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-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

    ${\displaystyle -\rho u_{i}^{\prime }u_{j}^{\prime }}$ = ${\displaystyle \mu t\left({\frac {\partial U_{i}}{\partial x_{j}}}+{\frac {\partial U_{i}}{\partial x_{i}}}\right)-{\frac {2}{3}}\rho k\delta _{ij}}$ = ${\displaystyle 2\mu tE_{ij}-{\frac {2}{3}}\rho k\delta _{ij}}$ 

The equation for the transport of kinematic Reynolds stress ${\displaystyle R_{ij}=u_{i}^{\prime }u_{j}^{\prime }=-\tau _{ij}/\rho }$ is ^[1]

  ${\displaystyle {\frac {DR_{ij}}{Dt}}=D_{ij}+P_{ij}+\Pi _{ij}+\Omega _{ij}-\varepsilon _{ij}}$

Rate of change of ${\displaystyle R_{ij}}$ + Transport of ${\displaystyle R_{ij}}$ by convection = Transport of ${\displaystyle R_{ij}}$ by diffusion + Rate of production of ${\displaystyle R_{ij}}$ + Transport of ${\displaystyle R_{ij}}$ due to turbulent pressure-strain interactions + Transport of ${\displaystyle R_{ij}}$ due to rotation + Rate of dissipation of ${\displaystyle R_{ij}}$.

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

   ${\displaystyle P_{ij}}$ =${\displaystyle -\left(R_{im}{\frac {\partial U_{j}}{\partial x_{m}}}+R_{jm}{\frac {\partial U_{i}}{\partial x_{m}}}\right)}$

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 ${\displaystyle \epsilon _{\rm {ij}}}$ assumes that the small dissipative eddies are isotropic. This term affects only the normal Reynolds stresses. ^[2]

          ${\displaystyle \epsilon _{\rm {ij}}}$ = ${\displaystyle 2/3\epsilon \delta _{ij}}$

where ${\displaystyle \epsilon }$ is dissipation rate of turbulent kinetic energy, and ${\displaystyle \delta _{ij}}$ = 1 when i = j and 0 when i ≠ j

Diffusion Term

The modelling of diffusion term ${\displaystyle D_{ij}}$ 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 ${\displaystyle D_{ij}}$ that is followed by commercial CFD codes is

${\displaystyle D_{ij}}$ = ${\displaystyle {\frac {\partial }{\partial x_{m}}}\left({\frac {v_{t}}{\sigma _{k}}}{\frac {\partial R_{ij}}{\partial x_{m}}}\right)}$ = ${\displaystyle div\left({\frac {v_{t}}{\sigma _{k}}}\nabla (R_{ij})\right)}$

where ${\displaystyle \upsilon _{t}}$ = ${\displaystyle C_{\mu }{\frac {k^{2}}{\epsilon }}}$ , ${\displaystyle \sigma _{k}}$ = 1.0 and ${\displaystyle C_{\mu }}$ = 0.9

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 ^[3]

${\displaystyle \Pi _{ij}=-C_{1}{\frac {\epsilon }{k}}\left(R_{ij}-{\frac {2}{3}}k\delta _{ij}\right)-C_{2}\left(P_{ij}-{\frac {2}{3}}P\delta _{ij}\right)}$

Rotational Term

The rotational term is given as ^[4]

 ${\displaystyle \Omega _{ij}=-2\omega _{k}\left(R_{jm}e_{ikm}+R_{im}e_{jkm}\right)}$

here${\displaystyle \omega _{k}}$ is the rotation vector, ${\displaystyle e_{ijk}}$=1 if i,j,k are in cyclic order and are different,${\displaystyle e_{ijk}}$=-1 if i,j,k are in anti-cyclic order and are different and ${\displaystyle e_{ijk}}$=0 in case any two indices are same.

Advantages of RSM

1)Compared with k-ε model, it is simple because of the use of an isotropic eddy viscosity.
2)It is the most general of all turbulence models and work 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 modelled, it can selectively damp the stresses due to buoyancy, curvature effects etc.

Disadvantages of RSM

1)It requires very large computing costs.
2)It is not very widely validated as the k-ε model and mixing length models.
3)Due to identical problems with the ε-equation modelling, it performs just as poorly as the k-ε model in some problems.
4)Because of being isotropic, it is not good in predicting normal stresses and is unable to account for irrotational strains.

See also

• Reynolds Stress
• Isotropy
• Turbulence Modeling
• Eddy
• k-epsilon turbulence model

References

Bibliography

• "An Introduction to Computational Fluid Dynamics",Second Edition by Versteeg & Malalasekera, published by Pearson Education Limited.
• "Turbulence : An Introduction for Scientists and Engineers" By P.A. Davidson.
• "Turbulence Models & Their Applications" By Tuncer Cebeci, published by Horizons Publications Inc.

Category:Turbulent Models