Reynolds stress equation model

Short description: Mathematical model of turbulence

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 (DNS) and Large Eddy Simulations.

## Contents

* 1 Shortcomings of Eddy-viscosity based models
* 2 Reynolds Stress Transport Equation
* 3 Production term
* 4 Rapid Pressure-Strain Correlation term
* 5 Slow Pressure-Strain Correlation term
* 6 Dissipation term
* 7 Diffusion term
* 8 Rotational term
* 9 Advantages of RSM
* 10 See also
* 11 See also
* 12 References
* 13 Bibliography

## Shortcomings of Eddy-viscosity based models

Eddy-viscosity based models like the k−ϵ and the k−ω 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 these 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 Rij=⟨ui′uj′⟩=−τij/ρ is [3]

DRijDt=Dij+Pij+Πij+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. While the Production term (Pij) is closed and does not require modelling, the other terms, like pressure strain correlation (Πij) and dissipation (εij), are unclosed and require closure models.

## Production term

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

Pij=−(Rim∂Uj∂xm+Rjm∂Ui∂xm)

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.

This is the only term that is closed in the Reynolds Stress Transport Equations. It requires no models for its direct evaluation. All other terms in the Reynolds Stress Transport Equations are unclosed and require closure models for their evaluation.

## Rapid Pressure-Strain Correlation term

The rapid pressure-strain correlation term redistributes energy among the Reynolds stresses components. This is dependent on the mean velocity gradient and rotation of the co-ordinate axes. Physically, this arises due to the interaction among the fluctuating velocity field and the mean velocity gradient field. The simplest linear form of the model expression is

ΠijRk=C2Sij+C3(bikSjk+bjkSik−23bmnSmnδij)+C4(bikWjk+bjkWik)

Here bij=uiuj‾2k−δij3 is the Reynolds stress anisotropy tensor, Sij is the rate of strain term for the mean velocity field and Wij is the rate of rotation term for the mean velocity field. By convention, C2,C3,C4 are the coefficients of the rapid pressure strain correlation model. There are many different models for the rapid pressure strain correlation term that are used in simulations. These include the Launder-Reece-Rodi model,[4] the Speziale-Sarkar-Gatski model,[5] the Hallback-Johanssen model,[6] the Mishra-Girimaji model,[7] the nonlocal model,[8] and others.

## Slow Pressure-Strain Correlation term

The slow pressure-strain correlation term redistributes energy among the Reynolds stresses. This is responsible for the return to isotropy of decaying turbulence where it redistributes energy to reduce the anisotropy in the Reynolds stresses. Physically, this term is due to the self-interactions amongst the fluctuating field. The model expression for this term is given as [9]

ΠijS=−C1εk(Rij−23kδij)−C2(Pij−23Pδij)

There are many different models for the slow pressure strain correlation term that are used in simulations. These include the Rotta model [10] , the Speziale-Sarkar model [11] , besides others.

## Dissipation term

The traditional modelling of the dissipation rate tensor εij assumes that the small dissipative eddies are isotropic. In this model the dissipation only affects the normal Reynolds stresses. [12]

εij=23εδij or eij=0

where ε is dissipation rate of turbulent kinetic energy, δij=1 when i = j and 0 when i ≠ j and eij is the dissipation rate anisostropy defined as eij=εijε−2δij3.

However, as has been shown by e.g. Rogallo,[13] Schumann & Patterson,[14] Uberoi,[15][16] Lee & Reynolds[17] and Groth, Hallbäck & Johansson[18] there exist many situations where this simple model of the dissipation rate tensor is insufficient due to the fact that even the small dissipative eddies are anisotropic. To account for this anisotropy in the dissipation rate tensor Rotta[19] proposed a linear model relating the anisotropy of the dissipation rate stress tensor to the anisotropy of the stress tensor.

εij=23εδij or eij=σaij

where aij=uiuj‾k−2δij3=2bij.

The parameter σ is assumed to be a function the turbulent Reynolds number, the mean strain rate etc. Physical considerations imply that σ should tend to zero when the turbulent Reynolds number tends to infinity and to unity when the turbulent Reynolds number tends to zero. However, the strong realizability condition implies that σ should be identically equal to 1.

