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12.2.3 Boussinesq Approach vs. Reynolds Stress Transport Models

The Reynolds-averaged approach to turbulence modeling requires that the Reynolds stresses in Equation  12.2-4 be appropriately modeled. A common method employs the Boussinesq hypothesis [ 142] to relate the Reynolds stresses to the mean velocity gradients:

 - \rho \overline{u'_i u'_j} = \mu_t \left ( \frac{\partial... ...u_t \frac{\partial u_k}{\partial x_k} \right ) \delta_{ij} (12.2-5)

The Boussinesq hypothesis is used in the Spalart-Allmaras model, the $k$- $\epsilon$ models, and the $k$- $\omega$ models. The advantage of this approach is the relatively low computational cost associated with the computation of the turbulent viscosity, $\mu_t$. In the case of the Spalart-Allmaras model, only one additional transport equation (representing turbulent viscosity) is solved. In the case of the $k$- $\epsilon$ and $k$- $\omega$ models, two additional transport equations (for the turbulence kinetic energy, $k$, and either the turbulence dissipation rate, $\epsilon$, or the specific dissipation rate, $\omega$) are solved, and $\mu_t$ is computed as a function of $k$ and $\epsilon$. The disadvantage of the Boussinesq hypothesis as presented is that it assumes $\mu_t$ is an isotropic scalar quantity, which is not strictly true.

The alternative approach, embodied in the RSM, is to solve transport equations for each of the terms in the Reynolds stress tensor. An additional scale-determining equation (normally for $\epsilon$) is also required. This means that five additional transport equations are required in 2D flows and seven additional transport equations must be solved in 3D.

In many cases, models based on the Boussinesq hypothesis perform very well, and the additional computational expense of the Reynolds stress model is not justified. However, the RSM is clearly superior for situations in which the anisotropy of turbulence has a dominant effect on the mean flow. Such cases include highly swirling flows and stress-driven secondary flows.

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