Womersley number#
Named after: John Ronald Womersley (1907-1958).
$$\alpha \stackrel{\text{def}}{=} \frac{L \sqrt{f} \sqrt{\rho}}{\sqrt{\mu}} \sim \frac{\text{characteristic length}}{\text{viscous penetration depth}}$$
Description#
Measures oscillatory inertia relative to viscous diffusion in unsteady internal flow. It indicates whether velocity profiles can adjust within each cycle.
Quantities#
| Name | Symbol | SI units | Dimension |
|---|---|---|---|
| characteristic length | \(L\) | \(\mathrm{m}\) | \(\text L\) |
| frequency | \(f\) | \(\mathrm{Hz}\) | \(\text T^{-1}\) |
| mass density | \(\rho\) | \(\mathrm{kg}\,\mathrm{m}^{-3}\) | \(\text L^{-3}\,\text M\) |
| dynamic viscosity | \(\mu\) | \(\mathrm{Pa}\,\mathrm{s}\) | \(\text L^{-1}\,\text M\,\text T^{-1}\) |
Regimes#
Oscillatory pipe flow
| Range | Regime | Description |
|---|---|---|
| 0 – 1 | quasi steady | Viscous diffusion spans the flow during each cycle. The velocity profile remains close to the steady laminar shape. |
| 1 – 10 | developing oscillatory | Unsteady inertia and viscosity are both important. Phase lag and profile distortion develop across the section. |
| 10 – ∞ | plug like oscillatory | Oscillation is too rapid for viscous diffusion to penetrate the full radius. The core moves nearly as a plug with thin oscillatory boundary layers. |