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雷诺平均纳维-斯托克斯（RANS）方程是描述湍流流体运动的时平均运动方程。这种方法被称为 雷诺兹 decomposition, separates flow variables into a mean and a fluctuating component. The averaging process introduces an additional term, the Reynolds stress tensor, which represents the effect of turbulence and must be modeled to achieve closure, making simulations computationally tractable.
RANS背后的核心思想是雷诺分解，即将瞬时量分解为其时间平均项和波动项。对于速度量，其表达式为：\overline{u}_i(x) + u’_i(x,t) 将此表达式代入纳维-斯托克斯方程并进行时间平均后，非线性对流项会产生新项：[latex] -\rho \overline{u’_i u’_j} [/latex]，即雷诺应力张量。该张量代表湍流波动引起的动量净转移。.
The appearance of this unknown tensor leads to the ‘closure problem’ of turbulence: there are more unknowns than equations. To solve the system, the Reynolds stresses must be related to the mean flow quantities through a turbulence model. The most common approach is the Boussinesq hypothesis, which assumes the Reynolds stresses are proportional to the mean strain rate, introducing an ‘eddy viscosity’ or ‘turbulent viscosity’. This is analogous to how molecular viscosity relates stress to strain rate in laminar flow. Turbulence models, such as the popular k-ε (k-epsilon) and k-ω (k-omega) models, are sets of additional transport equations used to compute this eddy viscosity throughout the flow field. For example, the k-ε model solves for the turbulent kinetic energy (k) and its rate of dissipation (ε). RANS provides a good balance of accuracy and computational cost for many engineering applications, as it avoids the prohibitive expense of resolving all turbulent eddies directly.
雷诺平均纳维-斯托克斯 (RANS) 方程
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