On the accuracy of eddy-viscosity and eddy-conductivity models in square duct flow

We carry out a priori tests of linear and nonlinear eddy-viscosity models using direct numerical simulation (DNS) data of square duct flow up to friction Reynolds number Re_τ = 2000. We focus on the ability of eddy-viscosity models to reproduce the anisotropic Reynolds stress tensor components a_ij responsible for turbulent secondary flows, namely the normal stress a₂₂ and the secondary shear stress a_23. We perform two types of tests: i) on constitutive relations and ii) on RANS models. A priori tests on constitutive relations for a_ij are performed using the tensor polynomial expansion of Pope, and they allow us to assess the maximum accuracy that one can achieve for different orders of the tensor polynomial, where one tensor base corresponds to the linear eddy-viscosity hypothesis and five bases return the exact representation of a_ij. Models performance are quantified using the mean correlation coefficient with respect to DNS data C_ij, which shows that the linear eddy-viscosity hypothesis always returns very accurate values of the primary shear stress (C₁₂>0.99), whereas two bases are sufficient to achieve good accuracy of the normal stress and secondary shear stress (C₂₂=0.911, C₂₃=0.743). Instead, a priori analysis carried out on popular RANS models, including k–ε and v²–f, reveals that none of them achieves ideal accuracy. The only model which approaches ideal performance is the quadratic correction of Spalart, which has an accuracy similar to models using four or more tensor bases. An equivalent approach, based on vector integrity bases, is used to assess the accuracy of eddy-conductivity models for approximating the turbulent-heat flux.