Modelling turbulence via numerical functional integration using Burgers' equation

TL;DR

This paper introduces a novel numerical method for modeling turbulence by transforming Burgers' equation into a functional integral, enabling parallel computation of correlation functions more efficiently than traditional simulations.

Contribution

The paper develops a functional integral approach for turbulence modeling based on Burgers' equation, with an implementation that allows parallelized computation across multiple processors.

Findings

Successful calculation of velocity cross correlation on a 10x2 lattice

Demonstrated parallelization of the integral computation

Analyzed computational resource requirements

Abstract

We investigate the feasibility of modelling turbulence via numeric functional integration. By transforming the Burgers' equation into a functional integral we are able to calculate equal-time spatial correlation of system variables using standard methods of multidimensional integration. In contrast to direct numerical simulation, our method allows for simple parallelization of the problem as the value of the integral within any region can be calculated separately from others. Thus the calculations required for obtaining one correlation data set can be distributed to several supercomputers and/or the cloud simultaneously. We present the mathematical background of our method and its numerical implementation. We are interested in a steady state system with isotropic and homogeneous turbulence, for which we use a lattice version of the functional integral used in the perturbative analysis of stochastic transport equations. The numeric implementation is composed of a fast serial program for evaluating the integral over a given volume and a parallel Python wrapper that divides the problem into subvolumes and distributes the work among available processes. The code is available at https://github.com/iljah/hdintegrator for anyone to download, use, study, modify and redistribute. We present velocity cross correlation for a 10x2 lattice in space and time respectively, and analyse the computational resources required for the integration. We also discuss potential improvements to the presented method.