Shear Velocity (Friction Velocity u*)

Divide a shear stress by a density and you get something with the units of velocity squared. Take its square root and you have the shear velocity, $u_* = \sqrt{\tau_0/\rho}$, also called the friction velocity. Nothing in the flow travels at this speed. It is a velocity SCALE — the natural size of the turbulent velocity fluctuations near a boundary — and it is the constant that makes the whole structure of a turbulent boundary layer collapse onto one curve.

Its importance is that the logarithmic velocity profile is written in terms of it: $u(z)/u_* = (1/\kappa)\ln(z/z_0)$. That single relation governs the wind profile above a field, the velocity profile in a river, the flow in a pipe, and the boundary layer on a ship's hull. It also gives you a way to measure $u_*$ without measuring a force at all: plot velocity against the logarithm of height above the bed, take the slope, multiply by von Kármán's constant, and you have the shear velocity. Squaring it and multiplying by density then gives the shear stress the profile implies, and comparing that against $\rho g R S$ is one of the very few genuine independent cross-checks available in field hydraulics. When the two disagree — and they often do — the disagreement is usually telling you about bedform drag or about non-uniform flow.

Some numbers for calibration. In an ordinary river $u_*$ comes out around 5 to 10 % of the depth-averaged velocity, so a stream running 1 m/s has a shear velocity near 5 to 10 cm/s. If your calculation gives a shear velocity a third of the mean velocity, go back and look at the shear stress that produced it — something is wrong. The same rule of thumb holds in pipes and in the atmospheric surface layer, which is a reminder that the turbulence does not much care what the fluid is or how big the channel.

The reason sediment work prefers $u_*$ to $\tau_0$ is that the two quantities that decide a grain's fate — its settling velocity and the turbulence trying to lift it — are then in the same units and can simply be divided. That ratio is the Rouse number, and it is the most useful single thing you can compute about a suspended load.