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The degenerate parabolic Generalized Porous Medium Equation (GPME) poses numerical challenges due to selfsharpening and its sharp corner solutions. For these problems, we show results for two subclasses of the GPME with differentiable k(p) with respect to p, namely the Porous Medium Equation (PME) and the superslow diffusion equation. Spurious temporal oscillations, and nonphysical locking and lagging have been reported in the literature. These issues have been attributed to harmonic averaging of the coefficient k(p) for small p, and arithmetic averaging has been suggested as an alternative. We show that harmonic averaging is not solely responsible and that an improved discretization can mitigate these issues. Here, we investigate the causes of these numerical artifacts using modified equation analysis. The modified equation framework can be used for any type of discretization. We show results for the second order finite volume method. The observed problems with harmonic averaging can be traced to two leading error terms in its modified equation. This is also illustrated numerically through a Modified Harmonic Method (MHM) that can locally modify the critical terms to remove the aforementioned numerical artifacts.

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.