### 1.2.1. Transport Equations

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:

#### 1.2.1.1. The Continuity Equation

| (1–83)

#### 1.2.1.2. The Momentum Equations

| (1–84)

Where the stress tensor, , is related to the strain rate by

| (1–85)

#### 1.2.1.3. The Total Energy Equation

| (1–86)

Where is the total enthalpy, related to the static enthalpy by:

| (1–87)

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.

#### 1.2.1.4. The Thermal Energy Equation

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 .

| (1–88)

The mechanical energy equation is derived by taking the dot product of with the momentum equation (Equation 1–84):

| (1–89)

Subtracting this equation from the total energy equation (Equation 1–86) yields the thermal energy equation:

| (1–90)

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:

| (1–91)

This equation can be derived from Equation 1–90 with two different sets of assumptions:

- If is actually interpreted as internal energy,

  | (1–92)

  then Equation 1–90 can be written as

  | (1–93)

  which is equivalent to Equation 1–91 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 1–90 then Equation 1–91 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.

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