Pressure strain rate modeling of homogeneous axisymmetric turbulence

Abstract

Recently several second order closure models have been proposed for closing the second moment equations, in which the velocity–pressure gradient tensor and the dissipation rate tensor are two of the most important terms. In the literature, the velocity–pressure gradient tensor is usually decomposed into a so called rapid term and a return-to-isotropy term. Models of these terms have been used in global flow calculations together with other modeled terms. However, their individual behavior in different flows has not been fully examined, because they are unmeasurable in the laboratory. Recently, the experimental data for axi-symmetric flow measurements of Ertunç (2008) (J.L. Lumley, 1975) have given us the opportunity to do this kind of study. In this article, we make direct comparisons of five representative rapid pressure strain rate models and eight return-to-isotropy models with the experimental data for axi-symmetric flow measurements of Ertunç (2008) (J.L. Lumley, 1975) and the theoretical RDT results of Sreenivasan (U. Schumann, 1977). The purpose of these direct comparisons is to explore the performance of these models at different contraction rates and identify the ones which give the best performance. The paper also describes the modeling procedure, model constraints, and the various evaluated models. The detailed results of the direct comparisons are discussed, and a few concluding remarks on turbulence models are given.

Keywords:

• Turbulence–transport model
• Reynolds stress
• realizability
• second-order closure

Acknowledgements

The first author is grateful to Prof. Ulrich Rüde for motivation to take part in the Bavarian Graduate school of Computational Engineering.

Notes

¹ In the current article, the definition of b _ij = /q ²− 1/3δ _ij .

² Since the Equation (10) is linear, the law of linear superposition should hold for its solution, i.e., if θ = θ₁ is the solution, and θ = θ₂ is another solution, then a θ₁ + b θ₂ is also another solution, where a and b are constants. This law of linear superposition fails with non linear models.

³a_ij = /k – 2/3δ_ij

⁴ There is some confusion in the literature about the correct values of these constants. The current values are from the thesis of Olof Grundestam, where a remark is made in the last paper of the thesis, that the paper of Johansson et al. [17] had a misprint and the correct values are given in the thesis.

⁵ When a principal Reynolds stress component vanishes, its time rate must also vanish and its second derivative must be positive. Pope [11] departed from the approach of Lumley in proposing what has come to be known as the weak form of realizability. Pope only required that when a principal Reynolds stress component vanishes, its time derivative becomes positive.

⁷The definition of is given in Chapter 7.