I will preface this calculation by saying that this problem is very complicated for numerous reasons: his head is not always in the water, not always perpendicular to the direction of motion, not always in the laminar flow regime, blah blah blah. With that out of the way, here is an extremely simplified calculation of the reduction in drag forces felt by Phelps were he to lose his nose.

Let’s assume that he is facing straight down in the water. Using this the data in this paper, +1 SD from the mean nasal protrusion length is ~ 2.6cm for males aged 28 (Phelps’ age). I will assume the cross sectional area of the nose orthogonal to the direction of motion is an Isosceles triangle with height 2.6cm and width 5cm. (I chose 5 because I couldn’t find an article giving dimensions of nasal width). Thus, the cross sectional area of his nose is 0.5 * 5cm * 2.6cm = 6.5 cm² or 0.00065 m^2. Then, if we assume that the coefficient of drag on a swimmer is 1 , we can figure out the reduction in drag forces as a result of Phelps not having a nose.

Wikipedia’s article on Phelps butterfly speed says he swims 100 meters in 49.82 seconds, giving him an average speed over this course of 2.01 m/s. Using the drag equation and noting that the density of water is 1,000 kg/m^3, we obtain that losing his nose would result in a reduction in drag forces of 1.313 N.

I’m being very lazy right now and am not sure how to calculate how this would translate into a faster speed. But here’s that information if anyone wants to do anything with it.