You are on the right track. You have reinvented the basic idea behind the "Reynolds-averaged Navier–Stokes" (RANS) equations, dating back to 1895. The goal is t

You are on the right track. You have reinvented the basic idea behind the "Reynolds-averaged Navier–Stokes" (RANS) equations, dating back to 1895. The goal is to predict an average velocity field, and this requires modeling unclosed terms (the "turbulence problem" in one respect). RANS and statistical methods in general are the most popular turbulence modeling approaches and much of turbulence theory is based around these ideas. My opinion is that RANS answers are typically what you want, but the models don't work that well. LES (large eddy simulation; applying a low pass filter to NS) is a newer approach that is gaining popularity, and makes physical sense, but is a lot more computationally expensive. And direct simulation (DNS) is an option too, but the computational costs are prohibitive outside of some relatively simple academic problems.