A robust k − ε − v2¯/k elliptic blending turbulence model applied to near-wall, separated and buoyant flows

This paper first reconsiders evolution over 20 years of the k − ε − v2¯ − f strand of eddy-viscosity models, developed since original proposal for a near-wall eddy viscosity model based on the physics of the full Reynolds stress transport models, but retaining only the wall-normal fluctuating velocity variance, v2¯, and its source, f, the redistribution by pressure fluctuations. Added to the classical k − ε (turbulent kinetic energy and dissipation) model, this resulted in three transport equations f or k, epsilon and v2¯, and one elliptic equation for f, which accurately reproduced the parabolic decay of v2¯/k down to the solid wall without introducing wall-distance or low-Reynolds number related damping functions in the eddy viscosity and k − ε equations. However, most v2¯ − f variants have suffered from numerical stiffness making them unpractical for industrial or unsteady RANS applications, while the one version available in major commercial codes tends to lead to degraded and sometimes unrealistic solutions. After considering the rationale behind a dozen variants and asymptotic behaviour of the variables in a number of zones (balance of terms in the channel flow viscous sublayer, logarithmic layer, and wake region, homogenous flows and high Reynolds number limits), a new robust version is proposed, which is applied to a number of cases involving flow separation and heat transfer. This k − ε type of model with v2¯/k anisotropy blends high Reynolds number and near-wall forms using two dimensionless parameters: the wall-normal anisotropy v2¯/k and a dimensionless parameter alpha resulting from an elliptic equation to blend the homogeneous and near-wall limiting expressions of f. The review of variants and asymptotic cases has also led to modifications of the epsilon equation: the second derivative of mean velocity is reintroduced as an extra sink term to retard turbulence growth in the transition layer (i.e. embracing the E term of the k − ε model), the homogeneous part of epsilon is now adopted as main transported variable (as it is less sensitive to the Reynolds number effects), and the excessive growth of the turbulent length-scale in the absence of production is corrected (leading to a better distinction between log layer and wake region of a channel flow). For each modification numerical stability implications are carefully considered and, after implementation in an industrial finite-volume code, the final model proved to be significantly more robust than any of the previous variants.