Mostly what seems to happen is that self-awareness falls away, in that I'm totally absorbed by what I'm doing and I forget to think about being me, i.e. to construct my identity out of what's happening.
For me, it's being completely focused on a task. To the extent that the task occupies all of my short-term memory, leaving nothing for anything else distracting.
This is, I think, why it's annoying to be disturbed while in the flow state. The whole house of cards falls down when a disturbance occurs (example: the phone rings) while in flow - something necessarily gets tossed out of short-term memory to accommodate the interruption.
I get flow while gaming, writing, and programming. Ironically I've never had it for a physical thing. Mostly it's a blur because flow doesn't lend itself to observation, but the few concrete memories I have are all pretty similar.
I'm laser focused, with no distractions. I feel powerful, like I can do anything I put my mind to. There's a tight feedback loop and a clear goal that I'm striving for, and the goal is complex enough to be challenging but not so hard that I have no idea what to do. My mind is fast. Thoughts stream through me and all shards of my mind are aligned with no internal strife. And I'm not aware of my body or where I am or how much time is passing; the task fills my entire consciousness.
I'm intrigued by this question because I consider "flow" to be more associated with mentally demanding tasks than physical activities. I most commonly experience flow when coding. It's a state of almost anti-distraction, because the only thing I want to do is just the task at hand, in the most extreme way. Even the need to sleep becomes an annoyance! The main thing is that the most fun, interesting, stimulating thing you want to do is the task at hand, you're totally absorbed in it, and it's really unpleasant to be removed from it.
That said I do recognise a slightly different concept from what you're saying about physical activities. There have been times where I've done longer periods of exercise of moderate intensity exercise (e.g. 45 mins on a run) and experienced that state of your mind being 'totally blank'. But I would say the experience is a bit similar to mindfulness breathing exercises, where focussing in one thing your body is doing - maintaining your cadence, or just focussing on your breaths - achieves an inner calmness, almost an absence of thought.
By contrast, when I'm in mental "flow" actually my mind is full of thoughts. I'm confronted with an interesting problem and I just have to solve it. If I stop working on it my mind is just spinning with stuff related to the task at hand.
Athletes often describe flow states with:
"My limbs just move on their own."
"Intense, totally absorbing focus."
"My mind is totally blank. I'm not thinking at all, just doing."
I'm very interested homing in on this "thinking" that is often reported absence in flow. It clearly points to something other than "you brain firing on all cylinders doing crazy computations" because that is certainly all happening while playing an intense sport. I figured it would be helpful to compare against flow states in more prototypical "mental" activites.
So with all that context, when you are in peak experience/flow doing anything highly mental (math, coding, chess, etc), what is it like? And what sorts of "thinking" aren't happening, and what sorts of "thinking are happening?