Artificial intelligence and turbulence modeling. Development of data-driven tools for RANS models in Turbomachinery Flows

To numerically simulate a turbulent flow through Computational Fluid Dynamics (CFD) has become a mandatory step in turbomachinery design and performance prediction. CFD users live in a constant compromise between accuracy of results and reasonable computational times. The most high-fidelity approach that is currently affordable is constituted by Large Eddy Simulations (LES), that grants the best representation of the physics of the flows. The tradeoff lies in the difficulty of generating proper inflow conditions, the complex numerical setups and the extreme grid refinements that are required. The latter is additionally worsened by the high Reynolds numbers that are normally involved in turbomachinery flows. Thus, the combination of long computational times with the complexity of the approach seriously impairs the applicability of LES simulations to internal flows. On the other hand, Reynolds Averaged Navier-Stokes approach (RANS), constitute the workhorse in the CFD practice. The less restrictive grid requirements lead to faster computations and an increase in robustness. The stationary assumption provides information on the time-averaged fields, that suffice in most of the cases, especially in performance prediction. Drawback of the RANS approach is constituted by the turbulence closure of the turbulent kinematic viscosity term, i.e. a set of equations or parameters that model the physics of turbulence. Such closure necessarily embeds a source of errors. Popular two-equation closures, e.g. k- or k-, correlates the turbulent viscosity to the transport of the Turbulent Kinetic Energy (TKE) and additional variables. A set of heuristically tuned coefficients are also included to maximize the effectiveness of the model. It is universally accepted that the previous hypothesis provide a fairly good agreement with physics of undisturbed flows, gradually losing consistency as long as we step away from the two-dimensional stationary flow assumption. RANS approach dramatically loses accuracy in presence of flows that are subject of unsteadiness such as adverse gradients of pressure, mixings, rotational forces or irregular geometries [1]. The correct prediction of the previous phenomena become critic in turbomachinery flows, where it is necessary to take in account their influence on the overall performance of the turbomachinery. Many strategies have been exploited to increase the fidelity of RANS simulations, like adding additional equations, e.g. v2-f model, including quadratic or cubic terms in the modeled Reynolds stress tensor (RST), e.g. Lien cubic k-, hybrid approaches, etc. Some of the previous solutions have been proven effective, but most of the RANS uncertainties remain yet unsolved. Despite that, we can conclude that research on the topic has reached a stagnation point, with no major breakthrough expected in the next years [2]. Machine-learning (ML) has found a deep consensus in the engineer community, becoming extremely popular in fields like robotics, maintenance and diagnosis. Nevertheless, the application of artificial intelligence to turbulence modeling is still a novelty, with the first comprehensive literature report only published in 2015 to Duraisamy et al [3]. In spite of the successful results, many questions remain unanswered and an exhaustive methodology has not been proposed so far. After an introduction to the topic, the manuscript articulates in three different sections. In the first part, “Turbulence Modeling in the Big Data Era”, we report a deep literature review of works on the topic, classifying them in different categories, based on the methodology. To the best of the authors’ knowledge, this is the first available attempt to organize the available contents. Three classes of approach were identified: field inversing, anisotropic modeling and derivative works. 1. Field inversing: a correction coefficient for the production or dissipation term in the TKE transport equation is inferred by means of ML. 2. Anisotropic modeling: the anisotropic part of the RST is corrected or predicted by neural networks / random forests. Turbulent invariants are exploited to this purpose. 3. Derivative application: isolated attempts or non-popular methodologies fall within this category. We analyze each of the classes and provide a mathematical formulation of the methodology. We also highlight some key issues that are still to be solved. In the second part of the manuscript, “Machine-learning Tools and Techniques”, after a brief historical introduction, we perform a dissertation on the currently available machine-learning algorithms. In particular, we focus on multi-layer multi-input neural networks that constitute our artificial intelligence further in the work. We give a mathematical representation of advanced non-trivial techniques, useful for our applications. The third section, “Development of Machine-Learning Assisted Tools for Turbulence Modeling” treats three distinct applications of neural-networks to turbomachinery flows. The first two are focused on the creation of ML-assisted wall treatments for internal flows, while the last part is centered on the identification of stagnation zones in Gas Turbines (GTs) enclosure. Wall treatment is a crucial and non-trivial task in numerical simulations. In fact, CFD results can be considered reliable if the grid is fine enough for a proper resolution of the boundary layers. However, grid refinement comes at the cost of a larger mesh that reflects in increased computational time. To reduce the impact of wall refinements, wall functions are universally accepted to model part of the boundary layers, returning values of dissipation rate, TKE, turbulent viscosity, etc.. These functions were derived by direct observations of turbulent flows over flat plates and come as polynomial expressions depending from Reynolds number and wall distance. In so doing, every deflection of the flow from the flat plate assumption is not considered and it can become an important source of errors. A Data-driven Wall