# Title: Estimation of the numerical viscosity
#
# Description:
#
# The velocity field is initialised with an exact stationary solution of
# the Euler equations in a periodic 2D domain. An exact Euler solver
# would not change this field, however any numerical solver will
# introduce numerical dissipation which will slowly dissipate the
# kinetic energy of the initial solution. By monitoring the evolution of
# the kinetic energy, the dissipative properties of the numerical scheme
# can be measured (see \cite{rider95} for details).
#
# Figures \ref{divmax} and figure \ref{divL2} illustrate the evolution
# of the divergence of the velocity field with time. This is a check of
# the stability of the approximate projection and should remain bounded.
#
# Figures \ref{kinetic} and \ref{reynolds} illustrates the evolution of
# the kinetic energy and the corresponding equivalent Reynolds number as
# a function of resolution. The higher the Reynolds number, the less
# dissipative the scheme.
#
# \begin{figure}[htbp]
# \caption{\label{divmax}Evolution of the maximum divergence.}
# \begin{center}
# \includegraphics[width=0.8\hsize]{divmax.eps}
# \end{center}
# \end{figure}
#
# \begin{figure}[htbp]
# \caption{\label{divL2}Evolution of the L2 norm of the divergence.}
# \begin{center}
# \includegraphics[width=0.8\hsize]{divL2.eps}
# \end{center}
# \end{figure}
#
# \begin{figure}[htbp]
# \caption{\label{kinetic}Evolution of the kinetic energy.}
# \begin{center}
# \includegraphics[width=0.8\hsize]{kinetic.eps}
# \end{center}
# \end{figure}
#
# \begin{figure}[htbp]
# \caption{\label{reynolds}Equivalent Reynolds number as a function of resolution.}
# \begin{center}
# \includegraphics[width=0.8\hsize]{reynolds.eps}
# \end{center}
# \end{figure}
#
# Author: St\'ephane Popinet
# Command: sh reynolds.sh reynolds.gfs 1
# Version: 0.6.4
# Required files: reynolds.sh div5.ref div6.ref div7.ref reynolds.ref
# Running time: 3 minutes
# Generated files: divmax.eps reynolds.eps divL2.eps kinetic.eps
#
1 2 GfsSimulation GfsBox GfsGEdge {} {
  Time { end = 2 }
  Refine LEVEL
  Init {} {
    U = (- cos (2.*M_PI*x)*sin (2.*M_PI*y))
    V = (sin (2.*M_PI*x)*cos (2.*M_PI*y))
  }
  ApproxProjectionParams { tolerance = 1e-6 }
  ProjectionParams { tolerance = 1e-6 }
  OutputScalarNorm { istep = 1 } divLEVEL { v = Divergence }
  OutputScalarSum { istep = 1 } kineticLEVEL { v = Velocity2 }
  OutputScalarSum { istep = 1 } stdout { v = Velocity2 }
}
GfsBox {}
1 1 right
1 1 top