An Efficient Numerical Method for the Generalised Kolmogorov Equation

TL;DR

This paper introduces a highly efficient numerical algorithm for solving the Generalised Kolmogorov Equation in plane channel flows, enabling detailed analysis of high-Reynolds-number turbulence with minimal computational resources.

Contribution

The paper presents a novel, parallel computing strategy-based algorithm that significantly accelerates GKE computations, allowing for the first-time analysis of full 4D domain channel flows at high Reynolds numbers.

Findings

Achieved 3-4 orders of magnitude speedup over previous methods.

Validated the algorithm by computing GKE residuals for Re_{τ}=200 and 1000.

Discovered changes in scale-energy flux properties with increasing Reynolds number.

Abstract

An efficient algorithm for computing the terms appearing in the Generalised Kolmogorov Equation (GKE) written for the indefinite plane channel flow is presented. The algorithm, which features three distinct strategies for parallel computing, is designed such that CPU and memory requirements are kept to a minimum, so that high-Re wall-bounded flows can be afforded. Computational efficiency is mainly achieved by leveraging the Parseval's theorem for the two homogeneous directions available in the plane channel geometry. A speedup of 3-4 orders of magnitude, depending on the problem size, is reported in comparison to a key implementation used in the literature. Validation of the code is demonstrated by computing the residual of the GKE, and example results are presented for channel flows at Re_{\tau}=200 and Re_{\tau}=1000, where for the first time they are observed in the whole four-dimensional domain. It is shown that the space and scale properties of the scale-energy fluxes change for increasing values of the Reynolds number. Among all scale-energy fluxes, the wall-normal flux is found to show the richest behaviour for increasing streamwise scales.