Understanding developing turbulence by a study of the nonlinear energy transfer in the Navier-Stokes equation

In the present work, we investigate a numerical one-dimensional solver to the Navier-Stokes equation that retains all terms, including both pressure and dissipation. Solutions to simple examples that illustrate the actions of the nonlinear term are presented and discussed. The calculations take the full 4D flow as its starting point and continuously projects the forces acting on the fluid at a fixed Eulerian point in a stationary coordinate system onto the direction of the instantaneous velocity.

Pressure is included through modeling. Adhering to the requirement that time must in general be considered an independent variable, the time development of the time records and power spectra of the velocity fluctuations are studied. It is found that the actions of the nonlinear term in the Navier-Stokes equation manifests itself by generating sharp pulses in the time traces, where the sharpness is bounded by the finite viscosity.

In the spectral domain, the sharp gradients in the pulses generate energy contributions at high frequencies that yields a $-2$ slope across the inertial range. The $-2$ (or $-6/3$) slope is explained through a simple example and the classically expected $-5/3$ slope in the inertial range can be recovered from the pressure fluctuations from the full flow field that can be considered a noise contribution at the point considered.

We also observe that the spectrum can in principle keep spreading to higher frequencies or wavenumbers without upper bound, as the viscosity is approaching the zero limit.