Integral length scale

The integral length scale measures the amount of time a process is correlated with itself. In essence, it looks at the overall memory of the process and how it is influenced by previous positions and parameters. An intuitive example would be the case in which you have very low Reynolds number flows (e.g., a Stokes flow), where the flow is fully reversible and thus fully correlated with previous particle positions. This concept may be extended to turbulence, where it may be thought of as the time during which a particle is influenced by its previous position.

$\Tau = \int_{0}^{\infty} \rho (\tau) d\tau$

Where $\tau$ is the time and $\rho$ the autocorrelation.

In isotropic homogeneous turbulence, the integral length scale $\ell$ is defined as the weighted average of the inverse wavenumber, i.e.,

$\ell=\int_{0}^{\infty} k^{-1} E(k) dk \left/ \int_{0}^{\infty} E(k) dk \right.$

where $E(k)$ is the energy spectrum.