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So I understand the principles of two-point time correlations (covariance) and how you can just normalise that to get the autocorrelation. I also understand the integral time scale which is quite intuitive really and tells us the largest scales in the flow.

Now Taylor's microscale is sightly trickier it seems. We look at the slope of how quickly the autocorrelation decays from 0-seperation which seems to be related to a short-time microscale $\lambda$ by an inverse-square law!? For clarity I am of course talking about

$$ -\frac{2}{{\lambda}^2} = \frac{d^2\sigma}{d{\tau}^2} $$ where $\tau$ is the time seperation and $\lambda$ is the Taylor's microscale. So can anyone explain what Taylor was thinking and how he got to this? Was it just a hypotheses which seemed to work or did he justify in theoretical ground? (probably a combination)

P.S. I am of course a mechanical engineering student and not a mathematician however still interested in these details so long as it could be explained simply.

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