# Inlet Turbulence Parameters

CFD simulations require specification of turbulence variables at inlet boundaries.

When measured turbulence data is available you can explicitly specify turbulence variables. For example, eddy viscosity for the Spalart-Allmaras (SA) model, turbulent kinetic energy and eddy frequency/dissipation rate for k-ε and k-ω based models. When measured data is not available, there are estimations for the turbulence values that are based on the turbulence intensity and turbulence length scale or eddy viscosity ratio.

## Turbulence Intensity

The turbulence intensity (I) is defined as the ratio of the root-mean-square of the turbulent velocity fluctuations ( $u\text{'}$ ) and the mean velocity ( $\overline{u}$ ):

where ${u}^{\prime}=\sqrt{\frac{\left(\overline{{u}_{x}^{\text{'}}{}^{2}}+\overline{{u}_{y}^{\text{'}}{}^{2}}+\overline{{u}_{z}^{\text{'}}{}^{2}}\right)}{3}}=\sqrt{\frac{2k}{3}}$ and $\overline{u}=\sqrt{\left({\overline{u}}_{x}^{2}+{\overline{u}}_{y}^{2}+{\overline{u}}_{z}^{2}\right)}$ .

For fully developed internal flows the turbulence intensity can be estimated as

where $R{e}_{h}=\frac{\rho \overline{u}{D}_{h}}{\mu}$ is the Reynolds number and ${D}_{h}$ is a hydraulic diameter.

- 5 percent < $I$ < 20 percent for rotating machineries, for example, turbines and compressors.
- 1 percent < $I$ < 5 percent for internal flows.
- $I$ ~0.05 percent for external flows.

## Turbulence Length Scale

The turbulence length scale represents the characteristic size of the turbulent eddies within a flow field. This parameter is often used to characterize the nature of the turbulence and appears in some form in nearly every turbulence model. RANS turbulence models have different definitions of the turbulence length scale based on the turbulence model and the type of application.

- $l=0.038{D}_{h}$ for fully developed internal flows.
- $l=0.22\delta $ for developing flows.

where $\delta \approx 0.382\frac{x}{R{e}_{x}^{1/5}}$ is the turbulent boundary layer thickness over a flat plate and $R{e}_{x}=\frac{\rho \overline{u}x}{\mu}$ is the Reynolds number and x is the distance from the start of the boundary layer.

## Eddy Viscosity Ratio

The eddy viscosity ratio ( ${\nu}_{r}$ ) is the ratio between the eddy viscosity ( ${\nu}_{t}$ ) and fluid kinematic viscosity ( $\nu $ ). The eddy viscosity ratio is set between 1 and 10 for internal flows, while it should be between 0.2 and 1.3 for external flows.

## Inlet Turbulence Specification for the SA Model

- Option 1: Eddy viscosity ${\nu}_{\text{t}}$
- Option 2: Turbulence intensity $I$ and length scale $l$ . For the eddy viscosity, ${\nu}_{t}=\sqrt{\frac{3}{2}}\overline{u}Il$ .
- Option 3: Viscosity ratio ${\nu}_{r}$ . For the eddy viscosity, ${\nu}_{t}=\nu {\nu}_{r}$

## Inlet Turbulence Specification for the k-ε Models

- Option 1: Kinetic energy $k$ and dissipation rate $\epsilon $
- Option 2: Turbulence intensity $I$ and length scale $l$ . For the kinetic energy, $k=\frac{3}{2}{(\overline{u}I)}^{2}$ . For the dissipation rate, $\epsilon ={C}_{\mu}\frac{{k}^{3/2}}{l}$ , where ${C}_{\mu}$ = 0.09.
- Option 3: Turbulence intensity $I$ and viscosity ratio ${\nu}_{r}$ . For the kinetic energy, $k=\frac{3}{2}{(\overline{u}I)}^{2}$ . For the dissipation rate, $\epsilon =\frac{{C}_{\mu}{k}^{2}}{\nu {\nu}_{r}}$ .

## Inlet Turbulence Specification for the k-ω Models

- Option 1: Kinetic energy $k$ and eddy frequency $\omega $
- Option 2: Turbulence intensity $I$ and length scale $l$ . For the kinetic energy, $k=\frac{3}{2}{(\overline{u}I)}^{2}$ . For the eddy frequency, $\omega =\frac{\sqrt{k}}{l{C}_{\mu}{}^{1/4}}$ .
- Option 3: Turbulence intensity $I$ and viscosity ratio ${\nu}_{r}$ . For the kinetic energy, $k=\frac{3}{2}{(\overline{u}I)}^{2}$ . For the dissipation rate, $\omega =\frac{k}{\nu {\nu}_{r}}$ .