I am looking for the equation of the velocity field $\textbf{u} = u(x,y)\hat{i} + v(x,y)\hat{j}$ of a two-dimensional steady flow, which streamlines look similarly to this image: example image of the desired streamlines.
The streamlines do not need to look exactly like the provided image, but I am looking for something that follows basic idea behind it (a bunch of differently sized vorticies scattered randomly with nice smooth flow between them).
So far I have found the equation $\textbf{u} = (0.1x^3 + y)\hat{i} + (-\sin{x} + \cos^2{x})\hat{j}$, which streamlines look like this: one vortex streamlines. It looks good, but there is only one vortex and not much happening around it.
From my fluid dynamics classes, I remember I can create an initial/boundary value problem for Euler/Bernoulli equation, but it appears like a too complex solution for a seemingly simple problem.
How can I derive the equation that satisfies my conditions in an simpler way or is there already a solution to this or similar problem?