Reynolds stress equation model

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.

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

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

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

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

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

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

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

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

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

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

Diffusion term

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

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

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

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

Rotational term

The rotational term is given as ^[4]

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

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

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.

References

↑ Bengt Andersson , Ronnie Andersson s (2012). Computational Fluid Dynamics for Engineers (First ed.). Cambridge University Press, New York. p. 97. ISBN 9781107018952.
↑ Peter S. Bernard & James M. Wallace (2002). Turbulent Flow: Analysis, Measurement & Prediction. John Wiley & Sons. p. 324. ISBN 0471332194.
↑ Magnus Hallback (1996). Turbulence and Transition Modelling (First ed.). Kluwer Academic Publishers. p. 117. ISBN 0792340604.
↑ H.Versteeg & W.Malalasekera (2013). An Introduction to Computational Fluid Dynamics (Second ed.). Pearson Education Limited. p. 96. ISBN 9788131720486.

Bibliography

"Turbulent Flows", S. B. Pope, Cambridge University Press (2000).
"Modelling Turbulence in Engineering and the Environment: Second-Moment Routes to Closure", Kemal Hanjalić and Brian Launder, Cambridge University Press (2011).

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