Second Order Closure Material Derivative Gradient terms Gradient

• the second term is negligible at high turbulence Reynolds numbers, and in

• In the Reynolds stress transport equations, closure hypotheses are necessary forthe following

• The source and sink terms can be modeled with reference to homogenous

Modelling viscous dissipation • Viscous dissipation occurs at the level of small eddies, in

Modelling turbulent diffusion terms Triple velocity correlations

(that being obtained by analogy with the approximation of the energy redistribution terms through

If the term (a) is neglected, then we recover approximation [6. 6]. Coefficient cs

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

• Very little is known about diffusion due to pressure fluctuations. These correlations

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

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

Determination of model constants • a) Constants c 1 and c 2 • Constants

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Second - Order Closure

Material Derivative

Gradient terms

Source and sink terms

Material derivative

Gradient terms

Source and sink terms

• 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.

• 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 Ret.

• The source and sink terms can be modeled with reference to homogenous turbulence. The other terms appear only in nonhomogenous 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.

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:

Anisotropy tensor

Modelling turbulent diffusion terms Triple velocity correlations

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

If the term (a) is neglected, then we recover approximation [6. 6]. Coefficient cs 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.

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

Other Proposals • Daly Harlow:

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

Other suggestions

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

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

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

Constants c’ 1 and c’ 2