On the Continuous Adjoint of Prominent Explicit Local Eddy Viscosity-based Large Eddy Simulation Approaches for Incompressible Flows

The manuscript deals with continuous adjoint companions of prominent explicit Large Eddy Simulation (LES) methods grounding on the eddy viscosity assumption for incompressible fluids. The subgrid-scale approximations considered herein address the classic Smagorinsky-Lilly, the Wall-Adapting Local Eddy-Viscosity (WALE), and the Kinetic Energy Subgrid-Scale (KESS) model, whereby only static implementations, i.e., those without dynamically adjusted model parameters, are considered. The associated continuous adjoint systems and resulting shape sensitivity expressions are derived. Information on the consistent discrete implementation is provided that benefits from the self-adjoint primal discretization of convective and diffusive fluxes via unbiased, symmetric approximations, frequently performed in explicit LES studies to minimize numerical diffusion. Algebraic primal subgrid-scale models yield algebraic adjoint LES relationships that resemble additional adjoint momentum sources. The KESS one equation model introduces an additional adjoint equation, which enlarges the resulting continuous adjoint KESS system with potentially increased numerical stiffness. The different adjoint LES methods are tested and compared against each other on a flow around a circular cylinder at \(\text{Re}_\text{D} = {140000\,}\) for a boundary (drag) and volume (deviation from target velocity distribution) based cost functional. Since all primal implementations predict similar flow fields, it is possible to swap the associated adjoint systems –i.e., applying an adjoint WALE method to a primal KESS result– and still obtain plausible adjoint results. Due to the LES’s inherent unsteady character, the adjoint solver requires the entire primal flow field over the cost-functional relevant time horizon. Even for the academic cases studied herein, the storage capacities are in the order of terabytes and refer to a practical bottleneck. However, in the case of suitable, time-averaged cost functional, the time-averaged primal flow field can be used directly in a steady adjoint solver, which results in a drastic effort reduction.