# Title: Time-reversed VOF advection in a shear flow
#
# Description:
#
# A test case initially presented by Rudman \cite{rudman97}.
# A circular patch of tracer is advected in a vortical shear flow. At t = 2.5
# the flow direction is reversed. An exact advection scheme would restore the 
# initial circular shape at t = 5.
#
# The VOF (Volume-Of-Fluid) advection scheme is not exact. The initial, intermediate
# and final shape of the interface are represented on Figure \ref{advection}. 
# Figure \ref{error} illustrates the error between the initial and final shapes. The
# corresponding error norms are given in Table \ref{norms}.
#
# Adaptive refinement is used with the gradient of the volume fraction as criterion.
# Eight levels of refinement are used on the interfaces and six away from the interface.
#
# \begin{figure}[htbp]
# \caption{\label{advection}Volume fraction field at times (a) 0, (b) 2.5 and (c) 5.}
# \begin{center}
# \begin{tabular}{ccc}
# \includegraphics[width=0.3\hsize]{t-0.eps} &
# \includegraphics[width=0.3\hsize]{t-2.5.eps} &
# \includegraphics[width=0.3\hsize]{t-5.eps} \\
# (a) & (b) & (c)
# \end{tabular}
# \end{center}
# \end{figure}
#
# \begin{figure}[htbp]
# \caption{\label{error}Difference between the initial and final volume fraction fields.}
# \begin{center}
# \includegraphics[width=0.4\hsize]{dt-5.eps}
# \end{center}
# \end{figure}
#
# \begin{table}[htbp]
# \caption{\label{norms}Norms of the error between the initial and final fields. The
# reference values are given in blue.}
# \begin{center}
# \begin{tabular}{|c|c|c|}\hline
# $L_1$ & $L_2$ & $L_\infty$ \\ \hline
# \input{norms.tex}
# \end{tabular}
# \end{center}
# \end{table}
#
# Author: St\'ephane Popinet
# Command: sh shear.sh shear.gfs
# Version: 091022
# Required files: shear.sh norms.ref
# Running time: 2 minutes
# Generated files: t-0.eps t-2.5.eps t-5.eps dt-5.eps norms norms.tex
#
# The type of the simulation is GfsAdvection which only solves the advection
# of passive tracers.
1 0 GfsAdvection GfsBox GfsGEdge {} {
    Time { end = 5 }
    Refine 8

    # Add tracer T, using a VOF advection scheme.
    # The default scheme is a Van-Leer limited, second-order upwind scheme.
    VariableTracerVOF T

    InitFraction T (ellipse (0, -.236338, 0.2, 0.2))

    # Maintain U and V with the vortical shear flow field defined by
    # its streamfunction
    VariableStreamFunction {
	# make sure that the entire field is reinitialised at t = 2.5
	step = 2.5 
    } Psi (t < 2.5 ? 1. : -1.)*sin((x + 0.5)*M_PI)*sin((y + 0.5)*M_PI)/M_PI
    
    # Adapt the mesh dynamically so that at any time the maximum of the gradient
    # of T is less than 1e-2 per cell length
    AdaptGradient { istep = 1 } { cmax = 1e-2 maxlevel = 8 } T
    
    OutputPPM { start = 0 } { convert ppm:- t-0.eps } { v = T }
    OutputPPM { start = 2.5 } { convert ppm:- t-2.5.eps } { v = T }
    OutputPPM { start = 5 } { convert ppm:- t-5.eps } { v = T }

    # Add a new variable 
    Variable Tref

    # Initialize Tref with the initial shape
    InitFraction { start = end } Tref (ellipse (0, -.236338, 0.2, 0.2))

    # Output the norms of the difference between T and Tref, stores that into
    # new variable DT
    OutputErrorNorm { start = end } norms { v = T } {
        s = Tref v = DT
    }

    OutputPPM { start = end } { convert ppm:- dt-5.eps } { v = DT }

    OutputScalarSum { istep = 1 } { awk '{ print $3,$5-8.743441e-01 }' > t } { v = T }
}
GfsBox {}