Laminar Flow in a Circular Pipe (Hagen-Poiseuille Flow)
Theory: the parabolic velocity profile
Overview
I want to check the setup of newly adopted CFD software on a simple problem. What should I use?
Laminar pipe flow — Hagen–Poiseuille flow. Fully developed, the velocity is a parabola from zero at the wall to twice the mean at the centre, and pressure drop and friction factor are known exactly. The geometry is simple, yet it checks boundary conditions, viscous treatment and near-wall grid effects together. There is even a formula for where the entrance region ends, so it checks domain length too.
Exact solution
$U$ is mean velocity, $R$ radius, $D$ diameter, $\mu$ viscosity and $Re = \rho U D/\mu$. Wall shear follows directly from the force balance on a length of pipe (pressure force = wall friction).
Entrance region
The distance from the inlet until the flow is fully developed: about 5 diameters at Re = 100, 50 at 1,000 and 100 at 2,000. Compare with the exact solution only downstream of this length.
A law born of blood vessels and water pipes
The law was found independently by the German engineer Hagen in 1839 from water-pipe experiments and by the French physician Poiseuille in the 1840s from experiments on blood flow in capillaries. Its hallmark is flow rate proportional to the fourth power of radius, which explains why slight narrowing of a blood vessel sharply cuts flow. The viscosity unit 'poise' honours Poiseuille. The theoretical derivation later became the most basic exact solution of the Navier–Stokes equations.
Worked example: radial finite volume
The radial equation for fully developed flow was solved by finite volume with 5, 10, 20 and 40 cells, and errors computed:
| Cells | Centreline error | Wall shear (from flux) | Wall shear (quadratic fit) |
|---|---|---|---|
| 5 | 1.00% | 0 (exact) | 6.7% |
| 10 | 0.25% | 0 (exact) | 3.3% |
| 20 | 0.063% | 0 (exact) | 1.7% |
| 40 | 0.016% | 0 (exact) | 0.83% |
The wall shear from the flux has zero error even with 5 cells?
Yes — thanks to the conservation property of finite volume. Each cell is solved so inflow and outflow balance, so summed over the domain the wall flux exactly balances the driving pressure force; even on a coarse grid the total friction is right. Fitting a quadratic through cell velocities to get the wall slope, however, divides velocity error (order spacing squared) by spacing, dropping to first order — which is why the right-hand column only halves with each halving. When post-processing wall shear, use the flux the solver actually used.
Example 2: checking pressure drop
Water (20 °C) at a mean 0.1 m/s in a 10 mm bore gives Re ≈ 1,000, laminar. Pressure drop over 1 m is 32 Pa and the friction factor 0.064 (= 64/Re). The entrance length is about 50 diameters (0.5 m), so for CFD comparison use a pipe at least 1 m long and read the pressure gradient downstream near the outlet.
Verification workflow
- Impose uniform inlet velocity, constant outlet pressure and no-slip walls — or solve only fully developed flow with periodic boundaries.
- Downstream of the entrance region, compare velocity profile, pressure gradient and wall shear with the exact solution.
- Check error reduction on three or more grids.
- For 3D pipes, cross-section meshing (e.g. O-grid topology) also matters; check its quality.
“Pressure drop comes out 10% high”
Verifying pipe pressure drop, the result came out about 10% above the exact solution — commonly because the overall inlet-to-outlet pressure difference was compared. Near the inlet the profile develops from flat to parabolic, and this entrance region has higher losses than fully developed flow; at Re = 1,000 in a 100-diameter pipe, half the length is affected. Comparing the pressure gradient in the downstream developed section matches well. In verification, what you compare and where must match the solution's assumptions.
Common mistakes
Mistakes and fixes
| Mistake | Effect | Fix |
|---|---|---|
| Comparing including the entrance region | Pressure drop overestimated | Compare in developed flow |
| Wall shear from fitted cell values | Drops to first order | Use the solver's flux |
| Judging from one grid | Order unknown | Three or more levels |
| Re too high | Turbulent; exact solution invalid | Verify below Re ≈ 2,000 |
| Pipe too short | No developed section | Longer than 0.05 Re D |
I'd like to know other verification problems.
Try the pipe flow calculator and Darcy vs Fanning friction factor for pipe pressure drop and the Richardson extrapolation tool for convergence order. Other exact and benchmark problems include Couette flow, lid-driven cavity and sphere drag; for problems without analytical solutions, use the method of manufactured solutions.
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