Session R10: Computational Fluid Dynamics: Uncertainty Quantification (5:00pm - 5:45pm CST)

Two-Stage Ensemble Kalman Filter Approach to Estimate Fracture Parameters in Sub-Surface Formations

The prime uncertainty in reservoir simulations lies in the permeability field, as only few measurements of reservoir properties are available. When fractures are present, their locations, orientations and sizes greatly influence the resulting flow field. Therefore, it is important to estimate these parameters as precisely as possible. Ensemble Kalman filters (EnKF) are widely used for data assimilation in the context of sub-surface flows in order to estimate parameters, reduce uncertainty and to improve simulation results. In this work, we consider large individual fractures with known orientation, which appear one after the other. We assume that location, length and hydraulic aperture of each fracture are uncertain and that we have prior probabilistic knowledge of those uncertain parameters, e.g. from seismic data. We reduce the uncertainty of the fracture parameters with an Ensemble Kalman filter using empirical information from measurements; here from a reference simulation. A two-stage data assimilation approach is devised. In a first stage, during fracture formation, pressure and flow at in- and outlet are used as measurements. In a second stage, once all fractures are created, a tracer is injected at the inlet and its concentrations at the outlets are used as measurements.   

Physics-constrained multi-fidelity convolutional neural networks for surrogate fluid modeling

Deep neural networks (DNN) have attracted increasing attention in surrogate modeling of fluid dynamics due to their strong expressivity and fast online inference speeds. In general, the performance of DNN-based models largely relies on a representative training set of high-fidelity (HF) data (from high-fidelity simulations or experiments), which, however, are too expensive to obtain sufficiently. On the other hand, low-fidelity (LF) data, although less accurate, often can be produced in a large amount with low costs. In this work, we develop a physics-informed multi-fidelity transfer learning strategy that leverages both HF and LF labeled information to effectively parameterize solutions of fluid flows in high dimensional parameter space. Moreover, PDE-based physics knowledge is incorporated into the training process to enforce learned solutions to conform to physical laws. The effectiveness of the proposed method is demonstrated on several canonical problems with transport phenomena, governed by classic PDEs, e.g., convection-diffusion equations and Navier-Stokes equations.   

Uncertainty Quantification in Complex Flows for Aeronautical {\&} Mechanical Engineering Applications

The classic trend in industrial activities involving fluid mechanics was to perform a decent amount of experimental tests even for prototyping, with few computational works involved. However, nowadays the situation is reversed, and Computational Fluid Dynamics (CFD) are an undeniable essential tool for designs in industry and academia. Whilst experimentalists normally provide uncertainty estimates, the vast majority of computational analysts omit any measure of uncertainty when providing results. To neglect the impact of uncertainty can be misleading and, under certain circumstances, a wrong approach. The effect of uncertainty can be actually exacerbated when several sources of inaccuracy (aleatoric and epistemic) are jointly analysed. This presentation intends to show and demonstrate the importance of uncertainty in CFD simulations of complex flows. Since a classical Monte-Carlo approach directly on the CFD simulations is usually unaffordable, other lower-cost options are applied to these problems. As relevant conclusion, it will be shown that despite a type of uncertainty may be influential for a certain region of the flow, another area may be insensitive. Additionally, the importance of amalgamating both aleatoric and epistemic uncertainty will be outlined.   

Influence of Reynolds Number and Flow Configuration on Turbulence Model Form Errors

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. In this work, an ``implied models'' approach is used to better understand how the sources and dynamics of turbulence model form errors for the Reynolds stresses vary with Reynolds number and flow configuration. In the ``implied models'' approach, a transport equation for the Reynolds stress model error is derived by taking the difference between the exact Reynolds stress transport equation and the transport equation implied by the Boussinesq hypothesis. Budgets of the model error transport equation are analyzed to determine how well the Boussinesq hypothesis captures the underlying physics and the contribution of error cancellation that may benefit a model's apparent predictiveness even if fundamentally physically inaccurate. Both turbulent channel flows (at various friction Reynolds numbers) and turbulent free-shear flows are considered.   

Quantifying the uncertainties of density self-correlation in RANS simulations for variable density flows

Variable density turbulent flows play important roles, both in natural phenomena such as supernova explosions and in industrial applications such as Inertial Confinement Fusion. Turbulence models for less expensive numerical simulation of variable density flows are therefore desirable. In second-moment turbulence closure models, e.g. those from the BHR model family, the density self-correlation, b, is very important for the production in the mass flux equation in variable density flows. From the conclusions of direct numerical simulations [Livescu et al. 2009,JoT] and experimental investigation [Tomkins et al. 2013,JFM], the production in the b evolution equation has significant value to drive the mixing initially, before it is balanced by the dissipation term when the flow is well mixed. These physics lead to significant errors and uncertainties in turbulence model predictions for such flows. In this study, we outline the framework to quantify the uncertainties from b in the BHR model. Several flows, including Rayleigh-Taylor mixing in a tilted rocket rig and variable density turbulent jets, are employed as test cases. These cases show that perturbations of b can be used to estimate the uncertainty for predictions of turbulent variable density flows.   

Epistemic uncertainty quantification of Reynolds stress models

Reynolds stress models (RSMs) account for the anisotropy of Reynolds stresses by solving individual transport equations for the terms in the Reynolds stress tensor. The additional equations and the many terms in these equations present a daunting task for modeling. In order to determine which terms are important for the modeling of a shear layer, we quantify the model form uncertainty of the SSG/LRR full Reynolds stress model. Specifically, we perturb the terms in the SSG/LRR model at conditions within a relevant parameter space (of, e.g., the Reynolds number and the Richardson number). Operationally, we employ Morris's one at a time method; but rather than sampling the parameter space randomly, we use more advanced sampling strategies like the minimax, the maximum-entropy and Latin hypercube. Each term in the SSG/LRR model is evaluated in terms of its overall effect on an objective function and whether that effect is consistent in the flow's parameter space.   

