Darcy-Brinkman Porous Pipe¶

Why this case matters¶

Porous media show up in wind engineering wherever the flow must pass through a permeable resistance: a vegetation canopy, a perforated screen, or a numerical sponge layer that damps outgoing waves before they reflect off the outlet. nassu models this with a volumetric momentum sink, a linear Darcy body force applied to the fluid inside a tagged region. Before relying on that sink to absorb pressure waves at an outlet, you need a case where the porous force is the only physics and an exact solution is available to check it against.

The Darcy-Brinkman porous pipe is that case. A pipe filled with a porous region under a constant body force has a closed-form axial velocity profile, given by a modified Bessel function, that the solver must reproduce. Validating against it confirms that the linear Darcy sink is applied with the correct magnitude and that the balance between the driving force, the porous resistance and the viscous diffusion is recovered. It is both an analytic 3-D benchmark for the volumetric porous-medium BC and a direct test of the outlet wave-damping use case.

Physical description¶

A circular pipe is filled with a porous medium and driven by a constant body force \(G\) along its axis. Inside the porous region the fluid feels a linear Darcy resistance proportional to the local velocity, a body force \(F_\alpha = -\alpha\, u_\alpha\) opposing the motion (the Brinkman extension of Darcy’s law, which retains the viscous diffusion term so the no-slip wall is honoured). The flow accelerates from rest until the driving force, the Darcy resistance and the viscous wall drag balance, settling into a steady axial profile.

Two limits bracket the behaviour. Far from the wall the viscous term is negligible and the velocity approaches the terminal value \(G / \alpha\), where the driving force exactly cancels the Darcy resistance. Near the wall the no-slip condition drives the velocity to zero through a Brinkman boundary layer whose thickness is set by the Darcy number.

Governing equations¶

For steady, fully developed flow the axial momentum balance is

(1)¶\[0 = G + \nu\, \nabla^2 u - \alpha\, u\]

between the constant driving force \(G\), the viscous diffusion and the linear Darcy sink with resistance coefficient \(\alpha\). In a pipe of radius \(R\) with no-slip at the wall this admits the closed-form solution

(2)¶\[u(r) = \frac{G}{\alpha} \left[1 - \frac{I_0(\beta\, r)}{I_0(\beta\, R)}\right], \qquad \beta = \sqrt{\frac{\alpha}{\nu}}\]

where \(I_0\) is the modified Bessel function of the first kind, order zero. The shape is governed by the Darcy number, the dimensionless permeability of the medium:

(3)¶\[\mathrm{Da} = \frac{\nu}{\alpha\, R^2}\]

At large \(\mathrm{Da}\) (weak resistance) the Bessel term is small and the profile approaches the parabolic Poiseuille shape; at small \(\mathrm{Da}\) (strong resistance) the core flattens to the terminal velocity \(u = G / \alpha\) and the velocity drops to zero only in a thin Brinkman layer at the wall.

Simulation setup¶

A straight pipe is filled with the porous region and driven by a constant axial body force. The Darcy resistance \(\alpha\) is swept to cover a range of Darcy numbers, from the near-parabolic regime to the strongly damped regime representative of an outlet sponge layer.

Parameter	Value
Geometry	circular pipe, porous region filling the cross-section
Driving force \(G\)	constant body force along the pipe axis
Porous resistance \(\alpha\)	swept to span the Darcy-number range
Darcy number \(\mathrm{Da} = \nu / (\alpha R^2)\)	swept (near-parabolic to strongly damped)
Fluid velocity set / operator	D3Q27 / RRBGK
Pipe wall	no-slip (halfway bounce-back)
Porous-medium treatment	volumetric region BC: linear Darcy momentum sink \(F_\alpha = -\alpha\, u_\alpha\) added to the Guo source term
Initial field	rest; the body force accelerates the flow to its terminal profile

The porous region is selected by a pos predicate and the resistance \(\alpha\) is baked into the kernel as a per-node Darcy body force, the same volumetric mechanism used to place a porous block at the outlet to damp pressure waves.

Reference and acceptance¶

The reference is the analytic Brinkman profile (2), following Brinkman[1] and the porous-media treatment of Nield and Bejan[2]. Two checks are applied across the Darcy sweep:

the radial axial-velocity profile \(u(r)\) matches the Bessel solution to within an \(L^2\) error of 2-3%;
the centreline terminal velocity matches \(G / \alpha\) to within 1% in the strongly damped (small-\(\mathrm{Da}\)) regime, confirming the driving force and the Darcy sink balance exactly where the viscous term is negligible.

Results¶

Note

The quantitative comparison notebook (the radial \(u(r)\) profile against the Bessel solution across the Darcy sweep and the terminal-velocity check) will be added once the GPU validation runs are committed, mirroring the workflow of the other v2.0 cases.