A non-hybrid method for the PDF equations of turbulent flows on unstructured grids

TL;DR

This paper introduces a non-hybrid, finite element-based method for solving PDF equations in turbulent flows on unstructured grids, improving efficiency and consistency without hybrid approaches.

Contribution

It presents a novel non-hybrid algorithm using FEM on unstructured grids for PDF equations, with an adaptive scalar modeling technique and detailed parallel implementation.

Findings

Efficient solution of PDF equations in complex geometries.

Validation on turbulent channel and street canyon flows.

Improved statistical homogenization without shape assumptions.

Abstract

In probability density function (PDF) methods of turbulent flows, the joint PDF of several flow variables is computed by numerically integrating a system of stochastic differential equations for Lagrangian particles. A set of parallel algorithms is proposed to provide an efficient solution of the PDF transport equation, modeling the joint PDF of turbulent velocity, frequency and concentration of a passive scalar in geometrically complex configurations. An unstructured Eulerian grid is employed to extract Eulerian statistics, to solve for quantities represented at fixed locations of the domain (e.g. the mean pressure) and to track particles. All three aspects regarding the grid make use of the finite element method (FEM) employing the simplest linear FEM shape functions. To model the small-scale mixing of the transported scalar, the interaction by exchange with the conditional mean model is adopted. An adaptive algorithm that computes the velocity-conditioned scalar mean is proposed that homogenizes the statistical error over the sample space with no assumption on the shape of the underlying velocity PDF. Compared to other hybrid particle-in-cell approaches for the PDF equations, the current methodology is consistent without the need for consistency conditions. The algorithm is tested by computing the dispersion of passive scalars released from concentrated sources in two different turbulent flows: the fully developed turbulent channel flow and a street canyon (or cavity) flow. Algorithmic details on estimating conditional and unconditional statistics, particle tracking and particle-number control are presented in detail. Relevant aspects of performance and parallelism on cache-based shared memory machines are discussed.