Centerline velocity decay ∝ 1/x, linear width growth, Gaussian profiles — turbulent entrainment
Far-field self-similar turbulent jet (Townsend 1956):
U_c(x) = B·√(M₀)/(x−x₀) — centerline velocity
b(x) = S·(x−x₀) — half-width (linear!)
U(x,r) = U_c(x)·f(η), η = r/b(x)
The similarity profile f(η) ≈ exp(−(ln2)·η²) (Gaussian). Momentum flux M = ∫ρU²dA is conserved. Entrainment: ∂Q/∂x = S·√(M₀) — ambient fluid engulfed at rate ∝ x.
Self-similarity means all downstream profiles collapse onto a single curve when scaled by U_c and b — a universal attractor of turbulence dynamics.