Evaluation of a polynomial-chaos based multi-fidelity simulation framework for predicting wind pressure loads on buildings.

Wind-resistant design of buildings and their components plays an important role to reduce losses, fatalities, and business discontinuities. Large-eddy simulations (LES) provide a powerful tool to calculate wind loads on buildings, but the computational cost remains too high for widespread use during the building design process. The objective of this study is to investigate if multi-fidelity simulation techniques could be used to reduce the computational cost while maintaining the high accuracy required for design. The approach considers the wind direction as a random parameter, and constructs high-order polynomial chaos expansions (PCEs) for the mean pressure coefficients on the building as a function of the wind direction based on 15 Reynolds-averaged Navier-Stokes (RANS) simulations. To improve the accuracy of these PCE surrogate models, a subset of 3 LES is performed to construct low-order PCEs for the discrepancy between the RANS and LES predictions for the pressure coefficients. The sum of both PCEs was shown to provide surrogate models for the pressure coefficients that have a similar accuracy as high-order PCEs constructed using 15 LES, while reducing the computational cost by 60{\%}.   

Machine-learning quasilinear Gaussian moment closures for uncertainty quantification of turbulent fluid flows

We present a novel nonlocal closure scheme for turbulent systems that utilizes deep recurrent neural networks. The dynamical equations of the turbulent systems of interest consist of a linear part, external forcing (potentially stochastic) and a quadratic and energy preserving term. We develop a closure scheme for the mean field of interest as well as the covariance of its perturbations. Our neural network architecture takes into account the energy-preserving properties of the nonlinear terms allowing for numerical stability during coarse-scale simulations, a feature lost when this constraints are not imposed. Spatial convolutions and time-delays are included in the deep learning network to incorporate nonlocal spatio- temporal information that enhances the accuracy of our predictions. For numerical results we focus on two different systems. A turbulent multiphase current where bubbles that act as passive tracers are being transported by an incompressible fluid, and high latitude turbulent quasi-geostrophic flows excited be surface wind forcing.   

Exploring turbulence model uncertainties in turbomachinery applications

The aerodynamic design of modern jet engines is highly affected by the prediction capabilities of the CFD-solver close to the operation limits. Since Reynolds-averaged Navier-Stokes (RANS) simulations are the workhorse in industrial development of jet engines, the accuracy of turbulence closure models is one of the main limitations in that process striving for future environmental friendly designs. Due to simplifying assumptions during the creation of turbulence models, the prediction accuracy of RANS computations is reduced in the presence of adverse pressure gradient, flow separation and bursting vortices. Consequently, these assumptions lead to a significant degree of epistemic uncertainty. The methodology to quantify these structural uncertainties based on eigenspace perturbations of the Reynolds stress tensor \footnote{Iaccarino et al., \textbf{Phy. Rev. Fluids} Vol.2, No.2, 2017} was implemented in the solver TRACE, being developed by DLR’s Institute of Propulsion Technology. In this investigation, we apply this implementation to turbulent flow cases pertinent to turbomachinery applications. Across these case studies, including DLR's 3D diffuser, the uncertainty bounds on such test case with respect to the closure model are presented in comparison with experimemental data.   

Full-scale validation of natural ventilation models using uncertainty quantification

Natural ventilation can significantly reduce building energy consumption, but the flow rates and corresponding cooling depend on highly variable weather and building operating conditions. To support robust design of natural ventilation systems, we need computational models with uncertainty quantification (UQ) that can account for this variability. The objective of the present study is to perform full-scale validation of natural ventilation models in Stanford’s Y2E2 building. Computational fluid dynamics (CFD) and UQ have been used to design the experiment and identify optimal temperature sensor locations under uncertain boundary and initial conditions. The resulting measurements are representative of the volume-averaged temperature, while also characterizing spatial variability in the temperature field.Validation of two models with different levels of fidelity will be considered. First, we will compare the measurements to predictions obtained with the CFD model. Second, we will validate predictions obtained with a fast, low-fidelity building thermal model, and investigate if surrogate models for the flow rates and heat transfer coefficients obtained from CFD can improve the thermal model accuracy.   

Field Sensitivity Analysis for Wind Energy Modeling

Wind energy systems are complex, necessitating turbulence models to approximate the true flow dynamics. However, designs based on inaccurate turbulence model parameters may yield unexpected turbine aerodynamics, extreme structural loads, or suboptimal energy production. A field sensitivity analysis can reveal crucial model parameters that introduce simulation errors leading to these undesirable outcomes. This study demonstrates field sensitivity analysis for a turbulent flow simulation relevant to wind energy --- flow over a NACA 0015 wing at 12 degrees angle of attack and Reynolds number of 1.5 million --- with respect to ten parameters in the 2003 Menter shear-stress transport turbulence model. Sensitivity is quantified using Sobol indices and the mean-squared gradient, which are estimated through polynomial chaos expansion and active subspace models, respectively. Two different sets of most sensitive turbulence model parameters are identified, corresponding to regions near and far from the wing. Simultaneous dimension reduction across several quantities of interest is also explored. This sensitivity analysis and dimension reduction will enable efficient model calibration for future wind energy studies.