Estimating pressure fields from planar velocity data around immersed bodies; a finite element approach

Abstract

A finite element approach for calculating pressure fields from planar velocity data is examined in this paper and compared with widely cited procedures. In the current approach, the order of velocity derivatives in the source function is reduced via a finite element procedure where the pressure Poisson equation is solved. This method benefits from the homogeneous spread of information in the Poisson solvers while preventing the magnification of velocity uncertainties. These combined features enable the method to limit the propagation of uncertainties into the computed pressure field. A cut-cell approach along fluid–solid boundaries is proposed, which enables information on moving, immersed boundaries to be passed into the pressure estimation procedure, thereby significantly reducing the pressure errors along the boundary. The finite element method is validated using three canonical laminar flows where the “ground-truth” pressure and velocity fields are obtained from analytical solutions and numerical simulations. The accuracy of the finite element approach is first investigated as a function of imposed noise in the velocity field data without any boundary treatments. The resulting pressure field is compared with existing pressure estimation procedures. The finite element method performs agreeably for all flow fields of interest, with approximately a 7% decrease in the pressure computation error for one of the flows, when compared to existing approaches. Next, the inclusion of boundary treatments is considered, where the implementation of the cut-cell approach is shown to reduce pressure estimation errors at solid–fluid boundaries when compared with existing boundary treatment methods. The findings are most pronounced for moving boundaries, where an order of magnitude decrease in the pressure estimation error along the moving surface is observed. This is because, when paired with the finite element approach, the cut-cell method incorporates the surface dynamics in the pressure solution, whereas existing solution approaches do not have this capability.