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Turbulence length scale

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

The turbulent length scale is often used to estimate the turbulent properties on the inlets of a CFD simulation. Since the turbulent length scale is a quantity which is intuitively easy to relate to the physical size of the problem it is easy to guess a reasonable value of the turbulent length scale. The turbulent 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.

In the k-epsilon model the turbulent length scale can be computed as:

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

Please note that some CFD codes (Fluent, Phoenics and CFD-ACE for example), use a different length scale definition based on the mixing-length. Then the following formula should be used instead:

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

This means that the turbulence length-scale variable used in these codes is about two times larger than the length-scale variable used in other codes.

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

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.

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