Session T02: Computational Fluid Dynamics: Uncertainty Quantification

A multi-fidelity approach to sensitivity estimation in large eddy simulation

An approach to compute approximate sensitivities in a large eddy simulation (LES) is proposed and assessed. The method is based on solving a linearized

mean equation, where the mean equation is defined based on the LES solution. This requires closure modeling which makes the computed sensitivities

approximate; the closure modeling is based on computing the inferred eddy viscosity from the LES data. The method is assessed for the flow over a

NACA0012 airfoil at different angles of attack, with the Reynolds number as the varying parameter and the lift and drag coefficients as the quantities-

of-interest. The results show the importance of accurate closure modeling, specifically that treating the eddy viscosity as “frozen” is insufficiently accurate.   

Grid Tailored Reduced-Order Models for Steady Hypersonic Aerodynamics

High-speed aerospace engineering applications rely heavily on computational fluid dynamics (CFD) models for design and analysis due to the expense and difficulty of flight tests and experiments. This reliance on CFD necessitates performing accurate and reliable uncertainty quantification (UQ). However, CFD for hypersonic flows is very computationally expensive due to high grid resolution requirements. Additionally, UQ approaches are “many-query” problems requiring many runs with a wide range of input parameters.

One way to enable computationally expensive models to be used in such many-query problems is to employ projection-based reduced-order models (ROMs) in lieu of the (high-fidelity) full-order model (FOM). However, the linear basis typically used in these approaches can be poorly suited for shock dominated flows.

This talk presents a grid tailored ROM, which uses snapshots from r-adapted FOMs in which the shock position is located at the same grid indices for all snapshots. Grid tailoring improves the ability of the ROM to significantly reduce computational costs while maintaining high levels of accuracy in any quantities of interest. Applications including the HIFiRE-1 flight vehicle will be presented.   

Uncertainty quantification and extreme event analysis for turbulent flows using energy-preserving data-driven closure schemes

We present a reduced-order modelling scheme for computing the statistics of incompressible turbulent flows. The scheme involves evolution equations for the mean and covariance of the system, in tandem with a neural-network modelling of the higher-order closure terms. The closure terms are approximated using spatio-temporally nonlocal neural-networks with appropriate constraints enforced during training. These constraints allow the reduced-order model to adhere to the physical properties of the reference system. In more detail, energy preservation constraints ensure that the energy transfer between scales, as modelled by the nonlinear terms of the full system, are properly included in our model. The constraints are essential for our approximate model to reach the appropriate statistical equilibrium that corresponds to the full reference system. We showcase that our computationally tractable model is able to robustly approximate effectively complicated and expensive turbulent flows in the ocean and atmosphere. We test the method in its ability to approximate important statistics of these systems such as its mean and energy spectrum. We also approximate the probability density functions of quantities of interest, showing that the model can estimate the probability of intermittent extreme events occuring. Turbulent flows with Gaussian and strongly non-Gaussian statistics are examined.   

Simulating natural ventilation in buildings using CFD: Importance of thermal boundary conditions

For CFD simulations of natural ventilation in buildings, far-field thermal boundary conditions (BC) are often taken from nearby weather stations, while indoor surfaces are often assumed adiabatic. This study aims to quantify the errors that can be introduced by these assumptions. LES is used to simulate buoyancy-driven ventilation in Stanford Y2E2 Building during a 3-hour night flush with an initial indoor-outdoor temperature difference of 4.0 K. Measured temperatures are used as BC and benchmark to quantify root mean square errors (RMSE) of LES results. The lowest RMSE of 0.36 K is achieved by prescribing the outdoor temperature measured on the building roof as the far-field temperature BC and prescribing the measured surface temperature as the thermal BC for the interior surfaces. Using the outdoor temperature from a weather station 1 km away as the far-field temperature BC increases the RMSE to 1.13 K. For the internal surfaces, the use of an adiabatic BC increases the RMSE to 1.73 K, showing that interior surfaces should not be assumed adiabatic. For prediction of natural ventilation using CFD, we suggest using outdoor temperature measured near a target building as the far-field thermal BC and measured or modeled surface temperatures as the thermal BC for internal surfaces.   

Data-driven Eigenspace Perturbations for RANS Uncertainty Quantification

Turbulence models are indispensable yet limited in their accuracy and therefore a significant source of model-form uncertainty in turbulent flow simulations. Data-free perturbations to the spectral decomposition of the Reynolds stress tensor can be used to obtain uncertainty envelopes for quantities of interest. Such data-free perturbations perturb at spatially uniform strength regardless of the expected local inaccuracy of the turbulence model. Since turbulence models work better in some situations than in others, this uniform approach tends to lead to an overestimation of the model-form uncertainty.

Non-uniform data-driven perturbations are able to account for a spatially varying degree of inaccuracy in the turbulence model predictions. A machine learning model trained on test cases using RANS and high-fidelity data is used to predict a local perturbation strength based on mean flow features. 

These data-driven perturbations are shown to give envelopes that are more characteristic of the true uncertainty in the predictions. Besides, a data-driven best estimate is presented to complement the uncertainty envelopes.

Generalization of this approach is studied over several training and test cases.   

Turbulence Model Form Errors in a Statistically Stationary Separation Bubble

Model form error arises from physical assumptions made in constructing models either to reduce the physical complexity or to model physical processes that are not well understood. In turbulence modeling, specifically Reynolds stress modeling, model form errors result from the Boussinesq hypothesis and other modeling choices, such as the specific form of the eddy viscosity. These linear eddy viscosity models have known points of failure in flows that introduce complex strain, such as separated flows. In this work, an "implied models" approach is used to better understand the sources and dynamics of model form error in separated flows. In the "implied models" approach, a transport equation is derived for the model error through the transport equation implied by the model for the quantity of interest. A boundary layer over a flat plate with a statistically stationary separation bubble is analyzed and shown to have two error modes corresponding to the qualitative behavior of turbulent wall-bounded and turbulent free-shear model form errors, which have previously been analyzed. These results indicate a complex picture of model error that changes through the flow but also that calibration of turbulence models against simpler canonical flows may capture the main modes of model failure.   

