High-Order Embedded Boundary Methods for Direct Numerical Simulations

Cut-cell methods for unsteady flow problems can greatly simplify the grid generation process and allow for high-fidelity simulations on complex geometries. Historically, however, cut-cell methods have been limited to low orders of accuracy. It is the conjecture of the authors that this has been driven, by the variety of procedures typically introduced to evaluate derivatives in a stable manner near the highly irregular embedded geometry. Indeed, even on a uniform mesh, it is non-trivial to derive high-order numerical boundary schemes to be used near the wall. Pursuing a finite-differences based cut-cell approach (also referred to as an "embedded boundary" or "Cartesian grid" method), allowed for formulating the small cell problem encountered by cut-cell methods as an optimization problem. Coupled with a truncation error matching idea, the optimization strategy has been successfully applied to cut-cells resulting in conservative 4th order approximations to hyperbolic problems without the addition of numerical dissipation as well as 8th order approximations for elliptic and parabolic systems. Generalizing our previous work to the Navier-Stokes equations highlights the need for optimizing interpolation stencils in addition to the derivative stencils. In the present work, we will discuss a unified framework to accomplish this. Several test cases will be showcased to highlight the stability of the resulting method and provide a path forward for pursuing higher-order stencils for general governing equations.