## Theses, Dissertations and Culminating Projects

8-2015

Thesis

#### Degree Name

Master of Science (MS)

#### College/School

College of Science and Mathematics

#### Department/Program

Mathematical Sciences

David Trubatch

Philip A. Yecko

Bogdan G. Nita

#### Subject(s)

Fluid dynamics, Korteweg-de Vries equation, Navier-Stokes equations, Solitons

#### Abstract

Tracking the behavior of an interface present between two fluids is critical to a wide array of different fields. Much can be learned through lab work and experimentation, however there are many limitations involved. Solving the Navier-Stokes equations with interfaces is very difficult, and in many cases intractable. One way to combat this is to make assumptions to help reduce the equation (e.g. KdV), and make it more easily studied. An entirely different approach is to use Direct Numerical Simulation on the N-S equation, weather it be by use of front-tracking, VOF, etc. In this thesis, we compare these two separate ways of studying solitons in an attempt to (i) verify the derivation of KdV, and (ii) verify the DNS code. In this thesis, we make use of a Volume of Fluid (VOF) multi-phase open source 3D simulator called PARIS, which is currently in active development and makes use of the Navier-Stokes PDE. Using this software we study KdV (Korteweg and de Vries) solitons, which are solitary non-linear waves that retain their shape and travel at a constant speed. The KdV equation is derived from Navier-Stokes under several assumptions, including small amplitude and long waves, with maximal balance in the asymptotic model, as well as zero viscosity in the bulk. An advantage of using DNS over wave-tank experimentation is our ability to implement parameters which are difficult or impossible to execute in the physical world (e.g. zero viscosity). Due to how it is derived, KdV solitons should be approximate solutions of the full set of equations, and should emerge from DNS with the appropriate initial conditions. We study KdV solitons by alterting the values of the small parameters, and comparing the measured velocity of the traveling peak to the velocity we would expect to see. An interpolation formula applied over three cells (the cell where the apex is located, and it’s two closest neighbors) is used to accurately reconstruct and measure the peak. The results presented display the correlation between the actual/measured values of the amplitude and velocity of the peak. As we increase the value of our small parameters, the error with which the simulator predicts the velocity becomes higher. We also show that as epsilon is increased, the apex starts to shrink as dispersion takes effect, which is what we would expect to see as the small parameters leave the KdV regime.

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