= Turbulence intensity
{title2=$I=u_{\rm rms}/U$}
\b[Turbulence intensity] expresses the size of a measured velocity fluctuation relative to a specified mean speed. For a streamwise velocity record $u(t)$, define
$$
U=\langle u\rangle,\qquad u'=u-U,\qquad
u_{\rm rms}=\sqrt{\langle (u')^2\rangle},\qquad
I=\frac{u_{\rm rms}}{|U|},\quad U\ne0.
$$
Here the symbol in the square-root definition is $u_{\rm rms}$, the <root mean square> fluctuation, and angle brackets denote the averaging operation. The dimensionless intensity is often reported as $100I$ percent. The averaging interval and measured velocity component are part of the definition.
For four equally weighted observations $9,11,9,11\,\mathrm{m\,s^{-1}}$, the mean is $10\,\mathrm{m\,s^{-1}}$ and each fluctuation is $\pm1\,\mathrm{m\,s^{-1}}$. Their record root mean square is therefore $1\,\mathrm{m\,s^{-1}}$, giving $I=0.10$, or 10 percent. This calculation describes that record; it does not estimate every fluctuation timescale from four observations.
A primary wind-tunnel investigation used <hot-wire anemometry>, recording mean and fluctuating signals to characterize the <wind-tunnel turbulence intensity>. Its bandwidth comparison demonstrates that integrating only part of the measured fluctuation spectrum changes the recovered root mean square.<Free-Stream Turbulence Intensity in the Langley 14- by 22-Foot Subsonic Tunnel by Dan H. Neuhart and Catherine B. McGinley>[NASA/TP-2004-213247, printed pp. 7–9: probes and signal analysis; Figure 20: restricted-bandwidth integration]{ref} An instrument's temporal response, filtering and record duration therefore affect the reported intensity. Slow changes of the mean can also contribute to a measured fluctuation unless the averaging convention separates them.
A three-component convention instead uses
$$
I_{3D}=\frac{\sqrt{\bigl(\langle (u')^2\rangle+\langle (v')^2\rangle+\langle (w')^2\rangle\bigr)/3}}{|U|}
=\frac{\sqrt{2k/3}}{|U|},
$$
where $k$ is the <turbulent kinetic energy> per unit mass. This agrees with the streamwise definition when the three fluctuation variances are equal, as in <isotropic turbulence>. A nonzero root mean square alone does not establish <turbulence>: a periodic laminar velocity oscillation also has a nonzero value. The ratio also becomes ill-conditioned near zero mean speed.
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