An UQ-ready finite element solver for a two-dimensional RANS model of free plane jets

TL;DR

This paper introduces a differentiable finite element solver for 2D RANS equations with a k-epsilon turbulence model, enabling scalable uncertainty quantification by providing forward, adjoint, and higher derivatives.

Contribution

The paper develops a UQ-ready finite element solver for 2D RANS models that is smoothly differentiable, addressing limitations of existing software that use non-differentiable perturbations.

Findings

Provides a differentiable solver suitable for UQ methods

Enables computation of higher-order derivatives for RANS models

Facilitates scalable Hessian-based uncertainty quantification

Abstract

Numerical solution of the system of partial differential equations arising from the Reynolds-Averaged Navier-Stokes (RANS) equations with $k-\epsilon$ turbulence model presents several challenges due to the advection dominated nature of the problem and the presence of highly nonlinear reaction terms. State-of-the-art software for the numerical solution of the RANS equations address these challenges by introducing non-differentiable perturbations in the model to ensure numerical stability. However, this approach leads to difficulties in the formulation of the higher-order forward/adjoint problems, which are needed for scalable Hessian-based uncertainty quantification (UQ) methods. In this note, we present the construction of a UQ-ready flow solver, i.e., one that is smoothly differentiable and provides not only forward solver capabilities but also adjoint and higher derivatives capabilities.