**7. Conclusion**

234 Advanced Fluid Dynamics

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Fig. 12. Computed vorticity field (*s*-1) at different phases of the flow. The lower levels of the


measurements over the ripple crest and trough are marked

Assuming that the fluid is in a randomly unsteady turbulent state and applying time averaging to the basic equations of motion, the fundamental equations of incompressible turbulent motion are obtained. A three-dimensional form of conservation equations for a single Reynolds stress and for the turbulent kinetic energy is derived. However, as the full three-dimensional form of equations is very complex and not easy to solve, with many unknown correlations to model, other much simpler one- and two-dimensional boundary layer forms of these relations are derived. A brief discussion about numerical models based on control volumes and finite difference approximations is presented to solve 1DV versions of the one- and two-equation rough turbulent bottom boundary layer model of the K-L type, and of the 2DV boundary layer model. These numerical models are then used to calibrate general parametric formulations for the instantaneous bottom shear stress due to both a wave and a wave-current interaction cases. They are still used to discuss some important aspects, like the *phase shift* and the *turbulence memory effects*. Mathematical formulations and parametric approaches are extended to include the effect of suspended non cohesive sediments. Comparisons with experimental results show that both 1DV and 2DV boundary layer models are able to predict quite well the complex flow properties. However, these models are strictly valid for permanent flows in the fully developed turbulent regime at high Reynolds numbers. When the flow is oscillatory, the condition of local equilibrium of the turbulence is no longer completely satisfied, particularly at the time when the velocity of the potential flow is small. Therefore, improvements are necessary to obtain more precise results for moderate Reynolds numbers.
