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

Abstract

Abstract 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 ν t -92 model. It detects transverse curvature in the distribution of the eddy viscosity $$ \tilde{\nu } $$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mover> <mml:mi>ν</mml:mi> <mml:mo>~</mml:mo> </mml:mover> </mml:math> , 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 λ 2 of the Hessian operator of $$ \tilde{\nu } $$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mover> <mml:mi>ν</mml:mi> <mml:mo>~</mml:mo> </mml:mover> </mml:math> . 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 $$ \tilde{\nu } $$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mover> <mml:mi>ν</mml:mi> <mml:mo>~</mml:mo> </mml:mover> </mml:math> distribution in the round jet. The quantity λ 2 is not infinitely differentiable, but it is free of singularities, and unlike the ν t -92 version, is not dependent on two large quantities cancelling. The core term added to the Lagrangian derivative $$ D\tilde{\nu }/Dt $$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mi>D</mml:mi> <mml:mover> <mml:mi>ν</mml:mi> <mml:mo>~</mml:mo> </mml:mover> <mml:mo>/</mml:mo> <mml:mi>D</mml:mi> <mml:mi>t</mml:mi> </mml:mrow> </mml:math> is simply c b3 λ 2 $$ \tilde{\nu } $$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mover> <mml:mi>ν</mml:mi> <mml:mo>~</mml:mo> </mml:mover> </mml:math> , where c b3 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 λ 2 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 c b3 constant is rather large, namely 6. The λ 2 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 λ 2 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 r SA parameter of the SA model, which is a measure of wall proximity. The λ 2 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.

Topics & Concepts