Reflections on Spalart-Allmarus Turbulence Model, 2024

The Spalart-Allmaras turbulence model, a one-equation turbulence model, was a response to the inadequacies observed in zero-equation models, particularly their lack of predictive accuracy in complex flow scenarios such as wakes, shear layers, and shock wave boundary layer interactions.

The creation of the Spalart-Allmaras model was influenced by multiple prior works, including the Baldwin Barth model and the insights gained from then developed two-equation models. Notably, the works of Soviet mathematicians played a pivotal role. Their contributions, though primarily published in Russian, provided a foundation for the development of the model by Spalart and Allmaras.

Central to the Spalart-Allmaras model is the equation for tracking eddy viscosity, which features a production term derived from vorticity magnitude, signifying the generation of turbulence. This is complemented by a diffusion term and a destruction term, tailored to account for the diffusion and dissipation of turbulence, respectively. The model underwent significant refinement to ensure its applicability to fully turbulent flows, highlighting the mathematical craftsmanship behind its formulation.

A distinctive feature of the model is its treatment of transitional flows through a source-like term, enabling the prediction of transition behaviors based on user inputs. This aspect, however, introduces a dependency on the distance to the wall parameter, which some later models have sought to mitigate.

The Spalart-Allmaras model’s appeal lies in its locality and independence from structured grid requirements, making it a versatile tool for various flow conditions, particularly in unstructured grid environments. Its simplicity, robustness, and ease of implementation are evident in the model’s formulation presented in the paper’s appendix, making it accessible to a wider audience within the computational fluid dynamics community. This is one of the reasons it is so successful

Despite its empirical foundations and term by term (e.g. terms inspired by previous models and knowledge of experiments) development approach in terms of term by term creation, the model resembles similarities to the $Re_t$ one-equation model (also from NASA Ames). While it may not rival direct models like the k-epsilon in terms of derivation from first principles, the Spalart-Allmaras model’s pragmatic approach and calibrations have set it apart in the field as one of the most popular models internationally. The model’s development was significantly aided by access to experimental data and collaborations with NASA’s high-performance computing research group at NASA Ames, among others.