Eigenvector perturbation methodology for uncertainty quantification of turbulence models

Reynolds-averaged Navier-Stokes (RANS) models are the primary numerical recourse to investigate complex engineering turbulent flows in industrial applications. However, to establish RANS models as reliable design tools, it is essential to provide estimates for the uncertainty in their predictions. In the recent past, an uncertainty estimation framework relying on eigenvalue and eigenvector perturbations to the modeled Reynolds stress tensor has been widely applied with satisfactory results. However, the methodology for the eigenvector perturbations is not well established. Evaluations using only eigenvalue perturbations do not provide comprehensive estimates of model form uncertainty, especially in flows with streamline curvature, recirculation, or flow separation. In this article, we outline a methodology for the eigenvector perturbations using a predictor-corrector approach, which uses the incipient eigenvalue perturbations along with the Reynolds stress transport equations to determine the eigenvector perturbations. This approach was applied to benchmark cases of complex turbulent flows. The uncertainty intervals estimated using the proposed framework exhibited substantial improvement over eigenvalue-only perturbations and are able to account for a significant proportion of the discrepancy between RANS predictions and high-fidelity data.

Figure 1

Schematic outline of the eigenvalue perturbations on the barycentric triangle, starting from an arbitrary state.

Figure 2

Flow geometry and profile markers for the convex channel of Arolla and Durbin [24].

Figure 3

Indices of performance, $\varphi$, associated with the tensor bases considered.

Figure 4

Schematic outlining the composition of uncertainty estimates for an asymmetric diffuser at $x/H=24$.

Figure 5

$X$ velocity ($U_x$) profiles along the radial direction $x$, with LES comparisons at (a) $0^{\circ }$, (b) ${30}^{\circ }$, and (c) ${60}^{\circ }$ locations along the channel. The LES data is represented by circles, the baseline (unperturbed) RANS prediction by the dark line and the uncertainty estimates by the gray-shaded zone. At each location, we exhibit uncertainty intervals from the proposed eigenvalue-eigenvector perturbation methodology contrasted against those from eigenvalue-only perturbations.

Figure 6

Turbulent kinetic energy profiles, along with LES comparisons at at (b) $0^{\circ }$, (c) ${30}^{\circ }$, (d) ${60}^{\circ }$, and (e) ${90}^{\circ }$ locations along the channel marked in (a). The LES data is represented by circles, the baseline (unperturbed) RANS prediction by the dark line and the uncertainty estimates by the gray-shaded zone.

Figure 7

Flow geometry for the asymmetric diffuser.

Figure 8

Coefficient of friction along the bottom wall of the diffuser.

Figure 9

Mean velocity profiles along the expansion section of the diffuser at locations (b) $x/H=24.4$, (c) $x/H=28.4$, and (d) $x/H=32.4$.