Robust experimental data assimilation for the Spalart-Allmaras turbulence model

TL;DR

This paper addresses robustly improving RANS closure accuracy for separated flows by calibrating the Spalart–Allmaras (SA) model using sparse, noisy experimental data via Ensemble Kalman Filtering (EnKF). The approach calibrates five SA coefficients, including a $r$-dependent $C_{b1}$ and a revised $f_w$ mapping through $C_{w2}$ and $C_{w3}$, by solving the inverse problem with the production, diffusion, and destruction terms. The calibrated model delivers improved predictions of $C_f$ and $C_p$ for BFS, 2D-WMH, and BFS2, while preserving SA behavior for attached/unbounded cases such as NACA0012 and axisymmetric jet, demonstrating transferability beyond the training case. The work demonstrates that EnKF-based, physics-informed coefficient updates can yield robust, generalizable improvements with sparse data, offering a data-assisted alternative to black-box ML for turbulence closure calibration.

Abstract

This study presents a methodology focusing on the use of computational model and experimental data fusion to improve the Spalart-Allmaras (SA) closure model for Reynolds-averaged Navier-Stokes solutions. In particular, our goal is to develop a technique that not only assimilates sparse experimental data to improve turbulence model performance, but also preserves generalization for unseen cases by recovering classical SA behavior. We achieve our goals using data assimilation, namely the Ensemble Kalman filtering approach (EnKF), to calibrate the coefficients of the SA model for separated flows. A holistic calibration strategy is implemented via the parameterization of the production, diffusion, and destruction terms. This calibration relies on the assimilation of experimental data collected in the form of velocity profiles, skin friction, and pressure coefficients. Despite using observational data from a single flow condition around a backward-facing step (BFS), the recalibrated SA model demonstrates generalization to other separated flows, including cases such as the 2D NASA wall mounted hump (2D-WMH) and modified BFS. Significant improvement is observed in the quantities of interest, i.e., skin friction coefficient ($C_f$) and pressure coefficient ($C_p$) for each flow tested. Finally, it is also demonstrated that the newly proposed model recovers SA proficiency for flows, such as a NACA-0012 airfoil and axisymmetric jet (ASJ), and that the individually calibrated terms in the SA model target specific flow-physics wherein the calibrated production term improves the re-circulation zone while destruction improves the recovery zone.

Figures (26)

• Figure 1: EnKF calibration loop, a.) An ensemble ($X$) of the SA coefficients - for the first iteration $X$ is sampled from an initial distribution determined by a parametric analysis in \ref{['senstivity']}. b.) The RANS solver and SA model calculate the flow variables based on the SA coefficients in $X$, the coefficients used in the ensemble are highlighted in red, and the coefficient $b$ is defined later. c.) The extracted quantities of interest (QOIs) from OpenFOAM at specified locations serve as the $HX$ for the EnKF. d.) The QOI in the current study are $C_f$ and $C_p$ along the bottom wall. e.) $D_{m\times N}$ matrix for the available data for QOI, $m$ is number of probe points $N$ is the ensemble size picked from the Gaussian distribution. g.) Updated $X^p$ based on the EnKF in f.
• Figure 2: Mean of the calibration data $D$, used for current study. The values of $C_f$ and $C_p$ are plotted along the bottom wall of BFS. The baseline data was obtained from the NASA turbulence repository nasa and a noise $\epsilon \sim \mathcal{N}(0, \sigma_D = 0.05)$ is added to formulate the $D$ matrix.
• Figure 3: $\Delta X$ vs. iterations for the calibration loop.
• Figure 4: a. $C_{b1}$, b. $f_w$ vs. $r$ obtained from the calibrated coefficients. The original values are also plotted in corresponding plots. For additional comparison, the $f_w$ obtained by Bin et al. bin2023data is also plotted.
• Figure 5: BFS: a.) $C_f$, b.) $C_p$ vs. $x/H$. The calibrated SA shows improvement in both the recovery zone and separation bubble as compared to the baseline model. The improvement in this case is defined as proximity to the experimental results.