I know a lot of people will either just shake the fermentor, or use a wand for 30 seconds or so, but I’d like a more controlled approach. Can anyone offer any guidance, or provide any literature/science on how much I should be adding specifically?
I’m very sparing with oxygenation. I use an in-line stone with about 20 ft of line that delivers wort from the kettle, through the chiller, and to the fermenter. I probably have 20 5-gal batches on the little red oxygen canister that I’m still using.
I pitch a sufficient amount of healthy yeast and don’t really aerate.
I don’t believe in one size fits all with the use of oxygen. Too many “it depends” situations. But my rule of thumb is about a minute of O2 in normal gravity beers. I pitch active starters by the way. With my high gravity beers I re-oxygenate after 3hrs and again after 12hrs.
As for science, 8ppm dissolved O2 is the most you can get using air, and 10ppm is about the most you’ll ever see recommended. BUT I think the most important thing is consistency, finding a procedure that gives you good results and then being able to repeat it. You can get cheap, in-line flow meters for your O2 or air line, and consult charts to determine ideal flow rate and time. But you can get equally good results from something like “X twists of the valve, Y Mississippis” (works fine for me.)
Yup, I count to 60 in my head while moving the wand around the bottom of the fermenter on each count. Highly scientific
I use dry yeast exclusively due to bad experiences with receiving DOA liquid cultures on delivery. I do not have a reliable LHBS that stocks heathy liquid yeast. So… in my world aeration is optional. During the drying process, the yeast manufacturer incorporates key nutrients needed for cell growth. Consequently, dry cultures can be pitched into oxygen-poor wort. Aeration won’t hurt, but it’s not necessary. YMMV
Even with high gravity beers?
Above 1.075ish, I’ll hit it with my MixStir for a minute. Below that I don’t bother. And I hardly ever make beers in excess of 1.075 any more.
I agree that if

I pitch a sufficient amount of healthy yeast and don’t really aerate.
is the best method.
Most the time I just shake my carboy, especially if I’m using a dry yeast, because I want everything to be kind of uniform before it settles out and ferments . I have a wand but noticed higher lag times because the yeast would work on the oxygen before the sugars. I rarely use my wand now and after I ran out of a can I never even bought another one. Seemed like more hassle to me.
With the little red O2 cannisters and O2 stone, I typically do about 90 seconds for an average gravity ale, 120 seconds for big daddy OG.
Lagers will get 2 minutes at 48F.
I have recently had better results using my O2 stone AND shaking the wort at the same time. I feel that more of the oxygen gets mixed in with the wort better leading to very short lag times.
I’d recommend aeration rather than oxygenation unless you have or can borrow a DO meter. Saturation levels are close enough to the recommendations for ales and lagers that I think the risk of using O2 blind outweigh the potential benefit.
Here is another take on oxygenation. I have considered going to an in-line stone, like Martin does, but have been using this procedure for years and it works fine.
On the end of my transfer hose that goes from the chiller to the fermenter, I have a small piece of 3/8" copper tubing that has about 20 holes drilled in it angled slightly toward the exit end of the tube. This allows air to be sucked in while transferring the wort. It works along the same principle as an aerator on the end of a kitchen faucet that keeps the water from splashing when it hits the sink. The tubing end of the hose is placed in the top opening of my conical and a sanitized paper towel is wrapped around the opening and the transfer hose to keep nasties out. It works well and is a cheap and easy way to get about 8 PPM of oxygen into the wort.
Just another idea.

I pitch a sufficient amount of healthy yeast and don’t really aerate.
Ditto. It splashes quite a bit going into my carboy anyway, I figure that’s probably good enough. Maybe I’m wrong. Maybe I’m not.
I’ve stopped aerating my wort in favor of aerating my yeast. While I’m chilling I dump my slurry into a jug along with a little wort and shake for about a minute until it’s all foam, let it stand for 30 min then pitch into fermenter. I have activity within an hour.
I stole grandma’s oxygen tank, she doesn’t need it anymore. 60 second blast, pitch the yeast.
Thanks Granny!

I’ve stopped aerating my wort in favor of aerating my yeast. While I’m chilling I dump my slurry into a jug along with a little wort and shake for about a minute until it’s all foam, let it stand for 30 min then pitch into fermenter. I have activity within an hour.
I’ve been thinking about going this route when repitching. I already do it to some extent shaking and swirling the yeast pack up into the liquid in the jar. My thought is then to do as you say, put my measured amount of slurry in a jug with some wort and “shake it like it owes me money” and let it settle. Makes more and more sense as I look into it. The result should be yeast in a condition analogous to dry yeast (that is, not requiring well-aerated wort,) which is becoming my preference for a first generation pitch. Over-aeration of wort has the potential to do way more harm than good, it seems. Somebody please correct me if I’m misguided.
Use a Speidel fermenter with caps so you can put it sideways.
Have 2/3 wort and 1/3 air.
Rock it sideways for 3 min.
If the temperature of the wort is constant, you will introduce a constant amount of oxygen in the wort that won’t over oxygenate.