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# Title: Spherical cap bubble
#
# Description:
#
# We now take Ar = 80 and Bo = 40. For these values, the bubble's shape is
# that of a spherical cap, for which the terminal velocity is given
# approximately as:
# \begin{equation}
# \frac{U}{U_c} = 0.71\left[ 1-\frac{\rho_b}{\rho_f} \right]^{1/2}
# \approx 0.71.
# \end{equation}
# The characteristic velocity is $U_c = \sqrt{g D}$ and the characteristic
# time is $t_c = D/U_c$.
# \begin{figure}\centering
# \includegraphics{vel.eps}
# \end{figure}
#
# Author: Dustin Langewisch
# Command: bash bubble.sh cap.gfs
# Version: 130112
# Required files: bubble.sh bubble.gfv
# Running time: 7 min (4 processors)
# Generated files: vel.eps
## Density Ratio (rho_f / rho_v)
Define RHOR 1000.0
##
## Viscosity Ratio (mu_f / mu_v)
Define MUR 100.0
##
## Archimedes Number
Define Ar 80.0
##
## Bond Number
Define Bo 40.0
##
## Domain half-width (WIDTH*Diameter)
Define WIDTH 6.0
##
## Min/Max Refinement levels
Define MINLEVEL 4
Define MIDLEVEL (MINLEVEL+2)
Define MAXLEVEL (MIDLEVEL+2)
Define VAR(T,min,max) (min + CLAMP(T,0,1)*(max-min))
##
## Density
Define RHO(T) VAR(T,1.0,1.0/RHOR)
##
## Viscosity (harmonic mean)
Define MUHARM(T) 1.0/VAR(T,1.0,MUR)
Define MAXTIME 35
8 7 GfsAxi GfsBox GfsGEdge { x = -1.0 } {
Time { end = MAXTIME }
PhysicalParams { L = WIDTH }
GModule hypre
Refine 6
VariableTracerVOFHeight T
VariableFiltered T1 T 1
VariableCurvature K T Kmax
InitFraction T ( -(x*x)-(y*y)+(.25) )
PhysicalParams { alpha = 1.0/RHO(T1) }
Source {} U -1.0
SourceViscosity MUHARM(T1)/Ar
SourceTension T 1.0/Bo K
AdaptGradient { istep = 1 } {
maxlevel = MAXLEVEL
minlevel = MINLEVEL
cmax = 1e-2
} T1
AdaptVorticity { istep = 1 } {
maxlevel = MIDLEVEL
minlevel = MINLEVEL
cmax = 1e-2
cfactor = 1
}
EventBalance { istep = 10 } 0.1
OutputProjectionStats { istep = 10 } stderr
OutputDiffusionStats { istep = 10 } stderr
OutputBalance { istep = 10 } stderr
OutputTime { istep = 10 } stderr
## compute bubble volume (needed to compute centroid)
SpatialSum { step = .1 } bubble_volume T
## write bubble position and velocity
OutputScalarSum { step = .1 } {
awk '{
if (NR == 1) {
## analytical solution (terminal velocity)
uex = 0.71
print $3, $5, 0.0, uex;
}
else {
print (t+$3)/2., $5, ($5-x)/($3-t), uex;
}
t = $3; x = $5;
fflush(stdout)
}' > xv
} { v = x*T/bubble_volume }
OutputSimulation { istep = 10 } stdout
GfsEventScript { start = end } {
gnuplot <<- EOF
set key bottom right
set xlabel "time"
set ylabel "velocity"
set term postscript eps lw 3 solid 20 colour
set output "vel.eps"
plot 'xv' u 1:3 title "computed", \
'' u 1:4 w l title "analytical approx."
EOF
}
}
GfsBox { bottom = Boundary }
GfsBox { bottom = Boundary }
GfsBox { bottom = Boundary }
GfsBox { bottom = Boundary }
GfsBox { bottom = Boundary }
GfsBox { bottom = Boundary }
GfsBox { bottom = Boundary }
GfsBox { bottom = Boundary }
1 2 right
2 3 right
3 4 right
4 5 right
5 6 right
6 7 right
7 8 right
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