The length scale corresponds, in a turbulent flow, to the size of the high energy eddies. Its value therefore depends on the configuration of the studied flow (flow around a NACA profile, a boat hull, a building, etc.). There is no general rule and the user will have to refer to the literature to estimate the value of \(l_t\).
Nevertheless, in the case of internal flow, it is possible to estimate the length scale as a percentage of the hydraulic diameter \(D_h\).
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In the case of a fully developed flow in pipe, it is generally common to use \(l_t = 0.038D_h\)
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In the case where the internal flow is delimited by walls, it is generally common to define \(l_t\) as being 22% of the boundary layer thickness.
Finally, the turbulence variables are calculated as follows:
\(k =\frac{3}{2}(U I_t)^2 \)
\(\epsilon =C_{\mu} \frac{k^{\frac{3}{2}}}{l_t}\)
\(\omega = \frac{\sqrt{k}}{l_t}\)
Where :
- \(I_t\) is the turbulence intensity,
- \(l_t\) is the turbulence length scale,
- \(U\) is the velocity scale,
- \(k\) is the turbulent kinetic energy,
- \(\epsilon\) is the turbulent dissipation,
- \(\omega\) is the turbulent dissipation rate.
The turbulence intensity or turbulence level is defined as the ratio between the turbulent fluctuation speed and the speed scale:
\(I_t = \frac{(\frac{1}{3}(u^2+v^2+w^2))^{\frac{1}{2}}}{U} \)
We generally assume that :
- the turbulence level is low when \(I_t <= 1 \)
- the turbulence level is medium when \( 1 <= I_t <= 5 \)
- the turbulence level is high when \( 5 <= I_t <= 10 \)
Like the scale of length, it will be necessary to refer, for each case study, to the literature to choose an adequate value of the intensity of turbulence. However, in the case of a fully developed turbulence flow in pipe we can use the following expression :