A continuous finite element framework for the pressure Poisson equation allowing non‐Newtonian and compressible flow behaviour,International Journal for Numerical Methods in Fluids

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A continuous finite element framework for the pressure Poisson equation allowing non‐Newtonian and compressible flow behaviour
International Journal for Numerical Methods in Fluids ( IF 1.7 ) Pub Date : 2020-11-16 , DOI: 10.1002/fld.4936
Douglas R. Q. Pacheco _{1,

2,

3} , Olaf Steinbach _{1,

2}

Affiliation

Computing pressure fields from given flow velocities is a task arising in various 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 generalised Newtonian fluids (e.g., blood) and compressible flows, while allowing standard finite element spaces to be employed for the given data and unknowns. This is accomplished 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.

中文翻译：

压力泊松方程的连续有限元框架允许非牛顿和可压缩流动行为

根据给定的流速计算压力场是各种工程、生物医学和科学计算应用中出现的一项任务。从线性动量平衡导出的所谓压力泊松方程 (PPE) 为此类任务提供了一个有吸引力的框架。然而，PPE 增加了对压力和速度空间的规律性要求，从而对其应用提出了理论和实践挑战。为了保持在拉格朗日有限元框架内，通常的做法是在计算压力时完全忽略粘度和可压缩性的影响，这限制了压力泊松方法的实际适用性。在这种情况下，我们提出了一个混合有限元框架，它能够将这种流行的技术与广义牛顿流体（例如，血液）和可压缩流，同时允许对给定数据和未知数采用标准有限元空间。这是通过使用适当的向量微积分恒等式和某些流量的简单投影来实现的。在可压缩的情况下，混合公式还包括一个额外的方程，用于从给定的速度中检索密度场，以便可以准确地确定压力。通过数值示例展示了这种新方法的潜力。混合公式还包括一个额外的方程，用于从给定的速度中检索密度场，以便可以准确地确定压力。通过数值示例展示了这种新方法的潜力。混合公式还包括一个额外的方程，用于从给定的速度中检索密度场，以便可以准确地确定压力。通过数值示例展示了这种新方法的潜力。

更新日期：2020-11-16

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