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

TL;DR

The paper develops a divergence-conforming HDG method for the incompressible RANS equations coupled with the Spalart–Allmaras one‑equation model, achieving pointwise divergence-free mean velocity and a balanced momentum–energy discretization. It introduces semi-discrete and fully discrete formulations with velocity, pressure, and turbulence-variable spaces, together with trace DOFs and static condensation to reduce global DOFs. The method is shown to be consistent, mass-conserving, globally momentum-balanced, and energy-stable for nonnegative turbulence viscosity, with optimal convergence demonstrated via manufactured solutions and strong agreement with DNS and CFL3D benchmarks for channel flow, backward-facing step, and a NACA0012 airfoil. The work furnishes a computationally efficient, high-order framework for incompressible turbulent flows and lays out avenues for extending to other turbulence models and coupled physics.

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.

Figures (12)

• Figure 1: Mesh Notation.
• Figure 2: Convergence for the method of manufactured solutions problem with $\nu =$ 1e-2, $\theta =$ 1e0, a mean velocity polynomial degree of $k$, a mean pressure polynomial degree of $k -1$, and a working viscosity polynomial degree of $k - 1$ for $k = 2, 3, 4$.
• Figure 3: Convergence for the method of manufactured solutions problem with $\nu =$ 1e-2, $\theta =$ 1e-2, a mean velocity polynomial degree of 4, a mean pressure polynomial degree of 3, and a working viscosity polynomial degree of $q$ for $q = 1, 2, 3$.
• Figure 4: Convergence for the method of manufactured solutions problem with $\nu =$ 1e-2, $\theta =$ 1e-1, a mean velocity polynomial degree of 4, a mean pressure polynomial degree of 3, and a working variable polynomial degree of $q$ for $q = 1, 2, 3$.
• Figure 5: Comparison of the velocity profile for DNS and RANS with the Spalart-Allmaras model for a turbulent channel flow at $\text{Re}_{\tau}$ = 550.