User:Reynolds stress equation model

Reynolds stress equation model (RSM), also known as second order or second moment closure model is the most complex classical turbulence model. Several shortcomings of k-ε 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


 * -ρui'uj' = μt(∂Ui/∂xj + ∂Uj/∂xi) - 2/3ρkδij = 2μtEij - 2/3ρkδij

The equation for the transport of kinematic Reynolds stress Rij = ui'uj' = -τij/ρ


 * DRij/Dt = Dij + Pij + ∏ij + Ωij - εij

Rate of change of Rij + Transport of Rij by convection = Transport of Rij by diffusion + Rate of production of Rij + Transport of Rij due to turbulent pressure-strain interactions + Transport of Rij due to rotation + Rate of dissipation of Rij

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
 * Pij = -(Rim ∂Uj/∂xm + Rjm ∂Ui/∂xm)

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 εij assumes that the small dissipative eddies are isotropic. This term affects only the normal Reynolds stresses.


 * εij = 2/3εij
 * where ε is dissipation rate of turbulent kinetic energy, and
 * δij = 1 when i = j and 0 when i ≠ j
 * δij = 1 when i = j and 0 when i ≠ j

Diffusion Term
The modelling of diffusion term Dij is based on the assumption that the rate of transport of Reynolds stresses by diffusion is proportional to the gradients of Reynolds stresses. The simplest form of Dij that is followed by commercial CFD codes is


 * Dij = ∂/∂xm(νt/σk ∗ ∂Rij/∂xm)= div(νt/σk grad(Rij))
 * where νt = Cμ k2/ε, σk = 1.0 and Cμ = 0.9
 * where νt = Cμ k2/ε, σk = 1.0 and Cμ = 0.9