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Derivation and Analysis of Pressure-Strain Correlation Tensor Models

This study explores the constraints and predictions of pressure-strain correlation tensor models in turbulence analysis. The research delves into spectral representation, differential Reynolds stress models, and the connection between spectral approach and closure models. Results offer insights into turbulence behavior and reveal areas for model improvement.

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Derivation and Analysis of Pressure-Strain Correlation Tensor Models

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  1. Constraints for the Pressure-Strain Correlation tensor derived from Spectral Representation S.R.Bogdanov T.J.Jongen EFMC-6, Stockholm, 2006

  2. Outline • DRSM, PSC models • The derivation of the restrictions on the “rapid” part of PSC. • Direct checking of the models for • Predicting the available areas of the parameters of stationary states • Others: (Implicit) checking of the models for non-linear part of PSC, non-uniform turbulence … EFMC-6, Stockholm, 2006

  3. Differential Reynolds stress models, DRSM Spectra of ideas, connected with the study of fully developed turbulence and calculations of the flows with the anisotropy is extremly wide: from the theory of fractals and renormalization group methods to direct numeric simulations and semiempirical modelling. At the same time nowadays the so-called first (with the turbulent viscosity as the key concept) and second-order closure (or full differential Reynolds stress models, DRSM). models are the most popular in practice. EFMC-6, Stockholm, 2006

  4. A lot of models were proposed for PSC. One of the most popular and well-known (quasi-linear) semi-empirical approximations for PSC looks like: EFMC-6, Stockholm, 2006

  5. Realizability criteria EFMC-6, Stockholm, 2006

  6. The derivation of the restrictions on the “rapid” part of PSC. EFMC-6, Stockholm, 2006

  7. Illustration of the basic idea EFMC-6, Stockholm, 2006

  8. EFMC-6, Stockholm, 2006

  9. EFMC-6, Stockholm, 2006

  10. EFMC-6, Stockholm, 2006

  11. Direct checking of the models for PSC EFMC-6, Stockholm, 2006

  12. EFMC-6, Stockholm, 2006

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  14. EFMC-6, Stockholm, 2006

  15. It's convenient to present the available areas for the set ofparameters B with graphic on B_3 - {B_2}/R plane. Available is the insidearea of the triangle ABC. 2-D turbulence corresponds to the sides ofthe triangle; points A,B,C correspond to 1-dimensional turbulence(with pulsations along axes 3,1,2).BE, CF, AD - present axisymmetric turbulence;At last, point O presents isotropic turbulence. The set of hyperbola (B_1 - isolines) isalso presented on the picture.When B_1 is fixed, the area of available values of theparameters B_2 и B_3/R is restricted with correspondent hyperbolaand BC- line. Y – axis: B3= -b33/2 X-axis: B2/R = (b11 – b22)/2 EFMC-6, Stockholm, 2006

  16. EFMC-6, Stockholm, 2006

  17. EFMC-6, Stockholm, 2006

  18. EFMC-6, Stockholm, 2006

  19. The available domain on invariant map EFMC-6, Stockholm, 2006

  20. B3 - isolines EFMC-6, Stockholm, 2006

  21. B2 - isolines, for B1=0 EFMC-6, Stockholm, 2006

  22. «Forbidden» areas on invariant map (SSG model, R=0, B1=0 and -0.1 correspondently) EFMC-6, Stockholm, 2006

  23. EFMC-6, Stockholm, 2006

  24. Stationary state analysis, SSG EFMC-6, Stockholm, 2006

  25. Conclusion • Possibilities of direct checking (and improving, hopefully: at least, through the proper choice of the set of the model constants) of PSC models • Direct analysis of the parameters of stationary states. • The link between spectral approach and one-point closures. • Most probably turbulence is more non-local than it’s usually assumed within second-order closures. EFMC-6, Stockholm, 2006

  26. Thank you EFMC-6, Stockholm, 2006

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