Based on extensive physical and numerical (DNS and EDQNM) experiments in combination with a strong adherence to fundamental physical and mathematical limitations and boundary conditions Groth, Hallbäck and Johansson proposed an improved model for the dissipation rate tensor.[20]

eij=[1+α(IIa2−23)]aij−α(aikakj−13IIaδij)

where IIa=aijaji is the second invariant of the tensor aij and α is a parameter that, in principle, could depend on the turbulent Reynolds number, the mean strain rate parameter etc.

However, Groth, Hallbäck and Johansson used rapid distortion theory to evaluate the limiting value of α which turns out to be 3/4.[21][22] Using this value the model was tested in DNS-simulations of four different homogeneous turbulent flows. Even though the parameters in the cubic dissipation rate model were fixed through the use of realizability and RDT prior to the comparisons with the DNS data the agreement between model and data was very good in all four cases.

The main difference between this model and the linear one is that each component of eij is influenced by the complete anisotropic state. The benefit of this cubic model is apparent from the case of an irrotational plane strain in which the streamwise component of aij is close to zero for moderate strain rates whereas the corresponding component of eij is not. Such a behaviour cannot be described by a linear model.[23]

## 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. This is an application of the concept of the gradient diffusion hypothesis to modeling the effect of spatial redistribution of the Reynolds stresses due to the fluctuating velocity field. The simplest form of Dij that is followed by commercial CFD codes is

Dij=∂∂xm(vtσk∂Rij∂xm)=div⁡(vtσk∇(Rij))

where υt=Cμk2ε, σk=1.0 and Cμ=0.09.

## Rotational term

The rotational term is given as [24]

Ωij=−2ωk(Rjmeikm+Rimejkm)

here ωk is the rotation vector, eijk=1 if i,j,k are in cyclic order and are different,eijk=-1 if i,j,k are in anti-cyclic order and are different and eijk=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.

