Theory for the Nonisothermal Flow and Conjugate Heat Transfer Interfaces
The following points of the theory of Nonisothermal Flow and Conjugate Heat Transfer are discussed in this part:
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.
The Nonisothermal Flow and Conjugate Heat Transfer Equations
In industrial applications it is common that the density of a process fluid varies. These variations can have a number of different sources but the most common one is the presence of an inhomogeneous temperature field. This module includes the Nonisothermal Flow predefined multiphysics coupling to simulate systems in which the density varies with temperature.
Other situations where the density might vary includes chemical reactions, for instance where reactants associate or dissociate.
The Nonisothermal Flow and Conjugate Heat Transfer interfaces contain the fully compressible formulation of the continuity and momentum equations:
(4-135)
where
  ρ is the density (SI unit: kg/m3)
  u is the velocity vector (SI unit: m/s)
  p is the pressure (SI unit: Pa)
τ is the viscous stress tensor (SI unit: Pa), equal for a compressible fluid to:
  μ is the dynamic viscosity (SI unit: Pa·s)
  F is the body force vector (SI unit: N/m3)
It also solves the heat equation, which for a fluid is given in Equation 4-18 by
where in addition to the quantities above
Cp is the specific heat capacity at constant pressure (SI unit: J/(kg·K))
T is the absolute temperature (SI unit: K)
q is the heat flux by conduction (SI unit: W/m2)
qr is the heat flux by radiation (SI unit: W/m2)
αp is the coefficient of thermal expansion (SI unit: 1/K):
Q contains heat sources other than viscous heating (SI unit: W/m3)
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.
Turbulent Nonisothermal Flow Theory
Turbulent energy transport is conceptually more complicated than energy transport in laminar flows because 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 and is defined by
where the bar denotes the usual Reynolds average. The full field is then decomposed as
With this notation the energy balance equation becomes
(4-136)
where H is the enthalpy. The vector
(4-137)
is the laminar conductive heat flux and
is the laminar, viscous stress tensor. Notice that the thermal conductivity is denoted λ.
The modeling assumptions are in large part analogous to those for incompressible turbulence modeling. The stress tensor
is modeled using the Boussinesq approximation:
(4-138)
where k is the turbulent kinetic energy, which in turn is defined by
(4-139)
The correlation between uj and H in Equation 4-136 is the turbulent transport of heat. It is modeled analogously to the laminar conductive heat flux
(4-140)
The molecular diffusion term,
and turbulent transport term,
are modeled by a generalization of the molecular diffusion and turbulent transport terms found in the incompressible k equation
(4-141)
Inserting Equation 4-137, Equation 4-138, Equation 4-139, Equation 4-140 and Equation 4-141 into Equation 4-136 gives
(4-142)
The Favre average can also be applied to the momentum equation, which, using Equation 4-138, can be written
(4-143)
Taking the inner product between and Equation 4-143 results in an equation for the resolved kinetic energy, which can be subtracted from Equation 4-142 with the following result:
(4-144)
where the relation
has been used.
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 4-144:
(4-145)
Larsson (Ref. 28) suggests to make the split
Since
for all applications of engineering interest, it follows that
and consequently
(4-146)
where
Equation 4-146 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):
which is the temperature equation solved in the turbulent Nonisothermal Flow and Conjugate Heat Transfer interfaces.
Turbulent Conductivity
Kays-Crawford
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
(4-147)
where the Prandtl number at infinity is PrT = 0.85 and λ is the conductivity.
Extended Kays-Crawford
Weigand and others (Ref. 30) suggested an extension of Equation 4-147 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.
Temperature Wall Functions
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.
Standard
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):
where in turn
λ is the thermal conductivity, and κ is the von Kármán constant equal to 0.41.
High viscous dissipation at wall
Launder and Spalding propose a temperature wall function that accounts for conduction as well as viscous dissipation. The heat flux is written:
Variables with an index “c” are evaluated at the critical distance, which corresponds to the switch from the laminar sublayer to the logarithmic layer. P is a measure of the resistance of the boundary layer to heat transport, and is estimated with Jayatilleke’s formula:
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.
Theory for the Nonisothermal Screen Boundary Condition
When the Nonisothermal Flow multiphysics coupling feature is active, the conditions that apply across a screen in isothermal flow are complemented by:
(4-148)
where H0 is the total enthalpy.
See Screen for the feature node details.
Also see Screen boundary condition described for the single-phase flow interfaces.
Theory for the Interior Fan Boundary Condition
When the Nonisothermal Flow multiphysics coupling feature is active, the conditions that apply across an interior fan are complemented by: