Bed shear stress, surface shape and velocity field near the tips of dam-breaks, tsunami and wave runup

An analytical model is presented for the 2DV flow-structure, bed shear stress and surface shape of the tips of dam-break waves, tsunami- or wave run-up. This model differs from previous analytical models, which have taken the usual 'hydraulics approach', describing bed-parallel velocities only and expressing the bed shear stress, tau in terms of a friction factor from steady, uniform flow and an ad-hoc velocity. The 2DV model presented here gives a simple, rational explanation for the fact that the boundary layer is very thin at the contact point. In turn, this explains the measurements of bed shear stresses, which decay with distance s from the tip, or at a fixed point, with time t since passage of the tip. The manner of tau-decay depends on the growth of the boundary layer thickness delta with distance from the tip. The details of this boundary layer growth depend on challenging turbulence features for unsteady, non-uniform flows over rough and usually mobile beds. For illustrative purposes, details are given for the simple example of delta = (v(t)t)(1/2) = (v(t)s/c)(1/2), where v(t) is the nominal eddy-viscosity and c is the speed of the tip, assumed to progress with constant form. With this variation of delta, the model gives tau similar to t(-1/2) = (s/c)(-1/2), which is in good qualitative agreement with measurements. Subsequently, this quasi steady model gives h similar to s(1/4) for the depth on a horizontal bed, also in good agreement with experiments. The 2DV flow pattern includes surface particles drifting towards the tip and eventually impacting on the bed at the contact point with full forward momentum and large bed-normal velocity. As well as large local bed shear stresses, this tip-flow pattern involves large vertical accelerations and associated large localized pressures, which are likely to be important for the sediment entrainment under the tip. The eddy viscosity is the only tuning parameter in this simple initial model. Based on shear-stress and depth measurements from laboratory settings, with tip propagation speeds of the order 2 m/s, one finds that the required eddy viscosity is in the range 1 x 10(-6) - 18 x 10(-6) m(2)/s increasing with increasing bed roughness in the range from smooth beds to beds of 2.85 mm fixed sand-grains.