A New “λ2” Term for the Spalart–Allmaras Turbulence Model, Active in Axisymmetric Flows

The new term belongs in the “basic,” free-shear-flow part of the Spalart–Allmaras (SA) model, and extends an idea of the Secundov team, incorporated in the νt-92 model. It detects transverse curvature in the distribution of the eddy viscosity ν~\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$ \tilde{\nu } $$\end{document}, so that it is passive in two-dimensional thin shear flows but potent especially in round jets. It eliminates the large over-prediction of the growth rate of such jets by the SA model, first detected by Birch in 1993. The originality is that the term is proportional to the middle eigenvalue λ2 of the Hessian operator of ν~\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$ \tilde{\nu } $$\end{document}. This is of course an empirical concept, but it is discriminating and rises when the distance r from the cylindrical axis becomes comparable with the length scale δ of the variations in the r direction. The inverted parabola is a prime example of such a distribution, and not unlike the ν~\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$ \tilde{\nu } $$\end{document} distribution in the round jet. The quantity λ2 is not infinitely differentiable, but it is free of singularities, and unlike the νt-92 version, is not dependent on two large quantities cancelling. The core term added to the Lagrangian derivative Dν~/Dt\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$ D\tilde{\nu }/Dt $$\end{document} is simply cb3 λ2ν~\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$ \tilde{\nu } $$\end{document}, where cb3 is a new constant. The computing cost of calculating and ordering the eigenvalues is moderate. We have no proof of well-posedness for the new equation set, but the evidence so far is favorable, both in structured and unstructured grids. The λ2 term is calibrated on a fully-developed round jet, and tested in nine other cases, either 2D flows or flows in which r » δ, finding that in the latter it is negligible as expected. This is although the cb3 constant is rather large, namely 6. The λ2 term is not strong enough to make a mature vortex fully relaminarize as would be desirable, but the eddy viscosity drops by 74%. The raw λ2 term reduces the eddy viscosity in pipe flow, where that is detrimental; therefore, in the final model, it is multiplied by a function of the rSA parameter of the SA model, which is a measure of wall proximity. The λ2 term appears to be a safe addition to the SA model, and its application in different codes and to a variety of flows to be desirable.