# A Divergence-Conforming Hybridized Discontinuous Galerkin Method for the Incompressible Reynolds Averaged Navier-Stokes Equations

## TL;DR

This paper presents a novel hybridized discontinuous Galerkin method for incompressible RANS equations with turbulence modeling, achieving divergence-free velocities and efficient computation through static condensation.

## Contribution

It introduces a divergence-conforming hybridized DG method that ensures point-wise divergence-free velocities and balances momentum and energy, with analysis of polynomial degree and mesh effects.

## Key findings

- Method produces point-wise divergence-free mean velocities.
- Static condensation reduces computational complexity.
- Numerical results confirm effectiveness of the approach.

## Abstract

We introduce a hybridized discontinuous Galerkin method for the incompressible Reynolds Averaged Navier-Stokes equations coupled with the Spalart-Allmaras one equation turbulence model. With a special choice of velocity and pressure spaces for both element and trace degrees of freedom, we arrive at a method which returns point-wise divergence-free mean velocity fields and properly balances momentum and energy. We further examine the use of different polynomial degrees and meshes to see how the order of the scalar eddy viscosity affects the convergence of the mean velocity and pressure fields, specifically for the method of manufactured solutions. As is standard with hybridized discontinuous Galerkin methods, static condensation can be employed to remove the element degrees of freedom and thus dramatically reduce the global number of degrees of freedom. Numerical results illustrate the effectiveness of the proposed methodology.