I haven't (and will not) invest any time doing the numbers but, a quick think about the Dyson fans leads me to conclude the whole thing is a big marketing scam. Kudos for pulling it off, but the thing ain't magic.
For starters, it has blades. It definitely has blades. They are simply concealed. It's like saying my computer has no fans when the fans are not visible from the outside. Nonsense.
Second, it takes a certain amount of power to accelerate a mass of air to a given speed. That does not change magically because the thing has a Dyson sticker on it. The most efficient way to accelerate a mass of air is, wait for it, a properly designed fan. It am pretty certain the Dyson contraption is actually much less efficient than a properly designed plain fan which produces the same airflow. It absolutely could not be more efficient, no way. Nowhere in conventional physics do you get more --of anything-- than you are entitled to.
In fact, the larger the fan the more efficiently it can accelerate a mass of air. Because the Dyson fans are constrained by the small space of the base they have to use screaming little fans to push against the back-pressure produced by the path the air has to follow.
Maybe not as sexy, but a really well designed fan with optimized airfoils and balanced rotor IS magical in the sense that it can be so much better than the stamped metal contraptions one usually finds at the store. I have seen fans like that come-up here and there but they are rare.
I did a project many years ago where we had to design and optimize an efficient cooling system for a 1500 Watt LED array. We ultimately ended-up going with a hybrid fluid/air system. However, for the bulk of the project we tried to make an air-only approach work because it was deemed simpler and easier to manufacture. I learned a lot from running tons and tons of FEA fluid and thermal simulations as making a number of prototypes. There is no free lunch (AKA: Conservation of Energy).