reasonable estimate on mean velocities measured by Jonsson and Carl sen, but 

 were not as successful in predicting the phase relationship. Multi-layered 

 (Kajiura, 1968) and time-dependent (Jonsson, 1980) eddy viscosities were used 

 to achieve reasonable agreement between computed and measuredon mean flow 

 variables. Drag coefficients varied from 0.01 to 0.08 when 

 amplitude-to-roughness ratio changed from 500 to 10. However, measurements of 

 Riedel et. al (1973) indicated that the drag coefficient was 50% smaller over 

 the entire range of amplitude-to-roughness ratio (Figure 6.5). This 

 discrepancy was not explained by Jonsson or Reidel et.al. The shear stress 

 value in Jonsson and Carlsen's experiments was not directly measured, but 

 obtained indirectly from the time variation of mean velocities. In addition, 

 none of the empirical analyses was able to accurately predict the thickness of 

 the wave boundary layer and its variation with time, although Jonsson and 

 Carl sen (1976) did measure the existence of a thin wave boundary layer within 

 which a relationship similar to Equation (6.4) should hold. 



The uncertainties found in the experimental and empirical analyses lead 

 us to perform a more rigorous analysis of the wave boundary layer. The 

 analysis was carried out with a dynamic turbulent transport model which does 

 not require any ad-hoc adjustments of the model parameters. Detailed 

 simulation of Jonsson and Carlsen's experiment can be found in Appendix D of 

 this report. The model was able to accurately predict the mean flow variables 

 throughout a complete wave cycle. Further, the phase relationship predicted 

 by our dynamic model agrees very well with data than did any ad-hoc 

 eddy-viscosity model. In addition, transient variation of the thickness of 

 the wave boundary layer can be predicted. Although our model prediction on 

 shear stress agrees well with Jonsson and Carlsen's estimate at 9-180°, it was 

 found that the model predicted shear stress is generally less than their 

 indirect estimates over most of the wave cycle. We believe this is due to the 

 error introduced in their indirect estimates of shear stresses arising from 

 the lack of time resolution in their mean flow data. In fact, their estimate 

 on shear stress averaged over a wave cycle contains approximately a twofold 

 overestimation. Empirical bottom stress formula derived from fitting their 

 data could lead to appreciable error. Although turbulent quantities were 

 computed by our turbulent transport model, they were not measured by Jonsson 

 and Carlsen. 



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