## Surface water waves on depth dependent flows

##### Doctoral thesis

##### View/Open

##### Permanent link

http://hdl.handle.net/11250/2456095##### Issue date

2017##### Metadata

Show full item record##### Collections

##### Abstract

The present thesis provides essential insights into surface water waves propagating
atop a horizontal current whose magnitude and direction may vary arbitrarily with
water depth. A comprehensive theory in this regard is developed in the framework
of linear wave theory in three dimensions, being readily applied to a wide range of
realistic circumstances. General theoretical solutions to different boundary value
problems are presented. In particular, explicit expressions with regard to the surface
elevation and the vertical velocity are derived. The boundary value problems
include the Cauchy-Poisson problem, surface disturbances generated by an initial
impulsive and a time-dependent pressure, and a steady pressure that normally
works as the model of moving vessels and oscillating travelling sources. Efforts
focus especially on the dispersion relation and the effects of a subsurface shear
current on surface waves. A subsurface shear current is most often found to have
significant effects on surface waves. In particular, the presence of a current of uniform
vorticity is analysed in detail for the problems of ship waves, an oscillating
advancing source and wave interferences.
A theory is especially presented to calculate waves from a general, time-dependent
applied surface pressure acting on the surface of a horizontally directed shear current
which may vary arbitrarily with depth in both direction and magnitude. It
is based on deriving the response function in the context of waves generated by
an impulsive applied pressure. Effective approaches to calculate wave resistance
without undue difficulty are presented. Strikingly, a lateral radiation force – that is
defined towards the starboard (right) – is firstly found apart from the well-known
wave resistance along the stern-wise direction due to the presence of a shear current
when a ship is making an oblique angle with the shear current. The lateral
radiation force may amount to 20 percent of the normal wave resistance in some
specific situations.
As for waves on a current in arbitrary variation of water depth, an implicit dispersion
relation is derived, which poses potential challenges in obtaining analytical
solutions. Several semi-analytical approaches to solve/approximate the dispersion
relation are hence derived. In particular, a direct integration approach – that solves
the linearised Rayleigh equation and implicit dispersion relation in a coupled way
– and approximations based on a perturbation method are presented. The proper
criteria under which different perturbed approximate dispersion relations are applicable
are determined. Furthermore, the analytical solutions of the dispersion
relation under limited circumstances are derived, e.g. for a shear current of uniform
vorticity and stationary waves for a specific class of shear profiles of non-zero
curvature.
Despite the fact that linear waves in the presence of a linear shear current have been
extensively analysed in two dimensions, studies in three dimensions are scarce.
This means that realistic three dimensional effects may be in some cases overlooked
or yet discovered. The present thesis hence attempts to fill this gap based
on theoretical as well as numerical analysis. Effects of a uniform vorticity are specially
analysed in the context of ship waves, waves generated by an oscillating travelling
source and interferences of waves generated by a two-point wavemaker of
monohull ships. Fascinating and novel features are found due to the uniform vorticity
S that is known either as the ’intrinsic shear Froude number’ Frsb = S
_
L/g
or the ’shear Froude number’ Frs = |V|S/g (L: the reference length; |V|: moving
speed of a wavemaker; g: the gravitational acceleration). In particular, asymmetrical
ship wave patterns, the critical shear velocity above which the transverse ship
waves vanish, the transitions between the sub-critical and supercritical situations
due to the complex interplays of the shear current and seabed, non-constant Kelvin
angles, and a somewhat similar effect as a finite water depth on wave interferences
are shown for the problem of ship wakes. All of those novel features would not
have been found if theoretical studies are constrained to 2D. Furthermore, the classical
Doppler resonance occurs when the non-dimensional frequency τ = |V|ω/g
(ω : the oscillating frequency) is equal to 1/4 in the absence of a shear current,
while there may be multiple Doppler resonances – as many as 4 – for Frs > 1/3 in
deep water due to the presence of a linear shear current. It is also indicated that the
Doppler resonance may be profoundly modified even for a linear current of weak
vorticity.

##### Has parts

Paper 1: Li, Y.; Ellingsen, S. Å. Direct integration method for surface waves on a depth dependent flow.Paper 2: Li, Y. (2017). Wave-interference effects on far-field ship waves in the presence of a shear current.

Paper 3: Li, Y., Smeltzer, B. K., and Ellingsen, S. Å. (2017). Transient wave resistance upon a real shear current © 2017. This manuscript version is made available under the CC-BY-NC-ND 4.0 license - Published version in European Journal of Mechanics - B/Fluids https://doi.org/10.1016/j.euromechflu.2017.08.012

Paper 4: Ellingsen, S. Å.; Li, Y. (2017). Approximate dispersion relations for waves on an arbitrary shear flow.

Paper 5: Li, Yan; Ellingsen, Simen Andreas Ådnøy. Multiple resonances of a moving oscillating surface disturbance on a shear current. Journal of Fluid Mechanics 2016 ;Volum 808. s. 668-689 Is not included due to copyright https://doi.org/10.1017/jfm.2016.637

Paper 6: Li, Yan; Ellingsen, Simen Andreas Ådnøy. Ship waves on uniform shear current at finite depth: Wave resistance and critical velocity. Journal of Fluid Mechanics 2016 ;Volum 791. s. 539-567 Is not included due to copyright

Paper 7: Li, Yan; Ellingsen, Simen Andreas Ådnøy. Effect of Anisotropic Shape on Ship Wakes in Presence of Shear Current of Uniform Vorticity. I: ASME 2016 35th International Conference on Ocean, Offshore and Arctic Engineering - Volume 7: Ocean Engineering. ASME Press 2016

Paper 8: Li, Yan; Ellingsen, Simen Andreas Ådnøy. Initial Value Problems for Water Waves in the Presence of a Shear Current. I: Proceedings of the twenty-fifth International Ocean and Polar Engineering Conference - ISOPE 2015. International Society of Offshore & Polar Engineers