Reynolds stress equation model

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 and Large Eddy Simulations.

Shortcomings of Eddy-viscosity based models

Eddy-viscosity based models like the $k-\epsilon$ and the $k-\omega$ 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 such models is unsatisfactory.

Such one- and two-equation based closures cannot account for the return to isotropy of turbulence,^[1] observed in decaying turbulent flows. Eddy-viscosity based models cannot replicate the behaviour of turbulent flows in the Rapid Distortion limit,^[2] 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 $R_{ij}=u_{i}^{\prime }u_{j}^{\prime }=-\tau _{ij}/\rho$ is ^[3]

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

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. While the Production term ( $P_{ij}$ ) is closed and does not require modelling, the other terms, like pressure strain correlation ( $\Pi _{ij}$ ) and dissipation ( $\varepsilon _{ij}$ ) , are unclosed and require closure models.

Production term

The Production term that is used in CFD computations with Reynolds stress transport equations is

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

Physically, the Production term represents the action of the mean velocity gradients working against the Reynolds stresses. This accounts for the transfer of kinetic energy from the mean flow to the fluctuating velocity field. It is responsible for sustaining the turbulence in the flow through this transfer of energy from the large scale mean motions to the small scale fluctuating motions.

Slow Pressure-strain correlation term

The slow pressure-strain correlation term redistributes energy amongst the normal Reynolds stresses. This is responsible for the return to isotropy of decaying turbulence. Physically, this arises due to the self-interactions amongst the fluctuating field. The correction term is given as ^[4]

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

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. ^[5]

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

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

Rotational term

The rotational term is given as ^[6]

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

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

↑ Lumley, John; Newman, Gary (1977). "The return to isotropy of homogeneous turbulence". Journal of Fluid Mechanics. 82: 161–178.
↑ Mishra, Aashwin; Girimaji, Sharath (2013). "Intercomponent energy transfer in incompressible homogeneous turbulence: multi-point physics and amenability to one-point closures". Journal of Fluid Mechanics. 731: 639–681.
↑ Bengt Andersson , Ronnie Andersson s (2012). Computational Fluid Dynamics for Engineers (First ed.). Cambridge University Press, New York. p. 97. ISBN 9781107018952.
↑ Magnus Hallback (1996). Turbulence and Transition Modelling (First ed.). Kluwer Academic Publishers. p. 117. ISBN 0792340604.
↑ Peter S. Bernard & James M. Wallace (2002). Turbulent Flow: Analysis, Measurement & Prediction. John Wiley & Sons. p. 324. ISBN 0471332194.
↑ 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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