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 $${\text {Re}}_\tau =1055$$ . 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 $$a_{22}$$ and the secondary shear stress $$a_{23}$$ . A priori tests on constitutive relations for $$a_{ij}$$ are performed using the tensor polynomial expansion of Pope (J Fluid Mech 72:331–340, 1975), whereby one tensor base corresponds to the linear eddy viscosity hypothesis and five bases return exact representation of $$a_{ij}$$ . We show that the bases subset has an important effect on the accuracy of the stresses and the best results are obtained when using tensor bases which contain both the strain rate and the rotation rate. Models performance is quantified using the mean correlation coefficient with respect to DNS data $${\widetilde{C}}_{ij}$$ , which shows that the linear eddy viscosity hypothesis always returns very accurate values of the primary shear stress $$a_{12}$$ ( $${\widetilde{C}}_{12}>0.99$$ ), whereas two bases are sufficient to achieve good accuracy of the normal stress and secondary shear stress ( $${\widetilde{C}}_{22}=0.911$$ , $${\widetilde{C}}_{23}=0.743$$ ). Unfortunately, RANS models rely on additional assumptions and a priori analysis carried out on popular models, including k– $$\varepsilon $$ and $$v^2$$ –f, reveals that none of them achieves ideal accuracy. The only model based on Pope’s expansion which approaches ideal performance is the quadratic correction of Spalart (Int J Heat Fluid Flow 21:252–263, 2000), which has similar accuracy to models using four or more tensor bases. Nevertheless, the best results are obtained when using the linear correction to the $$v^2$$ –f model developed by Pecnik and Iaccarino (AIAA Paper 2008-3852, 2008), although this is not built on the canonical tensor polynomial as the other models.

中文翻译：

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方管流涡粘性模型的先验检验

我们使用方形管道流动的直接数值模拟 (DNS) 数据进行线性和非线性涡粘性模型的先验测试，直到摩擦雷诺数 $${\text {Re}}_\tau =1055$$ 。我们专注于涡粘性模型再现各向异性雷诺应力张量分量 $$a_{ij}$$ 的能力，这些分量负责二次湍流，即法向应力 $$a_{22}$$ 和二次剪切应力 $$ a_{23}$$ . 使用 Pope (J Fluid Mech 72:331–340, 1975) 的张量多项式展开对 $$a_{ij}$$ 的本构关系进行先验测试，其中一个张量基对应于线性涡粘性假设，五个bases 返回 $$a_{ij}$$ 的精确表示。我们表明基子集对应力的准确性有重要影响，并且在使用同时包含应变速率和旋转速率的张量基时可以获得最佳结果。使用与 DNS 数据 $${\widetilde{C}}_{ij}$$ 相关的平均相关系数来量化模型性能，这表明线性涡流粘度假设总是返回非常准确的主剪切应力值 $$ a_{12}$$ ( $${\widetilde{C}}_{12}>0.99$$ )，而两个碱基足以实现正应力和二次剪切应力的良好精度 ( $${\widetilde{ C}}_{22}=0.911$$ , $${\widetilde{C}}_{23}=0.743$$ ）。不幸的是，RANS 模型依赖于额外的假设和对流行模型进行的先验分析，包括 k– $$\varepsilon $$ 和 $$v^2$$ –f，表明它们都没有达到理想的精度。基于 Pope 扩展的唯一接近理想性能的模型是 Spalart 的二次校正（Int J Heat Fluid Flow 21:252–263, 2000），其精度与使用四个或更多张量基的模型相似。尽管如此，当对 Pecnik 和 Iaccarino 开发的 $$v^2$$ -f 模型（AIAA Paper 2008-3852, 2008）使用线性校正时，可以获得最好的结果，尽管这不是建立在规范张量多项式上，如其他型号。