Shields parameter#

Also known as: Shields number.

Named after: Albert Frank Shields (1908-1974).

$$\theta \stackrel{\text{def}}{=} \frac{\tau}{(\rho_s - \rho_f) g L} \sim \frac{\text{bed shear stress}}{\text{submerged particle weight}}$$

Measures bed shear stress relative to submerged particle weight. It indicates whether grains on a bed remain stable or begin to move.

Name	Symbol	SI units	Dimension
shear stress	$\tau$	$\mathrm{Pa}$	$\text L^{-1}\,\text M\,\text T^{-2}$
solid-fluid density difference	$\rho_s - \rho_f$	$\mathrm{kg}\,\mathrm{m}^{-3}$	$\text L^{-3}\,\text M$
gravitational acceleration	$g$	$\mathrm{m}\,\mathrm{s}^{-2}$	$\text L\,\text T^{-2}$
characteristic length	$L$	$\mathrm{m}$	$\text L$

Sediment transport

Range	Regime	Description
0 – 0.03	stable bed	Bed shear is below the typical threshold for sustained grain motion.
0.03 – 0.06	incipient motion	Grains are near the onset of motion; transport depends sensitively on particle Reynolds number, packing, and turbulence.
0.06 – ∞	mobile bed	Bed shear is high enough for sustained sediment motion and bed-load transport.

Name	Symbol	SI units	Dimension
shear stress	\(\tau\)	\(\mathrm{Pa}\)	\(\text L^{-1}\,\text M\,\text T^{-2}\)
solid-fluid density difference	\(\rho_s - \rho_f\)	\(\mathrm{kg}\,\mathrm{m}^{-3}\)	\(\text L^{-3}\,\text M\)
gravitational acceleration	\(g\)	\(\mathrm{m}\,\mathrm{s}^{-2}\)	\(\text L\,\text T^{-2}\)
characteristic length	\(L\)	\(\mathrm{m}\)	\(\text L\)