Tsunami Generation and Propagation
Bed shear stress, surface shape and velocity field at the tips of dam-breaks, tsunami and wave runup.
School of Civil Engineering, The University of Queensland
Brisbane Australia, e-mail: email@example.com
An internally consisten 2DV model is proposed for the tip of dam-breaks, tsunami and wave runup.
This is rather badly needed, as measurements of any kind (bull-nose shape, bed shear stress, velocities or sediment transport) are scarce and the need to synthesize information from different studies with different subsets of measurements is needed. For this synthesis, a model of this kind is needed.
The surface shape h(s) is related to the bed shear stress (s) similarly toh Whitham (1955) quasi-steady balance:, where s is the distance from the tip. Whitham only considered the case of uniform u and , which gives h s1/2. But here the equation is integrated in general yieling and in particular, with the experimentally supported ~ s -1/2, one gets h s1/4. This compares much better with the most detailed surface shape dataset, namely that of Schoklitsch (1917) as shown in Figure 1.
Figure 1: The h ~ s1/4 surface shape which corresponds to ~ s -1/2 fits better near the tip than the h ~ s1/2 shape, which corresponds to uniform stress.
The s -1/2 shear stress corresponds to
which corresponds to the boundary layer displacement thicknes growing as s1/2, which is plausible considering the known behaviouers of boundary layer growth after abrupt start or for a flow coming over the edge of a plate. This boundary layer is very thin at the tip because of fluid with full forward momentum being injected to the contact point from above as per the observations of Baldock et al (2014) and the flow pattern from the present model in Figure 2.
Figure 2: Streamlines relative to the tip in a runup tip progressing towards the right at speed c. The spacing of the streamlines at the base correspons to the no slip condition, ie, speed c to the right in this frame of reference. Between the dotted line and the surface, motion is forward/downwards towards the contact point. The pattern corresponds to growing as in Eq(1) with vt = 0.002m2/s and c=1m/s.
The way forward is to calibrate the model against measurements of different kinds, the eddy viscosity vt being the only tuning parameter. With additional meesurements, refinements can be made to the velocity distribution v [1-exp(n/ used in Figure 2.
Baldock, T E, R Grayson, B Torr & H E Power (2014): Flow convergence at the tip and edges of a viscous swash front – Experimental and analytical modeling. Coastal Engineering, Vol 88, pp 123-130.
Schoklitsch, A (1917): Uber Dambruchwellen. KaiserlicheAkademie der Wissenschaften, Wien, Mathematische-Naturwissenschaftliche Klasse, IIA, pp1489-1514.
Whitham, G B (1955): The effects of hydraulic resistance in the dam-break problem. Proc Roy Soc Lond, A 227, pp 399-407.