/**
# Time-reversed VOF advection in a vortex
This classical test advects and stretches an initially circular
interface in a non-divergent vortical flow. The flow reverts in time
and the interface should come back to its original position. The
difference between the initial and final shapes is a measure of the
errors accumulated during advection.
This also checks that a constant "VOF concentration" remains constant
when it is advected with the VOF tracer.
We will need the advection solver combined with the VOF advection
scheme. */
#include "advection.h"
#include "vof.h"
/**
The volume fraction is stored in scalar field `f` which is listed as
an *interface* for the VOF solver. We do not advect any tracer with
the default (diffusive) advection scheme of the advection solver. */
scalar f[], cf[];
scalar * interfaces = {f}, * tracers = NULL;
int MAXLEVEL;
/**
We center the unit box on the origin and set a maximum timestep of 0.1 */
int main()
{
origin (-0.5, -0.5);
DT = .1;
/**
The scalar field `cf` is a "vof concentration" associated with phase
`f`. */
f.tracers = {cf};
/**
We then run the simulation for different levels of refinement. */
for (MAXLEVEL = 5; MAXLEVEL <= 7; MAXLEVEL++) {
init_grid (1 << MAXLEVEL);
run();
}
}
/**
The initial interface is a circle of radius 0.2 centered on
(-0.2,-0.236338) (for historical reasons). We use the levelset
function `circle()` to define this interface.
The period of the stretching cycle is set to 15, which will lead to
strong stretching. Milder conditions can be obtained by decreasing it. */
#define circle(x,y) (sq(0.2) - (sq(x + 0.2) + sq(y + .236338)))
#define T 15.
/**
We define the levelset function $\phi$ on each vertex of the grid and
compute the corresponding volume fraction field. */
event init (i = 0)
{
fraction (f, circle(x,y));
foreach()
cf[] = f[];
boundary ({cf});
}
event velocity (i++) {
/**
This event defines the velocity field.
On trees we first adapt the grid so that the estimated error on
the volume fraction is smaller than $5\times 10^{-3}$. We limit the
resolution at `MAXLEVEL` and we only refine the volume fraction field
`f` and associated tracer `cf`. */
#if TREE
adapt_wavelet ({f}, (double[]){5e-3}, MAXLEVEL, list = {f, cf});
#endif
/**
The velocity field is defined through a streamfunction $\psi$, defined
on the vertices of the grid. */
vertex scalar psi[];
foreach_vertex()
psi[] = - 1.5*sin(2.*pi*t/T)*sin((x + 0.5)*pi)*sin((y + 0.5)*pi)/pi;
/**
We can then differentiate the streamfunction to get the velocity
components. This guarantees that the velocity field is exactly
non-divergent. */
trash ({u});
struct { double x, y; } f = {-1.,1.};
foreach_face()
u.x[] = f.x*(psi[0,1] - psi[])/Delta;
boundary ((scalar *){u});
}
/**
At the start and end of the simulation we check the sum, min and max
values of the volume fraction field. The sum must be constant to
within machine precision and the volume fraction should be bounded by
zero and one. */
event logfile (t = {0,T}) {
stats s = statsf (f);
/**
We compute the minimum and maximum concentration. They should both
be equal to one. */
stats sc = statsf (cf);
double cmin = HUGE, cmax = 0.;
foreach (reduction(min:cmin) reduction(max:cmax))
if (f[] > 1e-12) { // round-off errors are a problem
double c = cf[]/f[];
if (c < cmin) cmin = c;
if (c > cmax) cmax = c;
}
fprintf (stderr, "# t\t\tf.sum\t\tf.min\t\tf.max\n");
fprintf (stderr, "# %f %.12f %.f %g\n", t, s.sum, s.min, s.max);
fprintf (stderr, "# t\t\tcf.sum\t\tc.min - 1\tc.max - 1\n");
fprintf (stderr, "# %f %.12f %.11f %.11f\n",
t, sc.sum, fabs(cmin - 1.), fabs(cmax - 1.));
}
/**
To compute the error, we reinitialise field `e` at the end of the
simulation with the initial shape and compute the difference with the
final shape. We output the norms as functions of the maximum
resolution `N`. */
event field (t = T) {
scalar e[];
fraction (e, circle(x,y));
foreach()
e[] -= f[];
norm n = normf (e);
fprintf (stderr, "%d %g %g %g\n", N, n.avg, n.rms, n.max);
}
/**
We also output the shape of the reconstructed interface at regular
intervals (but only on the finest grid considered). */
event shape (t += T/4.) {
if (N == 128)
output_facets (f);
}
/**
If we are using adaptivity, we also output the levels of refinement at
maximum stretching. */
#if TREE
event levels (t = T/2) {
if (N == 128) {
scalar l[];
foreach()
l[] = level;
output_ppm (l, file = "levels.png", n = 400, min = 0, max = 7);
}
}
#endif
#if 0
event movie (i += 10)
{
scalar l[];
foreach()
l[] = level;
output_ppm (l, 1 << MAXLEVEL, file = "level.mp4");
}
#endif
/**
## Results
We use gnuplot to compute the convergence rate of the error norms with
and without adaptation. The convergence rates are comparable.
~~~gnuplot Convergence rates for constant- and adaptive grids.
ftitle(a,b) = sprintf("%.0f/x^{%4.2f}", exp(a), -b)
f(x)=a+b*x
fit f(x) 'log' u (log($1)):(log($4)) via a,b
f2(x)=a2+b2*x
fit f2(x) 'log' u (log($1)):(log($2)) via a2,b2
fc(x)=ac+bc*x
fit fc(x) 'clog' u (log($1)):(log($4)) via ac,bc
fc2(x)=ac2+bc2*x
fit fc2(x) 'clog' u (log($1)):(log($2)) via ac2,bc2
set xlabel 'Maximum resolution'
set ylabel 'Maximum error'
set key bottom left
set logscale
set xrange [16:256]
set xtics 16,2,256
set grid ytics
set cbrange [1:1]
plot 'log' u 1:4 t 'max (adaptive)', exp(f(log(x))) t ftitle(a,b), \
'clog' u 1:4 t 'max (constant)', exp(fc(log(x))) t ftitle(ac,bc), \
'log' u 1:2 t 'norm1 (adaptive)', exp(f2(log(x))) t ftitle(a2,b2), \
'clog' u 1:2 t 'norm1 (constant)', exp(fc2(log(x))) t ftitle(ac2,bc2)
~~~
The shapes of the interface at $t=0$, $t=T/4$, $t=T/2$, $t=3T/4$ and
$t=T$ are displayed below for both sets of simulations (constant and
adaptive), for $N=128$. The shapes for $t=T/4$ should be identical to
those for $t=3T/4$ and similarly for $t=0$ and $t=T$ (for which we
measure the error). Note that the errors for $t=3T/4$ seem to be much
larger than those for $t=T$.
~~~gnuplot Shapes of the interface for $t=0$, $t=T/4$, $t=T/2$, $t=3T/4$ and $t=T$ for two sets of simulations.
reset
set size ratio -1
plot [-0.5:0.5][-0.5:0.5]'out' w l t "adaptive", 'cout' w l t "constant"
~~~
![Refinement levels for $t=T/2$ and $N=128$.](reversed/levels.png) */