Eddy viscosity models for pure waves over large roughness elements

Although the classical Grant and Madsen (1979 and 1986) eddy viscosity model was used successfully to translate wave energy dissipation measurements into an equivalent roughness by Mathisen (1993), it fails to resolve the details of the boundary layer velocity profile. This issue is addressed in the present study, in which three models are presented and their abilities to predict the details of the velocity profile are compared. First, a constant eddy viscosity model for wave boundary layer flows over twodimensional roughness elements simulating wave-generated, fully developed ripples is presented. M1athisen's (1993) measurements of energy dissipation for periodic waves over artificial bedforms are interpreted in terms of drag resistance, and a good correlation is obtained for the drag coefficient . CD, and the ratio of the ripple height, ra, to the maximum bottom excursion amplitude. Ab. This dependence of CD on r7/Ab is shown to be in general agreement with Sarpkava's (1981) analysis of the drag coefficient as a function of the Keulegan-Carpenter number. The drag law dissipation is similar in nature to that obtained for a constant eddy viscosity wave boundary laver model with vt ocq/(A2T), where A is the ripple length and T is the wave period. This eddy viscosity, as function of the inverse of the wave period, was also obtained by Sleath (1991). However, here the eddy viscosity is also a function of the ripple steepness (height and length), and not of any other flow parameter. The constant eddy viscosity model is applied with the linearized boundary layer equation to predict detailed velocity profiles (magnitudes and phases) and compare favorably with measurements from Mathisen (1993) and Barrantes (1996). The results of this study support the use of a constant eddy viscosity for the prediction of energy dissipation as well as the details of the velocity profile within the wave boundary layer for flows over rippled beds. Then a model based entirely on the classical GM model is presented, in which the no-slip condition is modified to be at the bottom and not at z = zo. With this change the predicted velocity profile agrees significantly better with the measurements than does the original model. Moreover, the analysis of the theoretical boundary layer thickness shows that the latter is proportional to the product of two monotonically increasing functions of the relative roughness, and not only of the boundary laver scale 1, as thought before. However, some features of the profile are not described accurately and this becomes the motivation to develop a model consisting of a combination of the first two. In the combined model the eddy viscosity is considered linearly varying with depth in the lower portion of the boundary layver and then constant above that. This model appears to describe the details of the velocity profile with the same accuracy as the constant eddy viscosity model. Additionally, its prediction of the energy dissipation, given a prescribed bottom roughness proportional to the ripple height, turns out to be in slightly better agreement with experiments performed over a rippled movable bed. than the predictions using the Constant eddy viscosity model. Thesis Supervisor: Ole S. Madsen Title: Professor, Department of Civil and Environmental Engineering