1. Second - Order Closure

4. Material Derivative

5. Gradient terms

8. Source and sink terms

12. Material derivative

13. Gradient terms

16. Source and sink terms

18. the second term is negligible at high turbulence Reynolds numbers, and in this case only the first term in fact represents the true dissipation rate.

19. In the Reynolds stress transport equations, closure hypotheses are necessary forthe following terms: viscous dissipation (f), redistribution of energy by pressure strain correlations (c), turbulent diffusion (d) + (e). In the present chapter, we shall consider only fully developed turbulence at high Reynolds numbers Re t.

20. The source and sink terms can be modeled with reference to homogenous turbulence. The other terms appear only in non- homogenous flows and in particular in wall flows. They are mainly turbulent diffusion terms and a part of the pressurestrain correlations and they will be considered separately.

21. Modelling viscous dissipation Viscous dissipation occurs at the level of small eddies, in the spectral zone of large wavenumbers in which turbulence is classically assumed to approach isotropy. If the Reynolds number is sufficiently high for the dissipation zone to be clearly separated from the production zone, the viscous dissipation process can be assumed to be isotropic. This is modeled using a second order isotropic tensor through the hypothesis:

23. Anisotropy tensor

24. Modelling turbulent diffusion terms Triple velocity correlations

28. (that being obtained by analogy with the approximation of the energy redistribution terms through pressure-strain correlations in the R ij transport equation

29. If the term (a) is neglected, then we recover approximation [6.6]. Coefficient c s in [6.6] is determined by referring to experimental data relative to various turbulent flows, the value c s = 0.11 is obtained in Launder B.E., Reece G.J., Rodi W.

30. Other proposals Donaldson: Where L is a macroscale. Mellor and Herring:

31. Other Proposals Daly Harlow:

32. Very little is known about diffusion due to pressure fluctuations. These correlations are not directly attainable by measurements using present time experimental means.

33. Other suggestions

34. Modelling pressure-strain PS=PHI(1)+PHI(2)+PHI(S) Purely turbulent İnteractions between turbulence and mean flow Wall effect

39. Launder B.E., Reece G.J. and Rodi W., suggest

44. Determination of model constants a) Constants c 1 and c 2 Constants c1 and c2 are determined by reference to the homogenous turbulent flow with uniform mean velocity gradient.

51. Constants c’1 and c’2