10 / CHAPTER 11
Turbulence, averaging, and modeling
Understand why the exact equations can still require closure assumptions.
ANCHOR EQUATION
Reynolds averaging
In symbols: u = ū + u′
Turbulence modeling separates mean flow from fluctuating motion.
Navier–Stokes assembles the earlier balances
The Navier–Stokes equation is not a new law pasted onto fluid mechanics; it is mass-weighted acceleration balanced by pressure, viscous stress, and body force for a Newtonian fluid.
Substitute the constitutive law
Momentum balance is ρDu/Dt = ∇·σ + ρg. Split stress into isotropic pressure and viscous parts: σ = −pI + τ. For constant-viscosity Newtonian incompressible flow, ∇·τ = μ∇²u, giving ρ(∂u/∂t + u·∇u) = −∇p + μ∇²u + ρg. Each term has units N/m³ and a distinct physical role.
For a 1D steady flow with negligible advection and body force, 0 = −dp/dx + μ d²u/dy². This is exactly the equation integrated in the Poiseuille derivation; the famous pipe formula is one special solution, not a separate law.
The equation does not close itself
Need pressure and velocity boundary conditions, a mass balance, viscosity law, and initial data. Turbulence may require averages and a closure model even though the instantaneous equation is known.
Work it: Nondimensionalize Navier–Stokes using U and L. Show the viscous term appears with coefficient 1/Re, then predict which term dominates for Re = 10 and Re = 10⁶.