Bed Shear Stress in an Open Channel

The derivation is a weight balance and takes three lines. Take a reach of channel of length $\Delta x$, flow area $A$ and wetted perimeter $P$. In steady uniform flow the water is not accelerating, so the downslope component of its weight, $\rho g A \Delta x S$, must be exactly balanced by the friction the boundary applies over the wetted area $P \Delta x$. Divide one by the other, write $R = A/P$, and you have $\tau_0 = \rho g R S$. No empiricism, no coefficient — just Newton's first law applied to a slab of water.

In a wide channel the hydraulic radius is very nearly the flow depth, so the relation is often written $\tau_0 = \rho g y S$ and called the depth-slope product. That approximation is good once the width exceeds about twenty times the depth, which most rivers comfortably satisfy and most laboratory flumes do not.

Three warnings, and the first is the one that matters most in practice. This is a REACH-AVERAGED stress, and grains respond to the local one. On the outside of a bend, boundary shear commonly runs one and a half to three times the reach average, which is exactly why bends scour and straights do not, and why bank protection design applies a bend factor rather than using the average. Around a bridge pier or an abutment the multiplication is larger still, which is what the pier scour equation exists to handle.

Second: this is the TOTAL boundary shear, and only part of it is available to move sediment. On a bed with dunes or ripples the flow expends much of its momentum on form drag against the bedforms themselves — pressure differences across the dune crests — and that portion never reaches the grains. Only the skin friction on the grain surfaces does. Over a well-developed dune field the skin-friction component can be half of the total or less, so feeding total shear into a Shields calculation overpredicts transport, sometimes by a lot. Separating the two is what bedform partitioning methods such as Einstein-Barbarossa and Engelund-Hansen exist for.

Third: $S$ is the ENERGY slope, not the bed slope. The two are equal only in uniform flow. In a backwater upstream of a weir, a drawdown into a steep reach, or the contraction through a bridge opening, they differ substantially, and it is the energy slope that carries the physics. Using a surveyed bed slope in a reach that is plainly not uniform is a common and quiet source of error.