Reynolds-averaged Navier–Stokes equations (RANS) is one of the approaches to turbulence description. Physical quantities, like for example velocity $u_i$, are represented as a sum of a mean and a fluctuating part:

$$ u_i = \overline{u_i} + u'_i $$

where the Reynolds averaging operator $\overline{\cdot}$ satisfies, among the others, relations: $$ \overline{\overline{u_i}} = \overline{u_i}, \qquad \overline{u'_i} = 0 $$ which distinguish it from other types of averaging. In fluid dynamics Reynolds operator is usually interpreted as time averaging: $$ \overline{u_i} = \lim_{T \to \infty} \frac{1}{T}\int_t^{t+T} u_i \,dt $$

The above construction seems to be universal for me and is likely to be used in other areas of physics. Where else does one encounter Reynolds averaging?

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    $\begingroup$ I don't have specific examples handy, but I would say anywhere that you only care about the mean of the temporal signal. The Reynolds averaging is just a low-pass filter on the time signal, so I would imagine any number of applications are possible from communications, electronics, control theory, etc.. Anybody who uses a low-pass filter on a time-varying signal. $\endgroup$ – tpg2114 Jul 24 '13 at 16:12
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    $\begingroup$ Oliver Penrose wrote a useful (and very detailed) article in Rep. Prog. Phys. in 1979 entitled, "Foundations of statistical mechanics." In that work, he has some very useful discussions on the differences between time-averages and ensemble averages (e.g., the last paragraph before section 1.2). $\endgroup$ – honeste_vivere Oct 18 '14 at 14:12
  • $\begingroup$ I should also mention that time-averages are not always appropriate. In some cases, a time-average amounts to a bad low-pass filter. I say bad because unlike a Fourier-based (or some other basis) low-pass filter, a time-average mixes neighboring data points potentially convolving two signals that are completely unrelated. In solar wind data analysis, for instance, using a time-average can be a bad idea because you start to mix things that can be hundreds of km apart and may be from completely separate structures. $\endgroup$ – honeste_vivere Oct 18 '14 at 14:15

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