Non-laminar flow is often hard to characterize with much mathematical rigor (though in certain special cases and geometries may be done with a little limitation). The phenomena depends heavily on channel geometry. For example, A backflow eddy can develop from a sudden stepwise increase in channel height/width. The Reynolds number of the fluid flow under examination will determine the strength and formation of the eddy. The formation of the eddy results from the water lamina closest to the wall being deformed by the sudden channel widening due to friction. Viscous interactions between adjacent lamina propagate away from the wall and result in the curling of local flow lines into an eddy. The Reynold's number will determine if an eddy forms, or if there is a smooth lamellar flow into the new channel width.
Ultimately, the Reynolds number is a very useful quantity for characterizing different types of fluid flow.
In the bulk of the fluid, most eddies will be locally co-rotating over a length scale that depends nontrivially on the fluid viscosity, as local lamina will tend to drag each other rather than slip. Of course, on some larger scale, this will appear as turbulent motion. Again, the Reynolds number is useful in this characterization.