# How do we measure mean part or time average of velocity field?

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?

The time average of anything is:

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

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:

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

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.