35 research outputs found
Relationships between a roller and a dynamic pressure distribution in circular hydraulic jumps
We investigated numerically the relation between a roller and the pressure
distribution to clarify the dynamics of the roller in circular hydraulic jumps.
We found that a roller which characterizes a type II jump is associated with
two high pressure regions after the jump, while a type I jump (without the
roller) is associated with only one high pressure region. Our numerical results
show that building up an appropriate pressure field is essential for a roller.Comment: 10 pages, 7 PS files. To appear in PR
Plane shearing waves of arbitrary form: exact solutions of the Navier-Stokes equations
We present exact solutions of the incompressible Navier-Stokes equations in a
background linear shear flow. The method of construction is based on Kelvin's
investigations into linearized disturbances in an unbounded Couette flow. We
obtain explicit formulae for all three components of a Kelvin mode in terms of
elementary functions. We then prove that Kelvin modes with parallel (though
time-dependent) wave vectors can be superposed to construct the most general
plane transverse shearing wave. An explicit solution is given, with any
specified initial orientation, profile and polarization structure, with either
unbounded or shear-periodic boundary conditions.Comment: 6 pages, 2 figures; version published in the European Physical
Journal Plu
Modeling water waves beyond perturbations
In this chapter, we illustrate the advantage of variational principles for
modeling water waves from an elementary practical viewpoint. The method is
based on a `relaxed' variational principle, i.e., on a Lagrangian involving as
many variables as possible, and imposing some suitable subordinate constraints.
This approach allows the construction of approximations without necessarily
relying on a small parameter. This is illustrated via simple examples, namely
the Serre equations in shallow water, a generalization of the Klein-Gordon
equation in deep water and how to unify these equations in arbitrary depth. The
chapter ends with a discussion and caution on how this approach should be used
in practice.Comment: 15 pages, 1 figure, 39 references. This document is a contributed
chapter to an upcoming volume to be published by Springer in Lecture Notes in
Physics Series. Other author's papers can be downloaded at
http://www.denys-dutykh.com
Steady water waves with multiple critical layers: interior dynamics
We study small-amplitude steady water waves with multiple critical layers.
Those are rotational two-dimensional gravity-waves propagating over a perfect
fluid of finite depth. It is found that arbitrarily many critical layers with
cat's-eye vortices are possible, with different structure at different levels
within the fluid. The corresponding vorticity depends linearly on the stream
function.Comment: 14 pages, 3 figures. As accepted for publication in J. Math. Fluid
Mec
A Book for the King:John Geddy's Methodi sive compendii mathematici (1586)
A remarkable little-known manuscript, Methodi sive compendii mathematici, was written by John Geddy (or perhaps Geddie) in St Andrews in 1586. This purports to be the first book of four, but no others are known to exist. It was composed for King James VI of Scotland, then aged twenty, who in 1603 would also become King James I of England. Prefaces extol the King and refer to Geddy's time in Paris, while the main text is devoted to arithmetic. The whole is written in beautiful calligraphy, with many letters highlighted in gold and with ornate capitals. This work and its context are described