Turbulent Non-Isothermal Flow Theory

Turbulent energy transport is conceptually more complicated than energy transport in laminar flows since the turbulence is also a form of energy.

Equations for compressible turbulence are derived using the Favre average. The Favre

˜

average of a variable T is denoted T and is defined by

where the bar denotes the usual Reynolds average. The full field is then decomposed as

The modeling assumptions are in large part analogous to incompressible turbulence modeling. The stress tensor – ui''u''j is model with the Boussinesq approximation:

where k is the turbulent kinetic energy which in turned is defined by

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The correlation between uj'' and h'' in Equation 12-5 is the turbulent transport of heat. It is model analogous to the laminar conductive heat flux

The molecular diffusion, ijui'' , and turbulent transport term, uj''ui''ui'' 2 , are modeled by a generalization of the molecular diffusion and turbulent transport term found in the incompressible k equation

The Favre average can also be applied to the momentum equation which, using Equation 12-7, can be written

Taking the inner product between ˜ and Equation 12-12 results in an equation for ui

the resolved kinetic energy which can be subtract from Equation 12-11 with the following result:

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According to Wilcox (Ref. 1), it is usually a good approximation to neglect the contributions of k for flows with Mach numbers up to the supersonic range. This gives the following approximation of Equation 12-13 is

Equation 12-15 is completely analogue to the laminar energy equation and can be expanded using the same theory (see for example Ref. 3):

which is the temperature equation solved in the turbulent Non-Isothermal Flow and Conjugate Heat Transfer interfaces.

T U R B U L E N T C O N D U C T I V I T Y

Kays-Crawford

This is a relatively exact model for PrT, still simple. In Ref. 4, it is compared to other models for PrT and found to be good for most kind of turbulent wall bounded flows except for liquid metals. The model is given by

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where the Prandtl number at infinity is PrT 0.85 and is the conductivity.

Extended Kays-Crawford

Weigand and others (Ref. 5) suggested an extension of Equation 12-16 to liquid metals by introducing

100 PrT = 0.85 + -------------------------------

Cp Re0.888

where Re , the Reynolds number at infinity must be provided either as a constant or as a function of the flow field. This is entered in the Model Inputs section of the Fluid feature.

T E M P E R A T U R E W A L L F U N C T I O N S

Analogous to the single-phase flow wall functions (see Wall Functions described for the Wall boundary condition), there is a theoretical gap between the solid wall and the computational domain of the fluid also for the temperature field. This gap is often ignored in so much that it is ignored when the computational geometry is drawn.

The heat flux between the fluid with temperature Tf and a wall with temperature Tw, a is:

where is the fluid density, Cp is the fluid heat capacity, C is a turbulence modeling constant and k is the turbulent kinetic energy. T is the dimensionless temperature and is given by (Ref. 6):

where in turn

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