Reynolds stress equation model (RSM), also referred to as second moment closures are the most complete classical turbulence model. In these models, the eddy-viscosity hypothesis is avoided and the individual components of the Reynolds stress tensor are directly computed. These models use the exact Reynolds stress transport equation for their formulation. They account for the directional effects of the Reynolds stresses and the complex interactions in turbulent flows. Reynolds stress models offer significantly better accuracy than eddy-viscosity based turbulence models, while being computationally cheaper than Direct Numerical Simulations (DNS) and Large Eddy Simulations.
Shortcomings of Eddy-viscosity based models Eddy-viscosity based models like the k − ϵ {\displaystyle k-\epsilon } and the k − ω {\displaystyle k-\omega } models have significant shortcomings in complex, real-life turbulent flows. For instance, in flows with streamline curvature, flow separation, flows with zones of re-circulating flow or flows influenced by mean rotational effects, the performance of these models is unsatisfactory. Such one- and two-equation based closures cannot account for the return to isotropy of turbulence, observed in decaying turbulent flows. Eddy-viscosity based models cannot replicate the behaviour of turbulent flows in the Rapid Distortion limit, where the turbulent flow essentially behaves as an elastic medium (instead of viscous).
Reynolds Stress Transport Equation Reynolds Stress equation models rely on the Reynolds Stress Transport equation. The equation for the transport of kinematic Reynolds stress R i j = ⟨ u i ′ u j ′ ⟩ = − τ i j / ρ {\displaystyle R_{ij}=\langle u_{i}^{\prime }u_{j}^{\prime }\rangle =-\tau _{ij}/\rho } is
D R i j D t = D i j + P i j + Π i j + Ω i j − ε i j {\displaystyle {\frac {DR_{ij}}{Dt}}=D_{ij}+P_{ij}+\Pi _{ij}+\Omega _{ij}-\varepsilon _{ij}}
Rate of change of R i j {\displaystyle R_{ij}} + Transport of R i j {\displaystyle R_{ij}} by convection = Transport of R i j {\displaystyle R_{ij}} by diffusion + Rate of production of R i j {\displaystyle R_{ij}} + Transport of R i j {\displaystyle R_{ij}} due to turbulent pressure-strain interactions + Transport of R i j {\displaystyle R_{ij}} due to rotation + Rate of dissipation of R i j {\displaystyle R_{ij}} . The six partial differential equations above represent six independent Reynolds stresses. While the Production term ( P i j {\displaystyle P_{ij}} ) is closed and does not require modelling, the other terms, like pressure strain correlation ( Π i j {\displaystyle \Pi _{ij}} ) and dissipation ( ε i j {\displaystyle \varepsilon _{ij}} ), are unclosed and require closure models.
Production term The Production term that is used in CFD computations with Reynolds stress transport equations is
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