TITLE: How does the eddy velocity scale with its time scale?
QUESTION [0 upvotes]: According to Kolmogorov's energy casacade model, if we have a flow with inertial velocity scale $\mathcal{V}$, inertial length scale $\mathcal{L}$ then we can calculate the eddy velocity of an eddy with length scale $l$ according to
$$v_l^3 \sim \epsilon l$$
Where $\epsilon =\mathcal{V}^3/\mathcal{L}$. Furthermore, this eddy has a timescale of $\tau_l \sim l/v_l$. Take now as example
\begin{equation}
\begin{split}
\mathcal{V}=1\\
\mathcal{L}=1\\
l=1
\end{split}
\end{equation}
Then also $v_l \sim 1$ m/s and $\tau_l \sim 1$ s. In the same flow, now consider an eddy that has $\tau_l\sim0.5$ s. What is the characteristic velocity of this eddy? The problem is that the eddy length scale also varies with the eddy time scale, so both $v_l$ and $l$ are unknown. Does anyone have an idea?

REPLY [1 votes]: We know that
$$
v_l^3\sim\epsilon l,
$$
so
$$
v_l^2\sim\epsilon\tau_l.
$$
In your case, $\epsilon=1$ and $\tau_l=0.5$s so $v_l=0.71$m/s.
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