A Reynolds-Semi-Robust, Globally Divergence-Free HDG Method for the Smagorinsky Model

We develop and analyze a fully discrete, globally divergence-free hybridizable discontinuous Galerkin (HDG) method for a gradient-based Smagorinsky model. The method combines backward Euler time stepping, interior-penalty discretizations of molecular and nonlinear eddy diffusion, and an upwind convective flux. The discrete velocity is $H(\operatorname{div})$-conforming and pointwise divergence-free, which yields pressure robustness. For sufficiently large penalty parameters, we prove unconditional energy stability and existence of a discrete solution, and establish uniqueness under additional smallness conditions. A velocity error estimate is derived without explicit inverse powers of the molecular viscosity. The nonlinear facet residuals are controlled using local trace-approximation estimates and a viscosity-independent facet penalty. We retain the dependence of the discrete Gronwall factor on the filter scale and the mesh size; a mesh-uniform bound follows under suitable solution regularity, a fixed time-step margin, and the scaling $δ=O(h)$ on quasi-uniform meshes. The reported manufactured-solution results are consistent with the resulting pre-asymptotic error bounds. Further flow examples illustrate the dissipative behavior of the method and are distinguished from the boundary conditions and parameter range covered by the analysis.