Exploring Machine Learning Strategies for RANS Uncertainty Quantification

This work builds the understanding of the critical machine learning considerations when predicting the uncertainty bounds for a Reynolds Averaged Naiver-Stokes (RANS) model. These bounds are created with eigenvalue perturbations to the Reynolds stress anisotropy tensor, predicted by a random forest trained on high-fidelity data that exhibits similar flow characteristics to those being studied with the low-fidelity approach. The first part of this work explores the hyperparameters of the random forest, the utilization of weak learners, and the optimal strategy for training applied to an asymmetric diffuser. This exploration provided insight into the random forests' sensitivity to different variables and dataset divisions, and the overall accuracy of eigenvalue perturbations. These insights led to restructuring the machine learning approach to two separate objectives summarized as a classification and regression problem. We seek to classify different areas of the flow field into universal regimes and then determine how much to perturb the corresponding fields from the classified insights and other flow characteristics. The second part of this work introduces the classification aspect with primary results applied to an asymmetric diffuser, jet-data, and other compressible flows.   

Potential flows: a playground for non-local and nonlinear inference problems

An accurate estimation of the flow field from limited and noisy observations is crucial in many fields of engineering. To tackle these inverse problems, classical localization schemes suppress correlations at long distances. However, these techniques are not well suited for incompressible fluid problems, in which the observations are typically obtained by non-local and nonlinear mappings of the state, e.g. the pressure Poisson equation. Instead these inference problems have low-rank structure in the sense that low-dimensional projections of the observations are most informative of a low-dimensional subspace of the state space. Thus, interactions between state and observation variables are best described as clusters of variables, reminiscent of the fast multipole method (FMM). To drive further research in this area, we show that potential flow problems constitute a nice playground to experiment with  non-local and nonlinear inference problems, while allievating the high-dimensionality of a standard incompressible fluid problem. Indeed, estimating the position and strength of a handful of singularities (e.g. vortices, sources) from pressure observations distills the main features of a realistic inverse problem. To exploit the low-rank structure, we present a systematic procedure to identify these clusters of variables from the sensitivity analysis of the observation operator. We also explore connections between the proposed sensitivity analysis and the FMM. Finally, we present recent advances in nonlinear prior-to-posterior transformations to perform consistent inference with nonlinear state-space models.   

A computationally affordable multi-fidelity approach to parametric studies

The focus of this work is to come up with a computationally affordable way to perform approximate parametric studies of high-fidelity (LES/DNS) data. The method relies on defining a linerized RANS problem around the LES/DNS mean flow fields. The method is assessed on a shock/boundary-layer interaction problem where the shock strength and wall temperature are varied.   

Multiphase flow applications of non-intrusive reduced-order models with Gaussian process emulation

Reduced-order models are computationally inexpensive simplifications of high-fidelity complex models. Such models can be found in computational fluid dynamics, e.g. in multiphase flow applications. We present a reduced-order model analysis framework where we couple compression techniques such as autoencoders with Gaussian process regression, and its natural extension: the Deep Gaussian Process (DGP) in the latent space. The mixture has some significant advantages as opposed to the standard encoding-decoding routine, primarily, the ability to interpolate or extrapolate in the initial conditions, which can offer results even when simulations are unavailable. We compare the methodology with alternative interpolation algorithms (long short-term memories) using exemplars from multiphase flow applications, and we examine how each variation affects our analysis.   

Hierarchical multifidelity models for the simulation of turbulent flows

Numerical simulations of turbulent flows are of utmost importance in physics and engineering. In practice, for a given computational budget an efficient strategy is required to address the so-called "outer-loop problem", such as flow prediction, uncertainty quantification and optimization, which typically demand multiple simulations. The present study reports our recent progress on further development and implementation of a class of hierarchical multifidelity models (MFMs) which allow for simultaneous calibration of uncertain parameters. In a Bayesian framework, at each fidelity level both model inadequacy and aleatoric uncertainties are considered. The developed MFM is applied to different flow problems as examples: i) prediction of wing polars for a standard airfoil where the angle of attack is the design parameter, and, ii) propagation of geometrical uncertainties into the quantities of interest of the turbulent flow over a periodic hill. In all problems, the predictions by the MFM are validated and the associated uncertainty is determined. The hierarchical MFM is shown to be compatible with the hierarchy of the turbulence modeling approaches and also capable of accurately handling the outer-loop problems with minimal computational cost.   

Reliable quantification of uncertainty in time averages of turbulence simulations

Estimation of uncertainties in time-averaged quantities of turbulence simulations requires statistical tools that take into account the temporal correlation within the data. This is the key result in our validation of the commonly used uncertainty estimators for turbulence time averages i.e., batch-means methods and autoregressive model (ARM) based estimators. The validation enabled us to derive novel guidelines for the choice of the hyperparameters intrinsic to these methods, such as the batch size in the batch-means methods and the order of the ARM model, in terms of turbulence timescales. Our study also revealed that the ARM estimator could be made to retain information through the autocorrelation exactly up to a given point in time by the combination of the model order and covered time lag. This property preserved the accuracy of the estimator even upon downsampling or batching the time series, thereby allowing computationally efficient implementations. We also address the need for including expected values of nonlinear components in the uncertainty quantification (UQ) of higher-order statistics, an aspect often neglected in the UQ/turbulence literature. The UQ analyses are performed using the time series of turbulent channel and periodic hill flow.