Difference between revisions of "Examples/2D/souto etal 2012 standingwave"

Revision as of 13:58, 12 March 2015

Introduction

In this examples we are reproducing the simulation of the standing wave carried out by Souto et all (2012) ^[1].

The standing wave is defined on top of an infinite periodic fluid domain composed by rectangular subdomains of dimensions ${\displaystyle L\times H}$ , with ${\displaystyle L=2H}$ . In such domain a wave with length ${\displaystyle\lambda=L}$ (i.e. the wave number is ${\displaystyle k=2\pi/L}$ ) and amplitude ${\displaystyle A}$ is imposed. Denoting the ratio ${\displaystyle 2A/H}$ by ${\displaystyle\varepsilon}$ the following velocity potential can be considered:

${\displaystyle\phi\left(x,y,t\right)=\phi_{0}\left(x,y\right)\,\cos(\omega t),}$

${\displaystyle\phi_{0}\left(x,y\right)=-\varepsilon{\frac{H\,g}{2\omega}}{\frac{\cosh\left(k\left(y+H\right)\right)}{\cosh\left(kH\right)}}\cos\left(kx\right),}$

valid for small values of ${\displaystyle\varepsilon}$ . In previous equations ${\displaystyle\omega}$ is the angular frequency given by the dispersion relation of gravity waves: ${\displaystyle\omega^{2}=g\,k\tanh\left(k\,H\right)}$ , and ${\displaystyle y=0}$ corresponds to the flat free surface (In Fig. 13 of Souto et all (2012)^[1] it is wrong described)

Therefore the velocity field can be computed as ${\displaystyle{\mathbf{u}}\left(x,y,t\right)=\nabla\phi\left(x,y,t\right)}$ . It should be noticed that for ${\displaystyle t=0}$ a flat free surface is obtained.

With such velocity potential, for a viscous fluid, it is possible to get an analytical solution for the kinetic energy decay^[2]:

${\displaystyle{\mathcal{E}}_{k}\left(t\right)=\varepsilon^{2}g{\frac{LH^{2}}{32}}e^{{-4\nu k^{2}t}}\left(1+\cos\left(2\omega t\right)\right),}$

where the kinematic viscosity can be computed from the Reynolds number:

${\displaystyle\nu={\frac{\mu}{\rho}}={\frac{H{\sqrt{gH}}}{{\mathcal{Re}}}}.}$

In our simulation ${\displaystyle g=1}$ , ${\displaystyle L=1}$ , ${\displaystyle\varepsilon=0.1}$ , ${\displaystyle\rho=1}$ and ${\displaystyle H/\Delta x=100}$ , where ${\displaystyle\Delta x}$ is the initial distance between the particles.

References

1. ↑ ^{1.0 1.1} Antonio Souto-Iglesias, Fabricio Macià, Leo M. González, Jose L. Cercos-Pita. On the consistency of MPS. Computer Physics Communications, 2012
2. ↑ J. Lighthill. Waves in Fluids. Cambridge University Press, 2001