A continuous finite element framework for the pressure Poisson equation allowing non‐Newtonian and compressible flow behavior

Abstract

Computing pressure fields from given flow velocities is a task frequently arising in engineering, biomedical, and scientific computing applications. The so‐called pressure Poisson equation (PPE) derived from the balance of linear momentum provides an attractive framework for such a task. However, the PPE increases the regularity requirements on the pressure and velocity spaces, thereby imposing theoretical and practical challenges for its application. In order to stay within a Lagrangian finite element framework, it is common practice to completely neglect the influence of viscosity and compressibility when computing the pressure, which limits the practical applicability of the pressure Poisson method. In this context, we present a mixed finite element framework which enables the use of this popular technique with generalized Newtonian fluids and compressible flows, while allowing standard finite element spaces to be employed for the unknowns and the given data. This is attained through the use of appropriate vector calculus identities and simple projections of certain flow quantities. In the compressible case, the mixed formulation also includes an additional equation for retrieving the density field from the given velocities so that the pressure can be accurately determined. The potential of this new approach is showcased through numerical examples.

A global residual‐based stabilization for equal‐order finite element approximations of incompressible flows

“…We hope that this new tool can offer the computational fluid dynamics community a practical, efficient alternative to some existing techniques such as PSPG and edge stabilization methods. Future and ongoing developments include a generalization to fluids with variable viscosity, 56 as well as a systematic numerical analysis to provide theoretical stability estimates. Also ongoing is the development of robust solvers for the stabilized system in general flow regimes, including suitable preconditioners and adaptive stepping for time‐dependent problems.…”

On the numerical treatment of viscous and convective effects in relative pressure reconstruction methods

“…with consistent Neumann boundary conditions already implied. 35 Since this is a pure Neumann problem, the pressure at each time is defined up to an arbitrary additive constant. An additional constraint is thus required, for instance enforcing that Z…”

“…Nonetheless, using the rotational description of the viscous forces ( 5) actually enables such a step. Notice that 42 This leads to the boundary vorticity or integrated pressure Poisson estimator (PPE ω ) recently proposed by Pacheco and Steinbach, 35 seeking p n X h such that…”

An Equal-Order Finite Element Framework for Incompressible Non-Newtonian Flow Problems

“…With these challenges in mind, this work presents, as its main contribution, a new residual-based framework for equal-order finite element approximations of quasi-Newtonian problems. The most important feature of our new method is overcoming the loss of consistency of standard PSPG-like stabilisation terms in the lowest-order case, by augmenting the continuity equation with a pressure Poisson equation [35,36] with consistent boundary conditions. In doing so, we achieve considerable improvements in robustness and accuracy with respect to state-of-the-art residual-based stabilisations.…”