A modeling and filtering framework for the semi-discretised Navier-Stokes equations

Spatial discretisation of fluid mechanical systems typically leads to descriptor systems consisting of large numbers of differential algebraic equations (DAEs). In an effort to apply standard control theory to such systems, physical insight is often used to analytically reformulate the DAE as an ordinary differential equation (ODE). In general, this is a difficult process that typically requires expert insight into specific systems, and so in this work we consider a more flexible numerical method that is straightforward to implement on any regular DAE. The numerical procedure is outlined and a new method for computing one of the steps is presented.With respect to Kalman filtering of descriptor systems, it is known in general that `process noise´ can sometimes not be added to all states owing to a violation of causality. In this paper we present a new method for computing the subspace of causal disturbances, suitable for large DAEs. Finally, the techniques developed in this paper are applied to the specific case of plane Poiseuille flow and it is shown how a standard state-space system is easily obtained that possesses a similar spectrum and pseudospectrum to the Orr-Somerfeld-Squire system.