty Quantification of Reacting Turbulence Closure Models using Bayesian Neural Networks
Oden Institute for Computational Engineering and Sciences, The University of Texas at Austin（德克萨斯大学奥斯汀分校奥登计算工程与科学学院）；Computational Science Center, National Renewable Energy Laboratory（美国国家可再生能源实验室计算科学中心）
Data-driven methods for improving turbulence modeling in Reynolds-Averaged Navier-Stokes (RANS) simulations have gained significant interest in the computational fluid dynamics community. Modern machine learning algorithms have opened up a new area of black-box turbulence models allowing for the tuning of RANS simulations to increase their predictive accuracy. While several data-driven turbulence models have been reported, the quantification of the uncertainties introduced has mostly been neglected. Uncertainty quantification for such data-driven models is essential since their predictive capa…
