Disclaimer: IANA physicist.
The Vornado people might have developed an extra-powerful design for a fan blade, but to my untutored eye, it’s still just a fan, no matter how badly the Vornado people want me to believe that it’s “not a fan, it’s an air circulator”. My Alert Consumer instincts tell me that ALL fans are air circulators, and I detect a certain amount of hype there.
For my money, a big $15 box fan is better for keeping cool, because it blows a bigger stream of air. You don’t need a small, concentrated stream of air; what you want is a big wave of air, hitting you all over your body, evaporating sweat as it goes, which a $15 box fan does admirably.
So I’ve never tried a Vornado, but as I said, it’s because my Alert Consumer instincts tell me it’s just hype, and I make do with less ambitious but just as effective mere “fans”.
My grandma always had one of those slow-moving floor-model “air circulators” that look like a round plastic ottoman. She placed great faith in the manufacturer’s assurance that she was moving cooler air from the floor into the room, but her house was always incredibly hot and stuffy, because there was never any air moving.
My experience, living in a 1920s two-story house with no air conditioning, is that there is really no detectable cooler air nearer the floor, and thus the height of the fan (or hyped “air circulator”) makes no difference. What you want is the fan blowing on you, where you are.
Our summer routine goes like this:
Have every possible window open all night, upstairs and downstairs, with four big air-moving fans set up in four upstairs windows, blowing out. This pulls cool night air through the entire house.
Then, get up early and close all the windows before 7 a.m. This is very important, because the longer you wait in the morning, the warmer the air becomes inside the house. Ya wanna preserve that lovely 70 degree air, not shut the windows after the house has already heated up to 80, which it will have done by 8 a.m.
And if you have storm windows that slide up and down, close the storm windows, too. This keeps even more hot air out.
And cover any possible windows with drapes, curtains, or shades, so as to exclude direct sunlight with its load of infrared.
Then the house will very, very slowly heat up to the low to mid-80s by late afternoon, when it’s in the mid- to upper-90s outside. I have actually measured this with my trusty minimum-maximum thermometer. As long as your comfort zone isn’t set to “I need it at 72 to be comfortable”, the mid-80s is actually quite comfortable. Humans evolved in the tropics, ya know, so temps in the 80s are not inimical for us. Remove some clothes.
My customary summertime lounging-around-the-house clothing consists of men’s cotton knit boxers and baggy t-shirts.
Then about 7:30 or 8 p.m. , you go around and open up all the windows and set the fans to exhausting again (no sooner than that or you’re just letting in a lot of 90s heat, when the inside is still only in the 80s).
Then you have an assortment of fans to sit in front of around the house, which increase the coolness sensation.
It’s a fair amount of management, opening and closing windows, but it does work.
Also, please note that pulling cool air into the house works MUCH better than attempting to push cool air into the house. Our upstairs master bedroom is always the hottest room in the house. And if you set up the fan at bedtime blowing into the room, even with the second window open, it’s still miserably hot in there. But if you set up the fan blowing out, and open the other window, it cools off in about 5 minutes.
As for window screens, go down to your friendly neighborhood TrueValue hardware store: they normally stock a selection of collapsible wire-and-wood window screens that you prop up in the open window. These are ready-to-use, unlike the screen kits Rick is talking about.
Places like Lowe’s and Menards might have them, too. Hint: measure the dang windows before you go down there, to save yourself an extra trip back home to measure. 
And no, no fan, not even a Vornado, can remove moisture from the air. All they can do is move the humid air around.