Nelkin scaling for the Burgers equation and the role of high-precision calculations

TL;DR

This paper demonstrates that Nelkin scaling for the Burgers equation can be effectively used at moderate Reynolds numbers with high-precision calculations, highlighting the importance of numerical accuracy in turbulence analysis.

Contribution

It shows both numerically and theoretically that Nelkin scaling works at Reynolds numbers around 100 for the Burgers equation, emphasizing the role of high-precision computations.

Findings

Nelkin scaling is valid at Reynolds numbers as low as 100.

High-precision calculations are essential for accurate scaling exponent determination.

The approach may be applicable to 3D Navier-Stokes turbulence simulations.

Abstract

Nelkin scaling, the scaling of moments of velocity gradients in terms of the Reynolds number, is an alternative way of obtaining inertial-range information. It is shown numerically and theoretically for the Burgers equation that this procedure works already for Reynolds numbers of the order of 100 (or even lower when combined with a suitable extended self-similarity technique). At moderate Reynolds numbers, for the accurate determination of scaling exponents, it is crucial to use higher than double precision. Similar issues are likely to arise for three-dimensional Navier--Stokes simulations.