How do we measure velocity field? How do we measure velocity field $u(r,t)$?
i know that how to measure ordinary velocity. $v=\frac{dr}{dt}$
but what about velocity field?
what is difference between them?
 A: Ordinary velocity is just the velocity of an individual particle; a velocity field $\vec{u}(\vec{r}, t)$ is a function whose first argument lets you pick out the particle whose velocity you want to measure.
If you follow an individual particle, you can write its position as $\vec{x}(t)$, a vector function of time.  Its velocity is just the (time) derivative of that function:
\begin{equation}
\vec{v}(t) = \frac{d}{dt} \vec{x}(t)~.
\end{equation}
Here, you can see that the velocity is generally a function of time.
On the other hand, the velocity field $\vec{u}(\vec{r}, t)$ is a function of space and time. The space part specifies the position of the particle (or infinitesimal fluid element) whose velocity you want to measure.  So if the particle above happens to be at position $\vec{r}$ at some time $t$, we have the following relationship between the field and the velocity:
\begin{equation}
\vec{u}(\vec{r}, t) = \vec{v}(t)~,
\end{equation}
where $\vec{v}$ here refers just to the velocity of that specific particle (or fluid element) at time $t$.
A: One technique commonly used to measure a velocity field is Particle Image Velocimetry (PIV). In PIV, a number of small particles are suspended in the fluid you're measuring, and pictures of the fluid are made with either a standard digital camera or something like MRI.
Parts of each image are then correlated to parts of the next image, with the greatest correlation corresponding to the displacement of that chunk of fluid over the time between the images. From this you can extract the velocity at that one point, and repeating this gives you a pretty good map of the velocity field of the whole fluid.
