Good question. Many Angelenos don't even know what the LA River is, or that LA even has a river. Many do assume that it's just a flood control channel or large rain gutter.
Regarding your question about water flow, there is usually a small or medium-sized trickle of water running down the center. After heavy rains, it has a much higher water level, with dangerously fast-rushing water. It's not uncommon for our local news media to report stories of people swept away and drowned in the LA River because they slipped and fell in; usually the media announces warnings during heavy rains to stay away from jogging and bicycle paths that line the river. I remember back in the early 1990s when there was a huge flood in one of the sections of the LA River that's left in its natural state, the Sepulveda Dam basin, in the San Fernando Valley. People had to be rescued from their cars as the floodwaters rose.
Before it was lined with concrete, the LA River was basically an arroyo or near-dry riverbed. For centuries, pre-Spanish colonization, it was a source of water for the native Tongva population. Because the river never had rushing water year-round, it never dug itself a permanent course. However, during very heavy rains, flooding would be a problem; water would overflow its banks, and sometimes the river itself would even change courses (think of drops of rainwater falling and flowing on a windowpane; you can see little streams of water running down but then occasionally the same streams will shift in one direction, and then yet another direction). A very disastrous flood in 1938 was the motivation to eventually line the river (and other rivers in Los Angeles County like the Rio Hondo and San Gabriel River) in concrete so that it would permanently follow a course. Of course back then, they didn't think of the environment. Even today, the LA River, and the other LA County rivers, aren't completely lined in concrete; there are sections that were left natural; these sections have become wildlife habitats with plants, species of waterfowl and fish. It's a shame that they couldn't have left these rivers unlined. I don't see why they just couldn't have dug a deeper channel for these rivers, rather than lining them with concrete, and then gradually plants and wildlife would have returned.
Here are some old photos.
This is from 1924. The bridge at the center of the photo is the Broadway Bridge. But look at the LA River. It's in its completely natural state, with plant life in it.
LAPL
Here's a view looking north from the Broadway Bridge in 1988.
LAPL
But here it is in the late 1930s, before the river was concreted.
LAPL
Here's a pic I gook on January 2, 2012, of the same general view. The metal truss railroad bridge has been replaced with a double-tracked concrete bridge for the Metro Gold Line light rail, and a maintenance yard for light rail trains has been built.
Photo by me
Here's an undated photo of the pre-concrete-lined LA River through the industrial section of LA County; you can clearly see here that the river could be volatile when filled with rushing water, and that it could dangerously change courses.
LAPL
Here's an undated photo, showing the 7th Street Bridge and a very dry non-concreted LA River. During this period, the LA River was a subject of jokes in Los Angeles because of its dry riverbed.
USC Archive
Here is the iconic 6th Street Bridge, back when it had more details and its original street lamps--and the unconcreted river. You can see some water flowing. I guess if the river were never lined with concrete, there would have been no drag race scene in the film "Grease."
USC Archive