High-order DG solvers for under-resolved turbulent incompressible flows: A comparison of $L^2$ and $H$(div) methods

TL;DR

This work benchmarks two high-order DG solvers for incompressible, under-resolved turbulent flows: a stabilized $L^{2}$-DG method and a divergence-free $H(div)$-conforming HDG method. By applying them to the Taylor–Green vortex and turbulent channel flow under varying resolutions and discretisation choices, it demonstrates that both approaches yield robust, dissipative behaviour and comparable accuracy when DOFs are matched, with exact mass conservation emerging as a key factor for stability. The study reveals that the particular treatment of convection and viscosity has a limited impact in under-resolved regimes, provided the divergence and normal-continuity constraints are respected, while over- or under-integration effects are managed via quadrature choices. Overall, both methods are viable candidates for no-model LES in practical applications, highlighting the primacy of divergence-free constraints and energy stability over specific flux formulations in high-order DG schemes.

Abstract

The accurate numerical simulation of turbulent incompressible flows is a challenging topic in computational fluid dynamics. For discretisation methods to be robust in the under-resolved regime, mass conservation as well as energy stability are key ingredients to obtain robust and accurate discretisations. Recently, two approaches have been proposed in the context of high-order discontinuous Galerkin (DG) discretisations that address these aspects differently. On the one hand, standard $L^2$-based DG discretisations enforce mass conservation and energy stability weakly by the use of additional stabilisation terms. On the other hand, pointwise divergence-free $H(\operatorname{div})$-conforming approaches ensure exact mass conservation and energy stability by the use of tailored finite element function spaces. The present work raises the question whether and to which extent these two approaches are equivalent when applied to under-resolved turbulent flows. This comparative study highlights similarities and differences of these two approaches. The numerical results emphasise that both discretisation strategies are promising for under-resolved simulations of turbulent flows due to their inherent dissipation mechanisms.

Figures (13)

• Figure 1: Comparisons of high-order ($k=8$) ${ \boldsymbol{L}^{2} }$- and ${ \boldsymbol{H}{{ \left({ \mathrm{div} }\right) }} }$-based simulations for the TGV. Evolution of kinetic energy (top left), total kinetic energy dissipation rate (top right) and kinetic energy spectra (bottom) at $t=10$ on different meshes with $N\in{ \left\{ 4,8,16 \right\} }$.
• Figure 2: Comparison of ${ \boldsymbol{H}{{ \left({ \mathrm{div} }\right) }} }$-based method of order $k$ with ${ \boldsymbol{L}^{2} }$-based method of order $k$ and $k+1$. Shown are the respective total dissipation rates.
• Figure 3: Dissipation mechanisms for $k=8$ under $h$-refinement (considering meshes with $N\in{ \left\{ 4,8,16 \right\} }$). The abscissa shows the time $t$.
• Figure 4: Dissipation mechanisms for fixed strong under-resolution under $k$-refinement. The abscissa shows the time $t$.
• Figure 5: Dissipation mechanisms for different convection fluxes for $k=8$, $N=4$. Stacked dissipation rates for ${ \boldsymbol{L}^{2} }$ results (top) and ${ \boldsymbol{H}{{ \left({ \mathrm{div} }\right) }} }$ results (bottom). Lax--Friedrichs form (left column), upwind form (middle column) and central flux/no stabilisation (right column). The abscissa shows the time $t$.