Turbulence length scale

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The turbulence length scale, $TuL$ , is a physical quantity describing the size of the large energy-containing eddies in a turbulent flow.

The turbulence length scale is often used to estimate the turbulent properties on the inlets of a CFD simulation. Since the turbulence length scale is a quantity which is intuitively easy to relate to the physical size of the problem it is sometimes possible to guess a reasonable value of the turbulence length scale. The turbulence length scale should normally not be larger than the dimension of the problem, since that would mean that the turbulent eddies are larger than the problem size. Setting a very low turbulence length scale at the inlet will quickly dissipate all turbulent energy and give a reduced turbulence intensity.

Note that there are several common definitions of the turbulence length scale:

The classical definition used in the k-epsilon model and also used in the classical book on Turbulence Modeling by Wilcox the turbulence length scale is defined as:

$TuL_{classical} = C_\mu \frac{k^\frac{3}{2}}{\epsilon}$

Many people using a mixing-length based derivation get a slightly different definition of the turbulence length scale. This definition is also used in for example Fluent and OpenFOAM.:

$TuL_{mixing} = C_\mu^{3/4} \, \frac{k^\frac{3}{2}}{\epsilon}$

There is also another definition of the turbulence length scale based purely only on dimensional argument. This definition is use in for example CFX:

$TuL_{dimensional} =\frac{k^\frac{3}{2}}{\epsilon}$

$C_\mu$ is a model constant which in the standard version of the k-epsilon model has a value of 0.09.

From the different definitions given above the following relations can be computed:

$TuL_{mixing} \approx 1.8 * TuL_{classical}$

$TuL_{dimensional} \approx 6 * TuL_{mixing} \approx 11 * TuL_{classical}$

Estimating the turbulence length scale

It is common to set the turbulence length scale to a certain percentage of a typical dimension of the problem. For example, at the inlet to a turbine stage a typical turbulence length scale could be say 5% of the channel height. In grid-generated turbulence the turbulence length scale is often set to something close to the size of the grid bars.

Fully developed pipe flow

In pipe flows the turbulence length scale can be estimated from the hydraulic diameter. In fully developed pipe flow the turbulence length scale is ~3.8% of the hydraulic diameter (in the case of a circular pipe the hydraulic diameter is the same as the diameter of the pipe). Hence:

$l = 0.038 \; d_h$

Where $d_h$ is the hydraulic diameter. For codes using a turbulence length-scale based on the mixing-length (Fluent, Phoenics and CFD-ACE for example) replace 0.038 and 3.8% with 0.07 and 7%.

Wall-bounded inlet flows

When the inlet flow is bounded by walls with turbulent boundary layers, the turbulence length scale can be estimated (approximately) from the inlet boundary layer thickness. Set $l$ to 0.22 of the inlet boundary layer thickness. For codes using a turbulence length-scale based on the mixing-length (Fluent, Phoenics and CFD-ACE for example) use 0.4 of the inlet boundary layer thickness.

Turbulence length scale

From CFD-Wiki

Estimating the turbulence length scale

Fully developed pipe flow

Wall-bounded inlet flows

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