A robust intermittency equation formulation for transition modeling in Spalart-Allmaras simulations of airfoil flows across a wide range of Reynolds numbers
Valerio D'Alessandro, Matteo Falone, Luca Giammichele, Renato Ricci
TL;DR
This work addresses robust laminar-to-turbulent transition prediction in airfoil flows by coupling the Spalart–Allmaras turbulence model to local correlation–based transition frameworks ($\gamma$ and $\gamma$–$\widetilde{R}_{\theta,t}$) within RANS. It introduces a logarithmic reformulation of the intermittency equation ($\tilde{\gamma}=\log\gamma$) paired with an energy‑limiting bound and a gradient‑driven artificial viscosity to suppress nonphysical growth and Gibbs‑like pressure oscillations, achieving stable solutions across wide Reynolds numbers. Two‑equation ($\gamma$ and $\Ret$) and one‑equation ($\log\gamma$) transition variants are explored, including SA‑R1 (rotation correction) and SA‑SMP (strain‑modulated production), and a single‑equation approach with algebraic onset correlations; all show improved robustness and reasonable agreement with experimental/LES data when stabilization is applied. The methodology demonstrates practical robustness for industrial RANS workflows and suggests straightforward extensions to $k$–$\omega$ formulations and potential DES applications for stalled regimes. Overall, the proposed stabilization strategies enable stable, accurate transitional predictions for airfoils over a broad $\mathrm{Re}$ range, with clear guidance on model variants and artificial viscosity tuning.
Abstract
This paper introduces a new robust formulation for local correlation-based laminar-to-turbulent transition models. This mechanism is incorporated into Reynolds-Averaged Navier-Stokes (RANS) equations, coupled with the Spalart-Allmaras (SA) turbulence model, considering both $γ$ and $γ$-${\widetilde{\mathrm{Re}}_{θ,t}}$ transition frameworks. In this context, special attention is placed on numerical stabilization of the $γ$ transport equation, which is identified as the root cause of instabilities observed in both $γ$ and $γ$-${\widetilde{\mathrm{Re}}_{θ,t}}$ based models. To this end, the intermittency equation is reformulated in logarithmic form and further stabilized through an energy--based limiting to bound excessively high positive values. In order to suppress unphysical pressure oscillations in the transition region, a gradient-driven artificial viscosity is also introduced. Additionally, the SA equation is augmented with strain-rate modulated production and rotation correction terms. The presented approach has demonstrated consistent effectiveness and robustness in the simulation of flow fields around airfoils over a wide range of Reynolds numbers, making it suitable for practical aerodynamic design applications.