Non-linear vorticity upsurge in Burgers' flow

TL;DR

This paper demonstrates that numerical solutions of Burgers' equation reveal a non-linear vorticity surge during flow evolution, showing localized shear concentration and turbulence onset without external disturbances, across finely-resolved scales.

Contribution

It introduces a scale-totality algorithm for solving Burgers' equation at very low viscosity, capturing the non-linear vorticity upsurge and turbulence features directly from the equations.

Findings

Vorticity surges are driven by non-linearity and instantaneous enstrophy production.

Localized shear concentration is crucial during flow evolution.

Long-term energy decay is independent of initial conditions and inversely proportional to time.

Abstract

We demonstrate that numerical solutions of Burgers' equation can be obtained by a scale-totality algorithm for fluids of small viscosity (down to one billionth). Two sets of initial data, modelling simple shears and wall boundary layers, are chosen for our computations. Most of the solutions are carried out well into the fully turbulent regime over finely-resolved scales in space and in time. It is found that an abrupt spatio-temporal concentration in shear constitutes an essential part during the flow evolution. The vorticity surge has been instigated by the non-linearity complying with instantaneous enstrophy production, while ad hoc disturbances play no role in the process. In particular, the present method predicts the precipitous vorticity re-distribution and accumulation, predominantly over localised regions of minute dimension. The growth rate depends on viscosity and is a strong function of initial data. Nevertheless, the long-time energy decay is history-independent and is inversely proportional to time. Our results provide direct evidence of the vorticity proliferation embedded in the equations of motion. The non-linear intensification is a robust feature, and is ultimately responsible for the drastic succession in boundary layer profiles over the intrinsic laminar-turbulent transition (Schubauer and Klebanoff 1955). The dynamical inception of turbulence can be decrypted by solving the full time-dependent Navier-Stokes equations which ascribe no instability stages.