Extended Data Fig. 9: Comparison of boundary shear stress estimates from different methods.

For a subset of the flume experiments with spheres, flow velocity was measured using laser particle image velocimetry (PIV). a, Profiles of fluid velocity in the downstream direction as a function of distance above the grain bed (blue dotted lines), offset on the x-axis for visual clarity, are fit with the law of the wall (black lines), u = (u*/κ)ln(30z/d_o), where κ = 0.4 is the von Karman constant, d_o is the grain diameter, and u* = √(τ/ρ) is the shear velocity, which yields an estimate of the shear stress. The law of the wall is fit to the part of each profile between 20% and 80% of the maximum velocity (solid blue lines). b, Plot of the nondimensional bed shear stress estimated from τ = ρgRS against the nondimensional shear stress estimated from the Law of the Wall (blue points) and the shear stress calculated by applying a wall correction factor⁵³ to the original estimates of τ = ρgRS (green points) for flume experiments with glass spheres. c, Same as b, but for the flume experiments with natural gravel, and without PIV-derived shear stress. Dashed lines are least-squares fits. The wall-corrected shear stress estimates for spheres and natural gravel are within error of each other and of the PIV-derived estimates. The average wall correction factor for the two grain types is (2.7 + 2.1)/2 = 2.41. Error bars show best estimate of uncertainty in shear stress estimates. For PIV-derived estimates this is the uncertainty of the log-linear fits in a; for the other estimates, it is the propagated standard error of the mean.