## Kinematics in Regular and Irregular Waves based on a Lagrangian Formulation

##### Abstract

Kinematics in two-dimensional regular and irregular swface waves is described based on the Lagrangian form of the equations of motion, with particular emphasis 011 the conditions in the so-called splash zone in irregular waves. A practical method for accurate calculation of kinematics in broad-banded irregular waves is developed based on Gerstner's wave theory, and theoretical calculations are compared with laborat01:y wave data. A review of basic hydrodynamics has also been called for, and is presented from a Lagrangian as well as Eulerian point of view. The results of the analytical study and the study of the wave data question the applicability of certain universally accepted fluid dynamical principles.
The basic equations of fluid motion are presented on Eulerian and Lagrangian form, including the general Lagrangian form of the Laplacian. The relations governing vortex motion are also presented, including the theorems of Helmholtz, Kelvin and others on the rate of change of vorticity and circulation. Rotation of fluid elements is also studied from a Lagrangian point of view, showing that vorticity is not suited to express how a fluid element actually rotates about itself. It is found reason to question the common Lagrangian form of the continuity equation, namely that the Jacobian must be constant and that it in general can be set equal to 1, since this requirement results in some ambiguities and fundamental inconsistencies. Further, when considered in a Lagrangian frame of reference, we have that the theorems of Helmholtz, Kelvin and others require that a given Lagrangian point always represents the same identifiable material "particle", i.e. that the Jacobian is constant and equals 1. Hence, there is also reason to question the common assumption of irrotational (potential) flow in motions generated by conservative (potential) forces only, since this requirement is based on a material (Lagrangian) interpretation of the theorems on vortex motion. The weaker requirement of zero curl of the acceleration in such flows still applies, irrespective of the behaviour of the Jacobian.
The Lagrangian wave theories of Gerstner and Miehe are presented, pertammg to regular waves in deep and intermediate water, respectively. These wave theories represent closed orbital particle motion, i.e. without any net transport of mass. They also contain vorticity (rotationality) at second order, and are therefore traditionally considered invalid beyond first order. The classical solution for surface waves is Stokes 211d order wave theory. The difference between this theory and the two above is Stokes drift; a second order forward transport of mass. Stokes waves and Stokes drift are here also studied from a Lagrangian point of view. It is found that Stokes waves violate continuity and cause a vorticity at second order within less than one wave period, even for waves of small amplitude. Stokes waves are therefore theoretically inconsistent in the Lagrangian frame of reference, which in turn questions the arguments rendering Gerstner's (and Miche's) theory invalid, i.e. the above-mentioned assumption of irrotational motion. The wave theories of Gerstner and Miehe are concluded to be applicable basic solutions for regular waves in the limit of negligible viscosity.
Irregular waves are here modelled as a sum of linear regular Gerstner or Miehe waves, superposed in the Lagrangian frame of reference. The Lagrangian approach is better suited to show the physics of the wave motions than the Eulerian approach, and the linear Lagrangian model of irregular waves automatically includes what are known as nonlinear interactions from an Eulerian point of view. The irregular approach presented here is still only a solution of the linearized Lagrangian problem; it is not a model for nonlinear irregular waves in a mathematical sense.
Iterative methods have been developed that determine which water particle occupies a specific spatial (Eulerian) position at a specific instant in time. This means that also Eulerian quantities can be calculated, in a practical manner, based on the Lagrangian solutions. The iteration methods apply to regular as well as broad-banded irregulm· waves, and yield theoretically consistent values everywhere, also in the splash zone. Since the models of irregular waves presented in this thesis are based on the linear (first order) parts of the regular solutions only, they are not affected by the above questions regarding continuity, vorticity and mass transport at second order.
The Lagrangian theories and models are compared with laboratory wave data for both regular and irregular wave cases. The wave data include measurements of the surface elevation and LDV-measurements of water particle velocities at different vertical positions, also above the still water level. The mean horizontal velocity in a vertical cross-section has been studied closely, and the instantaneous horizontal velocity in a vertical cross-section beneath individual crests and troughs has also been considered.
The analysis of the experimental data show that distinct transitions in the mean horizontal velocity in the flume take place after a relatively short period of time. These analysis, along with visual observations, also indicate that water particles actually move in more or less closed orbits, i.e. similar to Gerstner and Miehe waves, which is fundamentally different from the commonly assumed Stokes drift and associated return current. Again, this supports the above questioning of Stokes waves, Stokes drift and irrotational motion.
For regular waves, the wave theories of Gerstner and Miehe are found to compare exceptionally well with the measurements after the transitions have taken place and a relatively steady mean velocity profile has been established. For irregular waves, the Lagrangian models also compare well with the measurements, although these results are more subject to uncertainties. In particular, the horizontal velocity beneath crests and troughs predicted by the Lagrangian approach is compared with calculations according to the widely used Wheeler's method. The Lagrangian approach is generally found to compare better with the measurements than Wheeler's method does, and it accounts for the discrepancies typically observed when Wheeler's method is compared with wave flume measurements. It should be noted that Wheeler's "ad hoe" method does not satisfy the basic equations of motion, while the Lagrangian approach presented here does satisfy the basic equations consistently, even in the splash zone.
Hence, this study raises some fundamental theoretical questions with respect to continuity, vorticity and mass transport. For waves, it is of the utmost importance to resolve the issues of non-uniform mass transport when higher order solutions are sought. Caution should be taken when analyzing wave flume measurements, in particular for irregular wave cases. Such measurements may not be satisfactorily suited for comparisons with, or verification of, theoretical models of irregular ocean waves.
Anyhow, the Lagrangian approach presented herein should be of great practical and theoretical value, very well suited for simulations and design purposes. The potential for further development seems considerable, and may e.g. open for theoretically consistent superposition of nonlinear Lagrangian components and detailed modelling of wave-wave interactions and wave-current interactions.