## See also

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

## See also

* k-epsilon turbulence model
* Mixing length model

## References

1. ↑ Lumley, John; Newman, Gary (1977). "The return to isotropy of homogeneous turbulence". Journal of Fluid Mechanics 82: 161–178. doi:10.1017/s0022112077000585. Bibcode: 1977JFM....82..161L.
2. ↑ 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. doi:10.1017/jfm.2013.343. Bibcode: 2013JFM...731..639M.
3. ↑ Bengt Andersson, Ronnie Andersson s (2012). Computational Fluid Dynamics for Engineers (First ed.). Cambridge University Press, New York. p. 97. ISBN 9781107018952.
4. ↑ Launder, Brian Edward and Reece, G Jr and Rodi, W (1975). "Progress in the development of a Reynolds-stress turbulence closure". Journal of Fluid Mechanics 68 (3): 537–566. doi:10.1017/s0022112075001814. Bibcode: 1975JFM....68..537L.
5. ↑ Speziale, Charles G and Sarkar, Sutanu and Gatski, Thomas B (1991). "Modelling the pressure--strain correlation of turbulence: an invariant dynamical systems approach". Journal of Fluid Mechanics 227: 245–272. doi:10.1017/s0022112091000101. Bibcode: 1991JFM...227..245S.
6. ↑ Johansson, Arne V and Hallback, Magnus (1994). "Modelling of rapid pressure—strain in Reynolds-stress closures". Journal of Fluid Mechanics 269: 143–168. doi:10.1017/s0022112094001515. Bibcode: 1994JFM...269..143J.
7. ↑ Mishra, Aashwin A and Girimaji, Sharath S (2017). "Toward approximating non-local dynamics in single-point pressure--strain correlation closures". Journal of Fluid Mechanics 811: 168–188. doi:10.1017/jfm.2016.730. Bibcode: 2017JFM...811..168M.
8. ↑ Hamlington, Peter E and Dahm, Werner JA (2009). "Nonlocal form of the rapid pressure-strain correlation in turbulent flows". Physical Review E 80 (4). doi:10.1103/PhysRevE.80.046311. Bibcode: 2009PhRvE..80d6311H.
9. ↑ Magnus Hallback (1996). Turbulence and Transition Modelling (First ed.). Kluwer Academic Publishers. p. 117. ISBN 978-0792340607.
10. ↑ Rotta, J (1951). "Statistical theory of nonhomogeneous turbulence. ii". Z. Phys. 131: 51–77. doi:10.1007/BF01329645.
11. ↑ Sarkar, Sutanu and Speziale, Charles G (1990). "A simple nonlinear model for the return to isotropy in turbulence". Physics of Fluids A: Fluid Dynamics 2 (1): 84–93. doi:10.1063/1.857694. Bibcode: 1990PhFlA...2...84S.
12. ↑ Peter S. Bernard & James M. Wallace (2002). Turbulent Flow: Analysis, Measurement & Prediction. John Wiley & Sons. p. 324. ISBN 978-0471332190. https://archive.org/details/turbulentflowana00bern_387.
13. ↑ Rogallo, R S (1981). "Numerical experiments in homogeneous turbulence". NASA Tm 81315 81: 31508. Bibcode: 1981STIN...8131508R.
14. ↑ Schumann, U & Patterson, G S (1978). "Numerical study of the return of axisymmetric turbulence to isotropy". J. Fluid Mech. 88 (4): 711–735. doi:10.1017/S0022112078002359. Bibcode: 1978JFM....88..711S. https://elib.dlr.de/53666/1/78-num-stdy-ret.pdf. Retrieved 2019-07-07.
15. ↑ Uberoi, M S (1956). "Effect of wind-tunnel contraction on free-stream turbulence". Journal of the Aeronautical Sciences 23 (8): 754–764. doi:10.2514/8.3651.
16. ↑ Uberoi, M S (1978). "Equipartition of energy and local isotropy in turbulent flows". J. Appl. Phys. 28 (10): 1165–1170. doi:10.1063/1.1722600. https://deepblue.lib.umich.edu/bitstream/2027.42/70587/2/JAPIAU-28-10-1165-1.pdf.
17. ↑ Lee, M J & Reynolds, W C (1985). "Numerical experiments on the structure of homogeneous turbulence". Thermosciences Div., Dept. Of Mech. Engineering, Stanford University, Rep. No. TF-24.
18. ↑ Groth, J, Hallbäck, M & Johansson, A V (1989). Measurement and modelling of anisotropic turbulent flows. Advances in Turbulence 2. Springer-Verlag Berlin Heidelberg. p. 84. doi:10.1007/978-3-642-83822-4. ISBN 978-3-642-83822-4.
19. ↑ Rotta, J C (1951). "Statistische Theorie nichthomogener Turbulenz I". Z. Phys. 129 (6): 547–572. doi:10.1007/BF01330059. Bibcode: 1951ZPhy..129..547R.
20. ↑ Hallbäck, M, Groth, J & Johansson, A V (1989). A Reynolds stress closure for the dissipation in anisotropic turbulent flows. Symposium on Turbulent Shear Flows, 7th, Stanford, CA, Aug. 21-23, 1989, Proceedings. Stanford University.
21. ↑ Hallbäck, M, Groth, J & Johansson, A V (1990). "An algebraic model for nonisotropic turbulent dissipation rate in Reynolds stress clousers". Phys. Fluids A 2: 1859. doi:10.1063/1.857908.
22. ↑ Groth, J, Hallbäck, M & Johansson, A V (1990). A nonlinear model for the dissipation rate term in Reynolds stress models. Engineering Turbulence Modelling and Experiments: Proceedings of the International Symposium on Engineering Turbulence Modelling and Measurements. Elsevier. ISBN 978-0444015631.
23. ↑ Hallbäck, M, Groth, J & Johansson, A V (1991). Anisotropic Dissipation Rate - Implications for Reynolds Stress Models. Advances in Turbulence 3. Springer, Berlin, Heidelberg. p. 414. doi:10.1007/978-3-642-84399-0_45. ISBN 978-3-642-84401-0.
24. ↑ 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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