## 2.2. Eddy Viscosity Turbulence Models

One proposal suggests that turbulence consists of small eddies that are continuously forming and dissipating, and in which the Reynolds stresses are assumed to be proportional to mean velocity gradients. This defines an “eddy viscosity model”.

The eddy viscosity hypothesis assumes that the Reynolds stresses can be related to the mean velocity gradients and eddy (turbulent) viscosity by the gradient diffusion hypothesis, in a manner analogous to the relationship between the stress and strain tensors in laminar Newtonian flow:

| (2–9)

where is the eddy viscosity or turbulent viscosity, which must be modeled.

Analogous to the eddy viscosity hypothesis is the eddy diffusivity hypothesis, which states that the Reynolds fluxes of a scalar are linearly related to the mean scalar gradient:

| (2–10)

where is the eddy diffusivity, and this has to be prescribed. The eddy diffusivity can be written as:

| (2–11)

where is the turbulent Prandtl number. Eddy diffusivities are then prescribed using the turbulent Prandtl number.

The above equations can express turbulent fluctuations in terms of functions of the mean variables only if the turbulent viscosity, , is known. Both the - and - two-equation turbulence models use this variable.

Subject to these hypotheses, the Reynolds averaged momentum and scalar transport equations become:

| (2–12)

where is the sum of the body forces, and is the Effective Viscosity defined by:

| (2–13)

and is a modified pressure, defined by:

| (2–14)

The last term in Equation 2–14,

involves the divergence of velocity. It is neglected in ANSYS CFX, although this assumption is strictly correct only for incompressible fluids.

The treatment of the second term in the right-hand side of Equation 2–14 depends on the expert parameter `pressure value option`, which can take the following values:

Option | Description
--- | ---
1 | When is required (for example, to calculate material properties), it is derived from using Equation 2–14.
2 This is the default. | When is required (for example, to calculate material properties), it is approximated as being equal to . Option 2 is an approximation that may be numerically better-behaved than option 1.
3 | is not computed using the term . However, this term is incorporated directly in the momentum equation. This option is mathematically equivalent to option 1, but may differ numerically.

The Reynolds averaged energy equation becomes:

| (2–15)

Note that although the transformation of the molecular diffusion term may be inexact if enthalpy depends on variables other than temperature, the turbulent diffusion term is correct, subject to the eddy diffusivity hypothesis. Moreover, as turbulent diffusion is usually much larger than molecular diffusion, small errors in the latter can be ignored.

Similarly, the Reynolds averaged transport equation for Additional Variables (non-reacting scalars) becomes:

| (2–16)

Eddy viscosity models are distinguished by the manner in which they prescribe the eddy viscosity and eddy diffusivity.

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