|Golf-stroom interacties in een gegeneraliseerde Langrangiaans gemiddelde formulering|
Groeneweg, J. [s.d.]. Golf-stroom interacties in een gegeneraliseerde Langrangiaans gemiddelde formulering. Communications on Hydraulic and Geotechnical Engineering, 1999(october 99/5). [S.n.]: Delft.
Part of: Communications on Hydraulic and Geotechnical Engineering. Delft University of Technology. Department of Civil Engineering: Delft. ISSN 0169-6548, more
The generalized Lagrangian mean (GLM) formulation has been used to describe the interaction of waves and currents. In contrast to the more conventional Eulerian formulation, the GLM description enables splitting of the mean and oscillating motion over the entire depth in an unambiguous and unique way, also in the region between wave crest and trough.
Laboratory experiments by e.g. Kemp & Simons (1982, 1983) and Klopman (1994) showed that the effect of nonbreaking waves on a steady turbulent current over a rigid rough bed is significant. Unexpected wave-induced changes in the profile of the mean horizontal velocity have been observed. For following currents the mean horizontal velocity reaches a maximum and subsequently decreases towards the free surface. Opposing currents show the opposite feature. Compared to the logarithmic profile, the opposing current profile grows more rapidly towards the free surface.
To the authors' knowledge only two theoretical models have been presented to explain the wave-induced changes in the Eulerian-mean horizontal velocity profiles. The model of Nielsen & You (1996) is based on a local force balance in streamwise direction. Lateral variations have been neglected. Dingemans et al. (1996) presented a 2DV lateral model in which secondary circulations in the cross-sectional plane were held responsible for the mean velocity profile changes. The two explanations are in contrast with each other.
The aim of this study has actually been twofold. First, general three-dimensional flow equations have been derived in a GLM setting, in order to obtain a consistent description of the mean motion in an otherwise oscillating field. Secondly, development of a 1DV and a 2DV lateral model and comparison of model results with results obtained from measurements and other models must lead to a better understanding of the mechanisms that cause the changes in the mean horizontal velocity profiles.
Following the concept of the GLM theory, as described by Andrews & McIntyre (1978a), general three-dimensional flow equations have been derived in this formulation from the Reynolds-averaged Navier-Stokes equations. This permits the possibility to consider turbulent motion. Extension of Andrews & McIntyre's (1978a) original equations with viscous and turbulent stresses lead to equations that are far from comprehensive. Therefore, an alternative derivation has been proposed. The developed models are based on the latter GLM equations.
In this study only regular long-crested nonbreaking waves have been considered, interacting with a turbulent current. In order to develop the 1DV model a WKBJ perturbation series analysis has been applied to the 3D GLM-based flow equations. Assuming slow variation in time and horizontal direction of the amplitude functions of each quantity, a series of ordinary differential equations has been obtained. This series has been solved successively, resulting in the vertical distributions of the initial current velocity, the current-affected amplitude function of the carrier wave and the wave-induced second-order correction of the mean motion.
By allowing lateral variations, side-wall effects on the orbital and mean motion have been taken into account. The 1DV solution of the orbital motion has been extended with a laterally varying term. The variations in longitudinal direction have been neglected, resulting in a 2DV description of the waveinduced mean motion in a cross section of the flume. A numerical solver for non-hydrostatic flow has been applied to solve the 2DV GLM-based equations. Measurements in laboratory flumes have been used to validate both models. The wave-induced horizontal drift velocities, which have been obtained for the situation without currents, agreed with Longuet-Higgins' (1953) analytical conduction solution. Agreement with the measurements of Mei et al. (1972) has been obtained for situations in intermediate water depth. For both situations of waves following and opposing the turbulent current the computed profiles agree quantitatively with the measured profiles of Klopman (1994).
The computed cross-sectional distributions of the secondary circulations show qualitative agreement with Klopman's (1997) measurements. However, for following waves the reduction of the horizontal velocity towards the free surface is overpredicted, due to an overestimated downwelling in this region. Comparison of these results with those obtained for a significant wider flume and analysis of the 2DV model results of Dingemans et al. (1996) justify the conclusion that secondary circulations cannot be ruled out in laboratory experiments. However, the velocity profile changes purely due to these circulations are insufficient compared to the total changes of profile. Moreover, following the ideas of Nielsen & You (1996) a local force balance in longitudinal direction proved that the secondorder changes are mainly due to a combination of the wave-induced stress and the Stokes correction of the shear stress.
Although only one- and two-dimensional models have been developed to describe the combined motion of a turbulent current and regular nonbreaking waves, the general scheme followed in the development enables extension to threedimensional modelling. The desired application of the model to coastal areas and analysis of typical three-dimensional features of wave-current interaction in this area seem very well possible.