Abstract—

The main equations in analytical modeling of turbulence, used in open-channel flows, are the parabolic profile of the eddy viscosity and the exponentially decreasing turbulent kinetic energy function. However, when using the definition of the eddy viscosity as a product between velocity and length scales and by taking the velocity scale as the root-square of turbulent kinetic energy, we show that the parabolic eddy viscosity profile is incompatible with the turbulent kinetic energy function. Taking into account this shortcoming, we consider an eddy viscosity formulation which is in agreement with the turbulent kinetic energy profile in the equilibrium region. This eddy viscosity is written in a form that allows the calibration of the two \({\text{R}}{{{\text{e}}}_{{{\tau }}}}\)-dependent parameters which have a linear behavior (where \({\text{R}}{{{\text{e}}}_{{{\tau }}}}\) is the friction Reynolds number). All results were validated by both direct numerical simulation and experimental data in the same range of friction Reynolds numbers, respectively \(300 < {\text{R}}{{{\text{e}}}_{{{\tau }}}} < 5200\) for closed-channel flows and \(923 < {\text{R}}{{{\text{e}}}_{{{\tau }}}} < 6139\) for open-channel flows. Comparisons with the direct numerical simulation data of the eddy viscosity in closed-channel flows for eight different flow conditions show good agreement. Mean streamwise velocities are obtained from solving of the momentum equation. For closed-channel flows, mean velocity profiles show very good agreement. For open-channel flows, results confirm that the use of the parabolic eddy viscosity is unable to improve the velocities while the proposed method shows good agreement. These results show the ability of this analytical eddy viscosity model to predict accurately the velocities in the outer region for both closed- and open-channel flows, without any ad hoc function or parameter.

中文翻译：

---

封闭和明渠流外部速度剖面的分析涡粘度模型

摘要——

在明渠流动中使用的湍流分析建模中的主要方程是涡粘性的抛物线分布和指数递减的湍流动能函数。然而，当使用涡粘度的定义作为速度和长度尺度之间的乘积并通过将速度尺度作为湍动能的平方根时，我们表明抛物线涡粘度剖面与湍动能函数不相容. 考虑到这个缺点，我们考虑了一个涡流粘度公式，它与平衡区域的湍流动能分布一致。该涡流粘度以允许校准两个\({\text{R}}{{{\text{e}}}_{{{\tau }}}}\)具有线性行为的依赖参数（其中\({\text{R}}{{{\text{e}}}_{{{\tau }}}}\)是摩擦雷诺数）。所有结果均通过相同摩擦雷诺数范围内的直接数值模拟和实验数据分别验证\(300 < {\text{R}}{{{\text{e}}}_{{{\tau } }}} < 5200\)用于封闭通道流和\(923 < {\text{R}}{{{\text{e}}}}_{{{\tau }}}} < 6139\)用于明渠流动。与八种不同流动条件下封闭通道流动中涡粘性的直接数值模拟数据的比较显示出良好的一致性。平均流向速度是通过求解动量方程获得的。对于封闭通道流动，平均速度剖面显示出非常好的一致性。对于明渠流动，结果证实使用抛物线涡粘性无法提高速度，而所提出的方法显示出良好的一致性。这些结果表明，这种分析涡粘度模型能够准确预测封闭和明渠流动的外部区域的速度，而无需任何临时函数或参数。