See Theory for the Single-Phase Flow Interfaces and Theory for the Turbulent Flow Interfaces in the CFD Module User’s Guide for a description of the theory related to laminar and turbulent single-phase flow interfaces.

It also solves the heat equation, which for a fluid is given in Equation 4-18 by

The work done by pressure changes term

and the viscous heating term

are not included by default because they are usually negligible. These terms can, however, be added by selecting corresponding check boxes in the Nonisothermal Flow feature.

The physics interface also supports heat transfer in solids (Equation 4-16):

where Qted is the thermoelastic damping heat source (SI unit: W/(m3)). This term is not included by default but must be added by selecting the corresponding check box.

Equations for compressible turbulence are derived using the Favre average. The Favre average of a variable T is denoted and is defined by

where H is the enthalpy. The vector

is the laminar, viscous stress tensor. Notice that the thermal conductivity is denoted λ.

is modeled using the Boussinesq approximation:

where k is the turbulent kinetic energy, which in turn is defined by

The correlation between uj″ and H″ in Equation 4-140 is the turbulent transport of heat. It is modeled analogously to the laminar conductive heat flux

are modeled by a generalization of the molecular diffusion and turbulent transport terms found in the incompressible k equation

Inserting Equation 4-141, Equation 4-142, Equation 4-143, Equation 4-144 and Equation 4-145 into Equation 4-140 gives

The Favre average can also be applied to the momentum equation, which, using Equation 4-142, can be written

Taking the inner product between and Equation 4-147 results in an equation for the resolved kinetic energy, which can be subtracted from Equation 4-146 with the following result:

According to Wilcox (Ref. 27), 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 4-148:

Larsson (Ref. 28) suggests to make the split

Equation 4-150 is completely analogous to the laminar energy equation of Equation 4-15 and can be expanded using the same theory to get the temperature equation similar to Equation 4-18 (see for example Ref. 28):

This is a relatively exact model for PrT, while still quite simple. In Ref. 29, 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

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

Weigand and others (Ref. 30) suggested an extension of Equation 4-151 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.

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 for the fluid and temperature fields. This gap is often ignored when the computational geometry is drawn.

The standard temperature wall function takes into account conduction in the boundary layer. The heat flux between the fluid with temperature Tf and a wall with temperature Tw, is:

where ρ is the fluid density, Cp is the fluid heat capacity, and uτ is the friction velocity. T+ is the dimensionless temperature and is given by (Ref. 31):

λ is the thermal conductivity, and κ is the von Kármán constant equal to 0.41.

Variables with an index “c” are evaluated at the critical distance, which corresponds to the switch from the laminar sublayer to the logarithmic layer. The critical wall distance δw+c, which depends on the turbulent parameters κ and B, is estimated from a second-order accurate solution:

with κ0 = 0.41, B0 = 5.2, and X0 = 4.53554291157957.

The velocity at δw+c is then calculated from a velocity wall function, according to the turbulence model and wall treatment chosen to keep consistency. The velocity wall function used for the determination of Uc when the L-VEL or a LES turbulence model is chosen is based on a blending of the linear and logarithmic laws. It gives very similar results to the wall function actually used for the calculation of the velocity at the first cell which is based on the wall Reynolds number.

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 function, δw is at least hw/2 and can be bigger if necessary to keep δw+ higher than 11.06. The computational results 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.

where H0 is the total enthalpy.