Shields Parameter (Dimensionless Shear Stress)

In 1936 Albert Shields submitted a doctoral dissertation to the Preußische Versuchsanstalt für Wasserbau und Schiffbau in Berlin — volume 26 of their Mitteilungen — on the application of similarity mechanics to bed-load movement. He was asking when a grain on a river bed begins to move, and he answered it the way a fluid mechanician should: by finding the dimensionless group that governs it.

The reasoning is a force balance you can do on the back of an envelope. The flow drags on the grain with a force proportional to the shear stress $\tau$ times the grain's projected area, which goes as $d^2$. Holding it down is its submerged weight, proportional to $(\rho_s - \rho)g$ times its volume, which goes as $d^3$. Take the ratio, let the $d^2$ cancel, and you have $\theta = \tau/[(\rho_s-\rho)gd]$ — driving force over restoring force, dimensionless, and large when the grain is about to go. Shields plotted his measurements of the critical value of this ratio against the grain Reynolds number and drew a curve through them. Every sediment textbook since has reprinted it.

What matters most about that curve is how much scatter it hides. The data Shields drew through span roughly a factor of two either side of the line, and later compilations that added other investigators' work spread further still. Part of that is measurement, but most of it is definitional: different researchers meant different things by "movement". Some counted the first grain to shift anywhere in the test section. Some waited until a defined weak transport rate was sustained. Kramer defined four grades of movement and the critical value differed by a factor of two between the first and the last. A single number for the threshold is a convention, not a measurement.

And even with a perfect curve the threshold would not be sharp. The shear stress in the equation is a time-averaged, area-averaged quantity, but turbulence is neither: the instantaneous stress on one square centimetre of bed swings several times above and below the mean many times a second, and grains are not all equally exposed — one sitting proud on top of its neighbours goes at a fraction of the stress needed to shift one nested down between them. So the bed does not sit still and then start; it produces occasional single-grain movements at a third of the "critical" stress, a slow persistent trickle near it, and general transport somewhat above. Treat $\theta$ as a probability of motion, and treat a design that depends on being just under threshold as a design that will spend part of its life over it.

The customary critical values are 0.03 to 0.06 depending on where you sit on the curve, with 0.045 to 0.06 the usual choice for coarse sand and gravel, and around 0.03 in the dip near a grain Reynolds number of 10. For fine silt and clay the curve stops being useful altogether, because cohesion — not weight — holds those particles down, and a firm clay bed will resist a shear stress that strips gravel.