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# Title: Viscous folding of a fluid interface
#
# Description:
#
# An example of mixing between two viscous fluids at low Reynolds
# number. The two fluids are physically identical, and differentiated
# by a tracer.
#
# A sinusoidally varying Poiseuille flow applied in the upper half of
# the domain on the left boundary drives mixing between a cavity and
# the main flow. Over time the fluid in the cavity is replaced by
# lobes of the main fluid which are injected as the driving flow slows
# down.
#
# These flows are described by the Reynolds number $Re$ (which is low
# in this case) and the product of the Reynolds number and a Strouhal
# number, $Sr$, which defines the frequency with which the driving
# Pouseille flow oscillates. For more details, see
# \url{http://dx.doi.org/10.1017/S0022112001006917}{Horner et al
# (2002)}.
#
# These flows are simple systems which can represent a number of
# mixing processes in geological fluid mechanics, such as mixing two
# different kinds of magma. Similar flows have also been suggested as
# strategies for mixing (or preventing mixing) groundwater in
# underground aquifers, where the driving flow is provided by
# injecting and withdrawing water from pairs of wells.
#
# Adaptive refinement is used to resolve the interface and the
# velocity field. The simulation slows down over time as the interface
# stretches rapidly both in the cavity and within the main flow. A
# high degree of refinement is needed to resolve the interface at long
# times. In this example the lack of resolution results in the
# formation of droplets from the stretched and thinned lobes both in
# the cavity and in the channel.
#
# The movie which is generated shows the interface between the two
# fluids and the logarithm of the dimensionless velocity magnitude
# (Figure~\ref{viscmix}), which shows the weak counter-rotating flows
# which form in the cavity in response to the driving Poiseuille flow.
#
# \begin{figure}[htbp]
# \caption{\label{viscmix}MPEG movie of the tracer field and logarithm of the
# velocity magnitude.}
# \label{tracer}
# \begin{center}
# \htmladdnormallinkfoot{\includegraphics[width=0.8\hsize]{viscmix.eps}}{viscmix.mpg}
# \end{center}
# \end{figure}
#
# Author: Jess Robertson
# Command: sh run.sh
# Version: 120713
# Required files: run.sh viscmix.gfv
# Running time: 33 minutes in parallel on 4 processors
# Generated files: viscmix.eps viscmix.mpg
4 3 GfsSimulation GfsBox GfsGEdge {} {
Global {
// Parameters
static double Re = 50;
static double ReSr = 1.2;
static double cavity_depth = 1.0;
static double floor_depth = 0.5;
// Solver parameters
static int min_level = 4;
static int max_level = 8;
// Density functions
static double density(double tracer) {
return 1.0;
}
// Velocity distributions
static double velocity_bc(double y, double t) {
if (y > floor_depth) return (0.125*(2*y - 1)*(2*y - 1)*(cos(2*M_PI*t*(ReSr / Re)) + 1));
else return 0;
}
// Initial tracer distribution
static double tracer_init(double y) {
if (y <= floor_depth) return 1.0;
else return 0.0;
}
}
# Define the depth of the cavity by filling the region below it
# with solid. The cavity will be filled with tracer on
# initialisation. Gerris will remove the cells which are
# completely solid.
Solid ({
return y - (floor_depth - cavity_depth);
})
# Specify simulation times
Time { end = 250 }
# The viscosity is nu = 1/Re.
SourceViscosity (1/Re)
# Initialise the tracer location
VariableTracerVOF T
Init {} {
T = tracer_init(y)
U = velocity_bc(y, 0)
V = 0
}
# We want adaptivity around the fluid interface
AdaptFunction { istep = 1 } {
cmax = 0
minlevel = min_level
maxlevel = max_level
} (T > 0 && T < 1)
# We also need to control the error on the velocity field
AdaptError { istep = 1 } { cmax = 1e-2 maxlevel = max_level } U
AdaptError { istep = 1 } { cmax = 1e-2 maxlevel = max_level } V
# Balance the number of elements across parallel subdomains at
# every timestep if the imbalance is larger than 0.1 (i.e. 10%
# difference between the largest and smallest subdomains).
EventBalance { istep = 1 } 0.1
# Output of solution information/data
OutputTime { istep = 10 } stderr
OutputProjectionStats { istep = 10 } stderr
OutputDiffusionStats { istep = 10 } stderr
OutputTiming { start = end } stderr
# use gfsview to generate movies and figures
GModule gfsview
OutputView { step = 0.5 } { ppm2mpeg -s 400x300 > viscmix.mpg } {
width = 1280 height = 960
} viscmix.gfv
OutputView { start = end } { convert -colors 256 ppm:- viscmix.eps } {
width = 1280 height = 960
} viscmix.gfv
}
# Default boundary conditions are free-slip & impermeable
GfsBox {
bottom = Boundary {
BcDirichlet U 0
}
left = Boundary {
BcDirichlet V 0
}
right = Boundary {
BcDirichlet V 0
}
}
GfsBox {
left = Boundary {
BcDirichlet U velocity_bc(y, t)
}
bottom = Boundary {
BcDirichlet U 0
}
}
GfsBox {}
GfsBox {
bottom = Boundary {
BcDirichlet U 0
}
right = BoundaryOutflow
}
1 3 top
2 3 right
3 4 right
|
