Modulation equations and Reynolds averaging for finite-amplitude non-linear waves in an incompressible fluid

A formal perturbation scheme is developed to determine original modulation equations for laminar finite-amplitude non-linear waves in an incompressible fluid. Three idealized problems are analysed. The modulation equations comprise conservation of waves, averaged conditions for conservation of mass, momentum, kinetic energy and angular momentum and the averaged projection of the Navier–Stokes equations onto the vorticity vector. The last of these modulation equations, which is related to vortex stretching, only appears in 3D problems. The technique of Reynolds averaging is also employed to obtain equations for the mean velocities and pressure. The Reynolds-averaged Navier–Stokes equations correspond to the modulation equations for conservation of mass and momentum. However, the Reynolds stress transport equations are shown to be inconsistent with the other necessary modulation equations. In two further idealized problems, exact solutions of the Navier–Stokes equations are obtained by employing the modulation equations.