Function for Stationary Flows ASME GT 2019-91238 RANS approach fails to predict separating flows, regardless of the source of separation, i.e. geometrical or pressure-induced. We therefore created a training database through LES simulations of canonical flows, namely a 2D periodic hill, channel flows and backward-steps at different Re numbers. Simulations were validated against data available in literature. An Artificial Neural Network (ANN) is trained in Python 3.6 to predict TKE values on the first layer of cells near the walls. A deep analysis of data preprocessing, ANN training and testing phases shows the feasibility and accuracy of the approach. The model has been implemented in an open-source environment, OpenFoam v18, as a custom boundary condition. During run-time operation of the solver flow fields are extracted, manipulated and forwarded to the algorithm, that eventually predicts TKE boundary values. Coupling between the C++ based solver and Python is realized through the CAPI library. The framework is tested in two different cases and results from k- model are compared with standard wall treatment. No significate increase in computational time is found in the ML-enhanced framework. The first one is a two-dimensional periodic hill with three different grid resolutions, coarse, intermediate and fine. A similar geometry is included in the training dataset. ANN-based wall function corrects the overprediction of the model regardless of the mesh, showing that this approach is grid-insensitive. Reattachment length is compared with literature numerical results. A cross-validation is performed through the analysis of the flow around NACA profiles in a cascade arrangement. Two version are considered, a standard airfoil and a modified profile with leading edge bumps for stall control. While in the former results show no significative difference between the two wall treatments, in the modified airfoil the ANN-enhanced computations return more accurate results. In addition, flow fields and coefficient of pressure around the airfoils are reported. A Data-driven Wall Function for Rotating Passages ASME GT 2020-15353 The framework previously developed is here applied to rotating ducts. The effect of Coriolis and centrifugal forces are responsible of de- and stabilization of boundary layers. RANS approach, however, do not take in account rotating forces. In this work we create a wall treatment for rotating passages, commonly found in turbomachinery flows. A training database is created through LES simulation of a diverging diffuser with Rossby number = 0.41 and a diverging angle of 15deg., following experimental setup of Moore [4]. A comparison between LES, RANS and experimental results highlight the discrepancy of the RANS approach. Results from Exploratory Data Analysis (EDA) are reported. Data is filtered to obtain a zonal training with y+<400. EDA is used as a base to create input features that are also independent from the frame of reference. Training and Testing convergence histories show the quality of the training. The predictor returns a correction term for the TKE values on the first layer of cells. The derived model is tested with run-time computations of the same geometry with an inferior Rossby number, in cross-validation. On the last iteration of the solver, we report the prediction of the algorithm in different part of the domain. We report flow fields, accuracy and absolute relative errors for the whole domain, the inner layer and the first layer of cells. Results show the effect of the algorithm in the different zone of the domain, with an expected increase in accuracy as we move closer to the viscous sublayer. Identification of Poorly Ventilated Zones in GT Enclosures ASME GT 2019-91198 This part of the work has been conducted in cooperation with Baker Hughes, a GE company. Gas Turbines (GTs) ventilation systems of enclosures should grant the complete washing if gas leaking occurs. Given the extremely complex geometry that may lead to gas stagnation and the presence of high temperature surfaces, poor ventilation may lead to potential dangers. The aim is to build a model that, inferring from stationary fields provided by RANS simulations, classify portion of the computational domain through a Poorly Ventilated Index (PVI). PVI ranges from 1-safe to 16 – extremely dangerous and is based on the product of two term, function of temperature and concentration. The temperature term can be directly computed through numerical simulation, while the concentration of methane requires several days of URANS computations. The classifier therefore predicts the level of danger associated with the concentration of methane, reducing time requirements from several days to few seconds. A training database is generated through multi-phase simulations of the washing process of simplified geometries. Test geometries include high Reynolds computations of plain channel flow, backward facing step, and 2D sinusoidal hill flow, at different Reynolds number, axisymmetric hill flow and confined cylinder in cross-flow. Input features are constructed from the flow variables to assure independence from frame of reference and locally normalized. The ANN is trained to solve a four-class labeling problem, mapping local fields to the corresponding level of methane concentration. The algorithm was tested on all the training case, showing an extremely high accuracy (>95%). The model was validated on a model geometry of the GT enclosure. Temperature, concentration and velocity fields for different sampling planes are reported. A comparison between PVI levels given by the full URANS simulations and those derived by the RANS-derived classifier show a final accuracy of 91.3%. [1] J. D. Denton. “Some limitations of turbomachinery CFD.”, In: ASME GT 2010 - 22540. [2] J. P. Slotnick et al. CFD Vision 2030 Study: A Path to Revolutionary Computational Aerosciences. 2014. [3] H. Karthik Duraisamy. “A Framework for Turbulence Modeling using Big Data”. In: NASA Grant NNX15AN98A [4] Moore, J. "A Wake and an Eddy in a Rotating, Radial-Flow Passage—Part 1: Experimental Observations." (1973): 205-212.