Osborne Reynolds (1895) proposed decomposing any turbulent quantity into a time-averaged mean and a fluctuating part: u = Ū + u′. When substituted into the Navier-Stokes equations, the fluctuations generate the Reynolds stress tensor −ρ⟨u′ᵢu′ⱼ⟩, which acts like an additional stress on the mean flow. This closure problem — expressing Reynolds stresses in terms of mean quantities — is the central challenge of turbulence modeling (k-ε, k-ω, etc.).
u(x,t) = Ū(x) + u′(x,t) | −ρ⟨u′v′⟩ = Reynolds shear stress
Total velocity u(t) = Ū + u′(t)
Mean flow Ū (time-averaged)
Fluctuation u′(t) = u − Ū
Reynolds stress proxy −⟨u′v′⟩ (running average)