Artur Palha

Advection-Diffusion

[fruitful_sep]

Results

1. Gaussian hill with high deformation

This test case consists of the linear advection of a Gaussian hill function \omega_{0}(x,y)

    \[ \omega_{0}(x,y) = \frac{1}{2\pi\sigma^{2}}e^{-\frac{(x-x_{0})^{2}}{2\sigma^{2}}}e^{-\frac{(y-y_{0})^{2}}{2\sigma^{2}}}, \]

where \sigma = 0.2, x_{0}=0.01 - \pi, y_{0} = 2.0. The advecting vector field \vec{u} is given by

    \[ \vec{u} = \nabla\times\psi, \]

with

    \[ \psi(x,y) = -\cos(x-\pi) + \frac{1}{2}y^{2} + 1. \]

The computational domain is \Omega = [-2\pi,2\pi)\times[-3,3), with periodic boundaries.

The video below shows the evolution from t=0 to t=4 and then the advection velocity field is reversed from t=4 to t=8. With this test we can show the reversibility of the solver, since we can retrieve the initial Gaussian hill. For this simulation 40000 triangular elements of polynomial degree p=4 have been used. The time step is \Delta t = 10^{-2}.

[youtube video=https://www.youtube.com/watch?v=V92pBONnmDQ color=red suggested=0 showinfo=0 maxw=640]

Conservation of the total enstrophy \mathcal{E}(t):=\int_{\Omega}\omega(x,y,t)^{2} and total vorticity \mathcal{W}(t):=\int_{Omega}\omega(x,y,t) are shown in the figures below.

a
Enstrophy (left) and total vorticity (right) error for the passive advection of Gaussian hill with 40000 triangular finite elements, basis functions of polynomial degree p = 4, and time steps \Delta t = 0.01.