Implicit Large Eddy Simulation of Nearly Incompressible Flows with a Discontinuous Galerkin-Boltzmann Formulation

TL;DR

This work addresses the challenge of simulating turbulent flows in the nearly incompressible regime without explicit subgrid-scale models by developing a high-order nodal DG discretization of the Boltzmann–BGK equations. Velocity-space is discretized with tri-variate Hermite polynomials, producing a coefficient vector $\mathbf{q}=[q_1,\dots,q_{10}]$ that links to macroscopic variables via $p=\rho RT$ and $\nu=\tau RT$. Stability and accuracy are achieved through over-integration of the nonlinear collision term, a choice of flux (upwind or LLF) for surface coupling, and a third-order semi-analytic Runge–Kutta time integrator that handles stiff relaxation terms. Numerical experiments on the Taylor–Green vortex and flow over a sphere at $Re=3700$ demonstrate the method’s ability to capture laminar–turbulent transitions and multiscale vortex dynamics with accuracy comparable to reference DNS/LES, highlighting the framework as a robust ILES option for nearly incompressible turbulence.$

Abstract

We present a high-order implicit large eddy simulation (ILES) approach for simulating flows at the nearly incompressible regime. Our methodology based on utilization of a nodal discontinuous Galerkin (DG) discretization of the Boltzmann equations. The compactness and low-dissipative nature of the discontinuous Galerkin method are leveraged to mimic traditional large eddy simulations with subgrid-scale models. One of the key requirements of ILES is to provide dissipation only within a narrow band of high wavenumbers. This is validated through numerical experiments on the Taylor-Green Vortex problem in detail at a Reynolds number where varying scales of coherent turbulent structures are present. Furthermore, the approach is validated for external aerodynamic configurations by simulating the flow over a sphere at a Reynolds number of $Re=3700$, capturing the laminar-turbulent transition and the complex multiscale vortex dynamics characteristic of this regime. The results demonstrate the capability of the high-order DG-Boltzmann formulation to accurately capture transitional and turbulent flow features without the use of explicit sub-grid scale modeling, highlighting its potential as a robust and physically consistent framework for ILES of nearly incompressible turbulent flows.

Figures (9)

• Figure 1: Evolution of the kinetic energy for 3D TGV problem. Left: comparison of different orders of approximation. Right: comparison of different numbers of elements.
• Figure 2: Kinetic energy dissipation rate for 3D TGV problem. Comparison of different orders of approximation.
• Figure 3: Kinetic energy dissipation rate based on enstrophy for 3D TGV problem. Comparison of different orders of approximation.
• Figure 4: Numerical dissipation rate for increasing orders of approximation for 3D TGV Problem.
• Figure 5: Comparison of numerical dissipation rates for different numerical flux approaches with varying numbers of elements and orders of approximation for the 3D TGV problem. Left: for increasing number of elements, right: increasing orders of approximation.