# Vortex Ejection from a mode 3 instability

According to Kizner et al. (2013), a flow around a no-slip cylinder with radius R maybe unstable and eject three dipolar vortex pairs. We study the flow using embedded boundaries and the Navier-Stokes solver. Furthermore, we will view our results.

#include "embed.h"
#include "navier-stokes/centered.h"
#include "view.h"

The maximum resolution is set to \Delta_{min}=R/100. This allows to run on the sandbox server.

int maxlevel = 12;
double Re = 30000., ue = 0.003;
face vector muc[];

int main(){
init_grid(64);
L0 = 40;
mu = muc;
X0 = Y0 = -L0/2;
run();
}

event properties (i++){
foreach_face()
muc.x[] = fm.x[]/Re;
boundary ((scalar*){muc});
}

The cylinder is defined and the flow field is initialized c.f. Kizner et al. with an m=3 perturbation.

#define RAD (pow(sq(x) + sq(y), 0.5))
#define RADP(P, M) ((1 + P*sin(THETA(M)))/(pow(1 + 0.5*sq(P), 0.5)))
double a1 = 1.5, b1 = 2.25;
double P = 0.005, m = 3;

event init (t = 0){
double gamma = (sq(a1) - 1.)/(sq(b1) - sq(a1));
refine (RAD < b1  && level < (maxlevel - 1));
refine (RAD < 1.05 && RAD > 0.95  && level < maxlevel);
vertex scalar phi[];
foreach_vertex()
fractions (phi, cs, fs);
foreach(){
double vr;
if (r1 > 0.9 && r1 < a1)
vr = r1 - 1./r1;
else if (r1 >= a1 && r1 <= b1)
vr = -gamma*r1 + ((1 + gamma)*sq(a1) - 1.)/r1;
else // (0.9 > r || r > b)
vr = 0;
u.x[] = cm[]*0.5*vr*r*-y/(sq(x) + sq(y));
u.y[] = cm[]*0.5*vr*r* x/(sq(x) + sq(y));
}

The boundary conditions on the embedded boundary are set:

  u.t[embed] = dirichlet (0.);
u.n[embed] = dirichlet (0.);
boundary (all);
}

The grid is adaptedively refined and coarsened to properly represent the boundary and the evolution of the flow field. We set \zeta_{u,v}\approx U_{max}/150.

event adapt (i++)
adapt_wavelet ({cs, u.x, u.y}, (double[]){0.01, ue, ue}, maxlevel);

## Output

Movies are generated displaying the vorticity field and the adaptive-grid-cell structure.

event movie (t += 0.5){
scalar omega[];
vorticity (u, omega);
boundary ({omega});
view (fov = 8, width = 600, height = 600);
clear();
draw_vof ("cs", filled = -1, fc = {1., 1., 1.});
squares ("omega", min = -0.75, max = 0.75,
map = cool_warm, linear = true);
draw_vof ("cs", "fs", lw = 2);
save ("kizner.mp4");
clear();
view (fov = 5);
cells();
save ("kizner_cells.mp4");
clear();
view (fov = 2, width = 1080, height = 1080);
draw_vof ("cs", filled = -1, fc = {0.6, 0.6, 0.6});
squares ("omega", min = -0.75, max = 0.75,
map = cool_warm);
cells();
save ("kizner_cells_zoom.mp4");
}

Also there is this movie:

A zoom

We log some solver data:

event logger (t += 0.1){
fprintf (stderr, "%d %g %d %d %d %g %g\n",
i, t, mgp.i, mgu.i, mgpf.i, DT, dt);
int cn = 0;
foreach(reduction(+:cn))
cn++;
if (pid() == 0){
static FILE * fp = fopen ("data", "w");
fprintf (fp, "%g\t%d\t%d\t%g\t%g\n", t, i, cn, dt, DT);
fflush (fp);
}
}

Including the number of grid cells:

set yr [0 : 25000]
set xlabel 'time'
set ylabel 'Cells'
set key off
plot 'data' u 1:3 w l lw 2 These numbers may be compared against the millions of cells that Kizner et al. (2013) employed. (script)

event the_last_event (t = 100);

## Flow direction partitioning

We can study the radially-averaged absolute velocity in the azimutal and radial direction as a function of time and the distance from the centre. Therefore, we render movies of the radial profiles using profile6.h, gnuplot and ffmpeg.

A more quantative perspective on the instability. Mind the logaritmic vertical axis

#include "profile6.h"
FILE * gnuplotPipe;
event init (t = 0){
gnuplotPipe = popen ("gnuplot", "w");
fprintf(gnuplotPipe,
"set term pngcairo\n"
"set xr [1 : 6]\n"
"set yr [0.001 : 1]\n"
"set logscale y\n"
"set key top right\n"
"set grid\n"
"set xlabel 'r/R'\n"
"set ylabel 'E'\n"
"set size ratio 0.5\n");
}

int frame = 0;
event profs(t += 0.5){
scalar ur[], ua[] ,* list;
list = {ur, ua};
foreach(){
double r = sqrt(sq(x) + sq(y));
ur[] = fabs(u.x[]*x/r + u.y[]*y/r);
ua[] = fabs((u.y[]*x/r) - (u.x[]*y/r));
}
profile (list, sqrt(sq(x) + sq(y)), "prof.dat");
fprintf (gnuplotPipe, "set output 'plot%d.png'\n", frame);
fprintf (gnuplotPipe, "set title 't = %d' font ',25'\n",
(int)(t + 0.9));
fprintf (gnuplotPipe, "plot 'prof.dat' u 1:2 w l lw 5 lt rgb'#11BB11' t 'Radial',"
"'prof.dat' u 1:3 w l lw 5 lt rgb '#BB11BB' t 'Azimutal'\n");
fflush (gnuplotPipe);
frame++;
}

event stop (t = end){
system("ffmpeg -r 25 -f image2 -i plot%d.png -c:v libx264 -vf format=yuv420p -y mov.mp4 && "
"rm -f plot*");
return 1;
}

## Reference

Kizner, Z., Makarov, V., Kamp, L., & Van Heijst, G. (2013). Instabilities of the flow around a cylinder and emission of vortex dipoles. Journal of Fluid Mechanics, 730, 419-441. doi:10.1017/jfm.2013.345