Dynamically correct formulations of the linearised Navier–Stokes equations

Abstract

SUMMARYMotivated by the need to efficiently obtain low-order models of fluid flows around complex geometries for the purpose of feedback control system design, this paper considers the effect on system dynamics of basing plant models on different formulations of the linearised Navier-Stokes equations. We consider the dynamics of a single computational node formed by spatial discretisation of the governing equations in both primitive variables (momentum equation & continuity equation) and pressure Poisson equation (PPE) formulations. This reveals fundamental numerical differences at the nodal level, whose effects on the system dynamics at the full system level are exemplified by considering the corresponding formulations of a two-dimensional (2D) channel flow, subjected to a variety of different boundary conditions.

Global Stability Analysis of a Boundary Layer with Surface Indentations

“…In contrast, the LPPE formulation circumvents the checkerboard instability with additional information introduced for the pressure. Using a descriptor state-space model approach, Dellar and Jones [18] demonstrated the well-posedness of the formulation when the equations are discretized on a collocated grid. However, they commented that there is no guarantee that the LPPE formulation is entirely dynamically equivalent to the LNS equations formulated in primitive variables when spatially discretized.…”