Reynolds stress equation model (RSM), also referred to as second order or second moment closure model is the most complete classical turbulence model. Due to the disparate character of complex engineering flows, turbulence models must be robust so as to be applicable for most cases, yet possessing a high degree of fidelity in each. Furthermore, as the processes of analysis and engineering design involve repeated iterations, the predictive method must be computationally economical. In this light, Reynolds Averaged Navier Stokes (RANS)-based models represent the pragmatic approach for complex engineering flows as opposed to computationally intensive methods like Large Eddy Simulations or Direct Numerical Simulations. However, popular RANS-based modeling paradigms like one or two-equation models have significant shortcomings in all but the simplest turbulent flows. For instance, in flows with streamline curvature or a high preponderance of mean rotational effects, the performance of such models is highly unsatisfactory. In such flows, Reynolds stress based models can offer much better predictive fidelity. In summary, the second moment closure approach offers better accuracy than one or two equation turbulence models and yet is not as computationally demanding as Direct Numerical Simulations.
Contents
- Production term
- Pressure strain interactions
- Dissipation term
- Diffusion term
- Pressure strain correlation term
- Rotational term
- Advantages of RSM
- References
Several shortcomings of k-epsilon turbulence model were observed when it was attempted to predict flows with complex strain fields or substantial body forces. Under those conditions the individual Reynolds stresses were not found to be accurate while using formula
The equation for the transport of kinematic Reynolds stress
Rate of change of
The six partial differential equations above represent six independent Reynolds stresses. The models that we need to solve the above equation are derived from the work of Launder, Rodi and Reece (1975).
Production term
The Production term that is used in CFD computations with Reynolds stress transport equations is
Pressure-strain interactions
Pressure-strain interactions affect the Reynolds stresses by two different physical processes: pressure fluctuations due to eddies interacting with one another and pressure fluctuation of an eddy with a region of different mean velocity. This redistributes energy among normal Reynolds stresses and thus makes them more isotropic. It also reduces the Reynolds shear stresses.
It is observed that the wall effect increases the anisotropy of normal Reynolds stresses and decreases Reynolds shear stresses. A comprehensive model that takes into account these effects was given by Launder and Rodi (1975).
Dissipation term
The modelling of dissipation rate
where
Diffusion term
The modelling of diffusion term
where
Pressure-strain correlation term
The pressure-strain correlation term promotes isotropy of the turbulence by redistributing energy amongst the normal Reynolds stresses.The pressure-strain interactions is the most important term to model correctly. Their effect on Reynolds stresses is caused by pressure fluctuations due to interaction of eddies with each other and pressure fluctuations due to interaction of an eddy with region of flow having different mean velocity. The correction term is given as
Rotational term
The rotational term is given as
here
Advantages of RSM
1) Unlike the k-ε model which uses an isotropic eddy viscosity, RSM solves all components of the turbulent transport.
2) It is the most general of all turbulence models and works reasonably well for a large number of engineering flows.
3) It requires only the initial and/or boundary conditions to be supplied.
4) Since the production terms need not be modeled, it can selectively damp the stresses due to buoyancy, curvature effects etc.