Poiseuille Pipe Flow

Why this case matters

Pipe flow extends the laminar channel test to a curved, axisymmetric geometry, and is where the staircased representation of a round wall on a Cartesian lattice is first put under pressure. The parabolic profile in (1) has an exact analytical form, so deviations expose how faithfully the boundary conditions reproduce a circular no-slip wall and how much spurious anisotropy the velocity set introduces. Passing it in the laminar regime is a prerequisite for the turbulent pipe case that follows, where lattice isotropy becomes critical. The analytical solution is a classical result of viscous flow theory Pozrikidis[1].

Poiseuille pipe flow can be described as a pressure driven axissimetric flow that occurs in a circular cross-section pipe, as illustrated below.

../../../_images/poiseuille_pipe.svg

This results in an unidirectional flow to which an analytical solution of the velocity profile with \(0 \leq r \leq 1\) is given by (1)

(1)\[\frac{u(r)}{u_{\mathrm{avg}}}=2\left(1 - r^2\right)\]

In which \(u_{\mathrm{avg}}\) is the average velocity that can be obtained numerically, calculating its value using the simulation data, or using the analytical equation:

(2)\[u_{\mathrm{avg}}=\left(-\frac{\mathrm{d}p}{\mathrm{d}x}\right)\frac{\left(0.5d\right)^{2}}{8\rho\nu}\]

where the pressure gradient can be set through pressure inlet/outlet boundary conditions, or adding a external force to the bulk \(F_{x}=-\mathrm{d}p/\mathrm{d}x\).