In this section, the instantaneous equation of mass, momentum, and energy conservation are presented. For turbulent flows, the instantaneous equations are averaged leading to additional terms. These terms, together with models for them, are discussed in Turbulence and Wall Function Theory.
The instantaneous equations of mass, momentum and energy conservation can be written as follows in a stationary frame:
 | (1–85) |
where 
is the total
enthalpy, related to the static enthalpy 
by:
 | (1–86) |
The term 
represents the work due to viscous stresses and is called the viscous
work term. This models the internal heating by viscosity in the fluid,
and is negligible in most flows.
The term 
represents the work due to external momentum sources
and is currently neglected.
Note: A case that involves MFR (multiple frames of reference) and the Total Energy equation should have the Incl. Viscous Work Term option selected. If the option is not selected, the setup is inconsistent with the MFR interface implementation, and CFX-Pre and CFX-Solver issue a warning.
An alternative form of the energy equation, which is suitable
for low-speed flows, is also available. To derive it, an equation
is required for the mechanical energy 
.
 | (1–87) |
The mechanical energy equation is derived by taking the dot
product of 
with the momentum equation (Equation 1–83):
 | (1–88) |
Subtracting this equation from the total energy equation (Equation 1–85) yields the thermal energy equation:
 | (1–89) |
The term 
is always positive and is called
the viscous dissipation. This models the internal heating by viscosity
in the fluid, and is negligible in most flows.
With further assumptions discussed in a moment, we obtain the thermal energy equation:
 | (1–90) |
This equation can be derived from Equation 1–89 with two different sets of assumptions:
If
is actually interpreted as internal energy,
(1–91)
then Equation 1–89 can be written as
(1–92)
which is equivalent to Equation 1–90 if we neglect
and interpret 
as 
. This interpretation is appropriate for liquids,
where variable-density effects are negligible. Note that the principal
variable is still called 'Static Enthalpy' in CFD-Post, although it
actually represents internal energy. Note also that, for liquids that
have variable specific heats (for example, set as a CEL expression
or using an RGP table or Redlich Kwong equation of state) the solver
includes the 
contribution in the enthalpy tables. This
is inconsistent, because the variable is actually internal energy.
For this reason, the thermal energy equation should not be used in
this situation, particularly for subcooled liquids.On the other hand if
and 
are neglected in Equation 1–89 then Equation 1–90 follows directly. This interpretation is appropriate
for low Mach number flows of compressible gases.
The thermal energy equation, despite being a simplification, can be useful for both liquids and gases in avoiding potential stability issues with the total energy formulation. For example, the thermal energy equation is often preferred for transient liquid simulations. On the other hand, if proper acoustic behavior is required (for example, predicting sound speed), or for high speed flow, then the total energy equation is required.
For turbulent cases, the diffusion term in the energy transport equation has
a contribution from turbulent mixing as well as conduction. For the Eddy Diffusivity
option, the
diffusion term becomes
 | (1–93) |
where 
is the conductivity, 
is the
turbulent diffusivity (related to Eddy Viscosity), and 
is the specified Prandtl number.
For the Anisotropic Diffusion option, the diffusion term now contains a tensor for the effective
diffusivity:
 | (1–94) |
with tensor
 | (1–95) |
where 
is the
anisotropic diffusion coefficient (Ani. Diffusion Coeff.), 
is the density, 
and 
are turbulence kinetic energy
and dissipation, 
is the Reynolds Stress tensor, and 
is the cross derivative weighting (Cross Deriv. Coeff.) when
and 
when 
.