# Computing relative velocities,

When I compute a relative velocity vector to account for some background flow interaction with a rigid body, say, the flow is a horizontal wind, with wind gradient $\vec{u} = < u_x, u_y >$, I usually subtract its components from the velocity vector $\vec{v} = < v_x, v_y >$ to get a relative velocity vector

$$\vec{v}_{rel} = < v_x - u_x, v_y - u_y >$$

But what if the background flow were not simply horizontal and linear?

Let's say the flow were curvy -- for instance, a flow that looks something like the (nonlinear) sine wave.

How would I then compute the relative velocity of the rigid body in this flow?

Would it still be subtracting the wind gradient from the velocity vector?

That doesn't seem intuitive to me ...

Thanks,

It should still be just subtracting the wind velocity from the velocity of the body. However wind velocity itself will now be a function $x,y$. Effectively you would be doing this $$v_{rel}(x,y)=<v_x−u_x(x,y),v_y−u_y(x,y)>$$ The relative velocity will be a function of the coordinates.
This of it this way. The wind velocity as different points depend on $(x,y)$ in general. Your case of just horizontal/linear are simplest functions that the wind velocity can assume. The concepts involved do not change when the wind is curvy.