where h is the enthalpy. The vector
where k is the turbulent kinetic energy, which in turn is defined by
The correlation between and in Equation 5-3 is the turbulent transport of heat. It is modeled analogously to the laminar conductive heat flux
Taking the inner product between and Equation 5-10 results in an equation for the resolved kinetic energy, which can be subtracted from
Equation 5-9 with the following result:
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 5-11 is
Larsson (Ref. 2) suggests to make the split
Equation 5-13 is completely analogous to the laminar energy equation and can be expanded using the same theory (see for example
Ref. 3):
This is a relatively exact model for PrT, while still quite simple. In
Ref. 4, it is compared to other models for
PrT and found to be a good approximation for most kinds of turbulent wall bounded flows except for turbulent flow of liquid metals. The model is given by
Weigand and others (Ref. 5) suggested an extension of
Equation 5-14 to liquid metals by introducing
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.
where ρ is the fluid density,
Cp is the fluid heat capacity,
u∗ is the friction velocity given by the wall treatment (
u∗ for two-equation RANS models with automatic wall treatment and
uτ for all other cases).
T+ is the dimensionless temperature and is given by (
Ref. 6):
λ is the thermal conductivity, and
κ is the von Karman constant equal to 0.41.
The distance between the computational fluid domain and the wall, δw, is always
hw/2 for automatic wall treatment where
hw is the height of the mesh cell adjacent to the wall.
hw/2 is almost always very small compared to any geometrical quantity of interest, at least if a boundary layer mesh is used. For wall functions,
δw is at least
hw/2 and can be bigger if necessary to keep
δw+ higher than
11.06. So the computational results for wall functions should be checked so that the distance between the computational fluid domain and the wall,
δw, is everywhere small compared to any geometrical quantity of interest. The distance
δw is available for evaluation on boundaries.