Quantification of Turbulence Modeling Uncertainties Using Full Field Inversion

Abstract

Reynolds Averaged Navier-Stokes (RANS) turbulence models have historically been developed using a combination of theoretical/physical/mathematical arguments, modeler expertise, and empirical data-fitting. As a consequence of the loss of information incurred during the ensemble averaging process, RANS model development has always been data-driven, to an extent, out of necessity. With the profusion of high resolution simulations and experimental methods, there exists a prodigious opportunity to more comprehensively inform closure models with data and improve their utlility in practical predictions of complex turbulent flows. In contrast to inferring model parameters, this work uses inverse modeling to directly obtain corrective, spatially distributed terms that are consistent with the predictive RANS models, thus offering a route to address the issue of model discrepancy which has long plagued turbulence model development. The posterior distributions of the inferred terms are quantified using Bayesian inversion and sampling. Results are presented for turbulent channel flow and flows with wall curvature. While the source terms inferred in this work are problem dependent, an ongoing subset of our work is to use machine learning methodologies to obtain general functional forms of the type that are inferred in this work. This paper specifically focuses on the mechanics of Bayesian inversion and its implications on the quantification of errors and uncertainties in turbulence modeling.

Machine Learning Methods for Data-Driven Turbulence Modeling

“…where a (1) and w (1) ij are the activation function and weights associated with the first hidden layer, respectively. Similarly, the second layer of hidden nodes is constructed as…”

“…This work is part of a larger effort to improve closure models [1][2][3] in which inverse modeling is used to infer the functional forms of modeling discrepancies and machine learning (ML) is used to reconstruct the information from the inference process into modeling terms to be used in a predictive setting. In this paper, we focus on the latter aspect and explore existing and newly-developed machine learning techniques for use in turbulence modeling.…”

Inverse Turbulence Modeling of channel flow using Continuous Adjoint method

“…While the former strategy obtains the discrete form of adjoint equations directly from flow equations, in the latter, first a continuous set of adjoint equations are obtained which are discretized later. This study follows the work of Parish and Duraisamy [26] to lay the basic framework of inversion using the continuous adjoint approach.…”

“…First, the problem is set up as an inverse problem to extract the funcional form of deficiencies. Many sample test cases have been dealt with in references [26,9]. This step on its own, they argue, can provide valuable modeling insight.…”

“…This closely follows step 1 of the strategy outlined by Duraisamy et al. Continuous adjoint approach, instead of the discrete aproach used by Parish and Duraisamy [26], has been used to calculate the gradients. Inversion has first been performed for non-rotating channel flow using the DNS data of Moser et al [24].…”