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A finite element formulation for incompressible viscous flow based on the principle of minimum pressure gradient

Julian J. Rimoli

Abstract

We present a finite element formulation for incompressible viscous flow based on the principle of minimum pressure gradient (PMPG). This variational principle, recently established by Taha, Gonzalez & Shorbagy (Phys. Fluids, vol. 35, 2023), states that the Navier-Stokes equations are equivalent to determining the rate of change of velocity at each instant by minimizing the L2 norm of the implied pressure gradient, subject to incompressibility and boundary conditions. We discretize the PMPG functional directly using Q9 biquadratic finite elements and minimize over the nodal velocity rates (Rayleigh-Ritz). No pressure degrees of freedom appear; incompressibility and boundary conditions are enforced as linear equality constraints through a monolithic saddle-point system, whose Lagrange multipliers provide wall forces without pressure reconstruction. We verify the formulation against exact Poiseuille flow (machine-precision recovery) and the Kovasznay solution (convergence rate ~3.3), and validate it against published benchmarks for the lid-driven cavity, the backward-facing step, and flow past a circular cylinder. The formulation produces smooth, oscillation-free solutions on coarse meshes in the convection-dominated regime without stabilization. We further show that the element-wise PMPG functional density serves as a built-in error indicator for adaptive mesh refinement, and that the stationarity condition can be read backwards to estimate the kinematic viscosity directly from velocity field measurements.

A finite element formulation for incompressible viscous flow based on the principle of minimum pressure gradient

Abstract

We present a finite element formulation for incompressible viscous flow based on the principle of minimum pressure gradient (PMPG). This variational principle, recently established by Taha, Gonzalez & Shorbagy (Phys. Fluids, vol. 35, 2023), states that the Navier-Stokes equations are equivalent to determining the rate of change of velocity at each instant by minimizing the L2 norm of the implied pressure gradient, subject to incompressibility and boundary conditions. We discretize the PMPG functional directly using Q9 biquadratic finite elements and minimize over the nodal velocity rates (Rayleigh-Ritz). No pressure degrees of freedom appear; incompressibility and boundary conditions are enforced as linear equality constraints through a monolithic saddle-point system, whose Lagrange multipliers provide wall forces without pressure reconstruction. We verify the formulation against exact Poiseuille flow (machine-precision recovery) and the Kovasznay solution (convergence rate ~3.3), and validate it against published benchmarks for the lid-driven cavity, the backward-facing step, and flow past a circular cylinder. The formulation produces smooth, oscillation-free solutions on coarse meshes in the convection-dominated regime without stabilization. We further show that the element-wise PMPG functional density serves as a built-in error indicator for adaptive mesh refinement, and that the stationarity condition can be read backwards to estimate the kinematic viscosity directly from velocity field measurements.
Paper Structure (58 sections, 49 equations, 23 figures, 2 tables, 1 algorithm)

This paper contains 58 sections, 49 equations, 23 figures, 2 tables, 1 algorithm.

Table of Contents

  1. Introduction
  2. The principle of minimum pressure gradient
  3. Objectives and contributions
  4. Outline
  5. Formulation
  6. The PMPG variational principle
  7. Rayleigh--Ritz discretization
  8. Constrained instantaneous problem.
  9. Time discretization.
  10. Q9 finite element discretization
  11. Reference element
  12. Field interpolation
  13. Physical gradients
  14. Subparametric geometry mapping
  15. Divergence-free constraint
  16. ...and 43 more sections

Figures (23)

  • Figure 1: Q9 reference element (left) and a representative physical element (right) under the subparametric mapping $\bm{x}(\xi,\eta)$. Filled circles () denote the 4 corner nodes that define the bilinear geometry mapping; open circles () denote the 5 midside and center nodes used only for the biquadratic field interpolation. In the physical element, the midside nodes lie at the midpoints of the (straight) edges, and the center node lies at the intersection of the bimedians, as determined by the bilinear map.
  • Figure 2: Poiseuille flow: problem setup. Hatched boundaries denote no-slip walls ($\bm{v} = \bm{0}$), blue arrows indicate the prescribed parabolic inlet profile, and the dashed boundary marks the free outflow.
  • Figure 3: Poiseuille flow: computed velocity profiles (symbols) on the $40 \times 8$ mesh compared with the exact parabolic solution (solid line) at multiple $x$-stations.
  • Figure 4: Poiseuille flow: convergence history on the $40 \times 8$ mesh. The time-derivative norm $\|\partial_t \bm{v}\|$ converges to machine precision, confirming exact steady-state recovery.
  • Figure 5: Kovasznay flow: (a) problem setup: square domain $[-0.5, 1.5]^2$ with the exact Kovasznay solution \ref{['eq:kovasznay']} prescribed as Dirichlet conditions (blue) on all sides; gray arrows indicate the predominantly rightward mean flow; (b) representative unstructured Q9 mesh ($n = 16$).
  • ...and 18 more figures

Theorems & Definitions (8)

  • Remark 2.1
  • Remark 2.2
  • Remark 2.3
  • Remark 2.4
  • Remark 2.5
  • Remark 5.1
  • Remark 6.1
  • Remark A.1