Numerical simulations of high Reynolds number turbulent flows demand spatial discretizations that strictly conserve kinetic energy while preserving divergence-free velocity fields. We formulate an energy-stable, high-order Discontinuous Galerkin (DG) method equipped with interior penalty formulations and symmetric flux splitting. Simulations on canonical Taylor-Green vortex benchmarks demonstrate optimal spatial convergence up to polynomial degree p=7 without non-physical numerical dissipation. The proposed framework provides an exceptionally stable computational foundation for direct numerical simulation of complex turbulent boundary layers.
1. IntroductionNumerical simulations of high Reynolds number turbulent flows demand spatial discretizations that strictly conserve kinetic energy while preserving divergence-free velocity fields. Standard low-order finite volume schemes introduce artificial dissipation that smears small-scale turbulent vortices.2. Mathematical Discretization & Energy StabilityWe develop an energy-conserving high-order Discontinuous Galerkin formulation based on symmetric flux decomposition. Using Gauss-Legendre quadrature points, we prove unconditional nonlinear stability under exact polynomial integration.3. Numerical Convergence VerificationConvergence rates were verified on 3D Taylor-Green vortex flows up to polynomial order $p=7$. The kinetic energy dissipation rates matched analytical direct numerical simulation (DNS) spectra with zero unphysical numerical diffusion.