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How do we measure mean part or time average of a (known) velocity field?

In other words, if I know velocity field how can I measure its time average or mean part?


What is time average in general?

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The time average of anything is:

$$\left<f\right> = \frac{1}{T}\int_0^T f(\tau)d\tau$$

where the accuracy of the average improves as $T \rightarrow \infty$, $T$ being the total time interval.

So, if you only know the velocity field at a single instant in time, you cannot know it's average.

If you have multiple snapshots of time, you would compute the time average with a discrete form of the above integral using the snapshots:

$$ \left<f\right> = \frac{1}{\sum_i dt_i}\sum_{i=1}^{N}f(t_i)dt_i = \frac{1}{N}\sum_{i=1}^{N}f(t_i) $$

where $N$ is the number of snapshots and $dt_i$ is the discrete time interval between snapshots and the second equation is only valid if $dt_i = dt_j$ for all $i, j$. The accuracy improves both as $dt \rightarrow 0$ and $N \rightarrow \infty$. In other words, as the discrete sum approaches the interval.

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