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# Title: Boundary layer on a rotating disk
#
# Description:
#
# Von K\'arm\'an \cite{karman1921} showed that the steady flow of an
# incompressible liquid of kinematic viscosity $\nu$ induced by an
# infinite plane disk rotating at angular velocity $\Omega$ can be
# described by a similarity solution. In effect, using
# $\zeta=z\sqrt{\Omega/\nu}$ and setting the axial velocity $U$,
# radial velocity $V$ and azimuthal velocity $W$ as
# $$
# U=\sqrt{\nu \Omega} F(\zeta) \quad V=\Omega r H(\zeta)
# \quad W = \Omega r G(\zeta)
# $$
# the Navier-Stokes equations reduce to a couple of ODEs:
# $$
# F'''-F\, F'' +F'^2/2 +2G^2 = 0 \quad \mbox{and} \quad G''-F\,G'+G\,F' = 0
# $$
# with boundary conditions
# $$
# F(0)=F'(0)=0 \: G(0)=1. \quad \mbox{and} \quad F'(\infty)=G(\infty)=0.
# $$
# where the prime denotes differentiation with respect to $\zeta$.
#
# In figure \ref{veloc} the analytical dimensionless axial and
# azimuthal velocities of Von K\'arm\'an are compared with the
# numerical solutions obtained with the axisymmetric solver including
# azimuthal velocity (swirl).
#
# \begin{figure}[htbp]
# \caption{\label{veloc} Profiles of dimensionless axial and azimuthal velocities.}
# \begin{center}
# \includegraphics[width=0.8\hsize]{profiles.eps}
# \end{center}
# \end{figure}
#
# Author: J.M. L\'opez-Herrera
# Command: gerris2D swirl.gfs
# Version: 111123
# Required files: analytical
# Running time: 1 minute
# Generated files: profiles.eps
#
4 3 GfsAxi GfsBox GfsGEdge { x = 0.5 } {
PhysicalParams { L = 3 }
SourceViscosity 0.2
# the block below adds the azimuthal velocity (swirl) and
# associated terms
VariableTracer W
SourceDiffusion W 0.2
Init { istep = 1 } { W = W*(1. - dt*V/y) }
Source V W*W/y
Refine 5
Time { end = 12 dtmax = 2e-2 }
EventStop { step = 0.1 } U 1e-3 DU
# OutputTime { istep = 10 } stderr
# OutputProjectionStats { istep = 10 } stderr
# OutputSimulation { istep = 10 } stdout
OutputSimulation { start = end } end.txt { format = text variables = U,W }
OutputSimulation { start = end } end.gfs
EventScript { start = end } {
awk 'BEGIN{ nu = 0.2 } {
if ($2 != 0. && $1 != "#" && ($1*$1 + $2*$2) < 8.0)
print $1/sqrt(nu),-$4/sqrt(nu),$5/$2;
}' < end.txt > nu
if gnuplot <<EOF ; then :
set term postscript eps lw 3 color solid 20 enhanced
set output 'profiles.eps'
set xlabel '{/Symbol z}'
set key center right
plot [0:6]'analytical' u 1:2 w l t '-F({/Symbol z})', 'nu' u 1:2 t 'Gerris', \
'analytical' u 1:3 w l t 'G({/Symbol z})', 'nu' u 1:3 t 'Gerris'
EOF
else
exit $GFS_STOP;
fi
if python <<EOF ; then :
from check import *
from sys import *
if (Curve('nu',1,2) - Curve('analytical',1,2)).normi() > 8e-3 or \
(Curve('nu',1,3) - Curve('analytical',1,3)).normi() > 8e-3 :
print (Curve('nu',1,2) - Curve('analytical',1,2)).normi()
print (Curve('nu',1,3) - Curve('analytical',1,3)).normi()
exit(1)
EOF
else
exit $GFS_STOP;
fi
}
}
# box 1
GfsBox {
left = Boundary {
BcDirichlet U 0
BcDirichlet V 0
BcDirichlet W y
}
right = Boundary {
BcNeumann U 0
BcNeumann V 0
BcNeumann W 0
BcDirichlet P 0.
}
bottom = Boundary { BcDirichlet W 0 }
}
# box 2
GfsBox {
left = Boundary {
BcDirichlet U 0
BcDirichlet V 0
BcDirichlet W y
}
right = Boundary {
BcNeumann U 0
BcNeumann V 0
BcNeumann W 0
BcDirichlet P 0.
}
}
# box 3
GfsBox {
left = Boundary {
BcDirichlet U 0
BcDirichlet V 0
BcDirichlet W y
}
right = Boundary {
BcNeumann U 0
BcNeumann V 0
BcNeumann W 0
BcDirichlet P 0.
}
}
# box 4
GfsBox {
left = Boundary {
BcDirichlet U 0
BcDirichlet V 0
BcDirichlet W y
}
right = Boundary {
BcNeumann U 0
BcNeumann V 0
BcNeumann W 0
BcDirichlet P 0.
}
top = Boundary {
BcNeumann U 0
BcNeumann V 0
BcNeumann W 0
}
}
1 2 top
2 3 top
3 4 top
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