# Turbulence free-stream boundary conditions

In most CFD simulations it is necessary to specify values of the turbulence variables at the inlets. For example, if you are using a $k-\epsilon$ model you have to specify values of $k$ and $\epsilon$ at the inlets. This is often difficult and a source of uncertainty since the incoming turbulence is rarely known exactly. Most often you are forced to make a more or less educated guess of the incoming turbulence.

Estimating the turbulence model variables, like turbulent energy, dissipation or Reynolds stresses, directly is often difficult. Instead it is easier to think in terms of variables like the incoming turbulence intensity and turbulent length scale or eddy viscosity ratio. These properties are more intuitive to understand and can more easily be related to physical characteristics of the problem. For some guidelines on how these variables can be estimated see the respective CFD-Wiki page describing them.

## Formulas for computing the turbulence model variables

Once an appropriate turbulence intensity and turbulent length scale or eddy viscosity ratio has been estimated the primitive turbulence model variables can be computed from the following formulas:

### Modified turbulent viscosity

The modified turbulent viscosity, $\tilde{\nu}$, can be computed using the following formulas:

#### From the turbulence intensity and length scale

$\tilde{\nu} = \sqrt{\frac{3}{2}} \; (U \, I \, l)$

Where $U$ is the mean flow velocity, $I$ is the turbulence intensity and $l$ is the turbulent length scale.

Ideally with the Spalart-Allmaras model $\tilde{\nu}=0$, but some solvers can have problem with that so $\tilde{\nu}<=\frac{\nu}{2}$ can be used. This is if the trip term is used to "start up" the model. A convenient option is to set $\tilde{\nu}=5{\nu}$ in the freestream. The model then provides fully turbulent results and any regions like boundary layers that contain shear become fully turbulent.

### Turbulent energy

The turbulent energy, $k$, can be computed as:

$k = \frac{3}{2} \; (U \, I)^2$

Where $U$ is the mean flow velocity and $I$ is the turbulence intensity.

### Dissipation rate

The turbulent dissipation rate, $\epsilon$, can be computed using the following formulas:

#### From the turbulence length scale

$\epsilon = C_\mu^\frac{3}{4} \, \frac{k^\frac{3}{2}}{l}$

Where $C_\mu$ is a turbulence model constant which usually has a value of $0.09$, $k$ is the turbulent energy and $l$ is the turbulent length scale

#### From the eddy viscosity ratio

$\epsilon = C_\mu \, \frac{\rho \, k^2}{\mu} \, (\frac{\mu_t}{\mu})^{-1}$

Where $C_\mu$ is a turbulence model constant which usually has a value of $0.09$, $k$ is the turbulent energy, $\rho$ is the density, $\mu$is the molecular dynamic viscosity and $\frac{\mu_t}{\mu}$ is the eddy viscosity ratio.

### Specific dissipation rate

The specific turbulent dissipation rate, $\omega$, can be computed using the following formulas:

#### From the turbulence length scale

$\omega = C_\mu^{-\frac{1}{4}} \, \frac{\sqrt{k}}{l}$

Where $C_\mu$ is a turbulence model constant which usually has a value of $0.09$, $k$ is the turbulent energy and $l$ is the turbulent length scale

#### From the eddy viscosity ratio

$\omega = \frac{\rho \, k}{\mu} \, (\frac{\mu_t}{\mu})^{-1}$

Where $k$ is the turbulent energy, $\rho$ is the density, $\mu$is the molecular dynamic viscosity and $\frac{\mu_t}{\mu}$ is the eddy viscosity ratio.