A velocity-vorticity-pressure formulation for the steady Navier--Stokes--Brinkman--Forchheimer problem

TL;DR

The paper develops a non-augmented velocity–vorticity–Bernoulli pressure formulation for steady incompressible flow in highly permeable porous media, casting the problem as nested saddle-point systems and proving solvability under small data both continuously and numerically. It introduces a pressure-robust, nonconforming Galerkin discretization based on Crouzeix–Raviart elements with tangential and normal jump stabilization, together with a Raviart–Thomas interpolation to ensure robust velocity error control independent of the pressure. An explicit residual-based a posteriori error estimator is derived and shown to be reliable and efficient, supported by a novel inverse inequality for the Forchheimer nonlinearity and an octree-based adaptive solver with hanging-node constraints. Numerical experiments demonstrate optimal convergence, pressure-robustness, and effective adaptivity, validating the theoretical results and highlighting practical performance for nonlinear porous-flow regimes.

Abstract

The flow of incompressible fluid in highly permeable porous media in vorticity - velocity - Bernoulli pressure form leads to a double saddle-point problem in the Navier--Stokes--Brinkman--Forchheimer equations. The paper establishes, for small sources, the existence of solutions on the continuous and discrete level of lowest-order piecewise divergence-free Crouzeix--Raviart finite elements. The vorticity employs a vector version of the pressure space with normal and tangential velocity jump penalisation terms. A simple Raviart--Thomas interpolant leads to pressure-robust a priori error estimates. An explicit residual-based a posteriori error estimate allows for efficient and reliable a posteriori error control. The efficiency for the Forchheimer nonlinearity requires a novel discrete inequality of independent interest. The implementation is based upon a light-weight forest-of-trees data structure handled by a highly parallel set of adaptive mesh refining algorithms. Numerical simulations reveal robustness of the a posteriori error estimates and improved convergence rates by adaptive mesh-refining.

Figures (3)

• Figure 7.1: Approximate velocity (line integral contours and magnitude), vorticity, and Bernoulli pressure profiles computed with the modified $\mathbf{CR}-\mathbb{P}_0-\mathbb{P}_0$ scheme, with kinematic viscosity $\nu=10^{-4}$.
• Figure 7.2: Approximate velocity (streamlines), vorticity (streamlines), and Bernoulli pressure profiles on a mesh with $h=0.0383$, computed with the modified $\mathbf{CR}-\mathbb{P}_0-\mathbb{P}_0$ scheme (representing 241153 DoFs), with kinematic viscosity $\nu=1$.
• Figure 7.3: Sample of approximate solutions for the convergence test on an L-shaped domain (top rows), and coarse meshes produced after three, six, and nine steps of the adaptive refinement algorithm guided by the a posteriori error estimator.

Theorems & Definitions (16)

• Lemma 3.1
• Lemma 3.2
• Lemma 3.3
• Lemma 3.4: Unique solvability of the linearised problem
• Theorem 3.5: Unique solvability
• Lemma 4.1: Invertibility on the kernel
• Lemma 4.2
• Lemma 4.3
• Lemma 4.4: Wellposedness of the discrete linearised problem
• Theorem 4.5: Unique solvability of the discrete nonlinear problem