Renormalization of the Navier–Stokes Equation
A renormalization-group view of scale-dependent transport in turbulence.
This project explored how a turbulent flow changes when progressively smaller Fourier modes are eliminated from the incompressible Navier–Stokes equations. The calculation provides a statistical-physics view of turbulence in which the influence of unresolved scales appears as an effective, scale-dependent viscosity.
Renormalization follows how the effective dynamics change as successive bands of small-scale modes are removed.
Shell-by-shell coarse-graining
Starting from the Fourier-space Navier–Stokes equation, the wavenumber domain is divided into shells. Modes in the highest-wavenumber shell are integrated out, and their leading-order effect is evaluated with a Green’s-function treatment. Repeating this procedure produces a recursion for the renormalized viscosity.
The effective viscosity evolves with the coarse-graining scale and approaches a converged inertial-range form. This connects the nonlinear transfer of energy across scales with the transport coefficients that appear in reduced descriptions of turbulence.