A swamp cooler circulates water and evaporates some of it to cool some water. It then blows air through the cooler water to cool the air.

You won't benefit much by going through all the steps, and will get at least as good - probably more efficient - results by just spraying some water up into the air and letting it fall back into the pool.

"Cascading" the water back into the pool, and blowing some air through it to increase the amount of evaporation (and cooling) might get a little more cooling out of less pumping.

One of the side benefits of a "decorative waterfall" in any kind of pool is that it can provide some cooling, but also that it stirs the water so that temperature differences are minimized. In a deep enough pool, without too much circulation, the water at the bottom in the deepest part will tend toward the temp at which water density is the maximum (about 34 F = 1.1 C). Pumping this cooler water up and spraying (or dribbling) it back on top may help to even out the temp so the hot upper layers (where you swim?) get mixed in and don't "feel as hot."

In a typical swimming pool, the circulation pumps you need to maintain chlorination and filtering likely will produce enough circulation to eliminate all but minor temp gradients. If you have a "skimmer" type filter system, that sucks off the top layers it might be more likely that you have cooler water under the diving board that would help make the pool seem cooler with just a little more - or different - circulation.

If your pool is "ye ole swimmin' hole" you may have cooler water in the bottom of a deep spot that you could suck up and dribble down a waterfall to get a fair amount of actual cooling. (And the crawdads will appreciate a little aeration of that bottom water.)

A problem with evaporative cooling of large amounts of water is that you have to evaporate lots of water to significantly cool a little of it, so replenishment and treatment of the incoming water might cost more than putting a mechanical refrigeration unit in line with the filter pump.

As a SWAG (not expected to be too accurate) evaporating a gallon of water might cool a gallon by the difference between ambient air temp and the current dew point. (That probably overestimates what you can get.) Each gallon of water made that much cooler will bring your 20,000 gallons down by 1/20,000 of that many degrees.

Althugh contact with ambient air may produce a lot of heating of the pool water, it's likely that most of the load is from solar insolation. Shading the pool, especially during peak sun periods, might be as helpful as trying to cool it, if you can figure out a clever way to do it. Circulation of the water offers some prospects of benefits - and a nice waterfall could be pretty enough to take your mind off the heat.

John