Having never done this my initial guess is as follows. You have a source and starting from that source point on the terrain every 1 meter in 32 directions find the height (simple interpolation) and choose the lowest direction and go there. Then keep doing that same process and connect each point. When you're done build a spline. I'm sure at that point you can have your particles follow the spline like boids or something naive and not use real physics at all. I bet you could even just generate geometry over the spline to slope into the terrain slightly and just render a flowing water shader on it.

If you still want water particles maybe a weight tower the spline might help curve the flow.

At the very least you need to represent some form of friction/drag, otherwise the water particles will accelerate without limit and swoosh up the sides as in your screenshots.